Hájek–Le Cam convolution theorem: Difference between revisions

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Rossmo's formula is a geographic profiling formula to predict where a serial criminal lives. The formula was developed and patented^[1] by criminologist Kim Rossmo and integrated into a specialized crime analysis software product called Rigel. The Rigel product is developed by the software company Environmental Criminology Research Inc. (ECRI), which Rossmo co-founded.^[2]

Formula

Imagine a map with an overlaying grid of little squares named sectors. If this map is a raster image file on a computer, these sectors are pixels. A sector $S_{i, j}$ is the square on row i and column j, located at coordinates (X_i,Y_j). The following function gives the probability $p_{i, j}$ of the position of the serial criminal residing within a specific sector (or point) $(X_{i}, Y_{j})$ :^[3]

p_{i, j} = k \sum_{n = 1}^{(t o t a l c r i m e s)} [\underset{1^{s t} t e r m}{\underset{⏟}{\frac{ϕ_{i j}}{(| X_{i} - x_{n} | + | Y_{j} - y_{n} |)^{f}}}} + \underset{2^{n d} t e r m}{\underset{⏟}{\frac{(1 - ϕ_{i j}) (B^{g - f})}{(2 B - ∣ X_{i} - x_{n} ∣ - ∣ Y_{j} - y_{n} ∣)^{g}}}}],

where: $ϕ_{i j} = {\begin{cases} 1, & i f (∣ X_{i} - x_{n} ∣ + ∣ Y_{j} - y_{n} ∣) > B \\ 0, & e l s e \end{cases}$

Here the summation is over past crimes located at coordinates (x_n,y_n). $ϕ_{i j}$ is a characteristic function that returns 0 when a point $(X_{i}, Y_{j})$ is an element of the buffer zone B (the neighborhood of a criminal residence that is swept out by a radius of B from its center). $ϕ_{i j}$ allows p to switch between the two terms. If a crime occurs within the buffer zone, then $ϕ_{i j} = 0$ and, thus, the first term does not contribute to the overall result. This is a prerogative for defining the first term in the case when the distance between a point (or pixel) becomes equal to zero. When $ϕ_{i j} = 1$ , the 1st term is used to calculate $p_{i, j}$ .

$∣ X_{i} - x_{n} ∣ + ∣ Y_{j} - y_{n} ∣$ is the Manhattan distance between a point $(X_{i}, Y_{j})$ and the n-th crime site $(x_{n}, y_{n})$ .

Explanation

The summation in the formula consists of two terms. The first term describes the idea of decreasing probability with increasing distance. The second term deals with the concept of a buffer zone. The variable $ϕ$ is used to put more weight on one of the two ideas. The variable $B$ describes the radius of the buffer zone. The constant $k$ is empirically determined.

The main idea of the formula is that the probability of crimes first increases as one moves through the buffer zone away from the hotzone, but decreases afterwards. The variable $f$ can be chosen so that it works best on data of past crimes. The same idea goes for the variable $g$ .

The distance is calculated with the Manhattan distance formula.

Applications

The formula has been applied to fields other than forensics. Because of the buffer zone idea, the formula works well for studies concerning predatory animals such as white sharks. ^[4]

This formula and math behind it were used in crime detecting in the Pilot episode of the TV-series Numb3rs and in the 100th episode of the same show, called "Disturbed".

Notes

↑ Rossmo, D. K. (1996). U.S. Patent No. 5,781,704. Washington, DC: U.S. Patent and Trademark Office.
↑ Rich, T. and Shively, M (2004, December).P. 14. A Methodology for Evaluating Geographic Profiling Software. U.S. Department of Justice, Retrieved from https://www.ncjrs.gov/pdffiles1/nij/grants/208993.pdf
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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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References

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[1] Rossmo, D. K. (1996). U.S. Patent No. 5,781,704. Washington, DC: U.S. Patent and Trademark Office.

