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Biometrics & Biostatistics International Journal

Research Article Volume 9 Issue 6

Forecasting homicides, rapes and counterfeiting currency: A case study in Sri Lanka

Chathura B. Wickrama,1 Ruwan D. Nawarathna,2 Lakshika S. Nawarathna2

1Postgraduate Institute of Science, University of Peradeniya, Sri Lanka
2Department of Statistics and Computer Science, University of Peradeniya, Sri Lanka

Correspondence: Lakshika S. Nawarathna, Department of Statistics and Computer Science, University of Peradeniya, Sri Lanka, Tel +940767552223

Received: November 03, 2020 | Published: December 31, 2020

Citation: Nawarathna LS, Wickrama CB. Forecasting homicides, rapes and counterfeiting currency: A case study in Sri Lanka. Biom Biostat Int J. 2020;9(6):209-215. DOI: 10.15406/bbij.2020.09.00322

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Abstract

Crimes have been disturbing threats to all the Sri Lankans all over the country. Finding the main variables associated with crimes are very vital for policymakers. Our main goal in this study is to forecast of homicides, rapes and counterfeiting currency from 2013 to 2020 using auto-regressive conditional Poisson (ACP) and auto-regressive integrated moving average (ARIMA) models. All the predictions are made assuming that the prevailing conditions in the country affecting crime rates remain unchanged during the period. Moreover, multiple linear regression and Least Absolute Shrinkage and Selection Operator (LASSO) regression analysis were used to identify the key variables associated with crimes. Profiling of districts as safe or unsafe was performed based on the overall total crime rate of Sri Lanka which is to compare with individual district’s crime rates. Data were collected from the Department of Police and Department of Census and Statistics, Sri Lanka. It is observed that there are 14 safe and 11 unsafe districts in Sri Lanka. Moreover, it is found that the total migrant population and percentage of urban population is positively correlated with total crime. Besides, total migrant population, unemployment rate, mean household income and percentage of the urban population are significant variables for total crimes, and total migrant population, Gini index, mean household income and percentage of the urban population are significant variables for homicides. Random K-nearest neighbour (RKNN) algorithm classified districts as safe and unsafe with 84% of prediction accuracy.

Keywords: autoregressive conditional poisson model, autoregressive integrated moving average, crime analysis, gini index, random k-nearest neighbor algorithm.

Abbreviations

ACP, auto-regressive conditional Poisson; ARIMA, auto-regressive integrated moving average; LASSO, least absolute shrinkage and selection operator; RKNN, random K-nearest neighbour; CID, criminal investigating department

Introduction

Crime is one of the issues from which countries are suffered from the existence of mankind. These crimes have been disturbing threats to personalities, properties and lawful authorities of mankind. Reviews of the literature on this topic can be found in Louis et al.1 Crime began in the primitive days as a simple and less organized problem. Nowadays, due to the technological advancements, crimes are well organized and difficult to investigate and hence the situation is more complex.

The wave of crime is a key social problem in Sri Lanka and caused by the rising population and advancement of modern technology than earlier. Crimes such as homicides, rapes, child abuses, hitting, thefts, and illegal money printings are still threatening the Sri Lankan society. Due to this condition, a vast amount of harms have been occurred to people all over the country. Threats, suspicions, revenging, fear of the people, suicides are the major calamities resulting from the crimes.2 Crimes continue to attract the attention of all stakeholders, including the government and political leaders, the management and leadership of the Sri Lanka Police, individual citizens as well as the international community. Criminal Investigating Department (CID), criminal justice and law enforcement agencies exist to guarantee personal safety and security of property in Sri Lanka. The level of effectiveness of these agencies can be improved by information gained by crime analysis.

Crimes can be controlled by introducing new punishments such as the death penalty and finding the key factors affecting overall crimes and adjust those factors for positive changes by policy altering.3 It has been found that when an opportunity for crime is blocked, an offender has several other types of displacement. Therefore, this study facilitates for policy altering by identification of criminal factors. In order to find those factors, multivariate statistical tools can be applied and proved to be effective in many criminological explanations.4

Identification of trends in crimes is very important for policy makers to change their policies, for that, we look for possible trends of homicides, rapes and counterfeiting currency incidents. This study can answer the question of what factors significantly affect the total crimes and homicides by developing a model. Inorder to minimize crimes, it is important to know which factors mainly affect the crimes to determine what type of policy changes can be made. With the developed model, we predict the crimes for each district using significant factors. Moreover, associations between different crime types which can be used to lower the crimes will be assessed. Using the Random K-nearest neighbor (RKNN) algorithm, we profile districts of Sri Lanka as safe or unsafe without using the actual number of crimes committed in Sri Lankan districts. Furthermore, this article will provide effective guidance to help individuals better understanding of the factors associated with crimes and thus will be helpful in crime prevention.

The rest of this article is organized as follows. Section 2 presents the proposed methodology. In Section 3, we rank and classify the districts of Sri Lanka based on total crimes, land area and overall crime rate. Besides time series analysis is used to forecast crimes. Moreover, we propose a model for predicting total crimes and homicides. Further, the classification of crimes is performed using variables associated with the safeness. Finally, Section 4 concludes with a discussion.

Methodology

In this study, the required data are collected from the Department of Police and the Department of Census and Population, Sri Lanka. All the statistical analysis was done by using R statistical software version 3.5.1.5

The crime rate varies across individual districts and could be more or less than the overall crime rate of Sri Lanka. Therefore, districts are ranked and categorized as safe and unsafe districts. If a crime rate of a district is below the overall crime rate, it is considered as a safe district and if crime rate of a district is more than the overall crime rate, it is considered as an unsafe district. The Crime rate is calculated based on population and land area of a district.

