Introduction
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The developed package can be used to generate a spatial
population for different levels of relationships among the dependent and
auxiliary variables along with spatially varying model parameters. A
spatial layout is designed as a [0,k-1]x[0,k-1] square region on which
observations are collected at (k x k) lattice points with a unit
distance between any two neighbouring points along the horizontal and
vertical axes. Regression coefficients of the spatially varying
regression model are generated using both linear and non-linear function
of locations. In total eight different types of population can be
generated. Details of which are given below.
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Generation of simulated spatial population based on spatially varying regression model where the model parameters are generated based on both linear and non-linear functions of latitudes and longitudes
****
Generation of spatial coordinates of locations
The size of the population is N= k2. The spatial coordinates of the locations of observations can be expressed by the following equations
( Latitudei, Longitudei )= ( mod(i-1,k), [(i-1)/k] ), i= 1,…, k2
where, mod(i-1,k) is the remainder of (i-1) divided by k and [(i-1)/k] is the integer part of the number (i-1)/k
Generation of auxiliary variable from uniform distribution
Auxiliary variables has been generated independently from the uniform distribution i.e. U(0,1)
X =runif(N,0,1)
Error term drawn independently from normal distribution i.e. N(0,1)
e =rnorm(N, mean=0, sd=1)
For the function SpPOP_linear1
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and one explanatory variable (i.e. X). The auxiliary variable has been generated following U(0,1) and the regression coefficients are generated as linear function of latitudes and longitudes
Generation of spatially varying regression coefficients as linear function of latitudes and longitudes
B0=(Latitudei+Longitudei)/18
B1=(Latitudei/9)
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + B1( Latitudei,Longitudei )*Xi + ei ; i= 1,…, N
# Examples: Generate an uniform two dimensional grid of lattice points
library(SpPOP)
coords_grid=expand.grid(c(1:5),c(1:5))
coords_grid=as.data.frame(coords_grid)
names(coords_grid)=cbind("x","y")
coords_grid## x y
## 1 1 1
## 2 2 1
## 3 3 1
## 4 4 1
## 5 5 1
## 6 1 2
## 7 2 2
## 8 3 2
## 9 4 2
## 10 5 2
## 11 1 3
## 12 2 3
## 13 3 3
## 14 4 3
## 15 5 3
## 16 1 4
## 17 2 4
## 18 3 4
## 19 4 4
## 20 5 4
## 21 1 5
## 22 2 5
## 23 3 5
## 24 4 5
## 25 5 5plot(coords_grid)
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
coords_grid=expand.grid(c(1:5),c(1:5))
coords_grid=as.data.frame(coords_grid)
names(coords_grid)=cbind("x","y")
coords_grid## x y
## 1 1 1
## 2 2 1
## 3 3 1
## 4 4 1
## 5 5 1
## 6 1 2
## 7 2 2
## 8 3 2
## 9 4 2
## 10 5 2
## 11 1 3
## 12 2 3
## 13 3 3
## 14 4 3
## 15 5 3
## 16 1 4
## 17 2 4
## 18 3 4
## 19 4 4
## 20 5 4
## 21 1 5
## 22 2 5
## 23 3 5
## 24 4 5
## 25 5 5N<-nrow(coords_grid)
N## [1] 25k<-sqrt(nrow(coords_grid))
k## [1] 5sp_linear1<-SpPOP_linear1(25,5,c(1:5),c(1:5))
sp_linear1## Y X latitude longitude B0 B1
