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{{Machine learning bar}}
{{DISPLAYTITLE:''k''-means clustering}}
'''''k''-means clustering''' is a method of [[vector quantization]], originally from signal processing, that is popular for [[cluster analysis]] in [[data mining]]. ''k''-means clustering aims to [[partition of a set|partition]] ''n'' observations into ''k'' clusters in which each observation belongs to the cluster with the nearest [[mean]], serving as a [[prototype]] of the cluster. This results in a partitioning of the data space into [[Voronoi cell]]s.


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The problem is computationally difficult ([[NP-hard]]); however, there are efficient [[heuristic algorithm]]s that are commonly employed and converge quickly to a local optimum. These are usually similar to the [[expectation-maximization algorithm]] for [[Mixture model|mixtures]] of [[Gaussian distribution]]s via an iterative refinement approach employed by both algorithms. Additionally, they both use cluster centers to model the data; however, ''k''-means clustering tends to find clusters of comparable spatial extent, while the expectation-maximization mechanism allows clusters to have different shapes.


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Given a set of observations ('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, …, '''x'''<sub>''n''</sub>), where each observation is a ''d''-dimensional real vector, ''k''-means clustering aims to partition the ''n'' observations into ''k'' sets (''k'' ≤ ''n'') '''S'''&nbsp;=&nbsp;{''S''<sub>1</sub>,&nbsp;''S''<sub>2</sub>,&nbsp;…,&nbsp;''S''<sub>''k''</sub>} so as to minimize the within-cluster sum of squares (WCSS):


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:<math>\underset{\mathbf{S}} {\operatorname{arg\,min}}  \sum_{i=1}^{k} \sum_{\mathbf x_j \in S_i} \left\| \mathbf x_j - \boldsymbol\mu_i \right\|^2 </math>


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where '''''μ'''''<sub>''i''</sub> is the mean of points in ''S''<sub>''i''</sub>.


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== History ==
The term "''k''-means" was first used by James MacQueen in 1967,<ref name="macqueen1967">{{cite conference
|first=J. B. |last=MacQueen
|year=1967
|title=Some Methods for classification and Analysis of Multivariate Observations
|url=http://projecteuclid.org/euclid.bsmsp/1200512992 |accessdate=2009-04-07
|conference=Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability
|publisher=University of California Press
|volume=1 |pages=281&ndash;297
|mr=0214227
|zbl=0214.46201
}}</ref> though the idea goes back to [[Hugo Steinhaus]] in 1957.<ref>{{cite journal
|first=H. |last=Steinhaus |authorlink=Hugo Steinhaus
|title=Sur la division des corps matériels en parties
|journal=Bull. Acad. Polon. Sci.
|volume=4 |issue=12 |pages=801&ndash;804
|year=1957
|mr=0090073
|zbl=0079.16403 |language=French
}}</ref> The [[#Standard algorithm|standard algorithm]] was first proposed by Stuart Lloyd in 1957 as a technique for [[pulse-code modulation]], though it wasn't published outside of [[Bell Labs]] until 1982.<ref name="lloyd1957">{{cite journal
|first=S. P. |last=Lloyd
|title=Least square quantization in PCM
|journal=Bell Telephone Laboratories Paper
|year=1957
}} Published in journal much later:
{{cite journal
|first=S. P. |last=Lloyd.
|url=http://www.cs.toronto.edu/~roweis/csc2515-2006/readings/lloyd57.pdf |accessdate=2009-04-15
|title=Least squares quantization in PCM
|journal=[[IEEE Transactions on Information Theory]]
|volume=28 |issue=2 |pages=129&ndash;137
|year=1982
|doi=10.1109/TIT.1982.1056489
}}</ref> In 1965, E.W.Forgy published essentially the same method, which is why it is sometimes referred to as Lloyd-Forgy.<ref name="forgy65">{{Cite journal
|author=E.W. Forgy
|title=Cluster analysis of multivariate data: efficiency versus interpretability of classifications
|journal=Biometrics
|volume=21
|pages=768–769
|year=1965}}</ref> A more efficient version was proposed and published in Fortran by Hartigan and Wong in 1975/1979.<ref name="hartigan1975">{{Cite book
|title=Clustering algorithms
|author=J.A. Hartigan
|year=1975
|publisher=John Wiley & Sons, Inc.
}}</ref><ref name="hartigan1979" />


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== Algorithms ==


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=== Standard algorithm ===
The most common algorithm uses an iterative refinement technique. Due to its ubiquity it is often called the '''''k''-means algorithm'''; it is also referred to as '''[[Lloyd's algorithm]]''', particularly in the computer science community.
 
Given an initial set of ''k'' means ''m''<sub>1</sub><sup>(1)</sup>,…,''m''<sub>''k''</sub><sup>(1)</sup> (see below), the algorithm proceeds by alternating between two steps:<ref>{{Cite book
  | last =MacKay
  | first =David
  | authorlink =David MacKay (scientist)
  | title =Information Theory, Inference and Learning Algorithms
  | publisher = Cambridge University Press
  | year =2003
  | url =http://www.inference.phy.cam.ac.uk/mackay/itila/book.html
  | isbn = 0-521-64298-1
  | chapter=Chapter 20. An Example Inference Task: Clustering
  | chapterurl=http://www.inference.phy.cam.ac.uk/mackay/itprnn/ps/284.292.pdf
  | pages=284&ndash;292
  | ref=mackay2003
  | mr=2012999
}}</ref>
:'''Assignment step''': Assign each observation to the cluster whose mean yields the least within-cluster sum of squares (WCSS). Since the sum of squares is the squared [[Euclidean distance]], this is intuitively the "nearest" mean.<ref>Since the square root is a monotone function, this also is the minimum Euclidean distance assignment.</ref> (Mathematically, this means partitioning the observations according to the [[Voronoi diagram]] generated by the means).
::<math>S_i^{(t)} = \big \{ x_p : \big \| x_p - m^{(t)}_i \big \|^2 \le \big \| x_p - m^{(t)}_j \big \|^2 \ \forall j, 1 \le j \le k \big\},</math>
:: where each <math>x_p</math> is assigned to exactly one <math>S^{(t)}</math>, even if it could be is assigned to two or more of them.
:'''Update step''': Calculate the new means to be the [[centroids]] of the observations in the new clusters.
::<math>m^{(t+1)}_i = \frac{1}{|S^{(t)}_i|} \sum_{x_j \in S^{(t)}_i} x_j </math>
:: Since the arithmetic mean is a [[least-squares estimation|least-squares estimator]], this also minimizes the within-cluster sum of squares (WCSS) objective.
The algorithm has converged when the assignments no longer change. Since both steps optimize the WCSS objective, and there only exists a finite number of such partitionings, the algorithm must converge to a (local) optimum. There is no guarantee that the global optimum is found using this algorithm.
 