Crime rate per  100,000 population= Total crimes in a district Total population in that district *100,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qaiaabkhacaqGPbGaaeyBaiaabwgacaqGGaGaaeOCaiaabgga caqG0bGaaeyzaiaabccacaqGWbGaaeyzaiaabkhacaqGGcGaaeiiai aaigdacaaIWaGaaGimaiaacYcacaaIWaGaaGimaiaaicdacaqGGcGa aeiCaiaab+gacaqGWbGaaeyDaiaabYgacaqGHbGaaeiDaiaabMgaca qGVbGaaeOBaiabg2da9maalaaapaqaa8qacaqGubGaae4Baiaabsha caqGHbGaaeiBaiaabckacaqGJbGaaeOCaiaabMgacaqGTbGaaeyzai aabohacaqGGcGaaeyAaiaab6gacaqGGcGaaeyyaiaabckacaqGKbGa aeyAaiaabohacaqG0bGaaeOCaiaabMgacaqGJbGaaeiDaaWdaeaape Gaaeivaiaab+gacaqG0bGaaeyyaiaabYgacaqGGcGaaeiCaiaab+ga caqGWbGaaeyDaiaabYgacaqGHbGaaeiDaiaabMgacaqGVbGaaeOBai aabckacaqGPbGaaeOBaiaabckacaqG0bGaaeiAaiaabggacaqG0bGa aeiOaiaabsgacaqGPbGaae4CaiaabshacaqGYbGaaeyAaiaabogaca qG0baaaiaabQcacaaIXaGaaGimaiaaicdacaGGSaGaaGimaiaaicda caaIWaaaaa@947B@      (2.1)

and

Crime rate per  1  km 2 = Total crimes in a district Total area of a district in  1  km 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4qaiaabkhacaqGPbGaaeyBaiaabwgacaqGGaGaaeOCaiaabgga caqG0bGaaeyzaiaabccacaqGWbGaaeyzaiaabkhacaqGGaGaaeiOai aaigdacaqGGcGaae4Aaiaab2gapaWaaWbaaSqabeaapeGaaGOmaaaa kiabg2da9maalaaapaqaa8qacaqGubGaae4BaiaabshacaqGHbGaae iBaiaabckacaqGJbGaaeOCaiaabMgacaqGTbGaaeyzaiaabohacaqG GcGaaeyAaiaab6gacaqGGcGaaeyyaiaabckacaqGKbGaaeyAaiaabo hacaqG0bGaaeOCaiaabMgacaqGJbGaaeiDaaWdaeaapeGaaeivaiaa b+gacaqG0bGaaeyyaiaabYgacaqGGcGaaeyyaiaabkhacaqGLbGaae yyaiaabckacaqGVbGaaeOzaiaabckacaqGHbGaaeiOaiaabsgacaqG PbGaae4CaiaabshacaqGYbGaaeyAaiaabogacaqG0bGaaeiiaiaabM gacaqGUbGaaeiiaiaabckacaaIXaGaaeiiaiaabUgacaqGTbWdamaa CaaaleqabaWdbiaaikdaaaaaaOGaaiOlaaaa@849D@      (2.2)

Data from different crime types in 2012 were analyzed for each district. Further, annual total crime data ranges from 1973 to 2014 are used for time series analysis to predict homicides, rapes and counterfeiting currency. In ARIMA technique, the future value of a variable is a linear combination of past values and past errors, expressed as follows.

Y t = ϕ 0 + ϕ 1 Y t1 + ϕ 2 Y t2 ++ ϕ p Y tp + ε t θ 1 ε t1 θ 2 ε t2 θ q ε tq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywa8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH9aqpcqaH vpGzpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey4kaSIaeqy1dy 2damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaadMfapaWaaSbaaSqa a8qacaWG0bGaeyOeI0IaaGymaaWdaeqaaOWdbiabgUcaRiabew9aM9 aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaWGzbWdamaaBaaaleaa peGaamiDaiabgkHiTiaaikdaa8aabeaak8qacqGHRaWkcqWIVlctcq GHRaWkcqaHvpGzpaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaamyw a8aadaWgaaWcbaWdbiaadshacqGHsislcaWGWbaapaqabaGcpeGaey 4kaSIaeqyTdu2damaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabgkHi TiabeI7aX9aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqaH1oqzpa WaaSbaaSqaa8qacaWG0bGaeyOeI0IaaGymaaWdaeqaaOWdbiabgkHi TiabeI7aX9aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqaH1oqzpa WaaSbaaSqaa8qacaWG0bGaeyOeI0IaaGOmaaWdaeqaaOWdbiabgkHi TiabgAci8kabgkHiTiabeI7aX9aadaWgaaWcbaWdbiaadghaa8aabe aak8qacqaH1oqzpaWaaSbaaSqaa8qacaWG0bGaeyOeI0IaamyCaaWd aeqaaaaa@7898@      (2.3)

where Y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfadaWgaa WcbaGaamiDaaqabaaaaa@3911@  is the actual value, ε t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLnaaBa aaleaacaWG0baabeaaaaa@39DA@  is the random error at time t, φi and θj are the coefficients, p and q are integers that are often referred to as autoregressive and moving average, respectively. Optimal values of p, q and difference term (d) are determined using Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Given a time series of counts, Y 1 ,..., Y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfadaWgaa WcbaGaaeymaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaa dMfadaWgaaWcbaGaamiDaaqabaaaaa@3E4F@  where Y t 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfadaWgaa WcbaGaamiDaaqabaGcdaWgaaWcbaGaeyOeI0Iaaeymaaqabaaaaa@3AE8@ denote the information on the time series up to time t 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGHsi slcaqGGaGaaeymaaaa@3A4B@ , then for the ACP(1,1) model, the counts, conditioned on past observations, are modeled as

Y t | Y t1 Poisson( μ t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywa8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGG8bGaamyw a8aadaWgaaWcbaWdbiaadshacqGHsislcaaIXaaapaqabaGcpeGaey ipI4Naamiuaiaad+gacaWGPbGaam4CaiaadohacaWGVbGaamOBaiaa cIcacqaH8oqBpaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaaiykaa aa@4ADE@      (2.4)

with an autoregressive conditional mean given as

μ t =ω+α Y t1 +β μ t1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiVd02damaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabg2da9iab eM8a3jabgUcaRiabeg7aHjaadMfapaWaaSbaaSqaa8qacaWG0bGaey OeI0IaaGymaaWdaeqaaOWdbiabgUcaRiabek7aIjabeY7aT9aadaWg aaWcbaWdbiaadshacqGHsislcaaIXaaapaqabaaaaa@4ACC@       (2.5)

for ω > 0 and α, β ≥ 0. This can be extended to include additional lags.6 Provided the ACP (1,1) is stationary and has an unconditional mean and variance given by