## 1 -0.6385162 0.008670922 0 0 0.00000000 0.0000000
## 2 -0.3657937 0.555195503 1 0 0.05555556 0.1111111
## 3 -0.1238916 0.987145639 2 0 0.11111111 0.2222222
## 4 0.9924932 0.796818988 3 0 0.16666667 0.3333333
## 5 1.1909612 0.895889906 4 0 0.22222222 0.4444444
## 6 -2.0366106 0.272928117 0 1 0.05555556 0.0000000
## 7 0.2324620 0.460574525 1 1 0.11111111 0.1111111
## 8 -0.4950643 0.301175795 2 1 0.16666667 0.2222222
## 9 -0.6262052 0.433577891 3 1 0.22222222 0.3333333
## 10 0.3010648 0.800071618 4 1 0.27777778 0.4444444
## 11 1.3569661 0.870431042 0 2 0.11111111 0.0000000
## 12 0.6128895 0.778206456 1 2 0.16666667 0.1111111
## 13 -0.5661738 0.957810637 2 2 0.22222222 0.2222222
## 14 -1.4728098 0.739340476 3 2 0.27777778 0.3333333
## 15 -0.2885802 0.157180382 4 2 0.33333333 0.4444444
## 16 1.1159875 0.827919451 0 3 0.16666667 0.0000000
## 17 -0.9024278 0.135645042 1 3 0.22222222 0.1111111
## 18 -0.9148966 0.440048936 2 3 0.27777778 0.2222222
## 19 0.8294268 0.954400906 3 3 0.33333333 0.3333333
## 20 0.7075803 0.265156079 4 3 0.38888889 0.4444444
## 21 -0.2961651 0.134147050 0 4 0.22222222 0.0000000
## 22 -0.1983796 0.145500501 1 4 0.27777778 0.1111111
## 23 0.0149270 0.565660454 2 4 0.33333333 0.2222222
## 24 1.6122422 0.772888334 3 4 0.38888889 0.3333333
## 25 0.3779370 0.209692298 4 4 0.44444444 0.4444444For the function SpPOP_linear2
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and three explanatory variables (i.e. X1,X2 and X3). The auxiliary variables have been generated independently from the uniform distribution U(0,1) and the regression coefficients are generated as linear function of latitudes and longitudes
Generation of spatially varying regression coefficients as linear function of latitudes and longitudes
B0=(Latitudei+Longitudei)/6
B1=(Latitudei/3)
B2=(Longitudei/3)
B3=(2*Longitudei)
Generation of auxiliary variables independently from the uniform distribution
X1 =runif(N,0,1), X2 =runif(N,0,1) and X3 =runif(N,0,1)
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + B1( Latitudei,Longitudei )* X1i + B2(Latitudei,Longitudei)* X2i + B3( Latitudei,Longitudei )*X3i + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_linear2<-SpPOP_linear2(25,5,c(1:5),c(1:5))
sp_linear2## Y X1 X2 X3 latitude longitude B0
## 1 -1.82858131 0.5944992 0.311818000 0.76457523 0 0 0.0000000
## 2 2.15760008 0.9465491 0.443732792 0.24063934 1 0 0.1666667
## 3 1.89666619 0.8943838 0.883625589 0.63822624 2 0 0.3333333
## 4 0.09174716 0.4318616 0.069160961 0.62195066 3 0 0.5000000
## 5 2.38884324 0.4867799 0.298661276 0.38910061 4 0 0.6666667
## 6 2.11112629 0.4818445 0.978487346 0.25144887 0 1 0.1666667
## 7 1.56654299 0.8167164 0.892389085 0.68463087 1 1 0.3333333
## 8 1.43788016 0.4032495 0.728726941 0.23328465 2 1 0.5000000
## 9 3.90135865 0.2144529 0.504790984 0.67280208 3 1 0.6666667
## 10 3.05971600 0.5494753 0.350181414 0.53343253 4 1 0.8333333
## 11 0.15723008 0.6243017 0.376658351 0.06569530 0 2 0.3333333
## 12 5.74029141 0.2262922 0.276039757 0.80657806 1 2 0.5000000
## 13 2.32033140 0.9478098 0.046970449 0.09764455 2 2 0.6666667
## 14 2.41451164 0.1845203 0.005028286 0.42252495 3 2 0.8333333
## 15 2.24041893 0.9749800 0.082197745 0.06720287 4 2 1.0000000
## 16 4.47308182 0.9227784 0.082659904 0.69747700 0 3 0.5000000
## 17 3.66419513 0.2539389 0.615839009 0.35621910 1 3 0.6666667
## 18 4.66441275 0.7353120 0.057101715 0.31793006 2 3 0.8333333
## 19 3.01230682 0.5251119 0.670578606 0.40898542 3 3 1.0000000
## 20 4.47903237 0.9500805 0.986019709 0.02338269 4 3 1.1666667
## 21 4.66831813 0.4578746 0.379690734 0.65841758 0 4 0.6666667
## 22 7.83409460 