The algorithm is often presented as assigning objects to the nearest cluster by distance. This is slightly inaccurate: the algorithm aims at minimizing the WCSS objective, and thus assigns by "least sum of squares". Using a different distance function other than (squared) Euclidean distance may stop the algorithm from converging. It is correct that the smallest Euclidean distance yields the smallest squared Euclidean distance and thus also yields the smallest sum of squares. Various modifications of k-means such as spherical k-means and [[k-medoids]] have been proposed to allow using other distance measures.
 
====Initialization methods====
Commonly used initialization methods are Forgy and Random Partition.<ref name="hamerly">{{Cite conference
|author = Hamerly, G. and Elkan, C.
|year=2002
|title=Alternatives to the k-means algorithm that find better clusterings
|booktitle=Proceedings of the eleventh international conference on Information and knowledge management (CIKM)
|url=http://charlotte.ucsd.edu/users/elkan/cikm02.pdf
}}</ref>
The Forgy method randomly chooses ''k'' observations from the data set and uses these as the initial means. The Random Partition method first randomly assigns a cluster to each observation and then proceeds to the update step, thus computing the initial mean to be the centroid of the cluster's randomly assigned points. The Forgy method tends to spread the initial means out, while Random Partition places all of them close to the center of the data set. According to Hamerly et al.,<ref name="hamerly"/> the Random Partition method is generally preferable for algorithms such as the ''k''-harmonic means and fuzzy ''k''-means. For expectation maximization and standard ''k''-means algorithms, the Forgy method of initialization is preferable.
 
<gallery caption="Demonstration of the standard algorithm" widths="150px">
Image:K Means Example Step 1.svg|1) ''k'' initial "means" (in this case ''k''=3) are randomly generated within the data domain (shown in color).
Image:K Means Example Step 2.svg|2) ''k'' clusters are created by associating every observation with the nearest mean. The partitions here represent the [[Voronoi diagram]] generated by the means.
Image:K Means Example Step 3.svg|3) The [[centroid]] of each of the ''k'' clusters becomes the new mean.
Image:K Means Example Step 4.svg|4) Steps 2 and 3 are repeated until convergence has been reached.
</gallery>
As it is a heuristic algorithm, there is no guarantee that it will converge to the global optimum, and the result may depend on the initial clusters. As the algorithm is usually very fast, it is common to run it multiple times with different starting conditions. However, in the worst case, ''k''-means can be very slow to converge: in particular it has been shown that there exist certain point sets, even in 2 dimensions, on which ''k''-means takes exponential time, that is {{math|2<sup>Ω(<var>n</var>)</sup>}}, to converge.<ref>{{cite journal
|first=A. |last=Vattani.
|url=http://cseweb.ucsd.edu/users/avattani/papers/kmeans-journal.pdf
|title=k-means requires exponentially many iterations even in the plane
|journal=[[Discrete and Computational Geometry]]
|volume=45 |issue=4 |pages=596&ndash;616
|year=2011
|doi=10.1007/s00454-011-9340-1
}}</ref> These point sets do not seem to arise in practice: this is corroborated by the fact that the [[Smoothed analysis|smoothed]] running time of ''k''-means is polynomial.<ref name="Arthur, D.; Manthey, B.; Roeglin, H. 2009">{{cite conference | author=Arthur, D.; Manthey, B.; Roeglin, H. | year=2009 | title=k-means has polynomial smoothed complexity | booktitle=Proceedings of the 50th Symposium on Foundations of Computer Science (FOCS)}}</ref>
 
The "assignment" step is also referred to as '''expectation step''', the "update step" as '''maximization step''', making this algorithm a variant of the ''generalized'' [[expectation-maximization algorithm]].
 
=== Complexity ===
Regarding computational complexity, finding the optimal solution to the ''k''-means clustering problem for observations in ''d'' dimensions is:
*[[NP-hard]] in general Euclidean space ''d'' even for 2 clusters <ref>{{cite journal
|author=Aloise, D.; Deshpande, A.; Hansen, P.; Popat, P.
|title=NP-hardness of Euclidean sum-of-squares clustering
|journal=[[Machine Learning (journal)|Machine Learning]]
|year=2009
|volume=75 |pages=245&ndash;249
|doi=10.1007/s10994-009-5103-0}}</ref><ref>
{{cite journal
|title=Random Projection Trees for Vector Quantization
|author=Dasgupta, S. and Freund, Y.
|journal=Information Theory, IEEE Transactions on
|volume=55
|pages=3229&ndash;3242
|date=July 2009
|doi=10.1109/TIT.2009.2021326
|arxiv=0805.1390}}
</ref>
*[[NP-hard]] for a general number of clusters ''k'' even in the plane <ref>{{cite journal
|author=Mahajan, M.; Nimbhorkar, P.; Varadarajan, K.
|title=The Planar k-Means Problem is NP-Hard
|journal=[[Lecture Notes in Computer Science]]
|year=2009
|volume=5431 |pages=274&ndash;285
|doi=10.1007/978-3-642-00202-1_24}}
</ref>
*If ''k'' and ''d'' (the dimension) are fixed, the problem can be exactly solved in time '''''O(n<sup>dk+1</sup> log n)''''', where ''n'' is the number of entities to be clustered <ref>
{{cite conference
|author=Inaba, M.; Katoh, N.; Imai, H.
|year=1994
|title=Applications of weighted Voronoi diagrams and randomization to variance-based ''k''-clustering
|conference=[[Symposium on Computational Geometry|Proceedings of 10th ACM Symposium on Computational Geometry]]
|pages=332&ndash;339
|doi= 10.1145/177424.178042}}
</ref>
 
Thus, a variety of [[heuristic algorithm]]s such as Lloyds algorithm given above are generally used.
 
* Lloyd's <math>k</math>-means algorithm has polynomial smoothed running time. It is shown that <ref name="Arthur, D.; Manthey, B.; Roeglin, H. 2009"/> for arbitrary set of <math>n</math> points in <math>[0,1]^d</math>, if each point is independently perturbed by a normal distribution with mean <math>0</math> and variance <math>\sigma^2</math>, then the expected running time of <math>k</math>-means algorithm is bounded by <math>O( n^{34}k^{34}d^8 log^4(n)/ \sigma^6 )</math>, which is a polynomial in <math>n</math>, <math>k</math>, <math>d</math> and <math>1/\sigma</math>.
 
