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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>
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| ==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>'').
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| 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
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| | first = Kim D.
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| | last = Rossmo
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| | title = Geographic profiling: target patterns of serial murderers
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| | page = 225
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| | year = 1995
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| | publisher = [[Simon Fraser University]]
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| | url = http://lib-ir.lib.sfu.ca/bitstream/1892/8121/1/b17675819.pdf
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| }}
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| </ref>
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| :<math>
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| p_{i,j} = k \sum_{n=1}^{(\mathrm{total\;crimes})}
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| \left [
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| \underbrace{
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| \frac{\phi_{ij}}{
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| (|X_i-x_n| + |Y_j-y_n|)^f
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| }
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| }_{ 1^{\mathrm{st}}\mathrm{\;term} }
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| +
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| \underbrace{
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| \frac{(1-\phi_{ij})(B^{g-f})}{
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| (2B - \mid X_i - x_n \mid - \mid Y_j-y_n \mid)^g
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| }
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| }_{ 2^{\mathrm{nd}}\mathrm{\;term} }
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| \right ],
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| </math>
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| where:
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| <math>
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| \phi_{ij} =
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| \begin{cases}
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| 1, & \mathrm{\quad if\;} ( \mid X_i - x_n \mid + \mid Y_j - y_n \mid ) > B \quad \\
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| 0, & \mathrm{\quad else}
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| \end{cases}
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| </math>
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| Here the summation is over past crimes located at coordinates (''x<sub>n</sub>'',''y<sub>n</sub>'').
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| <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.
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| 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.
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| This is a prerogative for defining the first term in the case when the distance between a point (or pixel) becomes equal to zero.
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| When <math>\phi_{ij} = 1</math>, the 1st term is used to calculate <math>p_{i,j}</math>.
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| <math>\mid X_i - x_n \mid + \mid Y_j - y_n \mid</math>
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| 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>. | |
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| <!-- hiding this until the math gets corrected
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| ==Alternative Implementation==
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| <math>p_{i,j}</math> is not well suited for image processing because of the asymptotic behavior near the coordinates of a crime site.
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| Alternatively, Rossmo's function may use other [[distance decay]] functions instead of <math>\frac{1}{(\mathrm{Mathattan\;Distance})^f}</math>.
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| One method would be to use a probability distribution similar to the [[Gaussian Distribution]] as a distance decay function:
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| <math>
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| 1^{\mathrm{st}}\mathrm{\;term}(x,y) =
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| \left\lfloor
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| (\mathrm{\#\;of\;colors}) \times \sum_{n=1}^{(\mathrm{total\;crimes})}
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| \frac{1}{\sqrt{e^{(\mid x - C_{n}(x) \mid^{2} + \mid y - C_{n}(y) \mid^{2})}}}
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| \right\rfloor
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| </math>
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| 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.
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| Where:
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| <math>
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| J =
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| \begin{bmatrix}
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| p(n,0) & \cdots & \; & p(n,m) \\
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| \vdots & \ddots & \; & \vdots \\
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| p(x,0) & \cdots & p(x,y) & \vdots \\
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| \vdots & \; & \; & \vdots \\
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| p(0,0) & p(1,0) & p(2,0) & \cdots
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| \end{bmatrix}
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| </math>
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| -->
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| ==Explanation==
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| 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.
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| 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>.
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| The distance is calculated with the [[Taxicab geometry|Manhattan distance formula]].
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| ==Applications==
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| 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.
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| <ref>{{cite journal
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| | author = R. A. Martin; D. K. Rossmo; N. Hammerschlag
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| | title = Hunting patterns and geographic profiling of white shark predation
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| | journal = Journal of Zoology
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| | volume = 279
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| | pages = 111–118
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| | year = 2009
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| | doi = 10.1111/j.1469-7998.2009.00586.x
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| | url = http://www.rjd.miami.edu/scientific-publications/pdf/Martin_Rossmo_Hammerschlag_2009_JZool.pdf
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| }}
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| </ref>
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| This formula and math behind it were used in crime detecting in the [[Pilot (Numb3rs)|Pilot]] episode of the
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| TV-series [[Numb3rs]] and in the 100th episode of the same show, called "[[Disturbed (Numb3rs)|Disturbed]]".
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| <!-- hiding unused stuff
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| ==Distance formula==
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| Let C be a set of coordinates of sectors of crimes.
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| :<math>
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| C = \{
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| (x_1, y_1), (x_2, y_2), \dots , (x_c, y_c)
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| \} \qquad c \in \mathbb{N}
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| </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]]:
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| :<math>\sum_{i=1}^n |s-e_i| \qquad e_i \in C_s</math>
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| -->
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| ==Notes==
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| <references />
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| (fr) http://www.siteduzero.com/tutoriel-3-422405-profilage-geographique.html
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| ==References==
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| *{{cite book
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| | title = The numbers behind [[NUMB3RS]]: solving crime with mathematics
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| | last1 = Devlin
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| | first1 = Keith J.
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| | author2-link = Gary Lorden
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| | last2 = Lorden
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| | first2 = Gary
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| | isbn = 978-0-452-28857-7
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| | year = 2007
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| | publisher = Plumer
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| | edition = illustrated
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| | pages = 1–12
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| }}
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| *{{cite book
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| | title = Geographic profiling
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| | first = Kim D.
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| | last = Rossmo
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| | isbn = 978-0-8493-8129-4
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| | year = 2000
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| | publisher = CRC Press
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| | edition = illustrated
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| }}
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| [[Category:Offender profiling]]
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| [[Category:Criminology]]
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| [[Category:Crime mapping]]
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| [[Category:Spatial data analysis]]
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| [[Category:Forensic techniques]]
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