Simulación no condicional de una realización de un campo aleatorio espacio temporal no separable usando el modelo de covarianza cressie1

En primer lugar, se generar la grilla espacio temporal. Aquí suponemos n=6 ubicaciones espaciales y T=4 momentos en el tiempo, así en total son 24 ubicaciones espacio-tiempo. Se llevará a cabo la simulación y posteriormente se usará el predictor kriging con su respectiva estimación de varianza del error de predicción, en un punto no “observado”. Se asume conocida la función de covarianza. En la práctica esta matriz se puede estimar por métodos como maxima veorsimilitud, pseudoverosimilitud y métodos basados en mínimos cuadrados.

x1 <- seq(0,3,len = 3)
x2 <- seq(1,6,len = 2)
t <- 1:4
grillaSpT=expand.grid(x1,x2,t)
#matriz de distancias (rezagos) espaciales
matDistSp=as.matrix(dist(grillaSpT[,1:2]))
#matriz de distancias (rezagos) temporales
matDistT=as.matrix(dist(grillaSpT[,3:3]))
cressie1=function(h,u,p){(p[1]^2/((p[2]^2*u^2+1)))*exp(-(p[3]^2*h^2)/(p[2]^2*u^2+1))}
##parámetros p, mu, que en este caso son p=c(0.4,1.7,1.9) y mu=0
sigma=cressie1(matDistSp,matDistT,p=c(0.15,1.7,1.9))
sim1=rmvnorm(1,mean=rep(0,nrow(grillaSpT)), sigma=sigma)
datos1=cbind(grillaSpT,t(sim1))
names(datos1)=c("x","y","t","z((x,y),t)")
matDistSp
##           1        2        3        4        5        6        7        8
## 1  0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000
## 2  1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000
## 3  3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000
## 4  5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153
## 5  5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000
## 6  5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153
## 7  0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000
## 8  1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000
## 9  3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000
## 10 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153
## 11 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000
## 12 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153
## 13 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000
## 14 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000
## 15 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000
## 16 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153
## 17 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000
## 18 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153
## 19 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000
## 20 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000
## 21 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000
## 22 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153
## 23 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000
## 24 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153
##           9       10       11       12       13       14       15       16
## 1  3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000
## 2  1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153
## 3  0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952
## 4  5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000
## 5  5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000