* Better bounds are proved for simple cases. For example,<ref>{{Cite thesis| author=Arthur; Abhishek Bhowmick | year=2009 | title= A theoretical analysis of Lloyd's algorithm for k-means clustering }}[http://www.cse.iitk.ac.in/users/bhowmick/lloyd.pdf]{{dead link|date=January 2013}}</ref> showed that the running time of <math>k</math>-means algorithm is bounded by <math>O(dn^4M^2)</math> for <math>n</math> points in an integer lattice <math>\{1,\dots, M\}^d</math>.
 
=== Variations ===
* [[k-medians clustering]] uses the median in each dimension instead of the mean, and this way minimizes <math>L_1</math> norm ([[Taxicab geometry]]).
* [[k-medoids]] (also: Partitioning Around Medoids, PAM) uses the medoid instead of the mean, and this way minimizes the sum of distances for ''arbitrary'' distance functions.
* [[Fuzzy clustering#Fuzzy c-means clustering|Fuzzy C-Means Clustering]] is a soft version of K-means, where each data point has a fuzzy degree of belonging to each cluster.
* [[Mixture model#Gaussian mixture model|Gaussian mixture]] models trained with [[expectation-maximization algorithm]] (EM algorithm) maintains probabilistic assignments to clusters, instead of deterministic assignments, and multivariate Gaussian distributions instead of means.
* Several methods have been proposed to choose better starting clusters. One recent proposal is [[k-means++]].
* The filtering algorithm uses [[kd-tree]]s to speed up each k-means step.<ref>{{cite journal
|author=Kanungo, T.; [[David Mount|Mount, D. M.]]; [[Nathan Netanyahu|Netanyahu, N. S.]]; Piatko, C. D.; Silverman, R.; Wu, A. Y.
|doi=10.1109/TPAMI.2002.1017616
|url=http://www.cs.umd.edu/~mount/Papers/pami02.pdf |accessdate=2009-04-24
|title=An efficient k-means clustering algorithm: Analysis and implementation
|journal=IEEE Trans. Pattern Analysis and Machine Intelligence
|volume=24 |year=2002 |pages=881&ndash;892 }}
</ref>
* Some methods attempt to speed up each k-means step using [[coreset]]s<ref>{{Cite conference
|author=Frahling, G.; Sohler, C.
|year=2006
|title=A fast k-means implementation using coresets
|booktitle=[[Symposium on Computational Geometry|Proceedings of the twenty-second annual symposium on Computational geometry (SoCG)]]
|url=http://www.frahling.de/Gereon_Frahling/Publications_files/A%20fast%20k-means%20implementation%20using%20Coresets%20(Frahling,%20Sohler).pdf
}}
</ref> or the [[triangle inequality]].<ref>{{Cite conference
|author = Elkan, C.
|year=2003
|title=Using the triangle inequality to accelerate k-means
|booktitle=Proceedings of the Twentieth International Conference on Machine Learning (ICML)
|url=http://www-cse.ucsd.edu/~elkan/kmeansicml03.pdf
}}</ref>
* Escape local optima by swapping points between clusters.<ref name="hartigan1979">{{Cite journal
| first1 = J. A. |last1=Hartigan
| first2 = M. A. |last2=Wong
| year = 1979
| title =  Algorithm AS 136: A K-Means Clustering Algorithm
| journal = [[Journal of the Royal Statistical Society, Series C]]
| volume = 28
| issue = 1
| pages = 100&ndash;108
| jstor = 2346830
}}</ref>
 
* The [[Spherical k-means]] clustering algorithm is suitable for directional data.<ref>{{Cite journal
| first1 = I. S. |last1=Dhillon
| first2 = D. M. |last2=Modha
| year = 2001
| title =  Concept decompositions for large sparse text data using clustering
| journal = Machine Learning
| volume = 42
| issue = 1
| pages = 143&ndash;175
}}</ref>
 
* The [[Minkowski metric weighted k-means]] deals with irrelevant features by assigning cluster specific weights to each feature<ref>{{Cite journal
| first1 = R. C. |last1=Amorim
| first2 = B |last2=Mirkin
| year = 2012
| title =  Minkowski metric, feature weighting and anomalous cluster initializing in K-Means clustering
| journal = Pattern Recognition
| volume = 45
| issue = 3
| pages = 1061&ndash;1075
| doi=10.1016/j.patcog.2011.08.012
}}</ref>
 
== Discussion ==
[[File:K-means convergence to a local minimum.png|thumb|650px|A typical example of the k-means convergence to a local minimum. In this example, the result of k-means clustering (the right figure) contradicts the obvious cluster structure of the data set. The small circles are the data points, the four ray stars are the centroids (means). The initial configuration is on the left figure. The algorithm converges after five iterations presented on the figures, from the left to the right. The illustration was prepared  with the Mirkes Java applet.<ref name = "Mirkes2011"/>]]
[[File:Iris Flowers Clustering kMeans.svg|thumb|450px|''k''-means clustering result for the [[Iris flower data set]] and actual species visualized using [[Environment for DeveLoping KDD-Applications Supported by Index-Structures|ELKI]]. Cluster means are marked using larger, semi-transparent symbols.]]
[[File:ClusterAnalysis Mouse.svg|thumb|450px|''k''-means clustering and EM clustering on an artificial dataset ("mouse"). The tendency of ''k''-means to produce equi-sized clusters leads to bad results, while EM benefits from the Gaussian distribution present in the data set]]
 
The two key features of ''k''-means which make it efficient are often regarded as its biggest drawbacks:
* [[Euclidean distance]] is used as a [[metric (mathematics)|metric]] and [[variance]] is used as a measure of cluster scatter.
* The number of clusters ''k'' is an input parameter: an inappropriate choice of ''k'' may yield poor results.  That is why, when performing k-means, it is important to run diagnostic checks for [[determining the number of clusters in a data set|determining the number of clusters in the data set]].
* Convergence to a local minimum may produce counterintuitive ("wrong") results (see example in Fig.).
 
A key limitation of ''k''-means is its cluster model. The concept is based on spherical clusters that are separable in a way so that the mean value converges towards the cluster center. The clusters are expected to be of similar size, so that the assignment to the nearest cluster center is the correct assignment. When for example applying ''k''-means with a value of <math>k=3</math> onto the well-known [[Iris flower data set]], the result often fails to separate the three [[Iris (plant)|Iris]] species contained in the data set. With <math>k=2</math>, the two visible clusters (one containing two species) will be discovered, whereas with <math>k=3</math> one of the two clusters will be split into two even parts. In fact, <math>k=2</math> is more appropriate for this data set, despite the data set containing 3 ''classes''. As with any other clustering algorithm, the ''k''-means result relies on the data set to satisfy the assumptions made by the clustering algorithms. It works well on some data sets, while failing on others.
 
