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| | '''Bootstrap aggregating''' ('''bagging''') is a [[Ensemble learning|machine learning ensemble]] [[meta-algorithm]] designed to improve the stability and accuracy of [[machine learning]] algorithms used in [[statistical classification]] and [[Regression analysis|regression]]. It also reduces [[variance]] and helps to avoid [[overfitting]]. Although it is usually applied to [[Decision tree learning|decision tree]] methods, it can be used with any type of method. Bagging is a special case of the [[Ensemble learning|model averaging]] approach. |
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| | ==Description of the technique== |
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| | Given a standard [[training set]] ''D'' of size ''n'', bagging generates ''m'' new training sets <math>D_i</math>, each of size ''n′'', by [[Sampling (statistics)|sampling]] from ''D'' [[Probability distribution#With finite support|uniformly]] and [[Sampling (statistics)#Replacement of selected units|with replacement]]. By sampling with replacement, some observations may be repeated in each <math>D_i</math>. If ''n[[prime (symbol)|′]]''=''n'', then for large ''n'' the set <math>D_i</math> is expected to have the fraction (1 - 1/''[[e (mathematical constant)|e]]'') (≈63.2%) of the unique examples of ''D'', the rest being duplicates.<ref>Aslam, Javed A.; Popa, Raluca A.; and Rivest, Ronald L. (2007); [http://people.csail.mit.edu/rivest/pubs/APR07.pdf ''On Estimating the Size and Confidence of a Statistical Audit''], Proceedings of the Electronic Voting Technology Workshop (EVT '07), Boston, MA, August 6, 2007. More generally, when drawing with replacement ''n′'' values out of a set of ''n'' (different and equally likely), the expected number of unique draws is <math>n(1 - e^{-n'/n})</math>.</ref> This kind of sample is known as a [[bootstrap (statistics)|bootstrap]] sample. The ''m'' models are fitted using the above ''m'' bootstrap samples and combined by averaging the output (for regression) or voting (for classification). |
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| | Bagging leads to "improvements for unstable procedures" (Breiman, 1996), which include, for example, [[neural nets]], [[classification and regression tree]]s, and subset selection in [[linear regression]] (Breiman, 1994). An interesting application of bagging showing improvement in preimage learning is provided here.<ref>Sahu, A., Runger, G., Apley, D., Image denoising with a multi-phase kernel principal component approach and an ensemble version, IEEE Applied Imagery Pattern Recognition Workshop, pp.1-7, 2011.</ref> On the other hand, it can mildly degrade the performance of stable methods such as K-nearest neighbors (Breiman, 1996). |
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| | To illustrate the basic principles of bagging, below is an analysis on the relationship between [[ozone]] and temperature (data from [[Peter Rousseeuw|Rousseeuw]] and Leroy (1986), available at [[classic data sets]], analysis done in [[R (programming language)|R]]). |
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| | The relationship between temperature and ozone in this data set is apparently non-linear, based on the scatter plot. To mathematically describe this relationship, [[local regression|LOESS]] smoothers (with span 0.5) are used. |
| | Instead of building a single smoother from the complete data set, 100 [[bootstrap (statistics)|bootstrap]] samples of the data were drawn. Each sample is different from the original data set, yet resembles it in distribution and variability. For each bootstrap sample, a LOESS smoother was fit. Predictions from these 100 smoothers were then made across the range of the data. The first 10 predicted smooth fits appear as grey lines in the figure below. The lines are clearly very ''wiggly'' and they overfit the data - a result of the span being too low. |
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| | But taking the average of 100 smoothers, each fitted to a subset of the original data set, we arrive at one bagged predictor (red line). Clearly, the mean is more stable and there is less [[overfitting|overfit]]. |
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| | ==Bagging for nearest neighbour classifiers== |
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| | It is well known that the [[Bayes classifier|risk]] of a 1 nearest neighbour (1NN) classifier is at most twice the risk of the [[Bayes classifier]], but there are no guarantees that this classifier will be [[Bayes classifier|consistent]]. By careful choice of the size of the resamples, bagging can lead to substantial improvements of the performance of the 1NN classifier. By taking a large number of resamples of the data of size <math>n'</math>, the bagged nearest neighbour classifier will be consistent provided <math>n' \to \infty</math> diverges but <math>n'/n \to 0</math> as the sample size <math>n \to \infty</math>. |
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| == Buy Christian Louboutin Pigalle == | | Under infinite simulation, the bagged nearest neighbour classifier can be viewed as a [[weighted nearest neighbour classifier]]. Suppose that the feature space is <math>d</math> dimensional and denote by <math>C^{bnn}_{n,n'}</math> the bagged nearest neighbour classifier based on a training set of size <math>n</math>, with resamples of size <math>n'</math>. In the infinite sampling case, under certain regularity conditions on the class distributions, the [[Bayes classifier|excess risk]] has the following asymptotic expansion<ref name = "Samworth12">{{Cite Journal | author = Samworth R. J. |
| | | title = Optimal weighted nearest neighbour classifiers |
| | | journal = [[Annals of Statistics]] |
| | | volume = 40 |
| | | issue = 5 |
| | | pages = 2733-2763 |
| | | year = 2012 |
| | | doi = 10.1214/12-AOS1049 |
| | }} |
| | </ref> |
| | :<math>\mathcal{R}_\mathcal{R}(C^{bnn}_{n,n'}) - \mathcal{R}_{\mathcal{R}}(C^{Bayes}) = \left(B_1 \frac{n'}{n} + B_2 \frac 1 {(n')^{4/d}}\right) \{1+o(1)\},</math> |
| | for some constants <math>B_1</math> and <math>B_2</math>. The optimal choice of <math>n'</math>, that balances the two terms in the asymptotic expansion, is given by <math>n' = B n^{d/(d+4)}</math> for some constant <math>B</math>. |
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| | Bagging ('''B'''ootstrap '''agg'''regat'''ing''') was proposed by [[Leo Breiman]] in 1994 to improve the classification by combining classifications of randomly generated training sets. See Breiman, 1994. Technical Report No. 421. |
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| | == See also == |
|
| | *[[Boosting (meta-algorithm)]] |
| <li>?mod=viewthread&tid=65056</li>
| | *[[Bootstrapping (statistics)]] |
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| | *[[Cross-validation (statistics)]] |
| <li>?mod=viewthread&tid=2063713&extra=</li>
| | *[[Random forest]] |
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| == videos Abercrombie & Fitch Sydney == | | == References == |
| | {{Reflist}} |
| | * {{Cite journal |
| | | last = Breiman |
| | | first = Leo |
| | | authorlink = Leo Breiman |
| | | title = Bagging predictors |
| | | journal = [[Machine Learning (journal)|Machine Learning]] |
| | | volume = 24 |
| | | issue = 2 |
| | | pages = 123–140 |
| | | year = 1996 |
| | | id = {{citeseerx|10.1.1.121.7654}} |
| | | doi = 10.1007/BF00058655 |
| | }} |
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Bootstrap aggregating (bagging) is a machine learning ensemble meta-algorithm designed to improve the stability and accuracy of machine learning algorithms used in statistical classification and regression. It also reduces variance and helps to avoid overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the model averaging approach.