## 6  5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000
## 7  3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000
## 8  1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153
## 9  0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952
## 10 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000
## 11 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000
## 12 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000
## 13 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000
## 14 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153
## 15 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952
## 16 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000
## 17 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000
## 18 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000
## 19 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000
## 20 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153
## 21 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952
## 22 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000
## 23 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000
## 24 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000
##          17       18       19       20       21       22       23       24
## 1  5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952
## 2  5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153
## 3  5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000
## 4  1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000
## 5  0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000
## 6  1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000
## 7  5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952
## 8  5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153
## 9  5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000
## 10 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000
## 11 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000
## 12 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000
## 13 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952
## 14 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153
## 15 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000
## 16 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000
## 17 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000
## 18 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000
## 19 5.220153 5.830952 0.000000 1.500000 3.000000 5.000000 5.220153 5.830952
## 20 5.000000 5.220153 1.500000 0.000000 1.500000 5.220153 5.000000 5.220153
## 21 5.220153 5.000000 3.000000 1.500000 0.000000 5.830952 5.220153 5.000000
## 22 1.500000 3.000000 5.000000 5.220153 5.830952 0.000000 1.500000 3.000000
## 23 0.000000 1.500000 5.220153 5.000000 5.220153 1.500000 0.000000 1.500000
## 24 1.500000 0.000000 5.830952 5.220153 5.000000 3.000000 1.500000 0.000000
matDistT