E[ y t ]=μ= ω (1(α+β)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyraiaacUfacaWG5bWdamaaBaaaleaapeGaamiDaaWdaeqaaOWd biaac2facqGH9aqpcqaH8oqBcqGH9aqpdaWcaaWdaeaapeGaeqyYdC hapaqaa8qacaGGOaGaaGymaiabgkHiTiaacIcacqaHXoqycqGHRaWk cqaHYoGycaGGPaGaaiykaaaaaaa@4A7C@           (2.6)

 and

Var[ y t ]= μ(1 (α+β) 2 + α 2 )) (1 (α+β) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaiaadggacaWGYbGaai4waiaadMhapaWaaSbaaSqaa8qacaWG 0baapaqabaGcpeGaaiyxaiabg2da9maalaaabaGaeqiVd0Maaiikai aaigdacqGHsislcaGGOaGaeqySdeMaey4kaSIaeqOSdiMaaiyka8aa daahaaWcbeqaa8qacaaIYaaaaOGaey4kaSIaeqySde2damaaCaaale qabaWdbiaaikdaaaGccaGGPaGaaiykaaqaaiaacIcacaaIXaGaeyOe I0Iaaiikaiabeg7aHjabgUcaRiabek7aIjaacMcapaWaaWbaaSqabe aapeGaaGOmaaaaaaaaaa@5782@       (2.7)

Two Ordinary Least Squares (OLS) models are built as OLS total crime model and OLS homicide model. Total crime and homicide are dependent variables in OLS total crime model and OLS homicide model respectively. With OLS regression and LASSO regression analysis, this study can answer the question of what factors affect the total crimes and homicides and predict the future crimes for each district. A statistical model is created to predict total crimes for each district. All the variables utilized for the analysis are listed in the Table 1. Let

Y= β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β k X k +ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywaiabg2da9iabek7aI9aadaWgaaWcbaWdbiaaicdaa8aabeaa k8qacqGHRaWkcqaHYoGypaWaaSbaaSqaa8qacaaIXaaapaqabaGcpe Gaamiwa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRaWkcqaH YoGypaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaamiwa8aadaWgaa WcbaWdbiaaikdaa8aabeaak8qacqGHRaWkcqaHYoGypaWaaSbaaSqa a8qacaaIZaaapaqabaGcpeGaamiwa8aadaWgaaWcbaWdbiaaiodaa8 aabeaak8qacqGHRaWkcqWIVlctcqaHYoGypaWaaSbaaSqaa8qacaWG RbaapaqabaGcpeGaamiwa8aadaWgaaWcbaWdbiaadUgaa8aabeaak8 qacqGHRaWkcqaH1oqzaaa@57C0@          (2.8)

Variable No

Variable

Variable Description

1

Y

Total crimes

2

X1

Percentage of people between 15 and 24

3

X2

Total migrant population

4

X3

Unemployment rate

5

X4

Gini coefficient which describe income inequality

6

X5

No schooling percentage

7

X6

Mean household income

8

X7

Population density (People per square kilometer)

9

X8

Percentage of urban population

10

X9

Percentage of people below the poverty line

11

X10

Percentage of people divorced and separated

12

X11

Percentage difference between male and female

Table 1 Details of variables

where  s are the final selected variable using stepwise variable selection method,  is the error term,  is the intercept and  are coefficients for selected variables. In fitting a multiple regression model, it is much more convenient to express the mathematical operations using matrix notation. Suppose that there are k independent variables and n observations.

This model is a system of n equations that can be expressed in matrix notation as,

Y=Xβ+ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywaiabg2da9iaadIfacqaHYoGycqGHRaWkcqaH1oqzaaa@3E19@          (2.9)

where Y=[ Y 1 Y 2 . . . Y 25 ],β=[ β 0 β 1 . . . β k ],X=[ 1 x 11 . . . x 1k 1 x 12 . . . x 2k . . . . . . . . . 1 x 251 . . . x 25k ],ε=[ ε 1 ε 2 . . . ε 25 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywaiabg2da9maadmaapaqaauaabaqageaaaaqaa8qacaWGzbWd amaaBaaaleaapeGaaGymaaWdaeqaaaGcbaWdbiaadMfapaWaaSbaaS qaa8qacaaIYaaapaqabaaakeaapeGaaiOlaaWdaeaapeGaaiOlaaWd aeaapeGaaiOlaaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaaikdaca aI1aaapaqabaaaaaGcpeGaay5waiaaw2faaiaacYcacqaHYoGycqGH 9aqpdaWadaWdaeaafaqaaeGbbaaaaeaapeGaeqOSdi2damaaBaaale aapeGaaGimaaWdaeqaaaGcbaWdbiabek7aI9aadaWgaaWcbaWdbiaa igdaa8aabeaaaOqaa8qacaGGUaaapaqaa8qacaGGUaaapaqaa8qaca GGUaaapaqaa8qacqaHYoGypaWaaSbaaSqaa8qacaWGRbaapaqabaaa aaGcpeGaay5waiaaw2faaiaacYcacaWGybGaeyypa0ZaamWaa8aaba qbaeaabyGbaaaaaeaapeGaaGymaaWdaeaapeGaamiEa8aadaWgaaWc baWdbiaaigdacaaIXaaapaqabaaakeaapeGaaiOlaaWdaeaapeGaai OlaaWdaeaapeGaaiOlaaWdaeaapeGaamiEa8aadaWgaaWcbaWdbiaa igdacaWGRbaapaqabaaakeaapeGaaGymaaWdaeaapeGaamiEa8aada WgaaWcbaWdbiaaigdacaaIYaaapaqabaaakeaapeGaaiOlaaWdaeaa peGaaiOlaaWdaeaapeGaaiOlaaWdaeaapeGaamiEa8aadaWgaaWcba WdbiaaikdacaWGRbaapaqabaaakeaapeGaaiOlaaWdaeaapeGaaiOl aaWdaeaaaeaaaeaaaeaapeGaaiOlaaWdaeaapeGaaiOlaaWdaeaape GaaiOlaaWdaeaaaeaaaeaaaeaapeGaaiOlaaWdaeaapeGaaiOlaaWd aeaapeGaaiOlaaWdaeaaaeaaaeaaaeaapeGaaiOlaaWdaeaapeGaaG ymaaWdaeaapeGaamiEa8aadaWgaaWcbaWdbiaaikdacaaI1aGaaGym aaWdaeqaaaGcbaWdbiaac6caa8aabaWdbiaac6caa8aabaWdbiaac6 caa8aabaWdbiaadIhapaWaaSbaaSqaa8qacaaIYaGaaGynaiaadUga a8aabeaaaaaak8qacaGLBbGaayzxaaGaaiilaiabew7aLjabg2da9m aadmaapaqaauaabaqageaaaaqaa8qacqaH1oqzpaWaaSbaaSqaa8qa caaIXaaapaqabaaakeaapeGaeqyTdu2damaaBaaaleaapeGaaGOmaa WdaeqaaaGcbaWdbiaac6caa8aabaWdbiaac6caa8aabaWdbiaac6ca a8aabaWdbiabew7aL9aadaWgaaWcbaWdbiaaikdacaaI1aaapaqaba aaaaGcpeGaay5waiaaw2faaaaa@9003@