The result of ''k''-means can also be seen as the [[Voronoi diagram|Voronoi cells]] of the cluster means. Since data is split halfway between cluster means, this can lead to suboptimal splits as can be seen in the "mouse" example. The Gaussian models used by the [[Expectation-maximization algorithm]] (which can be seen as a generalization of ''k''-means) are more flexible here by having both variances and covariances. The EM result is thus able to accommodate clusters of variable size much better than ''k''-means as well as correlated clusters (not in this example).
 
== Applications ==
 
''k''-means clustering in particular when using heuristics such as Lloyd's algorithm is rather easy to implement and apply even on large data sets. As such, it has been successfully used in various topics, ranging from [[market segmentation]], [[computer vision]], [[geostatistics]],<ref>Honarkhah, M and Caers, J, 2010, ''[http://dx.doi.org/10.1007/s11004-010-9276-7 Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling]'', Mathematical Geosciences, 42: 487 - 517</ref> and [[astronomy]] to [[Data Mining in Agriculture|agriculture]]. It often is used as a preprocessing step for other algorithms, for example to find a starting configuration.
 
===Vector quantization===
{{Main|Vector quantization}}
 
[[File:Rosa Gold Glow 2 small noblue.png|frame|right|Two-channel (for illustration purposes -- red and green only) color image.]]
[[File:Rosa Gold Glow 2 small noblue color space.png|thumb|right|250px|Vector quantization of colors present in the image above into Voronoi cells using ''k''-means.]]
 
''k''-means originates from signal processing, and still finds use in this domain. For example in computer graphics, [[color quantization]] is the task of reducing the color palette of an image to a fixed number of colors ''k''. The ''k''-means algorithm can easily be used for this task and produces competitive results. Other uses of vector quantization include [[Sampling (statistics)|non-random sampling]], as ''k''-means can easily be used to choose ''k'' different but prototypical objects from a large data set for further analysis.
 
===Cluster analysis===
{{Main|Cluster analysis}}
 
In cluster analysis, the ''k''-means algorithm can be used to partition the input data set into ''k'' partitions (clusters).
 
However, the pure ''k''-means algorithm is not very flexible, and as such of limited use (except for when vector quantization as above is actually the desired use case!). In particular, the parameter ''k'' is known to be hard to choose (as discussed below) when not given by external constraints. In contrast to other algorithms, ''k''-means can also not be used with arbitrary distance functions or be use on non-numerical data. For these use cases, many other algorithms have been developed since.
 
=== Feature learning ===
''k''-means clustering has been used as a [[feature learning]] (or [[dictionary learning]]) step, which can be used in the for ([[semi-supervised learning|semi-]])[[supervised learning]] or [[unsupervised learning]].<ref name="Coates2012">{{cite encyclopedia
|last1 = Coates
|first1 = Adam
|last2 = Ng
|first2 = Andrew Y.
|title = Learning feature representations with k-means
|editors = G. Montavon, G. B. Orr, K.-R. Müller
|encyclopedia = Neural Networks: Tricks of the Trade
|publisher = Springer
|year = 2012
|url = http://www.stanford.edu/~acoates/papers/coatesng_nntot2012.pdf
}}</ref>
The basic approach is first to train a ''k''-means clustering representation, using the input training data (which need not be labelled). Then, to project any input datum into the new feature space, we have a choice of "encoding" functions, but we can use for example the thresholded matrix-product of the datum with the centroid locations, the distance from the datum to each centroid, or simply an indicator function for the nearest centroid,<ref name="Coates2012"/><ref>{{cite conference
|last1 = Csurka
|first1 = Gabriella
|last2 = Dance
|first2 = Christopher C.
|last3 = Fan
|first3 = Lixin
|last4 = Willamowski
|first4 = Jutta
|last5 = Bray
|first5 = Cédric
|title = Visual categorization with bags of keypoints
|conference = ECCV Workshop on Statistical Learning in Computer Vision
|year = 2004
|url = http://www.cs.cmu.edu/~efros/courses/LBMV07/Papers/csurka-eccv-04.pdf
}}</ref> or some smooth transformation of the distance.<ref name="coates2011"/> Alternatively, by transforming the sample-cluster distance through a [[Radial basis function|Gaussian RBF]], one effectively obtains the hidden layer of a [[radial basis function network]].<ref name="schwenker">{{cite journal
|last1 = Schwenker
|first1 = Friedhelm
|last2 = Kestler
|first2 = Hans A.
|last3 = Palm
|first3 = Günther
|title = Three learning phases for radial-basis-function networks
|journal = Neural Networks
|volume = 14
|pages = 439–458
|year = 2001
|url = http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.109.312&rep=rep1&type=pdf
}}</ref>
 
This use of ''k''-means has been successfully combined with simple, [[linear classifier]]s for semi-supervised learning in [[natural language processing|NLP]] (specifically for [[named entity recognition]])<ref>{{cite conference
|last1 = Lin
|first1 = Dekang
|last2 = Wu
|first2 = Xiaoyun
|title = Phrase clustering for discriminative learning
|conference = Annual Meeting of the [[Association for Computational Linguistics|ACL]] and IJCNLP
|year = 2009
|pages = 1030–1038
|url = http://www.aclweb.org/anthology/P/P09/P09-1116.pdf
}}</ref>
and in [[computer vision]]. On an object recognition task, it was found to exhibit comparable performance with more sophisticated feature learning approaches such as [[autoencoder]]s and [[restricted Boltzmann machine]]s.<ref name="coates2011">{{cite conference
|last1 = Coates
|first1 = Adam
|last2 = Lee
|first2 = Honglak
|last3 = Ng
|first3 = Andrew Y.
|title = An analysis of single-layer networks in unsupervised feature learning
|conference = International Conference on Artificial Intelligence and Statistics (AISTATS)
|year = 2011
|url = http://www.stanford.edu/~acoates/papers/coatesleeng_aistats_2011.pdf
}}</ref>
However, it generally requires more data than the sophisticated methods, for equivalent performance, because each data point only contributes to one "feature" rather than multiple.<ref name="Coates2012"/>
 
== Relation to other statistical machine learning algorithms ==
 
''k''-means clustering, and its associated [[Expectation–maximization algorithm|expectation-maximization algorithm]], is a special case of a [[Mixture model|Gaussian mixture model]], specifically, the limit of taking all covariances as diagonal, equal, and small.  It is often easy to generalize a ''k''-means problem into a Gaussian mixture model.<ref>{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 16.1. Gaussian Mixture Models and k-Means Clustering | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=842}}</ref>  Another generalization of the k-means algorithm is the [[K-SVD]] algorithm, which estimates data points as a sparse linear combination of "codebook vectors".  K-means corresponds to the special case of using a single codebook vector, with a weight of 1.<ref name="
K-SVD">[http://intranet.daiict.ac.in/~ajit_r/IT530/KSVD_IEEETSP.pdf].</ref>
 