##    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
## 1  0 0 0 0 0 0 1 1 1  1  1  1  2  2  2  2  2  2  3  3  3  3  3  3
## 2  0 0 0 0 0 0 1 1 1  1  1  1  2  2  2  2  2  2  3  3  3  3  3  3
## 3  0 0 0 0 0 0 1 1 1  1  1  1  2  2  2  2  2  2  3  3  3  3  3  3
## 4  0 0 0 0 0 0 1 1 1  1  1  1  2  2  2  2  2  2  3  3  3  3  3  3
## 5  0 0 0 0 0 0 1 1 1  1  1  1  2  2  2  2  2  2  3  3  3  3  3  3
## 6  0 0 0 0 0 0 1 1 1  1  1  1  2  2  2  2  2  2  3  3  3  3  3  3
## 7  1 1 1 1 1 1 0 0 0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
## 8  1 1 1 1 1 1 0 0 0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
## 9  1 1 1 1 1 1 0 0 0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
## 10 1 1 1 1 1 1 0 0 0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
## 11 1 1 1 1 1 1 0 0 0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
## 12 1 1 1 1 1 1 0 0 0  0  0  0  1  1  1  1  1  1  2  2  2  2  2  2
## 13 2 2 2 2 2 2 1 1 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  1
## 14 2 2 2 2 2 2 1 1 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  1
## 15 2 2 2 2 2 2 1 1 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  1
## 16 2 2 2 2 2 2 1 1 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  1
## 17 2 2 2 2 2 2 1 1 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  1
## 18 2 2 2 2 2 2 1 1 1  1  1  1  0  0  0  0  0  0  1  1  1  1  1  1
## 19 3 3 3 3 3 3 2 2 2  2  2  2  1  1  1  1  1  1  0  0  0  0  0  0
## 20 3 3 3 3 3 3 2 2 2  2  2  2  1  1  1  1  1  1  0  0  0  0  0  0
## 21 3 3 3 3 3 3 2 2 2  2  2  2  1  1  1  1  1  1  0  0  0  0  0  0
## 22 3 3 3 3 3 3 2 2 2  2  2  2  1  1  1  1  1  1  0  0  0  0  0  0
## 23 3 3 3 3 3 3 2 2 2  2  2  2  1  1  1  1  1  1  0  0  0  0  0  0
## 24 3 3 3 3 3 3 2 2 2  2  2  2  1  1  1  1  1  1  0  0  0  0  0  0
sigma
##               1            2            3            4            5
## 1  2.250000e-02 6.677680e-06 1.745640e-16 1.435838e-41 4.261364e-45
## 2  6.677680e-06 2.250000e-02 6.677680e-06 4.261364e-45 1.435838e-41
## 3  1.745640e-16 6.677680e-06 2.250000e-02 1.113981e-55 4.261364e-45
## 4  1.435838e-41 4.261364e-45 1.113981e-55 2.250000e-02 6.677680e-06
## 5  4.261364e-45 1.435838e-41 4.261364e-45 6.677680e-06 2.250000e-02
## 6  1.113981e-55 4.261364e-45 1.435838e-41 1.745640e-16 6.677680e-06
## 7  5.784062e-03 7.168131e-04 1.364348e-06 4.857107e-13 6.019365e-14
## 8  7.168131e-04 5.784062e-03 7.168131e-04 6.019365e-14 4.857107e-13
## 9  1.364348e-06 7.168131e-04 5.784062e-03 1.145697e-16 6.019365e-14
## 10 4.857107e-13 6.019365e-14 1.145697e-16 5.784062e-03 7.168131e-04
## 11 6.019365e-14 4.857107e-13 6.019365e-14 7.168131e-04 5.784062e-03
## 12 1.145697e-16 6.019365e-14 4.857107e-13 1.364348e-06 7.168131e-04
## 13 1.791401e-03 9.382886e-04 1.348240e-04 1.356956e-06 7.107376e-07
## 14 9.382886e-04 1.791401e-03 9.382886e-04 7.107376e-07 1.356956e-06
## 15 1.348240e-04 9.382886e-04 1.791401e-03 1.021269e-07 7.107376e-07
## 16 1.356956e-06 7.107376e-07 1.021269e-07 1.791401e-03 9.382886e-04
## 17 7.107376e-07 1.356956e-06 7.107376e-07 9.382886e-04 1.791401e-03
## 18 1.021269e-07 7.107376e-07 1.356956e-06 1.348240e-04 9.382886e-04
## 19 8.330248e-04 6.166746e-04 2.501787e-04 2.947989e-05 2.182348e-05
## 20 6.166746e-04 8.330248e-04 6.166746e-04 2.182348e-05 2.947989e-05
## 21 2.501787e-04 6.166746e-04 8.330248e-04 8.853568e-06 2.182348e-05
## 22 2.947989e-05 2.182348e-05 8.853568e-06 8.330248e-04 6.166746e-04
## 23 2.182348e-05 2.947989e-05 2.182348e-05 6.166746e-04 8.330248e-04
## 24 8.853568e-06 2.182348e-05 2.947989e-05 2.501787e-04 6.166746e-04