We wish to find the vector of least square estimators (L),  minimizes the least squares estimator where  is the solution for  in the equations.

L β =0 and  β ^ = ( X T X ) 1 X T Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabgkGi2kaadYeaa8aabaWdbiabgkGi2kabek7a IbaacqGH9aqpcaaIWaWdaiaacckapeGaaeyyaiaab6gacaqGKbWdai aacckapeGafqOSdiMbaKaacqGH9aqpdaqadaWdaeaapeGaaeiwa8aa daahaaWcbeqaa8qacaqGubaaaOGaaeiwaaGaayjkaiaawMcaa8aada ahaaWcbeqaa8qacqGHsislcaaIXaaaaOGaamiwa8aadaahaaWcbeqa a8qacaWGubaaaOGaamywaaaa@4FE6@         (2.10)

LASSO technique is useful as it improves the quality of predictions by shrinking regression coefficients, compared to predictions based on a model fitted via unpenalized maximum likelihood.7 Given a set of input measurements x 1 , x 2 ,..., x p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaeymaaqabaGccaGGSaGaamiEamaaBaaaleaacaqGYaaabeaa kiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamiEamaaBaaaleaaca WGWbaabeaaaaa@4121@  and an outcome measurement y, the LASSO fits a linear model

y ^ = β 0 + β 1 x 1 + β 2 x 2 +...+ β p x p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabmyEayaajaGaeyypa0JaeqOSdi2damaaBaaaleaapeGaaGimaaWd aeqaaOWdbiabgUcaRiabek7aI9aadaWgaaWcbaWdbiaaigdaa8aabe aak8qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgUca Riabek7aI9aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaWG4bWdam aaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabgUcaRiaac6cacaGGUaGa aiOlaiabgUcaRiabek7aI9aadaWgaaWcbaWdbiaadchaa8aabeaak8 qacaWG4bWdamaaBaaaleaapeGaamiCaaWdaeqaaaaa@50FF@

We minimize j=1 p | β j |λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaybCaeqal8aabaWdbiaadQgacqGH9aqpcaaIXaaapaqaa8qacaWG Wbaan8aabaWdbiabggHiLdaakmaaemaabaGaeqOSdi2aaSbaaSqaai aadQgaaeqaaaGccaGLhWUaayjcSdGaeyizImQaeq4UdWgaaa@46D5@  subject to      i=1 N ( Y i β 0 j=1 p x ij β j ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaqahabaaeaa aaaaaaa8qadaqadaqaaiaadMfapaWaaSbaaSqaa8qacaWGPbaapaqa baGcpeGaeyOeI0IaeqOSdi2damaaBaaaleaapeGaaGimaaWdaeqaaO GaeyOeI0YaaabCaeaacaWG4bWaaSbaaSqaaiaadMgacaWGQbaabeaa kiabek7aInaaBaaaleaacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaG ymaaqaaiaadchaa0GaeyyeIuoaaOWdbiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaaapaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOtaa qdcqGHris5aaaa@5203@    

where the bound λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@38C2@  is a tuning parameter. The sum is taken over observations in the data set. When λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaWGHaeq4UdW gaaa@390D@  is large enough, the constraint has no effect and the solution is just the usual multiple linear least squares regression of y on x 1 , x 2 ,..., x p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaeymaaqabaGccaGGSaGaamiEamaaBaaaleaacaqGYaaabeaa kiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamiEamaaBaaaleaaca WGWbaabeaaaaa@4121@ . However when for smaller values of λ( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSnaabm aabaGaeyyzImRaaGimaaGaayjkaiaawMcaaaaa@3CCB@  the solutions are shrunken versions of the least squares estimates. Often, some of the coefficients b j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkgadaWgaa WcbaGaamOAaaqabaaaaa@3910@ ’s are zero. Choosing λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@38C2@  is like choosing the number of predictors to use in a regression model, and cross-validation is used for estimating the best value for λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@38C2@ .7