[2] Rich, T. and Shively, M (2004, December).P. 14. A Methodology for Evaluating Geographic Profiling Software. U.S. Department of Justice, Retrieved from https://www.ncjrs.gov/pdffiles1/nij/grants/208993.pdf

[3] 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

[4] 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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+'''Rossmo's formula''' is a [[geographic profiling]] formula to predict where a serial criminal lives. The formula was developed and  patented<ref>Rossmo, D. K. (1996). U.S. Patent No. 5,781,704. Washington, DC: U.S. Patent and Trademark Office.</ref> by [[criminologist]] [[Kim Rossmo]] and integrated into a specialized crime analysis software product called [http://www.ecricanada.com/rigel/ Rigel].  The Rigel product is developed by the software company [[Environmental Criminology Research Inc.]] (ECRI), which Rossmo co-founded.<ref>Rich, T. and Shively, M (2004, December).P. 14. A Methodology for Evaluating Geographic Profiling Software. U.S. Department of Justice, Retrieved from https://www.ncjrs.gov/pdffiles1/nij/grants/208993.pdf</ref>
+==Formula==
+Imagine a map with an overlaying grid of little squares named sectors. If this map is a [[raster image]] file on a computer, these sectors are pixels. A sector <math>S_{i,j}</math> is the square on row ''i'' and column ''j'', located at coordinates (''X<sub>i</sub>'',''Y<sub>j</sub>'').
+The following function gives the probability <math>p_{i,j}</math> of the position of the serial criminal residing within a specific sector (or point) <math>(X_{i},Y_{j})</math>:<ref>{{cite journal
+| first = Kim D.
+| last = Rossmo
+| title = Geographic profiling: target patterns of serial murderers
+| page = 225
+| year = 1995
+| publisher = [[Simon Fraser University]]
+| url = http://lib-ir.lib.sfu.ca/bitstream/1892/8121/1/b17675819.pdf
+}}
+</ref>
+:<math>
+p_{i,j} = k \sum_{n=1}^{(\mathrm{total\;crimes})}
+\left [
+\underbrace{
+\frac{\phi_{ij}}{
+(|X_i-x_n| + |Y_j-y_n|)^f
+}
+}_{ 1^{\mathrm{st}}\mathrm{\;term} }
++
+\underbrace{
+\frac{(1-\phi_{ij})(B^{g-f})}{
+(2B - \mid X_i - x_n \mid - \mid Y_j-y_n \mid)^g
+}
+}_{ 2^{\mathrm{nd}}\mathrm{\;term} }
+\right ],
+</math>
+where:
+<math>
+\phi_{ij} =
+\begin{cases}
+, & \mathrm{\quad if\;} ( \mid X_i - x_n \mid + \mid Y_j - y_n \mid ) > B \quad \\
+, & \mathrm{\quad else}
+\end{cases}
+</math>
+Here the summation is over past crimes located at coordinates (''x<sub>n</sub>'',''y<sub>n</sub>'').
+<math>\phi_{ij}</math> is a [[characteristic function (probability theory)|characteristic function]] that returns 0 when a point <math>(X_{i},Y_{j})</math> is an element of the buffer zone B (the neighborhood of a criminal residence that is swept out by a radius of B from its center). <math>\phi_{ij}</math> allows ''p'' to switch between the two terms.
+If a crime occurs within the buffer zone, then <math>\phi_{ij}=0</math> and, thus, the first term does not contribute to the overall result.
+This is a prerogative for defining the first term in the case when the distance between a point (or pixel) becomes equal to zero.
+When <math>\phi_{ij} = 1</math>, the 1st term is used to calculate <math>p_{i,j}</math>.
+<math>\mid X_i - x_n \mid + \mid Y_j - y_n \mid</math>
+is the [[Manhattan distance]] between a point <math>(X_{i},Y_{j})</math> and the ''n''-th crime site <math>(x_{n},y_{n})</math>.
+<!-- hiding this until the math gets corrected
+==Alternative Implementation==
+<math>p_{i,j}</math> is not well suited for image processing because of the asymptotic behavior near the coordinates of a crime site.
+Alternatively, Rossmo's function may use other [[distance decay]] functions instead of <math>\frac{1}{(\mathrm{Mathattan\;Distance})^f}</math>.
+One method would be to use a probability distribution similar to the [[Gaussian Distribution]] as a distance decay function:
+<math>