##               6            7            8            9           10
## 1  1.113981e-55 5.784062e-03 7.168131e-04 1.364348e-06 4.857107e-13
## 2  4.261364e-45 7.168131e-04 5.784062e-03 7.168131e-04 6.019365e-14
## 3  1.435838e-41 1.364348e-06 7.168131e-04 5.784062e-03 1.145697e-16
## 4  1.745640e-16 4.857107e-13 6.019365e-14 1.145697e-16 5.784062e-03
## 5  6.677680e-06 6.019365e-14 4.857107e-13 6.019365e-14 7.168131e-04
## 6  2.250000e-02 1.145697e-16 6.019365e-14 4.857107e-13 1.364348e-06
## 7  1.145697e-16 2.250000e-02 6.677680e-06 1.745640e-16 1.435838e-41
## 8  6.019365e-14 6.677680e-06 2.250000e-02 6.677680e-06 4.261364e-45
## 9  4.857107e-13 1.745640e-16 6.677680e-06 2.250000e-02 1.113981e-55
## 10 1.364348e-06 1.435838e-41 4.261364e-45 1.113981e-55 2.250000e-02
## 11 7.168131e-04 4.261364e-45 1.435838e-41 4.261364e-45 6.677680e-06
## 12 5.784062e-03 1.113981e-55 4.261364e-45 1.435838e-41 1.745640e-16
## 13 1.021269e-07 5.784062e-03 7.168131e-04 1.364348e-06 4.857107e-13
## 14 7.107376e-07 7.168131e-04 5.784062e-03 7.168131e-04 6.019365e-14
## 15 1.356956e-06 1.364348e-06 7.168131e-04 5.784062e-03 1.145697e-16
## 16 1.348240e-04 4.857107e-13 6.019365e-14 1.145697e-16 5.784062e-03
## 17 9.382886e-04 6.019365e-14 4.857107e-13 6.019365e-14 7.168131e-04
## 18 1.791401e-03 1.145697e-16 6.019365e-14 4.857107e-13 1.364348e-06
## 19 8.853568e-06 1.791401e-03 9.382886e-04 1.348240e-04 1.356956e-06
## 20 2.182348e-05 9.382886e-04 1.791401e-03 9.382886e-04 7.107376e-07
## 21 2.947989e-05 1.348240e-04 9.382886e-04 1.791401e-03 1.021269e-07
## 22 2.501787e-04 1.356956e-06 7.107376e-07 1.021269e-07 1.791401e-03
## 23 6.166746e-04 7.107376e-07 1.356956e-06 7.107376e-07 9.382886e-04
## 24 8.330248e-04 1.021269e-07 7.107376e-07 1.356956e-06 1.348240e-04
##              11           12           13           14           15
## 1  6.019365e-14 1.145697e-16 1.791401e-03 9.382886e-04 1.348240e-04
## 2  4.857107e-13 6.019365e-14 9.382886e-04 1.791401e-03 9.382886e-04
## 3  6.019365e-14 4.857107e-13 1.348240e-04 9.382886e-04 1.791401e-03
## 4  7.168131e-04 1.364348e-06 1.356956e-06 7.107376e-07 1.021269e-07
## 5  5.784062e-03 7.168131e-04 7.107376e-07 1.356956e-06 7.107376e-07
## 6  7.168131e-04 5.784062e-03 1.021269e-07 7.107376e-07 1.356956e-06
## 7  4.261364e-45 1.113981e-55 5.784062e-03 7.168131e-04 1.364348e-06
## 8  1.435838e-41 4.261364e-45 7.168131e-04 5.784062e-03 7.168131e-04
## 9  4.261364e-45 1.435838e-41 1.364348e-06 7.168131e-04 5.784062e-03
## 10 6.677680e-06 1.745640e-16 4.857107e-13 6.019365e-14 1.145697e-16
## 11 2.250000e-02 6.677680e-06 6.019365e-14 4.857107e-13 6.019365e-14
## 12 6.677680e-06 2.250000e-02 1.145697e-16 6.019365e-14 4.857107e-13
## 13 6.019365e-14 1.145697e-16 2.250000e-02 6.677680e-06 1.745640e-16
## 14 4.857107e-13 6.019365e-14 6.677680e-06 2.250000e-02 6.677680e-06
## 15 6.019365e-14 4.857107e-13 1.745640e-16 6.677680e-06 2.250000e-02
## 16 7.168131e-04 1.364348e-06 1.435838e-41 4.261364e-45 1.113981e-55
## 17 5.784062e-03 7.168131e-04 4.261364e-45 1.435838e-41 4.261364e-45
## 18 7.168131e-04 5.784062e-03 1.113981e-55 4.261364e-45 1.435838e-41
## 19 7.107376e-07 1.021269e-07 5.784062e-03 7.168131e-04 1.364348e-06
## 20 1.356956e-06 7.107376e-07 7.168131e-04 