Feature selection is performed in order to find the importance of the variables and RKNN algorithm is run in order to classify the districts. The random Forest package8 and rknn package in R9 are used in this purpose. The Random Forest algorithm is used for variable selection. The relative rank (i.e. depth) of a feature used as a decision node in a tree are used to assess the relative importance of that feature with respect to the predictability of the target variable. Features used at the top of the tree used to contribute to the final prediction decision of a larger fraction of the input samples. The expected fraction of the samples is used as an estimate of the relative importance of the features. By averaging those expected activity rates over several randomized trees, one can reduce the variance of such an estimate and use it for feature selection.10 After selecting the best variables, for model building, RKNN algorithm is used and RKNN constitutes of an ensemble of base k-nearest neighbor models, each built from a random subset of the input variables.11 Random KNN method was introduced using some techniques used in random forest method and is similar in the method of random subspace selection used for decision forests. Random KNN uses KNN as base classifiers, with no hierarchical structure involved. Compared with decision trees, KNN is simple to implement and is stable.12 Thus, Random KNN is stabilized with a small number of base KNN’s and hence only a small number of important variables will be needed. This implies that the final model with Random KNN will be simpler than that with random forest or decision forests. Specifically, a collection of r different KNN classifiers will be generated. Each one takes a random subset of the input variables. Since KNN is stable, bootstrapping is not necessary for KNN. Each KNN classifier classifies a test point by its majority, or weighted majority class, of its k-nearest neighbors. The final classification in each case is determined by majority voting of , KNN classifications. This can be viewed as a sort of voting by a majority of a majority.

Let F= { f 1 , f 2 ,..., f p } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacqGH9a qpcaqGGaWaaiWaaeaacaWGMbWaaSbaaSqaaiaabgdaaeqaaOGaaiil aiaadAgadaWgaaWcbaGaaeOmaaqabaGccaGGSaGaaiOlaiaac6caca GGUaGaaiilaiaadAgadaWgaaWcbaGaamiCaaqabaaakiaawUhacaGL 9baaaaa@459A@ be the p input features, and X be the n original input data vectors of length p, (an n x p matrix). For a given integer m < p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gacaqGGa GaeyipaWJaaeiiaiaadchaaaa@3B3F@ , denote F m = f j 1 , f j 2 ,..., f jm | f jl F,lm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaahaa Wcbeqaaiaad2gaaaGccqGH9aqpcaWGMbWaaSbaaSqaaiaadQgaaeqa aOWaaSbaaSqaaiaabgdaaeqaaOGaaiilaiaadAgadaWgaaWcbaGaam OAaaqabaGcdaWgaaWcbaGaaeOmaaqabaGccaGGSaGaaiOlaiaac6ca caGGUaGaaiilaiaadAgadaWgaaWcbaGaamOAaiaad2gaaeqaaOGaai iFaiaadAgadaWgaaWcbaGaamOAaiaadYgaaeqaaOGaeyicI4SaamOr aiaacYcacaqGXaGaaeiiaiabgsMiJkaadYgacqGHKjYOcaWGTbaaaa@54C9@  a random subset drawn from F with equi-probability. Similarly, let X m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaahaa Wcbeqaaiaad2gaaaaaaa@390A@  be the data vectors in the subspace defined by F m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaahaa Wcbeqaaiaad2gaaaaaaa@38F8@ , i.e., an n m matrix. Then a KNN (m) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabUeacaqGob GaaeOtamaaCaaaleqabaGaaiikaiaad2gacaGGPaaaaaaa@3BF6@ classifier is constructed by applying the basic KNN algorithm to the random collection of features in X m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaahaa Wcbeqaaiaad2gaaaaaaa@390A@ . A collection of r such base classifiers is then combined to build the final random KNN classifier.

Results and Discussion

Figure 1 illustrates the pie chart of different crime types in percentages. This pie chart shows that the majority of crimes in 2012 is related to property crimes in which home break and theft represents 49% and robbery represents 19%. Hurt by knife is recorded as the highest number of crimes against persons which is 8% while counts of rapes account 6 %.

Figure 1 Percentages of crime types.

Further, box plots in Figure 2 are used to study the distributions of different crime rates per 100,000 population. Districts with rates of a low number of home breaks and thefts are more condensed than the districts with rates of a higher number of home breaks and thefts. Moreover, it can be observed that Colombo and Gampaha districts are outliers for many crime types. Further, Gampaha and Colombo districts are outliers for homicide and drug-related crimes respectively. Besides Gampaha district is an outlier for abduction/kidnapping, home break and thefts and robbery. Child abuses are prevalent in Mannar and Pollonnaruwa districts.

Figure 2 Box-plots of different crime types.

District ranks based on total crimes and homicides

Table 2 shows the ranks of districts based on total crimes per 100,000 population (i.e. population criteria) and per 1 square Kilometer (i.e. area criteria) basis. Total crimes of each district were used for this analysis.

Rank

Population criteria (Per 100,000 people)

Area criteria
(Per 1 km 2)

District

Rate

Dis trict

Rate

1

Colombo

47.40

Colombo

16.20

2

Gampaha

39.36

Gampaha

6.74

3

Killinochchi

39.03

Kalutara

2.04

4

Kegalle

31.57

Kegalle

1.57

5

Anuradhapura

31.08

Galle

1.55

6

Vavunia

30.22

Kandy

1.55

7

Polonnaruwa

26.75

Matara

1.32

8

Hambantota

26.63

Jaffna

1.04

9

Mannar

26.54

Rathnapura

0.83

10

Kalutara

26.41

Kurunegala

0.72

11

Rathnapura

24.89

Hambantota

0.64

12

Galle

23.71

Puttalam

0.52

13

Kandy

21.72

Matale

0.49

14

Monaragala

21.24

Nuwara Eliya

0.49

15

Matara

20.65

Badulla

0.49

16

Batticaloa

20.65

Batticaloa

0.42

17

Kurunegala

20.57

Anuradhapura

0.40

18

Trincomale

20.44

Killinochchi

0.37

19

Matale

19.96

Polonnaruwa

0.35

20

Puttalam

19.77

Vavunia

0.28

21

Mullativu

17.50

Ampara

0.26

22

Badulla

16.91

Trincomale

0.20

23

Ampara

16.83

Monaragala

0.17

24

Jaffna

16.59

Mannar

0.14

25

Nuwara Eliya

11.79

Mullativu

0.07

Table 2 Ranks of districts based on total crimes

According to the results, Colombo and Gampaha have the highest crime rates based on both population and area criteria and have been ranked in first and second positions respectively, whereas Nuwara Eliya district records the lowest based on the population criteria (per 100,000 people). Based on the area criteria, Mullativu district records the lowest. It is found that a resident in Nuwara Eliya district have experienced nearly 4 times fewer crimes than a resident in Colombo district based on the population criteria and a resident in Colombo district could see 231.4 times of more crimes than a resident in Mullativu district based on area criteria. A heat map of a total crimes based on area criteria is indicated in Figure 3. It shows that crimes are more prevalent in Western Province of Sri Lanka. It also shows that Kegalle, Galle, Kandy, Matara and Jaffna districts have significant number of total crimes per area.