+^{\mathrm{st}}\mathrm{\;term}(x,y) =
+\left\lfloor
+(\mathrm{\#\;of\;colors}) \times \sum_{n=1}^{(\mathrm{total\;crimes})}
+\frac{1}{\sqrt{e^{(\mid x - C_{n}(x) \mid^{2} + \mid y - C_{n}(y) \mid^{2})}}}
+\right\rfloor
+</math>
+If implementing on a computer, the maximum value of p() matches the maximum value of a set of colors being used to create the n by m '''Jeopardy Surface''' matrix J. The elements of the matrix J may represent the pixel values of an image.
+Where:
+<math>
+J =
+\begin{bmatrix}
+p(n,0)   & \cdots   & \;       & p(n,m) \\
+\vdots   & \ddots   & \;       & \vdots \\
+p(x,0)   & \cdots   & p(x,y)   & \vdots \\
+\vdots   & \;       & \;       & \vdots \\
+p(0,0)   & p(1,0)   & p(2,0)   & \cdots
+\end{bmatrix}
+</math>
+-->
+==Explanation==
+The summation in the formula consists of two terms. The first term describes the idea of ''decreasing probability with increasing distance''. The second term deals with the concept of a ''buffer zone''. The variable <math>\phi</math> is used to put more weight on one of the two ideas. The variable <math>B</math> describes the radius of the buffer zone. The constant <math>k</math> is empirically determined.
+The main idea of the formula is that the probability of crimes first increases as one moves through the buffer zone away from the ''hotzone'', but decreases afterwards. The variable <math>f</math> can be chosen so that it works best on data of past crimes. The same idea goes for the variable <math>g</math>.
+The distance is calculated with the [[Taxicab geometry|Manhattan distance formula]].
+==Applications==
+The formula has been applied to fields other than forensics. Because of the buffer zone idea, the formula works well for studies concerning predatory animals such as white sharks.
+<ref>{{cite journal
+| author = R. A. Martin; D. K. Rossmo; N. Hammerschlag
+| title = Hunting patterns and geographic profiling of white shark predation
+| journal = Journal of Zoology
+| volume = 279
+| pages = 111–118
+| year = 2009
+| doi = 10.1111/j.1469-7998.2009.00586.x
+| url = http://www.rjd.miami.edu/scientific-publications/pdf/Martin_Rossmo_Hammerschlag_2009_JZool.pdf
+}}
+</ref>
+This formula and math behind it were used in crime detecting in the [[Pilot (Numb3rs)|Pilot]] episode of the
+TV-series [[Numb3rs]] and in the 100th episode of the same show, called "[[Disturbed (Numb3rs)|Disturbed]]".
+<!-- hiding unused stuff
+ ==Distance formula==
+Let C be a set of coordinates of sectors of crimes.
+:<math>
+C = \{
+(x_1, y_1), (x_2, y_2), \dots ,  (x_c, y_c)
+\} \qquad c \in \mathbb{N}
+</math>
+<math>x \in C</math> is an element in a two-dimensional vector space.
+The distance from a given sector ''s'' to all other sectors <math>e \in C_s</math> is calculated with the [[Taxicab geometry|Manhattan distance formula]]:
+:<math>\sum_{i=1}^n |s-e_i| \qquad e_i \in C_s</math>
+-->
+==Notes==
+<references />
+(fr) http://www.siteduzero.com/tutoriel-3-422405-profilage-geographique.html
+==References==
+*{{cite book
+| title = The numbers behind [[NUMB3RS]]: solving crime with mathematics
+| last1 = Devlin
+| first1 = Keith J.
+| author2-link = Gary Lorden
+| last2 = Lorden
+| first2 = Gary
+| isbn = 978-0-452-28857-7
+| year = 2007
+| publisher = Plumer
+| edition = illustrated
+| pages = 1–12
+}}
+*{{cite book
+| title = Geographic profiling
+| first = Kim D.
+| last = Rossmo
+| isbn = 978-0-8493-8129-4
+| year = 2000
+| publisher = CRC Press
+| edition = illustrated
+}}
+[[Category:Offender profiling]]
+[[Category:Criminology]]
+[[Category:Crime mapping]]
+[[Category:Spatial data analysis]]
+[[Category:Forensic techniques]]

Hájek–Le Cam convolution theorem: Difference between revisions

Revision as of 07:56, 23 August 2013

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Hájek–Le Cam convolution theorem: Difference between revisions

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