=== Mean shift clustering ===
Basic [[mean shift]] clustering algorithms maintain a set of data points the same size as the input data set. Initially, this set is copied from the input set. Then this set is iteratively replaced by the mean of those points in the set that are within a given distance of that point. By contrast, ''k''-means restricts this updated set to ''k'' points usually much less than the number of points in the input data set, and replaces each point in this set by the mean of all points in the ''input set'' that are closer to that point than any other (e.g. within the Voronoi partition of each updating point). A mean shift algorithm that is similar then to ''k''-means, called ''likelihood mean shift'', replaces the set of points undergoing replacement by the mean of all points in the input set that are within a given distance of the changing set.<ref name="Little2011">{{cite journal|last=Little|first= M.A.|coauthors=Jones, N.S.|title=Generalized Methods and Solvers for Piecewise Constant Signals: Part I| journal=[[Proceedings of the Royal Society A]]|url=http://www.maxlittle.net/publications/pwc_filtering_arxiv.pdf|year = 2011 }}</ref> One of the advantages of mean shift over ''k''-means is that there is no need to choose the number of clusters, because mean shift is likely to find only a few clusters if indeed only a small number exist. However, mean shift can be much slower than ''k''-means, and still requires selection of a bandwidth parameter. Mean shift has soft variants much as ''k''-means does.
 
=== Principal component analysis (PCA) ===
It was asserted in <ref>{{cite journal|authors=H. Zha, C. Ding, M. Gu, X. He and H.D. Simon|title=Spectral Relaxation for K-means Clustering|journal=Neural Information Processing Systems vol.14 (NIPS 2001)|pages=1057–1064|location=Vancouver, Canada|date=Dec 2001|url=http://ranger.uta.edu/~chqding/papers/Zha-Kmeans.pdf}}</ref><ref>{{cite journal|authors=Chris Ding and Xiaofeng He|title=K-means Clustering via Principal Component Analysis|work=Proc. of Int'l Conf. Machine Learning (ICML 2004)|pages=225–232|date=July 2004|url=http://ranger.uta.edu/~chqding/papers/KmeansPCA1.pdf}}</ref> that the relaxed solution of {{math|<var>k</var>}}-means clustering, specified by the cluster indicators, is given by the PCA ([[principal component analysis]]) principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of k-means clustering was not a new result (see, for example,<ref>{{cite journal | title = Clustering large graphs via the singular value decomposition | journal = Machine learning | year = 2004 | first = P. | last = Drineas | coauthors = A. Frieze, R. Kannan, S. Vempala, V. Vinay | volume = 56 | pages = 9–33| id = | url = http://www.cc.gatech.edu/~vempala/papers/dfkvv.pdf | accessdate = 2012-08-02}}</ref>), and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions{{Citation needed|date=January 2014}}.
 
=== Bilateral filtering ===
''k''-means implicitly assumes that the ordering of the input data set does not matter. The [[bilateral filter]] is similar to K-means and [[mean shift]] in that it maintains a set of data points that are iteratively replaced by means. However, the bilateral filter restricts the calculation of the (kernel weighted) mean to include only points that are close in the ordering of the input data.<ref name="Little2011"/> This makes it applicable to problems such as image denoising, where the spatial arrangement of pixels in an image is of critical importance.
 
==Similar problems==
The set of squared error minimizing cluster functions also includes the [[k-medoids|{{math|<var>k</var>}}-medoids]] algorithm, an approach which forces the center point of each cluster to be one of the actual points, i.e., it uses [[medoids]] in place of [[centroids]].
 
== Software ==
 
=== Free ===
* [[Apache Mahout]] [http://cwiki.apache.org/MAHOUT/k-means-clustering.html k-Means]
* [[CrimeStat]] implements two spatial K-means algorithms, one of which allows the user to define the starting locations.
* [[ELKI]] contains k-means (with Lloyd and MacQueen iteration, along with different initializations such as k-means++ initialization) and various more advanced clustering algorithms
* [[MLPACK (C++ library)|MLPACK]] contains a C++ implementation of k-means
* [[R (programming language)|R]] [http://stat.ethz.ch/R-manual/R-patched/library/stats/html/kmeans.html kmeans] implements a variety of algorithms<ref name="macqueen1967"/><ref name="lloyd1957"/><ref name="hartigan1979"/>
* [[SciPy]] [http://docs.scipy.org/doc/scipy/reference/cluster.vq.html vector-quantization]
* [[Scikit-learn]] implements a popular python machine-learning library which contains various clustering algorithms
* [http://www.codeding.com/?article=14 Silverlight widget demonstrating k-means algorithm]
* [http://pgxn.org/dist/kmeans/ PostgreSQL extension for k-means]
* [http://graphlab.org/toolkits/clustering/ CMU's GraphLab Clustering library] Efficient multicore implementation for large scale data.
* [[Weka (machine learning)|Weka]] contains k-means and a few variants of it, including k-means++ and x-means.
* [http://spectralpython.sourceforge.net/algorithms.html#k-means-clustering Spectral Python] contains methods for [[unsupervised classification]] including a K-means clustering method.
* [http://scikit-learn.org/dev/modules/generated/sklearn.cluster.KMeans.html scikit learn] machine learning in Python contains a K-Means implementation
* [[OpenCV]] contains a [http://docs.opencv.org/modules/core/doc/clustering.html?highlight=kmeans#cv2.kmeans K-means] implementation under [[Bsd licence|BSD licence]].
* [http://gforge.inria.fr/projects/yael/ Yael] includes an efficient multi-threaded C implementation of k-means, with C, Python and Matlab interfaces.
 