0.6541902 0.785046917 0.91688492 1 4 0.8333333
## 23 10.27319050 0.3367476 0.348909366 0.99233891 2 4 1.0000000
## 24 7.68753889 0.3527206 0.820356217 0.76678176 3 4 1.1666667
## 25 8.14228855 0.6702242 0.970578888 0.71428470 4 4 1.3333333
## B1 B2 B3
## 1 0.0000000 0.0000000 0
## 2 0.3333333 0.0000000 0
## 3 0.6666667 0.0000000 0
## 4 1.0000000 0.0000000 0
## 5 1.3333333 0.0000000 0
## 6 0.0000000 0.3333333 2
## 7 0.3333333 0.3333333 2
## 8 0.6666667 0.3333333 2
## 9 1.0000000 0.3333333 2
## 10 1.3333333 0.3333333 2
## 11 0.0000000 0.6666667 4
## 12 0.3333333 0.6666667 4
## 13 0.6666667 0.6666667 4
## 14 1.0000000 0.6666667 4
## 15 1.3333333 0.6666667 4
## 16 0.0000000 1.0000000 6
## 17 0.3333333 1.0000000 6
## 18 0.6666667 1.0000000 6
## 19 1.0000000 1.0000000 6
## 20 1.3333333 1.0000000 6
## 21 0.0000000 1.3333333 8
## 22 0.3333333 1.3333333 8
## 23 0.6666667 1.3333333 8
## 24 1.0000000 1.3333333 8
## 25 1.3333333 1.3333333 8For the function SpPOP_linear3
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and one explanatory variable (i.e. X). The auxiliary variable has been generated from the uniform distribution U(0,2) and the regression coefficients are generated as linear function of locations
Generation of spatially varying regression coefficients as linear function of latitudes and longitudes
B0=(2*(Latitudei+Longitudei))/6
B1=(Latitudei/3)
Generation of auxiliary variable from the uniform distribution
X =runif(N,0,2),
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + B1( Latitudei,Longitudei )* Xi + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_linear3<-SpPOP_linear3(25,5,c(1:5),c(1:5))
sp_linear3## Y X latitude longitude B0 B1
## 1 -0.71849330 1.8789632 0 0 0.0000000 0.0000000
## 2 -0.04776410 1.8544099 1 0 0.3333333 0.3333333
## 3 2.00033861 1.9035184 2 0 0.6666667 0.6666667
## 4 1.53753266 1.7442356 3 0 1.0000000 1.0000000
## 5 1.71374611 0.6079337 4 0 1.3333333 1.3333333
## 6 1.93900661 0.1530089 0 1 0.1666667 0.0000000
## 7 1.41147766 1.8450406 1 1 0.5000000 0.3333333
## 8 2.34690898 1.8592073 2 1 0.8333333 0.6666667
## 9 2.96400402 0.4472633 3 1 1.1666667 1.0000000
## 10 2.04576275 1.1197784 4 1 1.5000000 1.3333333
## 11 0.78821032 0.7714012 0 2 0.3333333 0.0000000
## 12 -0.32017246 1.8813707 1 2 0.6666667 0.3333333
## 13 1.54828369 1.4999020 2 2 1.0000000 0.6666667
## 14 2.27817515 1.4539635 3 2 1.3333333 1.0000000
## 15 4.23432658 1.4461893 4 2 1.6666667 1.3333333
## 16 1.20254710 1.5870078 0 3 0.5000000 0.0000000
## 17 2.27092989 0.5242652 1 3 0.8333333 0.3333333
## 18 2.86804285 1.9928006 2 3 1.1666667 0.6666667
## 19 1.89067803 0.1305419 3 3 1.5000000 1.0000000
## 20 0.08090745 0.3207972 4 3 1.8333333 1.3333333
## 21 -0.60789192 0.2081725 0 4 0.6666667 0.0000000
## 22 0.16001054 0.1811717 1 4 1.0000000 0.3333333
## 23 0.53142405 0.4343316 2 4 1.3333333 0.6666667
## 24 2.70059083 1.8530458 3 4 1.6666667 1.0000000
## 25 3.97953774 1.7701464 4 4 2.0000000 1.3333333For the function SpPOP_linear4
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and one explanatory variable (i.e. X). The auxiliary variable has been generated from the uniform distribution U(0,2) and the regression coefficients are generated as linear function of locations. In this function, we have used non-linear model for generating the response variable i.e. the relationship between the response and auxiliary variable is non-linear in nature.