Figure 3 Heat map of total crimes based on area criteria.

Moreover, most of the crimes can be observed in Colombo and Gampaha districts and spread over to the down-south in decreasing magnitude. Further, crimes are decreasing in rate from Colombo, Gampaha, and Kegalle to Kandy. When the distance from Colombo to other adjacent districts increases, crime rates tend to be lower.

Table 3 shows the ranking of districts based on homicides per 100,000 populations and per 1 km2 basis. Homicides of each district were used for this analysis.

Rank

Population criteria (Per 100,000 people)

Area criteria
(Per 1 km 2)

District

Rate

Dis trict

Rate

1

Vavunia

5.83

Colombo

0.0917

2

Monaragala

5.35

Gampaha

0.0567

3

Galle

4.72

Galle

0.0309

4

Rathnapura

4.25

Jaffna

0.0248

5

Mannar

4.04

Matara

0.0228

6

Jaffna

3.94

Kalutara

0.0222

7

Kurunegala

3.60

Kegalle

0.0161

8

Matara

3.58

Rathnapura

0.0142

9

Killinochchi

3.54

Kurunegala

0.0125

10

Hambantota

3.52

Kandy

0.0104

11

Trincomale

3.44

Nuwara Eliya

0.0100

12

Badulla

3.43

Badulla

0.0099

13

Gampaha

3.31

Hambantota

0.0084

14

Mullativu

3.30

Puttalam

0.0076

15

Kegalle

3.11

Matale

0.0072

16

Matale

2.90

Vavunia

0.0054

17

Puttalam

2.89

Trincomale

0.0051

18

Kalutara

2.87

Monaragala

0.0044

19

Anuradhapura

2.80

Batticaloa

0.0042

20

Colombo

2.68

Ampara

0.0038

21

Ampara

2.47

Anuradhapura

0.0036

22

Nuwara Eliya

2.41

Killinochchi

0.0033

23

Polonnaruwa

2.23

Polonnaruwa

0.0029

24

Batticaloa

2.09

Mannar

0.0021

25

Kandy

1.46

Mullativu

0.0012

Table 3 Ranks of districts based on total homicides

Vavunia and Monaragala districts have the highest homicide rates per 100,000 people and have been ranked in first and second positions respectively. Based on the population criteria (per 100,000 people), Kandy district records the lowest. According to the area criteria, Mullativu district records the lowest. It is found that a resident in Kandy district has 4 times less chance of being killed compared to a resident in a Vavunia district based on the population criteria. In one square kilometer, a resident in Colombo district could see 76.4 more homicides than a resident in Mullativu district. Figure 4 shows the 3-D representation of total crimes and homicides.

Figure 4 3-D representation of total crimes and homicides.

Table 4 describes the status of districts as safe or unsafe based on country’s total crime rate in which safe districts have its crime rate below the overall total crime rate and unsafe districts have its crime rate higher than the overall total crime rate.

No

Safe districts

Unsafe districts

1

Galle

Colombo

2

Kandy

Gampaha

3

Monaragala

Killinochchi

4

Matara

Kegalle

5

Batticaloa

Anuradhapura

6

Kurunegala

Vavunia

7

Trincomale

Polonnaruwa

8

Matale

Hambantota

9

Puttalam

Mannar

10

Mullativu

Kalutara

11

Badulla

Rathnapura

12

Ampara

 

13

Jaffna

 

14

Nuwara Eliya

 

Table 4 Classification of districts as safe and unsafe

According to the classification, there are 14 safe and 11 unsafe districts in Sri Lanka. It should be noted that the Central Province is a safe as its all districts (Kandy, Matale, and Nuwara Eliya) are safe and also Western province is an unsafe province as its crime rates of all representing districts are much higher than the overall crime rate.

Time series analysis for crime data

Time series analysis of homicides, rapes and counterfeiting currency was performed separately to find any underlying model. Time series analysis of homicides was done by developing ARIMA and ACP models using data from 1973 to 2012. Two outliers of homicide data were detected in 1988 and 1989 and those data points were cleaned and replaced by the linear interpolation. The Linear interpolation concerns the act of predicting or estimating extreme values based on their relationship to one or more other variables. Besides, it concerns estimation within ranges already measured. ACP models of homicides, rapes and counterfeiting currency were selected over ARIMA models as they had low AIC and BIC values. Selected ACP models for homicides, rapes and counterfeiting currency are shown in Table 5. All the coefficients of models are significant at 5% significant level. Forecasts were made using selected ACP models. Figure 5 shows the forecast of homicides, rapes and counterfeiting currency for 2013-2020. It seems that homicide counts are increasing from 2015 to 2020. The trend of increasing rape counts continues until 2020. Counterfeiting currency incidents will be stable until 2020. But a constant forecast for counterfeiting was observed for 2013 -2015.

Model

Coefficient

Estimate

Standard
Error

t-value

p-value

Homicides

ω α

195.80
0.9610

25.24
0.0233

7.7547
41.1857

<0.0001
<0.0001

 

β

-0.1064

0.0232

-4.5927

<0.0001

Rapes

ω α

5.66
1.1237

0.75
0.024

7.54
45.9972

<0.0001
<0.0001

 

β

-0.08

0.0238

-2.5497

0.0151

Counterfeitin g currency

ω α β

41.86
0.34
-0.077

4.34
0.037
0.007

9.63
9.16
-1.0588

<0.0001
<0.0001
<0.0001

Table 5 Coefficient estimates of ACP models for homicides, rapes and counterfeiting currency

Figure 5 Forecast of homicides, rapes and counterfeiting currency for 2013-2020.