=== Commercial ===
* IDL Cluster, Clust_Wts
* [http://reference.wolfram.com/mathematica/ref/ClusteringComponents.html  ''Mathematica'' ClusteringComponents function]
* [[MATLAB]] [http://www.mathworks.com/access/helpdesk/help/toolbox/stats/kmeans.html kmeans]
* [[SAS System|SAS]] [http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/fastclus_toc.htm FASTCLUS]
* [[Stata]] [http://www.stata.com/help13.cgi?cluster+kmeans kmeans]
* [http://www.visumap.com/index.aspx?p=Products VisuMap kMeans Clustering]
 
===Source code===
* [[ELKI]] and [[Weka]] are written in Java and include k-means and variations
* K-means application in PHP,<ref>http://www25.brinkster.com/denshade/kmeans.php.htm</ref> using VB,<ref>[http://people.revoledu.com/kardi/tutorial/kMean/download.htm K-Means Clustering Tutorial: Download<!-- Bot generated title -->]</ref> using Perl,<ref>[http://www.lwebzem.com/cgi-bin/k_means/test3.cgi Perl script for Kmeans clustering<!-- Bot generated title -->]</ref> using C++,<ref>[http://codingplayground.blogspot.com/2009/03/k-means-in-c.html Antonio Gulli's coding playground: K-means in C<!-- Bot generated title -->]</ref> using Matlab,<ref>[http://people.revoledu.com/kardi/tutorial/kMean/matlab_kMeans.htm K-Means Clustering Tutorial: Matlab Code<!-- Bot generated title -->]</ref> using Ruby,<ref>[http://ai4r.org/index.html AI4R :: Artificial Intelligence for Ruby<!-- Bot generated title -->]</ref><ref>[http://github.com/reddavis/K-Means/tree/master reddavis/K-Means · GitHub<!-- Bot generated title -->]</ref> using Python with scipy,<ref>[http://docs.scipy.org/doc/scipy/reference/cluster.vq.html K-means clustering and vector quantization (scipy.cluster.vq) — SciPy v0.11 Reference Guide (DRAFT)<!-- Bot generated title -->]</ref> using X10<ref>http://dist.codehaus.org/x10/applications/samples/KMeansDist.x10</ref>
*  A parallel out-of-core implementation in C<ref>http://www.cs.princeton.edu/~wdong/kmeans/</ref>
* An open-source collection of clustering algorithms, including k-means, implemented in Javascript.<ref>http://code.google.com/p/figue/ FIGUE</ref>  Online demo.<ref>http://jydelort.appspot.com/resources/figue/demo.html</ref>
 
===Visualization, animation and examples===
* [[ELKI]] can visualize k-means using [[Voronoi diagram|Voronoi cells]] and [[Delaunay triangulation]] for 2D data. In higher dimensionality, only cluster assignments and cluster centers are visualized
* Demos of the K-means-algorithm<ref>[http://home.dei.polimi.it/matteucc/Clustering/tutorial_html/AppletKM.html Clustering - K-means demo<!-- Bot generated title -->]</ref><ref>[http://siebn.de/other/yakmeans/ siebn.de - YAK-Means<!-- Bot generated title -->]</ref><ref>[http://informationandvisualization.de/blog/kmeans-and-voronoi-tesselation-built-processing k-Means and Voronoi Tesselation: Built with Processing | Information & Visualization<!-- Bot generated title -->]</ref><ref>[http://www.javaworld.com/javaworld/jw-11-2006/jw-1121-thread.html Hyper-threaded Java - JavaWorld<!-- Bot generated title -->]</ref><ref>[http://www.leet.it/home/lale/clustering/ Color clustering<!-- Bot generated title -->]</ref><ref>[http://www.onmyphd.com/?p=k-means.clustering Interactive step-by-step examples in Javascript of good and bad k-means clustering]</ref>
* K-means and K-medoids (Applet), [[University of Leicester]]<ref name = "Mirkes2011">E.M. Mirkes, [http://www.math.le.ac.uk/people/ag153/homepage/KmeansKmedoids/Kmeans_Kmedoids.html K-means and K-medoids applet]. University of Leicester, 2011.</ref>
* Clustergram - cluster diagnostic plot - for visual diagnostics of choosing the number of (k) clusters ([[R (programming language)|R]] code)<ref>[http://www.r-statistics.com/2010/06/clustergram-visualization-and-diagnostics-for-cluster-analysis-r-code/ Clustergram: visualization and diagnostics for cluster analysis (R code) | R-statistics blog<!-- Bot generated title -->]</ref>
 
==See also==
* [[Canopy clustering algorithm]]
* [[Centroidal Voronoi tessellation]]
* [[k q-flats]]
* [[Linde–Buzo–Gray algorithm]]
* [[Nearest centroid classifier]]
* [[Self-organizing map]]
* [[silhouette (clustering)|Silhouette clustering]]
 
==References==
{{Reflist|2}}
 
{{DEFAULTSORT:K-Means Clustering}}
[[Category:Data clustering algorithms]]
[[Category:Statistical algorithms]]

Revision as of 03:28, 26 February 2013

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k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.

The problem is computationally difficult (NP-hard); however, there are efficient heuristic algorithms that are commonly employed and converge quickly to a local optimum. These are usually similar to the expectation-maximization algorithm for mixtures of Gaussian distributions via an iterative refinement approach employed by both algorithms. Additionally, they both use cluster centers to model the data; however, k-means clustering tends to find clusters of comparable spatial extent, while the expectation-maximization mechanism allows clusters to have different shapes.

Description

Given a set of observations (x1, x2, …, xn), where each observation is a d-dimensional real vector, k-means clustering aims to partition the n observations into k sets (kn) S = {S1S2, …, Sk} so as to minimize the within-cluster sum of squares (WCSS):

argmin𝐒i=1k𝐱jSi𝐱j𝝁i2

where μi is the mean of points in Si.

History

The term "k-means" was first used by James MacQueen in 1967,[1] though the idea goes back to Hugo Steinhaus in 1957.[2] The standard algorithm was first proposed by Stuart Lloyd in 1957 as a technique for pulse-code modulation, though it wasn't published outside of Bell Labs until 1982.[3] In 1965, E.W.Forgy published essentially the same method, which is why it is sometimes referred to as Lloyd-Forgy.[4] A more efficient version was proposed and published in Fortran by Hartigan and Wong in 1975/1979.[5][6]

Algorithms

Standard algorithm

The most common algorithm uses an iterative refinement technique. Due to its ubiquity it is often called the k-means algorithm; it is also referred to as Lloyd's algorithm, particularly in the computer science community.

Given an initial set of k means m1(1),…,mk(1) (see below), the algorithm proceeds by alternating between two steps:[7]

Assignment step: Assign each observation to the cluster whose mean yields the least within-cluster sum of squares (WCSS). Since the sum of squares is the squared Euclidean distance, this is intuitively the "nearest" mean.[8] (Mathematically, this means partitioning the observations according to the Voronoi diagram generated by the means).
Si(t)={xp:xpmi(t)2xpmj(t)2 j,1jk},
where each xp is assigned to exactly one S(t), even if it could be is assigned to two or more of them.
Update step: Calculate the new means to be the centroids of the observations in the new clusters.
mi(t+1)=1|Si(t)|xjSi(t)xj
Since the arithmetic mean is a least-squares estimator, this also minimizes the within-cluster sum of squares (WCSS) objective.