Generation of spatially varying regression coefficients as linear function of latitudes and longitudes
B0=(2*(Latitudei+Longitudei))/6
B1=(Latitudei/3)
Generation of auxiliary variable from the uniform distribution
X =runif(N,0,2),
Spatially varying regression model for generating the response variable The relation between Y and X is non-linear in the model
Yi = B0( Latitudei,Longitudei ) + exp[B1( Latitudei,Longitudei )* Xi] + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_linear4<-SpPOP_linear4(25,5,c(1:5),c(1:5))
sp_linear4## Y X latitude longitude B0 B1
## 1 1.63022656 0.3084330 0 0 0.0000000 0.0000000
## 2 2.27253969 0.6275131 1 0 0.3333333 0.3333333
## 3 3.09709199 1.4976805 2 0 0.6666667 0.6666667
## 4 1.92356590 0.6874736 3 0 1.0000000 1.0000000
## 5 3.60919324 0.7524658 4 0 1.3333333 1.3333333
## 6 -0.01102871 0.7088710 0 1 0.1666667 0.0000000
## 7 2.63170364 0.8592169 1 1 0.5000000 0.3333333
## 8 5.20839998 1.8881164 2 1 0.8333333 0.6666667
## 9 6.84022703 1.4723306 3 1 1.1666667 1.0000000
## 10 1.21419820 0.4023375 4 1 1.5000000 1.3333333
## 11 1.97560689 0.9812003 0 2 0.3333333 0.0000000
## 12 2.24961790 1.0121873 1 2 0.6666667 0.3333333
## 13 1.57966116 0.9306789 2 2 1.0000000 0.6666667
## 14 3.99729274 1.1797239 3 2 1.3333333 1.0000000
## 15 2.43677286 0.1351487 4 2 1.6666667 1.3333333
## 16 0.61539390 1.6809876 0 3 0.5000000 0.0000000
## 17 3.17996103 0.0224011 1 3 0.8333333 0.3333333
## 18 4.87331724 1.9785420 2 3 1.1666667 0.6666667
## 19 6.01895268 1.6893461 3 3 1.5000000 1.0000000
## 20 6.13318725 1.0783261 4 3 1.8333333 1.3333333
## 21 0.46171371 0.8753324 0 4 0.6666667 0.0000000
## 22 1.40784204 1.1912744 1 4 1.0000000 0.3333333
## 23 2.76548131 1.5661801 2 4 1.3333333 0.6666667
## 24 2.23214454 0.4521896 3 4 1.6666667 1.0000000
## 25 10.94825762 1.7000948 4 4 2.0000000 1.3333333For the function SpPOP_linear5
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and two explanatory variables (i.e. X1 and X2). The auxiliary variables are drawn independently from the uniform distribution U(0,2) and normal distribution N(1,1) and the regression coefficients are generated as linear function of locations. In this function, we have used non-linear model for generating the response variable i.e. the relationship between the response and auxiliary variables are non-linear in nature.
Generation of spatially varying regression coefficients as linear function of latitudes and longitudes
B0=(Latitudei+Longitudei)/6
B1=(Latitudei/3)
B2=(Longitudei/3)
Generation of auxiliary variables independently from the uniform and normal distribution
X1 =runif(N,0,2) and X2 =rnorm(N,1,1)
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + exp[(B1( Latitudei,Longitudei )* X1i)+(B2( Latitudei,Longitudei )* X2i)] + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_linear5<-SpPOP_linear5(25,5,c(1:5),c(1:5))
sp_linear5## Y X1 X2 latitude longitude B0 B1
## 1 0.90774682 1.1080677 1.52696062 0 0 0.0000000 0.0000000
## 2 1.70331851 1.8936248 1.68113877 1 0 0.1666667 0.3333333
## 3 0.08851388 0.1805360 -0.03672761 2 0 0.3333333 0.6666667
## 4 0.47529815 0.2032689 1.78728039 3 0 0.5000000 1.0000000
## 5 7.09171840 1.3640243 -0.15527361 4 0 0.6666667 1.3333333
## 6 0.38831157 1.1744001 0.95787190 0 1 0.1666667 0.0000000
## 7 1.00858864 0.4235612 1.79129414 1 1 0.3333333 0.3333333
## 8 4.02695986 0.7081248 1.49971810 2 1 0.5000000 0.6666667
## 9 1.50682081 0.1290912 0.54455424 3 1 0.6666667 1.0000000
## 10 6.62702025 0.8683968 1.98848239 4 1 0.8333333 1.3333333
## 11 1.21260586 1.6719326 0.60883034 0 2 0.3333333 0.0000000
## 12 3.21561787 0.5113690 0.91528869 1 2 0.5000000 0.3333333
## 13 3.64738730 1.1225266 0.33997247 2 2 0.6666667 0.6666667
## 14 2.58721852 0.1728036 0.46247247 3 2 0.8333333 1.0000000
## 15 2.12968904 0.2192020 -0.18311113 4 2 1.0000000 1.3333333
## 16 4.36459759 1.2268703 1.24627612 0 3 0.5000000 0.0000000
## 17 9.35188471 1.8713079 1.52332969 1 3 0.6666667 0.3333333
## 18 2.13055974 0.1486552 0.66468157 2 3 0.8333333 0.6666667
## 19 3.04084144 1.5372691 -0.79163116 3 3 1.0000000 1.0000000
## 20 3.41228055 0.3638633 0.29967309 4 3 1.1666667 1.3333333
## 21 2.09439697 1.0949643 0.76750081 0 4 0.6666667 0.0000000
## 22 0.94136046 0.3418208 0.49284326 1 4 0.8333333 0.3333333
## 23 71.03603286 1.7925546 2.28157204 2 4 1.0000000 0.6666667
## 24 6.62100931 0.1177230 1.20395688 3 4 1.1666667 1.0000000
## 25 9.77152049 0.1903301 1.53282444 4 4 1.3333333 1.3333333
## B2
## 1 0.0000000
## 2 0.0000000
## 3 0.0000000
## 4 0.0000000
## 5 0.0000000
## 6 0.3333333
## 7 0.3333333
## 8 0.3333333
## 9 0.3333333
## 10 0.3333333
## 11 0.6666667
## 12 0.6666667
## 13 0.6666667
## 14 0.6666667
## 15 0.6666667
## 16 1.0000000
## 17 1.0000000
## 18 1.0000000
## 19 1.0000000
## 20 1.0000000
## 21 1.3333333
## 22 1.3333333
## 23 1.3333333
## 24 1.3333333
## 25 1.3333333For the function SpPOP_nonlinear1
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and one explanatory variable (i.e. X). The auxiliary variable has been generated from the uniform distribution U(0,1) and the regression coefficients are generated as non-linear function of locations.
