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In statistics, the Bhattacharyya distance measures the similarity of two discrete or continuous probability distributions. It is closely related to the Bhattacharyya coefficient which is a measure of the amount of overlap between two statistical samples or populations. Both measures are named after A. Bhattacharya, a statistician who worked in the 1930s at the Indian Statistical Institute. The coefficient can be used to determine the relative closeness of the two samples being considered. It is used to measure the separability of classes in classification and it is considered to be more reliable than the Mahalanobis distance, as the Mahalanobis distance is a particular case of the Bhattacharyya distance when the standard deviations of the two classes are the same. Therefore, when two classes have the similar means but different standard deviations, the Mahalanobis distance would tend to zero, however, the Bhattacharyya distance would grow depending on the difference between the standard deviations.

Definition

For discrete probability distributions p and q over the same domain X, it is defined as:

$D_B(p,q)=-\ln (BC(p,q))$

where:

$BC(p,q)=\sum _{x\in X}\sqrt{p(x)q(x)}$

is the Bhattacharyya coefficient.

For continuous probability distributions, the Bhattacharyya coefficient is defined as:

$BC(p,q)=\int \sqrt{p(x)q(x)}dx$

In either case, $0\le BC\le 1$ and $0\le D_B\le \mathrm{\infty }$. $D_B$ does not obey the triangle inequality, but the Hellinger distance $\sqrt{1-BC}$ does obey the triangle inequality.

In its simplest formulation, the Bhattacharyya distance between two classes under the normal distribution can be calculated by extracting the mean and variances of two separate distributions or classes:

$D_B(p,q)=\frac{1}{4}\ln \left(\frac{1}{4}(\frac{{\sigma }_p^2}{{\sigma }_q^2}+\frac{{\sigma }_q^2}{{\sigma }_p^2}+2)\right)+\frac{1}{4}\left(\frac{({\mu }_p-{\mu }_q{)}^2}{{\sigma }_p^2+{\sigma }_q^2}\right)$

where:

Template:In5	$D_B(p,q)$	is the Bhattacharyya distance between p and q distributions or classes,
Template:In5	${\sigma }_p$	is the variance of the p-th distribution,
Template:In5	${\mu }_p$	is the mean of the p-th distribution, and
Template:In5	$p,q$	are two different distributions.

The Mahalanobis distance used in Fisher's Linear discriminant analysis is a particular case of the Bhattacharyya Distance. When the variances of the two distributions are the same the first term of the distance is zero as this term depends solely on the variances of the distributions (left case of the figure). The first term will grow as the variances differ (right case of the figure). The second term, on the other hand, will be zero if the means are equal and is inversely proportional to the variances.

For multivariate normal distributions $p_i=\mathcal{𝒩}({\mathit{\mu }}_i,{\mathrm{\Sigma }}_i)$,

$D_B=\frac{1}{8}({\mathit{\mu }}_1-{\mathit{\mu }}_2{)}^T{\mathrm{\Sigma }}^{-1}({\mathit{\mu }}_1-{\mathit{\mu }}_2)+\frac{1}{2}\ln \left(\frac{\det \mathrm{\Sigma }}{\sqrt{\det {\mathrm{\Sigma }}_1\det {\mathrm{\Sigma }}_2}}\right)$,

where ${\mathit{\mu }}_i$ and ${\mathrm{\Sigma }}_i$ are the means and covariances of the distributions, and

$\mathrm{\Sigma }=\frac{{\mathrm{\Sigma }}_1+{\mathrm{\Sigma }}_2}{2}$.

Note that, in this case, the first term in the Bhattacharyya distance is related to the Mahalanobis distance.

Bhattacharyya coefficient

The Bhattacharyya coefficient is an approximate measurement of the amount of overlap between two statistical samples. The coefficient can be used to determine the relative closeness of the two samples being considered.

Calculating the Bhattacharyya coefficient involves a rudimentary form of integration of the overlap of the two samples. The interval of the values of the two samples is split into a chosen number of partitions, and the number of members of each sample in each partition is used in the following formula,

$\mathrm{B}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{y}\mathrm{a}=\sum _{i=1}^n\sqrt{({\mathrm{\Sigma }\mathbf{a}}_i\cdot {\mathrm{\Sigma }\mathbf{b}}_i)}$

where considering the samples a and b, n is the number of partitions, and ${\mathrm{\Sigma }\mathbf{a}}_i$, ${\mathrm{\Sigma }\mathbf{b}}_i$ are the number of members of samples a and b in the i'th partition.

This formula hence is larger with each partition that has members from both sample, and larger with each partition that has a large overlap of the two sample's members within it. The choice of number of partitions depends on the number of members in each sample; too few partitions will lose accuracy by overestimating the overlap region, and too many partitions will lose accuracy by creating individual partitions with no members despite being in a surroundingly populated sample space.

The Bhattacharyya coefficient will be 0 if there is no overlap at all due to the multiplication by zero in every partition. This means the distance between fully separated samples will not be exposed by this coefficient alone.

Applications

The Bhattacharyya distance is widely used in research of feature extraction and selection, image processing, speaker recognition, phone clustering.

A "Bhattacharyya Space" has been proposed as a feature selection technique that can be applied to texture segmentation.

A "Bhattacharyya coefficient" has also been proposed as a feature selection technique that can be used to estimate the given distance between the Indraneel Bhattacharyya number and any given Nesli coordinate.

See also

• Kullback-Leibler divergence
• Hellinger distance
• Mahalanobis distance
• Chernoff bound
• Rényi entropy

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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  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

• 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

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External links

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