The algorithm has converged when the assignments no longer change. Since both steps optimize the WCSS objective, and there only exists a finite number of such partitionings, the algorithm must converge to a (local) optimum. There is no guarantee that the global optimum is found using this algorithm.

The algorithm is often presented as assigning objects to the nearest cluster by distance. This is slightly inaccurate: the algorithm aims at minimizing the WCSS objective, and thus assigns by "least sum of squares". Using a different distance function other than (squared) Euclidean distance may stop the algorithm from converging. It is correct that the smallest Euclidean distance yields the smallest squared Euclidean distance and thus also yields the smallest sum of squares. Various modifications of k-means such as spherical k-means and k-medoids have been proposed to allow using other distance measures.

Initialization methods

Commonly used initialization methods are Forgy and Random Partition.[9] The Forgy method randomly chooses k observations from the data set and uses these as the initial means. The Random Partition method first randomly assigns a cluster to each observation and then proceeds to the update step, thus computing the initial mean to be the centroid of the cluster's randomly assigned points. The Forgy method tends to spread the initial means out, while Random Partition places all of them close to the center of the data set. According to Hamerly et al.,[9] the Random Partition method is generally preferable for algorithms such as the k-harmonic means and fuzzy k-means. For expectation maximization and standard k-means algorithms, the Forgy method of initialization is preferable.

As it is a heuristic algorithm, there is no guarantee that it will converge to the global optimum, and the result may depend on the initial clusters. As the algorithm is usually very fast, it is common to run it multiple times with different starting conditions. However, in the worst case, k-means can be very slow to converge: in particular it has been shown that there exist certain point sets, even in 2 dimensions, on which k-means takes exponential time, that is Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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The "assignment" step is also referred to as expectation step, the "update step" as maximization step, making this algorithm a variant of the generalized expectation-maximization algorithm.

Complexity

Regarding computational complexity, finding the optimal solution to the k-means clustering problem for observations in d dimensions is:

  • NP-hard in general Euclidean space d even for 2 clusters [12][13]
  • NP-hard for a general number of clusters k even in the plane [14]
  • If k and d (the dimension) are fixed, the problem can be exactly solved in time O(ndk+1 log n), where n is the number of entities to be clustered [15]

Thus, a variety of heuristic algorithms such as Lloyds algorithm given above are generally used.

  • Lloyd's k-means algorithm has polynomial smoothed running time. It is shown that [11] for arbitrary set of n points in [0,1]d, if each point is independently perturbed by a normal distribution with mean 0 and variance σ2, then the expected running time of k-means algorithm is bounded by O(n34k34d8log4(n)/σ6), which is a polynomial in n, k, d and 1/σ.
  • Better bounds are proved for simple cases. For example,[16] showed that the running time of k-means algorithm is bounded by O(dn4M2) for n points in an integer lattice {1,,M}d.

Variations

  • k-medians clustering uses the median in each dimension instead of the mean, and this way minimizes L1 norm (Taxicab geometry).
  • k-medoids (also: Partitioning Around Medoids, PAM) uses the medoid instead of the mean, and this way minimizes the sum of distances for arbitrary distance functions.
  • Fuzzy C-Means Clustering is a soft version of K-means, where each data point has a fuzzy degree of belonging to each cluster.
  • Gaussian mixture models trained with expectation-maximization algorithm (EM algorithm) maintains probabilistic assignments to clusters, instead of deterministic assignments, and multivariate Gaussian distributions instead of means.
  • Several methods have been proposed to choose better starting clusters. One recent proposal is k-means++.
  • The filtering algorithm uses kd-trees to speed up each k-means step.[17]
  • Some methods attempt to speed up each k-means step using coresets[18] or the triangle inequality.[19]
  • Escape local optima by swapping points between clusters.[6]

Discussion

A typical example of the k-means convergence to a local minimum. In this example, the result of k-means clustering (the right figure) contradicts the obvious cluster structure of the data set. The small circles are the data points, the four ray stars are the centroids (means). The initial configuration is on the left figure. The algorithm converges after five iterations presented on the figures, from the left to the right. The illustration was prepared with the Mirkes Java applet.[22]
k-means clustering result for the Iris flower data set and actual species visualized using ELKI. Cluster means are marked using larger, semi-transparent symbols.
k-means clustering and EM clustering on an artificial dataset ("mouse"). The tendency of k-means to produce equi-sized clusters leads to bad results, while EM benefits from the Gaussian distribution present in the data set

The two key features of k-means which make it efficient are often regarded as its biggest drawbacks:

  • Euclidean distance is used as a metric and variance is used as a measure of cluster scatter.
  • The number of clusters k is an input parameter: an inappropriate choice of k may yield poor results. That is why, when performing k-means, it is important to run diagnostic checks for determining the number of clusters in the data set.
  • Convergence to a local minimum may produce counterintuitive ("wrong") results (see example in Fig.).

A key limitation of k-means is its cluster model. The concept is based on spherical clusters that are separable in a way so that the mean value converges towards the cluster center. The clusters are expected to be of similar size, so that the assignment to the nearest cluster center is the correct assignment. When for example applying k-means with a value of k=3 onto the well-known Iris flower data set, the result often fails to separate the three Iris species contained in the data set. With k=2, the two visible clusters (one containing two species) will be discovered, whereas with k=3 one of the two clusters will be split into two even parts. In fact, k=2 is more appropriate for this data set, despite the data set containing 3 classes. As with any other clustering algorithm, the k-means result relies on the data set to satisfy the assumptions made by the clustering algorithms. It works well on some data sets, while failing on others.

The result of k-means can also be seen as the Voronoi cells of the cluster means. Since data is split halfway between cluster means, this can lead to suboptimal splits as can be seen in the "mouse" example. The Gaussian models used by the Expectation-maximization algorithm (which can be seen as a generalization of k-means) are more flexible here by having both variances and covariances. The EM result is thus able to accommodate clusters of variable size much better than k-means as well as correlated clusters (not in this example).

Applications

k-means clustering in particular when using heuristics such as Lloyd's algorithm is rather easy to implement and apply even on large data sets. As such, it has been successfully used in various topics, ranging from market segmentation, computer vision, geostatistics,[23] and astronomy to agriculture. It often is used as a preprocessing step for other algorithms, for example to find a starting configuration.