5.784062e-03 7.168131e-04
## 21 7.107376e-07 1.356956e-06 1.364348e-06 7.168131e-04 5.784062e-03
## 22 9.382886e-04 1.348240e-04 4.857107e-13 6.019365e-14 1.145697e-16
## 23 1.791401e-03 9.382886e-04 6.019365e-14 4.857107e-13 6.019365e-14
## 24 9.382886e-04 1.791401e-03 1.145697e-16 6.019365e-14 4.857107e-13
##              16           17           18           19           20
## 1  1.356956e-06 7.107376e-07 1.021269e-07 8.330248e-04 6.166746e-04
## 2  7.107376e-07 1.356956e-06 7.107376e-07 6.166746e-04 8.330248e-04
## 3  1.021269e-07 7.107376e-07 1.356956e-06 2.501787e-04 6.166746e-04
## 4  1.791401e-03 9.382886e-04 1.348240e-04 2.947989e-05 2.182348e-05
## 5  9.382886e-04 1.791401e-03 9.382886e-04 2.182348e-05 2.947989e-05
## 6  1.348240e-04 9.382886e-04 1.791401e-03 8.853568e-06 2.182348e-05
## 7  4.857107e-13 6.019365e-14 1.145697e-16 1.791401e-03 9.382886e-04
## 8  6.019365e-14 4.857107e-13 6.019365e-14 9.382886e-04 1.791401e-03
## 9  1.145697e-16 6.019365e-14 4.857107e-13 1.348240e-04 9.382886e-04
## 10 5.784062e-03 7.168131e-04 1.364348e-06 1.356956e-06 7.107376e-07
## 11 7.168131e-04 5.784062e-03 7.168131e-04 7.107376e-07 1.356956e-06
## 12 1.364348e-06 7.168131e-04 5.784062e-03 1.021269e-07 7.107376e-07
## 13 1.435838e-41 4.261364e-45 1.113981e-55 5.784062e-03 7.168131e-04
## 14 4.261364e-45 1.435838e-41 4.261364e-45 7.168131e-04 5.784062e-03
## 15 1.113981e-55 4.261364e-45 1.435838e-41 1.364348e-06 7.168131e-04
## 16 2.250000e-02 6.677680e-06 1.745640e-16 4.857107e-13 6.019365e-14
## 17 6.677680e-06 2.250000e-02 6.677680e-06 6.019365e-14 4.857107e-13
## 18 1.745640e-16 6.677680e-06 2.250000e-02 1.145697e-16 6.019365e-14
## 19 4.857107e-13 6.019365e-14 1.145697e-16 2.250000e-02 6.677680e-06
## 20 6.019365e-14 4.857107e-13 6.019365e-14 6.677680e-06 2.250000e-02
## 21 1.145697e-16 6.019365e-14 4.857107e-13 1.745640e-16 6.677680e-06
## 22 5.784062e-03 7.168131e-04 1.364348e-06 1.435838e-41 4.261364e-45
## 23 7.168131e-04 5.784062e-03 7.168131e-04 4.261364e-45 1.435838e-41
## 24 1.364348e-06 7.168131e-04 5.784062e-03 1.113981e-55 4.261364e-45
##              21           22           23           24
## 1  2.501787e-04 2.947989e-05 2.182348e-05 8.853568e-06
## 2  6.166746e-04 2.182348e-05 2.947989e-05 2.182348e-05
## 3  8.330248e-04 8.853568e-06 2.182348e-05 2.947989e-05
## 4  8.853568e-06 8.330248e-04 6.166746e-04 2.501787e-04
## 5  2.182348e-05 6.166746e-04 8.330248e-04 6.166746e-04
## 6  2.947989e-05 2.501787e-04 6.166746e-04 8.330248e-04
## 7  1.348240e-04 1.356956e-06 7.107376e-07 1.021269e-07
## 8  9.382886e-04 7.107376e-07 1.356956e-06 7.107376e-07
## 9  1.791401e-03 1.021269e-07 7.107376e-07 1.356956e-06
## 10 1.021269e-07 1.791401e-03 9.382886e-04 1.348240e-04
## 11 7.107376e-07 9.382886e-04 1.791401e-03 9.382886e-04
## 12 1.356956e-06 1.348240e-04 9.382886e-04 1.791401e-03
## 13 1.364348e-06 4.857107e-13 6.019365e-14 1.145697e-16
## 14 7.168131e-04 6.019365e-14 4.857107e-13 6.019365e-14
## 15 5.784062e-03 1.145697e-16 6.019365e-14 4.857107e-13
## 16 1.145697e-16 5.784062e-03 7.168131e-04 1.364348e-06
## 17 6.019365e-14 7.168131e-04 5.784062e-03 7.168131e-04
## 18 4.857107e-13 1.364348e-06 7.168131e-04 5.784062e-03
## 19 1.745640e-16 1.435838e-41 4.261364e-45 1.113981e-55
## 20 6.677680e-06 4.261364e-45 1.435838e-41 4.261364e-45
## 21 2.250000e-02 1.113981e-55 4.261364e-45 1.435838e-41
## 22 1.113981e-55 2.250000e-02 6.677680e-06 1.745640e-16
## 23 4.261364e-45 6.677680e-06 2.250000e-02 6.677680e-06
## 24 1.435838e-41 1.745640e-16 6.677680e-06 2.250000e-02
datos1