Comparison of actual and forecast values of homicides, rapes and counterfeiting currency was illustrated in Table 6. It is observed that homicides, rapes and counterfeiting currency actual values are approximately same as the predicted values.

Crime

Year

Actual value

Forecast

Difference value

Homicides

2013
2014

586
548

732
681

146
133

Rapes

2013
2014

2181
2008

2372
2114

191
106

 

2015

2033

2125

92

Counterfeiting currency

2013
2014

59
52

53
58

6
6

Table 6 Actual and forecasted values of homicides, rapes and counterfeiting currency

Regression analysis for total crimes and homicides

In the regression analysis for total crimes, a model with Total migrant population, Unemployment rate, Mean Household income, Percentage of urban population and Percentage of people below poverty line are significant a 5% significance level and the following model was selected as the best model.

Total Crimes= 6050 + 0.01458Total migrant population+ 329.9Unemployment rate 507.8Mean MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfacaWGVb GaamiDaiaadggacaWGSbGaaeiiaiaadoeacaWGYbGaamyAaiaad2ga caWGLbGaam4Caiabg2da9iaabccacaqG2aGaaGimaiaabwdacaaIWa GaaeiiaiabgUcaRiaabccacaaIWaGaaiOlaiaaicdacaqGXaGaaein aiaabwdacaqG4aGaey4fIOIaamivaiaad+gacaWG0bGaamyyaiaadY gacaqGGaGaamyBaiaadMgacaWGNbGaamOCaiaadggacaWGUbGaamiD aiaabccacaWGWbGaam4BaiaadchacaWG1bGaamiBaiaadggacaWG0b GaamyAaiaad+gacaWGUbGaey4kaSIaaeiiaiaabodacaqGYaGaaeyo aiaac6cacaqG5aGaey4fIOIaamyvaiaad6gacaWGLbGaamyBaiaadc hacaWGSbGaam4BaiaadMhacaWGTbGaamyzaiaad6gacaWG0bGaaeii aiaadkhacaWGHbGaamiDaiaadwgacqGHsislcaqGGaGaaeynaiaaic dacaqG3aGaaiOlaiaabIdacqGHxiIkcaWGnbGaamyzaiaadggacaWG Ubaaaa@83CF@

 Household income+ 40.81Percentage of urban population+ 36.42Percentage of people below poverty line MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacckacaWGib Gaam4BaiaadwhacaWGZbGaamyzaiaadIgacaWGVbGaamiBaiaadsga caqGGaGaamyAaiaad6gacaWGJbGaam4Baiaad2gacaWGLbGaey4kaS IaaeiiaiaabsdacaaIWaGaaiOlaiaabIdacaqGXaGaey4fIOIaamiu aiaadwgacaWGYbGaam4yaiaadwgacaWGUbGaamiDaiaadggacaWGNb GaamyzaiaabccacaWGVbGaamOzaiaabccacaWG1bGaamOCaiaadkga caWGHbGaamOBaiaabccacaWGWbGaam4BaiaadchacaWG1bGaamiBai aadggacaWG0bGaamyAaiaad+gacaWGUbGaey4kaSIaaeiiaiaaboda caqG2aGaaiOlaiaabsdacaqGYaGaey4fIOIaamiuaiaadwgacaWGYb Gaam4yaiaadwgacaWGUbGaamiDaiaadggacaWGNbGaamyzaiaabcca caWGVbGaamOzaiaabccacaWGWbGaamyzaiaad+gacaWGWbGaamiBai aadwgacaqGGaGaamOyaiaadwgacaWGSbGaam4BaiaadEhacaqGGaGa amiCaiaad+gacaWG2bGaamyzaiaadkhacaWG0bGaamyEaiaabccaca WGSbGaamyAaiaad6gacaWGLbaaaa@90FE@

Model validation was done, comparing actual values with the predicted values for the best model and results are shown in Figure 6. The predicted crimes go fairly well with the actual crimes and display almost the same pattern. This reveals the estimated model adequately utilizes the data. Total crime model has higher adjusted R-squared value of 0.9712. This means that the independent variables included in the total crime model can explain 97.12% of variation around the mean of total crimes.

Figure 6 Actual and predicted total crimes.

Moreover, a regression analysis was conducted to find the best model for homicides. Total migrant population, Gini coefficient and percentage of the urban population are significant a 5% significance level. The resulting model is as follows.

The actual and estimated value of crimes were compared to validate the model and the resulting plot is shown in Figure 7. The predicted crimes and actual crimes are overlapping and shows almost the same pattern. This reveals the estimated model is adequate to utilize the sample. Total homicide model has an adjusted R-squared value of 0.83. According to the model coefficients, total migrant population, Gini-coefficient, mean household income and percentage of the urban population are significant variables. Gini-index describes the income inequality of society. This variable found to have significant at the 5% significance level. Gini coefficient is a very large factor in crime rate and finds it to have a positive coefficient. This suggests for policy makers that government should try to reduce the income inequality. They can do this by making the income distribution more even which will reduce the amount of poverty and in turn reduce the amount of crime in their districts. City planners should be concerned about their town planning, as crowded streets and sidewalks could be effective deterrents to criminal behavior. Studies done by Schuessler and Galle et al.13,14 found positively correlated relationships between crime and population density and matched with our findings.

Figure 7 Actual and predicted values plot for homicides.

Forecasts of Kurunegala and Anuradhapura districts for total crimes and Colombo and Gampaha districts for homicides indicate in Table 7. This shows that all the predicted values are in the 95% prediction interval range.