Vector quantization

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Two-channel (for illustration purposes -- red and green only) color image.
Vector quantization of colors present in the image above into Voronoi cells using k-means.

k-means originates from signal processing, and still finds use in this domain. For example in computer graphics, color quantization is the task of reducing the color palette of an image to a fixed number of colors k. The k-means algorithm can easily be used for this task and produces competitive results. Other uses of vector quantization include non-random sampling, as k-means can easily be used to choose k different but prototypical objects from a large data set for further analysis.

Cluster analysis

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In cluster analysis, the k-means algorithm can be used to partition the input data set into k partitions (clusters).

However, the pure k-means algorithm is not very flexible, and as such of limited use (except for when vector quantization as above is actually the desired use case!). In particular, the parameter k is known to be hard to choose (as discussed below) when not given by external constraints. In contrast to other algorithms, k-means can also not be used with arbitrary distance functions or be use on non-numerical data. For these use cases, many other algorithms have been developed since.

Feature learning

k-means clustering has been used as a feature learning (or dictionary learning) step, which can be used in the for (semi-)supervised learning or unsupervised learning.[24] The basic approach is first to train a k-means clustering representation, using the input training data (which need not be labelled). Then, to project any input datum into the new feature space, we have a choice of "encoding" functions, but we can use for example the thresholded matrix-product of the datum with the centroid locations, the distance from the datum to each centroid, or simply an indicator function for the nearest centroid,[24][25] or some smooth transformation of the distance.[26] Alternatively, by transforming the sample-cluster distance through a Gaussian RBF, one effectively obtains the hidden layer of a radial basis function network.[27]

This use of k-means has been successfully combined with simple, linear classifiers for semi-supervised learning in NLP (specifically for named entity recognition)[28] and in computer vision. On an object recognition task, it was found to exhibit comparable performance with more sophisticated feature learning approaches such as autoencoders and restricted Boltzmann machines.[26] However, it generally requires more data than the sophisticated methods, for equivalent performance, because each data point only contributes to one "feature" rather than multiple.[24]

Relation to other statistical machine learning algorithms

k-means clustering, and its associated expectation-maximization algorithm, is a special case of a Gaussian mixture model, specifically, the limit of taking all covariances as diagonal, equal, and small. It is often easy to generalize a k-means problem into a Gaussian mixture model.[29] Another generalization of the k-means algorithm is the K-SVD algorithm, which estimates data points as a sparse linear combination of "codebook vectors". K-means corresponds to the special case of using a single codebook vector, with a weight of 1.[30]

Mean shift clustering

Basic mean shift clustering algorithms maintain a set of data points the same size as the input data set. Initially, this set is copied from the input set. Then this set is iteratively replaced by the mean of those points in the set that are within a given distance of that point. By contrast, k-means restricts this updated set to k points usually much less than the number of points in the input data set, and replaces each point in this set by the mean of all points in the input set that are closer to that point than any other (e.g. within the Voronoi partition of each updating point). A mean shift algorithm that is similar then to k-means, called likelihood mean shift, replaces the set of points undergoing replacement by the mean of all points in the input set that are within a given distance of the changing set.[31] One of the advantages of mean shift over k-means is that there is no need to choose the number of clusters, because mean shift is likely to find only a few clusters if indeed only a small number exist. However, mean shift can be much slower than k-means, and still requires selection of a bandwidth parameter. Mean shift has soft variants much as k-means does.

Principal component analysis (PCA)

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Bilateral filtering

k-means implicitly assumes that the ordering of the input data set does not matter. The bilateral filter is similar to K-means and mean shift in that it maintains a set of data points that are iteratively replaced by means. However, the bilateral filter restricts the calculation of the (kernel weighted) mean to include only points that are close in the ordering of the input data.[31] This makes it applicable to problems such as image denoising, where the spatial arrangement of pixels in an image is of critical importance.

Similar problems

The set of squared error minimizing cluster functions also includes the Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "[http://www.blitzkappers.com/?option=com_k2&view=itemlist&task=user&id=64019 public listed property developers In singapore] agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-medoids
algorithm, an approach which forces the center point of each cluster to be one of the actual points, i.e., it uses medoids in place of centroids.

Software

Free

Commercial

Source code

  • ELKI and Weka are written in Java and include k-means and variations
  • K-means application in PHP,[35] using VB,[36] using Perl,[37] using C++,[38] using Matlab,[39] using Ruby,[40][41] using Python with scipy,[42] using X10[43]
  • A parallel out-of-core implementation in C[44]
  • An open-source collection of clustering algorithms, including k-means, implemented in Javascript.[45] Online demo.[46]

Visualization, animation and examples

See also

References

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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  6. 6.0 6.1 6.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  7. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  8. Since the square root is a monotone function, this also is the minimum Euclidean distance assignment.
  9. 9.0 9.1 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  10. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  11. 11.0 11.1 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  12. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  13. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  14. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  15. 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  16. Template:Cite thesis[1]Template:Dead link
  17. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  18. 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  19. 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  20. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  21. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  22. 22.0 22.1 E.M. Mirkes, K-means and K-medoids applet. University of Leicester, 2011.
  23. Honarkhah, M and Caers, J, 2010, Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, Mathematical Geosciences, 42: 487 - 517
  24. 24.0 24.1 24.2

    My name is Dorris Bosanquet but everybody calls me Dorris. I'm from Iceland. I'm studying at the university (3rd year) and I play the Bass Guitar for 5 years. Usually I choose music from my famous films ;).
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  25. 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

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  26. 26.0 26.1 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  27. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  28. 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

    You can view that web-site... ccleaner free download
  29. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  30. [2].
  31. 31.0 31.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  32. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  33. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  34. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  35. http://www25.brinkster.com/denshade/kmeans.php.htm
  36. K-Means Clustering Tutorial: Download
  37. Perl script for Kmeans clustering
  38. Antonio Gulli's coding playground: K-means in C
  39. K-Means Clustering Tutorial: Matlab Code
  40. AI4R :: Artificial Intelligence for Ruby
  41. reddavis/K-Means · GitHub
  42. K-means clustering and vector quantization (scipy.cluster.vq) — SciPy v0.11 Reference Guide (DRAFT)
  43. http://dist.codehaus.org/x10/applications/samples/KMeansDist.x10
  44. http://www.cs.princeton.edu/~wdong/kmeans/
  45. http://code.google.com/p/figue/ FIGUE
  46. http://jydelort.appspot.com/resources/figue/demo.html
  47. Clustering - K-means demo
  48. siebn.de - YAK-Means
  49. k-Means and Voronoi Tesselation: Built with Processing | Information & Visualization
  50. Hyper-threaded Java - JavaWorld
  51. Color clustering
  52. Interactive step-by-step examples in Javascript of good and bad k-means clustering
  53. Clustergram: visualization and diagnostics for cluster analysis (R code) | R-statistics blog