Generation of spatially varying regression coefficients as non-linear function of latitudes and longitudes
B0= 2* sin((pi * Latitudei)/6)
B1=(Latitudei2+Longitudei2)/18
Generation of auxiliary variable from the uniform distribution
X =runif(N,0,1),
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + B1( Latitudei,Longitudei )* Xi + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_nonlinear1<-SpPOP_nonlinear1(25,5,c(1:5),c(1:5))
sp_nonlinear1## Y X latitude longitude B0 B1
## 1 -0.12962621 0.741273956 0 0 0.000000 0.00000000
## 2 0.86173341 0.969768110 1 0 1.000000 0.05555556
## 3 0.88873585 0.202640370 2 0 1.732051 0.22222222
## 4 3.45582866 0.467743181 3 0 2.000000 0.50000000
## 5 -0.19538739 0.124330212 4 0 1.732051 0.88888889
## 6 -0.93423003 0.641858309 0 1 0.000000 0.05555556
## 7 0.98241639 0.641460546 1 1 1.000000 0.11111111
## 8 3.33614187 0.695222820 2 1 1.732051 0.27777778
## 9 1.75314964 0.685712544 3 1 2.000000 0.55555556
## 10 3.00499945 0.884925370 4 1 1.732051 0.94444444
## 11 -1.03611254 0.116913621 0 2 0.000000 0.22222222
## 12 1.69564386 0.988615921 1 2 1.000000 0.27777778
## 13 3.31021212 0.184704783 2 2 1.732051 0.44444444
## 14 1.84340945 0.008165184 3 2 2.000000 0.72222222
## 15 1.85492790 0.811728082 4 2 1.732051 1.11111111
## 16 0.47787864 0.740558583 0 3 0.000000 0.50000000
## 17 1.82490518 0.382840886 1 3 1.000000 0.55555556
## 18 1.93683394 0.600314991 2 3 1.732051 0.72222222
## 19 3.21588692 0.067088166 3 3 2.000000 1.00000000
## 20 2.06960087 0.195909230 4 3 1.732051 1.38888889
## 21 -0.04528709 0.592952824 0 4 0.000000 0.88888889
## 22 2.42877854 0.478905111 1 4 1.000000 0.94444444
## 23 2.36525657 0.675122472 2 4 1.732051 1.11111111
## 24 1.33254549 0.207335375 3 4 2.000000 1.38888889
## 25 0.98849700 0.070341430 4 4 1.732051 1.77777778For the function SpPOP_nonlinear2
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and one explanatory variable (i.e. X). The auxiliary variable has been generated from the uniform distribution U(0,1) and the regression coefficients are generated as non-linear function of locations.