District

OLS model

Actual

Predicted

Confidence level

Difference

Lower

Upper

Kurunegala
Anuradhapura

Total crime

3314
2662

3422.65
2412.71

1809.47 745.76

3843.87
2765.65

108.65
249.29

Colombo
Gampaha

Homicide

62
76

97.16 81.21

31.96 24.27

167.98
134.67

35.16 5.21

Table 7 Forecasting of total crimes and homicides with OLS models

Moreover, assumptions of homoscedasticity, auto correlation, multicolinearity, normality and linear relationship are not violated in homicide and total crime OLS models. Two separate model were fitted for predicting total crimes and homicides using the Lasso regression technique as follows.

Total crimes= 181.69 + 0.00685Total migrant population+ 0.942population density 45.32Mean household income MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfacaWGVb GaamiDaiaadggacaWGSbGaaeiiaiaadogacaWGYbGaamyAaiaad2ga caWGLbGaam4Caiabg2da9iaabccacqGHsislcaqGXaGaaeioaiaabg dacaGGUaGaaeOnaiaabMdacaqGGaGaey4kaSIaaeiiaiaaicdacaGG UaGaaGimaiaaicdacaqG2aGaaeioaiaabwdacqGHxiIkcaWGubGaam 4BaiaadshacaWGHbGaamiBaiaabccacaWGTbGaamyAaiaadEgacaWG YbGaamyyaiaad6gacaWG0bGaaeiiaiaadchacaWGVbGaamiCaiaadw hacaWGSbGaamyyaiaadshacaWGPbGaam4Baiaad6gacqGHRaWkcaqG GaGaaGimaiaac6cacaqG5aGaaeinaiaabkdacqGHxiIkcaWGWbGaam 4BaiaadchacaWG1bGaamiBaiaadggacaWG0bGaamyAaiaad+gacaWG UbGaaeiiaiaadsgacaWGLbGaamOBaiaadohacaWGPbGaamiDaiaadM hacqGHsislcaqGGaGaaeinaiaabwdacaGGUaGaae4maiaabkdacqGH xiIkcaWGnbGaamyzaiaadggacaWGUbGaaeiiaiaadIgacaWGVbGaam yDaiaadohacaWGLbGaamiAaiaad+gacaWGSbGaamizaiaabccacaWG PbGaamOBaiaadogacaWGVbGaamyBaiaadwgaaaa@96A9@

Homicides= 2.314 + 4.79e 05Total migrant population+ 0.0334Gini coefficient+ 0.0246Population density MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaWGVb GaamyBaiaadMgacaWGJbGaamyAaiaadsgacaWGLbGaam4Caiabg2da 9iaabccacqGHsislcaqGYaGaaiOlaiaabodacaqGXaGaaeinaiaabc cacqGHRaWkcaqGGaGaaeinaiaac6cacaqG3aGaaeyoaiaadwgacqGH sislcaqGGaGaaGimaiaabwdacqGHxiIkcaWGubGaam4Baiaadshaca WGHbGaamiBaiaabccacaWGTbGaamyAaiaadEgacaWGYbGaamyyaiaa d6gacaWG0bGaaeiiaiaadchacaWGVbGaamiCaiaadwhacaWGSbGaam yyaiaadshacaWGPbGaam4Baiaad6gacqGHRaWkcaqGGaGaaGimaiaa c6cacaaIWaGaae4maiaabodacaqG0aGaey4fIOIaam4raiaadMgaca WGUbGaamyAaiaabccacaWGJbGaam4BaiaadwgacaWGMbGaamOzaiaa dMgacaWGJbGaamyAaiaadwgacaWGUbGaamiDaiabgUcaRiaabccaca aIWaGaaiOlaiaaicdacaqGYaGaaeinaiaabAdacqGHxiIkcaWGqbGa am4BaiaadchacaWG1bGaamiBaiaadggacaWG0bGaamyAaiaad+gaca WGUbGaaeiiaiaadsgacaWGLbGaamOBaiaadohacaWGPbGaamiDaiaa dMhaaaa@91D8@

In comparison to the OLS homicide model, the percentage of urban population is not significant and population density is significant in LASSO homicide model.

Variable importance is done by measuring the total decrease in node impurities and the results are shown in Table 8. No schooling percentage, percentage of people below the poverty line and population density, mean household income and Gini coefficient are the most important variables in determining the safeness of districts and those variables are used to run the RKNN algorithm. Only Badulla district is wrongly categorized with the error rate is 16.6. If all 25 districts are categorized using the above selected variables, four districts as safeness results could be erroneous in general. Therefore, if a comparison is made with OLS regression and LASSO technique, Total migrant population is a common variables in OLS regression and LASSO regression for both total crimes and homicides. Population density is a key factor for total crimes in OLS and LASSO regressions, and Safeness. Gini Coefficient is common in OLS homicide model and LASSO homicide model.

District

Actual

Predicted

Anuradhapura

Unsafe

Unsafe

Polonnaruwa

Unsafe

Unsafe

Badulla

Safe

Unsafe

Monaragala

Safe

Safe

Rathnapura

Unsafe

Unsafe

Kegalle

Unsafe

Unsafe

Table 8 Comparison of actual safeness and predicted safeness

Conclusion

Colombo district has the highest total crime rate based on per 100,000 population and per 1km2. Vavunia district has the highest homicide rate per 100,000 population and Colombo district has the highest homicide rate per 1km2. It is evident that all the districts in Western Province are unsafe in relation to other districts. Nuwara Eliya district has the lowest total crime rate per 100,000 population and Mullativu district has the lowest total crime rate per 1km2. Kandy district has the lowest homicide rate per 100,000 population and Mullativu district has the lowest homicide rate per 1km2. There are 14 safe and 11 unsafe districts in Sri Lanka. All the districts in Central Province are safe with other districts. Spearman correlation analysis suggests that minimizing of one type of crime causes to reduce another type of crime by their positive correlation. Therefore, at first, policy makers should try to reduce crimes which are easily controllable and less costly. Also, they should take actions to reduce migrations as this leads to more crimes and concentrate their efforts on stopping crimes in highly crowded districts.

Acknowledgments

Authors wish to thank the Department of Police and Department of Census & Statistics, Sri Lanka for providing the data sets used in this paper.

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