##      x y t    z((x,y),t)
## 1  0.0 1 1  0.1921248779
## 2  1.5 1 1  0.0309070884
## 3  3.0 1 1 -0.2683372216
## 4  0.0 6 1  0.0141522127
## 5  1.5 6 1 -0.0930716131
## 6  3.0 6 1  0.1131714253
## 7  0.0 1 2 -0.0039375425
## 8  1.5 1 2  0.0882744591
## 9  3.0 1 2  0.0009303811
## 10 0.0 6 2  0.0609770452
## 11 1.5 6 2  0.1974399457
## 12 3.0 6 2 -0.0592810460
## 13 0.0 1 3  0.0349119815
## 14 1.5 1 3  0.0478862053
## 15 3.0 1 3  0.1609222747
## 16 0.0 6 3 -0.2746997974
## 17 1.5 6 3  0.1919877383
## 18 3.0 6 3 -0.0749448000
## 19 0.0 1 4  0.0541006174
## 20 1.5 1 4  0.0597481119
## 21 3.0 1 4 -0.1363482454
## 22 0.0 6 4 -0.3271711154
## 23 1.5 6 4  0.1920837837
## 24 3.0 6 4 -0.1114832654

Se requiere predecir predecir en el tiempo t=2.3 y en el lugar s0=(1.5,2.7).Nótese que tanto el dominio espacial como el dominio temporal con continuos y fijos. A continuación se presenta el procedimiento para llevar a cabo Kriging simple con su respectiva varianza de error de predicción estimada

grillaSpT0=rbind(expand.grid(x1,x2,t),c(1.5,2.7,2.3))
matDistSp0=as.matrix(dist(grillaSpT0[,1:2]))
matDistT0=as.matrix(dist(grillaSpT0[,3:3]))
sigma0=cressie1(matDistSp0,matDistT0,p=c(0.15,1.7,1.9))
#vector de covarianzas entre la coordenada a predecir y las observadas
sigma0
##               1            2            3            4            5
## 1  2.250000e-02 6.677680e-06 1.745640e-16 1.435838e-41 4.261364e-45
## 2  6.677680e-06 2.250000e-02 6.677680e-06 4.261364e-45 1.435838e-41
## 3  1.745640e-16 6.677680e-06 2.250000e-02 1.113981e-55 4.261364e-45
## 4  1.435838e-41 4.261364e-45 1.113981e-55 2.250000e-02 6.677680e-06
## 5  4.261364e-45 1.435838e-41 4.261364e-45 6.677680e-06 2.250000e-02
## 6  1.113981e-55 4.261364e-45 1.435838e-41 1.745640e-16 6.677680e-06
## 7  5.784062e-03 7.168131e-04 1.364348e-06 4.857107e-13 6.019365e-14
## 8  7.168131e-04 5.784062e-03 7.168131e-04 6.019365e-14 4.857107e-13
## 9  1.364348e-06 7.168131e-04 5.784062e-03 1.145697e-16 6.019365e-14
## 10 4.857107e-13 6.019365e-14 1.145697e-16 5.784062e-03 7.168131e-04
## 11 6.019365e-14 4.857107e-13 6.019365e-14 7.168131e-04 5.784062e-03
## 12 1.145697e-16 6.019365e-14 4.857107e-13 1.364348e-06 7.168131e-04
## 13 1.791401e-03 9.382886e-04 1.348240e-04 1.356956e-06 7.107376e-07
## 14 9.382886e-04 1.791401e-03 9.382886e-04 7.107376e-07 1.356956e-06
## 15 1.348240e-04 9.382886e-04 1.791401e-03 1.021269e-07 7.107376e-07
## 16 1.356956e-06 7.107376e-07 1.021269e-07 1.791401e-03 9.382886e-04
## 17 7.107376e-07 1.356956e-06 7.107376e-07 9.382886e-04 1.791401e-03
## 18 1.021269e-07 7.107376e-07 1.356956e-06 1.348240e-04 9.382886e-04
## 19 8.330248e-04 6.166746e-04 2.501787e-04 2.947989e-05 2.182348e-05
## 20 6.166746e-04 8.330248e-04 6.166746e-04 2.182348e-05 2.947989e-05
## 21 2.501787e-04 6.166746e-04 8.330248e-04 8.853568e-06 2.182348e-05
## 22 2.947989e-05 2.182348e-05 8.853568e-06 8.330248e-04 6.166746e-04
## 23 2.182348e-05 2.947989e-05 2.182348e-05 6.166746e-04 