Generation of spatially varying regression coefficients as non-linear function of latitudes and longitudes
B0= 2 * sin(pi * (Latitudei+Longitudei)/6)
B1=(1/2)* exp[Latitudei+Longitudei]
Generation of auxiliary variable from the uniform distribution
X =runif(N,0,1),
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + B1( Latitudei,Longitudei )* Xi + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_nonlinear2<-SpPOP_nonlinear2(25,5,c(1:5),c(1:5))
sp_nonlinear2## Y X latitude longitude B0 B1
## 1 -0.2623128 0.2136018802 0 0 0.000000e+00 0.500000
## 2 3.6063024 0.0347995891 1 0 1.000000e+00 1.359141
## 3 2.1258446 0.0808213637 2 0 1.732051e+00 3.694528
## 4 5.6889865 0.3585180207 3 0 2.000000e+00 10.042768
## 5 5.8825229 0.1180463440 4 0 1.732051e+00 27.299075
## 6 2.3612643 0.1174956451 0 1 1.000000e+00 1.359141
## 7 3.9155667 0.2803148762 1 1 1.732051e+00 3.694528
## 8 4.6119542 0.1854201185 2 1 2.000000e+00 10.042768
## 9 9.6893354 0.3499171450 3 1 1.732051e+00 27.299075
## 10 61.5363914 0.8047664363 4 1 1.000000e+00 74.206580
## 11 2.5665627 0.1674575754 0 2 1.732051e+00 3.694528
## 12 2.6452148 0.0795166050 1 2 2.000000e+00 10.042768
## 13 13.0161041 0.4819087794 2 2 1.732051e+00 27.299075
## 14 58.3441234 0.7615233329 3 2 1.000000e+00 74.206580
## 15 -0.1757544 0.0002595691 4 2 2.449213e-16 201.714397
## 16 5.6183821 0.2604585702 0 3 2.000000e+00 10.042768
## 17 22.5556072 0.7594652874 1 3 1.732051e+00 27.299075
## 18 19.3824061 0.2626759058 2 3 1.000000e+00 74.206580
## 19 11.9894343 0.0579528441 3 3 2.449213e-16 201.714397
## 20 498.8457553 0.9120622594 4 3 -1.000000e+00 548.316579
## 21 6.6865163 0.1865091843 0 4 1.732051e+00 27.299075
## 22 5.1039199 0.0536370645 1 4 1.000000e+00 74.206580
## 23 108.2835400 0.5439063066 2 4 2.449213e-16 201.714397
## 24 112.3453052 0.2068959472 3 4 -1.000000e+00 548.316579
## 25 968.7352693 0.6514479788 4 4 -1.732051e+00 1490.478994For the function SpPOP_nonlinear3
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and three explanatory variables (i.e. X1,X2 and X3). The auxiliary variables have been generated independently from the uniform distribution U(0,1) and the regression coefficients are generated as non-linear function of latitudes and longitudes
Generation of spatially varying regression coefficients as non-linear function of latitudes and longitudes
B0= 2* sin((pi * Latitudei)/6)
B1=(Latitudei2+Longitudei2)/18
B2= 4 * sin(pi * (Latitudei+Longitudei)/6)
B3=(1/2)* exp[Latitudei+Longitudei]
Generation of auxiliary variables independently from the uniform distribution
X1 =runif(N,0,1), X2 =runif(N,0,1) and X3 =runif(N,0,1)
Spatially varying regression model for generating the response variable
Yi = B0( Latitudei,Longitudei ) + B1( Latitudei,Longitudei )* X1i + B2(Latitudei,Longitudei)* X2i + B3( Latitudei,Longitudei )*X3i + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_nonlinear3<-SpPOP_nonlinear3(25,5,c(1:5),c(1:5))
sp_nonlinear3## Y X1 X2 X3 latitude longitude B0
## 1 -0.9182138 0.33320843 0.353142677 0.6147018 0 0 0.000000
## 2 2.0599607 0.76750836 0.664317156 0.4671899 1 0 1.000000
## 3 5.9395778 0.34249101 0.199650929 0.5837977 2 0 1.732051
## 4 14.8853313 0.93272172 0.652078782 0.9594945 3 0 2.000000
## 5 11.4938456 0.94825165 0.104067099 0.3395415 4 0 1.732051
## 6 1.3777097 0.58647576 0.813394757 0.1774095 0 1 0.000000
## 7 4.6824405 0.88651445 0.369619058 0.1307069 1 1 1.000000
## 8 7.6032416 0.83138303 0.446163624 0.5089649 2 1 1.732051
## 9 19.8955613 0.57165250 0.499292862 0.6409255 3 1 2.000000
## 10 44.4453160 0.63847773 0.370049337 0.5853078 4 1 1.732051
## 11 4.2456951 0.76315328 0.474677670 0.3176453 0 2 0.000000
## 12 9.1865769 0.19094042 0.853151965 0.6166145 1 2 1.000000
## 13 15.0977615 0.08367446 0.002866959 0.5002576 2 2 1.732051
## 14 40.6172229 0.95984891 0.339944619 0.4849133 3 2 2.000000
## 15 26.5007960 0.87298681 0.799043830 0.1205405 4 2 1.732051
## 16 4.4745776 0.51734989 0.843996886 0.1486959 0 3 0.000000
## 17 8.6581817 0.42473847 0.470353034 0.1285846 1 3 1.000000
## 18 74.0552837 0.45822004 0.732084861 0.9472868 2 3 1.732051
## 19 145.1338644 0.70347839 0.590421190 0.7063045 3 3 2.000000
## 20 367.3100693 0.09848853 0.660449098 0.6680832 4 3 1.732051
## 21 22.9646200 