Description of the technique
Given a standard training set D of size n, bagging generates m new training sets
, each of size n′, by sampling from D uniformly and with replacement. By sampling with replacement, some observations may be repeated in each
. If n′=n, then for large n the set
is expected to have the fraction (1 - 1/e) (≈63.2%) of the unique examples of D, the rest being duplicates.[1] This kind of sample is known as a bootstrap sample. The m models are fitted using the above m bootstrap samples and combined by averaging the output (for regression) or voting (for classification).
Bagging leads to "improvements for unstable procedures" (Breiman, 1996), which include, for example, neural nets, classification and regression trees, and subset selection in linear regression (Breiman, 1994). An interesting application of bagging showing improvement in preimage learning is provided here.[2] On the other hand, it can mildly degrade the performance of stable methods such as K-nearest neighbors (Breiman, 1996).
Example: Ozone data
Template:Essay-like
To illustrate the basic principles of bagging, below is an analysis on the relationship between ozone and temperature (data from Rousseeuw and Leroy (1986), available at classic data sets, analysis done in R).
The relationship between temperature and ozone in this data set is apparently non-linear, based on the scatter plot. To mathematically describe this relationship, LOESS smoothers (with span 0.5) are used.
Instead of building a single smoother from the complete data set, 100 bootstrap samples of the data were drawn. Each sample is different from the original data set, yet resembles it in distribution and variability. For each bootstrap sample, a LOESS smoother was fit. Predictions from these 100 smoothers were then made across the range of the data. The first 10 predicted smooth fits appear as grey lines in the figure below. The lines are clearly very wiggly and they overfit the data - a result of the span being too low.
But taking the average of 100 smoothers, each fitted to a subset of the original data set, we arrive at one bagged predictor (red line). Clearly, the mean is more stable and there is less overfit.
Bagging for nearest neighbour classifiers
It is well known that the risk of a 1 nearest neighbour (1NN) classifier is at most twice the risk of the Bayes classifier, but there are no guarantees that this classifier will be consistent. By careful choice of the size of the resamples, bagging can lead to substantial improvements of the performance of the 1NN classifier. By taking a large number of resamples of the data of size
, the bagged nearest neighbour classifier will be consistent provided
diverges but
as the sample size
.
Under infinite simulation, the bagged nearest neighbour classifier can be viewed as a weighted nearest neighbour classifier. Suppose that the feature space is
dimensional and denote by
the bagged nearest neighbour classifier based on a training set of size
, with resamples of size
. In the infinite sampling case, under certain regularity conditions on the class distributions, the excess risk has the following asymptotic expansion[3]
![{\displaystyle {\mathcal {R}}_{\mathcal {R}}(C_{n,n'}^{bnn})-{\mathcal {R}}_{\mathcal {R}}(C^{Bayes})=\left(B_{1}{\frac {n'}{n}}+B_{2}{\frac {1}{(n')^{4/d}}}\right)\{1+o(1)\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2757fe1688d73c3b3e8c4c9e3cee7dc2bea709f4)
for some constants
and
. The optimal choice of
, that balances the two terms in the asymptotic expansion, is given by
for some constant
.
History
Bagging (Bootstrap aggregating) was proposed by Leo Breiman in 1994 to improve the classification by combining classifications of randomly generated training sets. See Breiman, 1994. Technical Report No. 421.
See also
References
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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
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
- 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
- ↑ Aslam, Javed A.; Popa, Raluca A.; and Rivest, Ronald L. (2007); On Estimating the Size and Confidence of a Statistical Audit, Proceedings of the Electronic Voting Technology Workshop (EVT '07), Boston, MA, August 6, 2007. More generally, when drawing with replacement n′ values out of a set of n (different and equally likely), the expected number of unique draws is
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- ↑ Sahu, A., Runger, G., Apley, D., Image denoising with a multi-phase kernel principal component approach and an ensemble version, IEEE Applied Imagery Pattern Recognition Workshop, pp.1-7, 2011.
- ↑ Template:Cite Journal