Tricritical point

Topological data analysis (TDA) is a new area of study aimed at having applications in areas such as data mining and computer vision. The main problems are:

1. how one infers high-dimensional structure from low-dimensional representations; and
2. how one assembles discrete points into global structure.

The human brain can easily extract global structure from representations in a strictly lower dimension, i.e. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting discrete data into continuous images, e.g. dot-matrix printers and televisions communicate images via arrays of discrete points.

The main method used by topological data analysis is:

1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
2. Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homology.
3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number which is called a barcode.

Point cloud data

Data is often represented as points in a Euclidean n-dimensional space E_n. The global shape of the data may provide information about the phenomena that the data represent.

One type of data set for which global features are certainly present is the so-called point cloud data coming from physical objects in 3D. E.g. a laser can scan an object at a set of discrete points and the cloud of such points can be used in a computer representation of the object. Point cloud data refers to any collection of points in E_n or a (perhaps noisy) sample of points on a lower-dimensional subset.

For point clouds in low-dimensional spaces there are numerous approaches for inferring features based on planar projections in the fields of computer graphics and statistics. Topological data analysis is needed when the spaces are high-dimensional or too twisted to allow planar projections to faithfully represent the features of the point cloud.

To convert a point cloud in a metric space into a global object, use the point cloud as the vertices of a graph whose edges are determined by proximity, then turn the graph into a simplicial complex and use algebraic topology to study it. An alternative approach is the minimum spanning tree-based method in the geometric data clustering. If a group of data points forms a cluster, then the geometry of this point cloud can be determined.

Background

1. Topology
2. Simplicial Complex
3. Nerve and Cover

Topological Data analysis refers to different methods and representations in order to cluster variegated data via a point cloud stated above. The following are various methods to do so.

Combinatorial Representations

1. Cech Complex. The Cech complex $C_{ϵ}$ is the nerve of the cover of balls of radius $ϵ$ around each point in a set. Since balls are convex and convex sets are contractible, its nerve captures the topology of the cover. The Cech Complex is not computed in practice due to its computational complexity. The uniform ball radii imply an assumption of uniform sampling on the input, which is not valid in a real world dataset. Non-uniform radii methods can also be used, such as in the case of the Alpha Simplex.
2. Alpha Complex. The Voronoi diagram is the set of all Voronoi regions for the points in $S\subseteq Y$. This diagram is considered a closed cover for $Y$. The Delaunay complex is the nerve of the Voronoi diagram. The Voronoi cover and its nerve are fundamental geometric objects and have been extensively studied within computational geometry. Alpha complexes are constructed by first building the Delaunay complex. For each simplex of the Delaunay complex, we compute the minimum scale at which each simplex enters the alpha complex. Then the simplices are sorted by their minimum scale to get a partial order of simplices. The alpha complex is not formed with any scale $ϵ$ using this ordering. Efficient algorithms and software exist for computing Delaunay complexes, and in turn, alpha complexes in 2 and 3 dimensions. However, the construction of the Delaunay complex is difficult in higher dimensions.
3. Vietoris-Rips complex

Topological Invariants

1. Definition. Topological Invariants
2. Euler characteristic
3. Simplicial homology

Multiscale Invariants

1. Multifiltration Model. Morse Theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
2. Persistent homology. See homology for an introduction to the notation.

Persistent homology essentially calculates homology groups at different spatial resolutions to see which features persist over a wide range of length scales. It is assumed that important features and structures are the ones that persist. We define persistent homology as follows: Let $K^l$ be a filtration. The p-persistent kth homology group of $K^l$ is $H_k^{l,p}=Z_k^l/(B_k^{l+p}\cap Z_k^l)$.

Let $z$ be a nonbounding $k$-cycle created at time $I$ by simplex $\sigma$ and let $z^{\prime }\sim z$ be a homologous $k$-cycle that becomes a boundary cycle at time $J$ by simplex $\tau$. Then we can define the persistence interval associated to $z$ as $(I,J)$. We call $\sigma$ the creator of $z$ and $\tau$ the destroyer of $z$. If $z$ does not have a destroyer, its persistence is $\mathrm{\infty }$. Instead of using an index-based filtration, we can use a time-based filtration. Let $K$ be a simplicial complex and $K^{\rho }=\{{\sigma }^i\in K|\rho ({\sigma }^i)\le \rho \}$ be a filtration defined for an associated map $\rho :S(K)\to \mathbb{ℝ}$ that maps simplices in the final complex to real numbers. Then for all real numbers $\pi \ge 0$, the $\pi$-persistent kth homology group of $K^{\rho }$ is $H_k^{\rho ,\pi }=Z_k^{\rho }/(B_k^{\rho +\pi }\cap Z_k^{\rho })$. The persistence of a $k$-cycle created at time ${\rho }_i$ and destroyed at ${\rho }_j$ is ${\rho }_j-{\rho }_i$.

There are various software packages for computing persistence intervals of a finite filtration, such as javaPlex, Dionysus, Perseus, and PHAT.

See also

• Dimensionality reduction
• Data mining
• Computer vision
• Computational topology
• Digital topology
• Digital Morse theory
• Shape analysis
• Size theory
• Structured data analysis (statistics)

Further reading

• Topological Methods in Scientific Computing, Statistics and Computer Science Stanford group
• BARCODES: THE PERSISTENT TOPOLOGY OF DATA
• Topological Data Analysis: the algebraic topology of point data clouds?
• 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
• TOPOLOGY AND DATA, GUNNAR CARLSSON, BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 46, Number 2, April 2009, Pages 255–308, Article electronically published on January 29, 2009
• Computational Topology: An Introduction, Herbert Edelsbrunner, John L. Harer, AMS Bookstore, 2010, ISBN 978-0-8218-4925-5
• Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, Editors Valerio Pascucci, Hans Hagen, Xavier Tricoche, Julien Tierny, Springer, 2010, ISBN 978-3-642-15013-5
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Ayasdi Resources on Topological Data Analysis for Big Data
• Software package for computing persistent homology