0.95622267 0.489628288 0.7994466 0 4 0.000000
## 22 22.6661120 0.85342770 0.816861999 0.2449077 1 4 1.000000
## 23 132.3450919 0.22334179 0.891990443 0.6520130 2 4 1.732051
## 24 514.3874813 0.06718166 0.053868327 0.9328562 3 4 2.000000
## 25 1211.6669216 0.62739128 0.917154031 0.8130110 4 4 1.732051
## B1 B2 B3
## 1 0.00000000 0.000000e+00 0.500000
## 2 0.05555556 2.000000e+00 1.359141
## 3 0.22222222 3.464102e+00 3.694528
## 4 0.50000000 4.000000e+00 10.042768
## 5 0.88888889 3.464102e+00 27.299075
## 6 0.05555556 2.000000e+00 1.359141
## 7 0.11111111 3.464102e+00 3.694528
## 8 0.27777778 4.000000e+00 10.042768
## 9 0.55555556 3.464102e+00 27.299075
## 10 0.94444444 2.000000e+00 74.206580
## 11 0.22222222 3.464102e+00 3.694528
## 12 0.27777778 4.000000e+00 10.042768
## 13 0.44444444 3.464102e+00 27.299075
## 14 0.72222222 2.000000e+00 74.206580
## 15 1.11111111 4.898425e-16 201.714397
## 16 0.50000000 4.000000e+00 10.042768
## 17 0.55555556 3.464102e+00 27.299075
## 18 0.72222222 2.000000e+00 74.206580
## 19 1.00000000 4.898425e-16 201.714397
## 20 1.38888889 -2.000000e+00 548.316579
## 21 0.88888889 3.464102e+00 27.299075
## 22 0.94444444 2.000000e+00 74.206580
## 23 1.11111111 4.898425e-16 201.714397
## 24 1.38888889 -2.000000e+00 548.316579
## 25 1.77777778 -3.464102e+00 1490.478994For the function SpPOP_nonlinear4
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and one explanatory variable (i.e. X). The auxiliary variable has been generated from the uniform distribution U(0,2) and the regression coefficients are generated as non-linear function of locations. In this function, we have used non-linear model for generating the response variable i.e. the relationship between the response and auxiliary variable is non-linear in nature.
Generation of spatially varying regression coefficients as non-linear function of latitudes and longitudes
B0= 2* sin((pi * Latitudei)/6)
B1=(Latitudei2+Longitudei2)/18
Generation of auxiliary variable from the uniform distribution
X =runif(N,0,2),
Spatially varying regression model for generating the response variable The relation between Y and X is non-linear in the model
Yi = B0( Latitudei,Longitudei ) + exp[B1( Latitudei,Longitudei )* Xi] + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_nonlinear4<-SpPOP_nonlinear4(25,5,c(1:5),c(1:5))
sp_nonlinear4## Y X latitude longitude B0 B1
## 1 -0.3696235 1.4409775 0 0 0.000000 0.00000000
## 2 0.3605322 0.5804171 1 0 1.000000 0.05555556
## 3 3.5153521 0.8417020 2 0 1.732051 0.22222222
## 4 3.7441876 1.0005980 3 0 2.000000 0.50000000
## 5 3.7884874 0.5496977 4 0 1.732051 0.88888889
## 6 1.5157027 0.8237628 0 1 0.000000 0.05555556
## 7 2.2305095 1.5783105 1 1 1.000000 0.11111111
## 8 3.0830630 1.2933834 2 1 1.732051 0.27777778
## 9 4.5098572 1.8153169 3 1 2.000000 0.55555556
## 10 1.9304841 0.2982768 4 1 1.732051 0.94444444
## 11 0.1901395 1.4421258 0 2 0.000000 0.22222222
## 12 2.2915563 1.5851315 1 2 1.000000 0.27777778
## 13 4.6069955 1.8556445 2 2 1.732051 0.44444444
## 14 1.3902122 0.5467772 3 2 2.000000 0.72222222
## 15 8.4926218 1.6470423 4 2 1.732051 1.11111111
## 16 0.1134475 0.9620176 0 3 0.000000 0.50000000
## 17 4.5185826 0.1337762 1 3 1.000000 0.55555556
## 18 4.7197435 1.9792287 2 3 1.732051 0.72222222
## 19 6.0081672 0.7038567 3 3 2.000000 1.00000000
## 20 2.8899071 0.6359357 4 3 1.732051 1.38888889
## 21 6.8000526 1.9671486 0 4 0.000000 0.88888889
## 22 2.6481501 0.7462832 1 4 1.000000 0.94444444
## 23 4.7645310 1.0055878 2 4 1.732051 1.11111111
## 24 6.2816722 0.9383703 3 4 2.000000 1.38888889
## 25 3.7243283 0.4797880 4 4 1.732051 1.77777778For the function SpPOP_nonlinear5
The developed function returns a spatial population consist of simulated response variable (i.e. Y) along with their spatial coordinates,spatially varying coefficients and two explanatory variables (i.e. X1 and X2). The auxiliary variables are drawn independently from the uniform distribution U(0,2) and normal distribution N(1,1) and the regression coefficients are generated as non-linear function of locations. In this function, we have used non-linear model for generating the response variable i.e. the relationship between the response and auxiliary variable is non-linear in nature.
Generation of spatially varying regression coefficients as non-linear function of latitudes and longitudes
B0= 2* sin((pi * Latitudei)/6)
B1=(Latitudei2+Longitudei2)/18
B2= 4 * sin(pi * (Latitudei+Longitudei)/6)
Generation of auxiliary variable from the uniform and normal distribution
X1 =runif(N,0,2) and X2 =rnorm(N,1,1)
Spatially varying regression model for generating the response variable The relation between Y and X’s is non-linear in the model
Yi = B0( Latitudei,Longitudei ) + exp[(B1( Latitudei,Longitudei )* X1i)+(B2( Latitudei,Longitudei )* X2i)] + ei ; i= 1,…, N
# Examples: Simulated population along with spatial coordinates and spatially varying model parameters
library(SpPOP)
sp_nonlinear5<-SpPOP_nonlinear5(25,5,c(1:5),c(1:5))
sp_nonlinear5## Y X1 X2 latitude longitude B0 B1
## 1 8.227128e-02 1.18429871 0.9075744 0 0 0.000000 0.00000000
## 2 4.068243e+00 0.97126260 0.6751864 1 0 1.000000 0.05555556
## 3 2.119790e+00 1.98198479 -0.2424008 2 0 1.732051 0.22222222
## 4 2.416104e+00 0.08431764 -0.8172399 3 0 2.000000 0.50000000
## 5 9.475765e+00 0.88648261 0.3306336 4 0 1.732051 0.88888889
## 6 2.564479e+01 0.79968165 1.6028736 0 1 0.000000 0.05555556
## 7 9.294188e+01 1.01014365 1.2733994 1 1 1.000000 0.11111111
## 8 1.305297e+00 0.79246110 -0.5043042 2 1 1.732051 0.27777778
## 9 1.302245e+00 1.19535064 -0.8388222 3 1 2.000000 0.55555556
## 10 3.585096e+00 0.85012206 -0.4133791 4 1 1.732051 0.94444444
## 11 7.991012e+02 0.63664388 1.8888545 0 2 0.000000 0.22222222
## 12 2.634243e+01 1.57577768 0.7058088 1 2 1.000000 0.27777778
## 13 2.282557e+00 0.65062688 -0.6047667 2 2 1.732051 0.44444444
## 14 6.502995e+00 0.05448695 0.5906315 3 2 2.000000 0.72222222
## 15 6.448941e+00 1.26281235 2.3175286 4 2 1.732051 1.11111111
## 16 2.854499e+05 0.59050570 3.0666421 0 3 0.000000 0.50000000
## 17 1.725124e+03 1.87665493 1.8503290 1 3 1.000000 0.55555556
## 18 1.612427e+02 0.99627158 2.1773306 2 3 1.732051 0.72222222
## 19 3.458355e+00 0.67306760 0.3495986 3 3 2.000000 1.00000000
## 20 2.481131e+00 0.25112831 1.4154481 4 3 1.732051 1.38888889
## 21 9.046575e+03 1.28200224 2.3009003 0 4 0.000000 0.88888889
## 22 1.142145e+01 0.89027777 0.7992260 1 4 1.000000 0.94444444
## 23 1.056977e+01 1.77225611 1.5967825 2 4 1.732051 1.11111111
## 24 9.096007e-01 0.01240777 2.1065627 3 4 2.000000 1.38888889
## 25 1.958584e+01 0.57993075 -0.5319461 4 4 1.732051 1.77777778
## B2
## 1 0.000000e+00
## 2 2.000000e+00
## 3 3.464102e+00
## 4 4.000000e+00
## 5 3.464102e+00
## 6 2.000000e+00
## 7 3.464102e+00
## 8 4.000000e+00
## 9 3.464102e+00
## 10 2.000000e+00
## 11 3.464102e+00
## 12 4.000000e+00
## 13 3.464102e+00
## 14 2.000000e+00
## 15 4.898425e-16
## 16 4.000000e+00
## 17 3.464102e+00
## 18 2.000000e+00
## 19 4.898425e-16
## 20 -2.000000e+00
## 21 3.464102e+00
## 22 2.000000e+00
## 23 4.898425e-16
## 24 -2.000000e+00
## 25 -3.464102e+00