8.330248e-04
## 24 8.853568e-06 2.182348e-05 2.947989e-05 2.501787e-04 6.166746e-04
## 25 1.632912e-04 6.493361e-04 1.632912e-04 1.206101e-06 4.796124e-06
##               6            7            8            9           10
## 1  1.113981e-55 5.784062e-03 7.168131e-04 1.364348e-06 4.857107e-13
## 2  4.261364e-45 7.168131e-04 5.784062e-03 7.168131e-04 6.019365e-14
## 3  1.435838e-41 1.364348e-06 7.168131e-04 5.784062e-03 1.145697e-16
## 4  1.745640e-16 4.857107e-13 6.019365e-14 1.145697e-16 5.784062e-03
## 5  6.677680e-06 6.019365e-14 4.857107e-13 6.019365e-14 7.168131e-04
## 6  2.250000e-02 1.145697e-16 6.019365e-14 4.857107e-13 1.364348e-06
## 7  1.145697e-16 2.250000e-02 6.677680e-06 1.745640e-16 1.435838e-41
## 8  6.019365e-14 6.677680e-06 2.250000e-02 6.677680e-06 4.261364e-45
## 9  4.857107e-13 1.745640e-16 6.677680e-06 2.250000e-02 1.113981e-55
## 10 1.364348e-06 1.435838e-41 4.261364e-45 1.113981e-55 2.250000e-02
## 11 7.168131e-04 4.261364e-45 1.435838e-41 4.261364e-45 6.677680e-06
## 12 5.784062e-03 1.113981e-55 4.261364e-45 1.435838e-41 1.745640e-16
## 13 1.021269e-07 5.784062e-03 7.168131e-04 1.364348e-06 4.857107e-13
## 14 7.107376e-07 7.168131e-04 5.784062e-03 7.168131e-04 6.019365e-14
## 15 1.356956e-06 1.364348e-06 7.168131e-04 5.784062e-03 1.145697e-16
## 16 1.348240e-04 4.857107e-13 6.019365e-14 1.145697e-16 5.784062e-03
## 17 9.382886e-04 6.019365e-14 4.857107e-13 6.019365e-14 7.168131e-04
## 18 1.791401e-03 1.145697e-16 6.019365e-14 4.857107e-13 1.364348e-06
## 19 8.853568e-06 1.791401e-03 9.382886e-04 1.348240e-04 1.356956e-06
## 20 2.182348e-05 9.382886e-04 1.791401e-03 9.382886e-04 7.107376e-07
## 21 2.947989e-05 1.348240e-04 9.382886e-04 1.791401e-03 1.021269e-07
## 22 2.501787e-04 1.356956e-06 7.107376e-07 1.021269e-07 1.791401e-03
## 23 6.166746e-04 7.107376e-07 1.356956e-06 7.107376e-07 9.382886e-04
## 24 8.330248e-04 1.021269e-07 7.107376e-07 1.356956e-06 1.348240e-04
## 25 1.206101e-06 7.188593e-09 4.529707e-06 7.188593e-09 8.000712e-19
##              11           12           13           14           15
## 1  6.019365e-14 1.145697e-16 1.791401e-03 9.382886e-04 1.348240e-04
## 2  4.857107e-13 6.019365e-14 9.382886e-04 1.791401e-03 9.382886e-04
## 3  6.019365e-14 4.857107e-13 1.348240e-04 9.382886e-04 1.791401e-03
## 4  7.168131e-04 1.364348e-06 1.356956e-06 7.107376e-07 1.021269e-07
## 5  5.784062e-03 7.168131e-04 7.107376e-07 1.356956e-06 7.107376e-07
## 6  7.168131e-04 5.784062e-03 1.021269e-07 7.107376e-07 1.356956e-06
## 7  4.261364e-45 1.113981e-55 5.784062e-03 7.168131e-04 1.364348e-06
## 8  1.435838e-41 4.261364e-45 7.168131e-04 5.784062e-03 7.168131e-04
## 9  4.261364e-45 1.435838e-41 1.364348e-06 7.168131e-04 5.784062e-03
## 10 6.677680e-06 1.745640e-16 4.857107e-13 6.019365e-14 1.145697e-16
## 11 2.250000e-02 6.677680e-06 6.019365e-14 4.857107e-13 6.019365e-14
## 12 6.677680e-06 2.250000e-02 1.145697e-16 6.019365e-14 4.857107e-13
## 13 6.019365e-14 1.145697e-16 2.250000e-02 6.677680e-06 1.745640e-16
## 14 4.857107e-13 6.019365e-14 6.677680e-06 2.250000e-02 6.677680e-06
## 15 6.019365e-14 4.857107e-13 1.745640e-16 6.677680e-06 2.250000e-02
## 16 7.168131e-04 1.364348e-06 1.435838e-41 4.261364e-45 1.113981e-55
## 17 5.784062e-03 7.168131e-04 4.261364e-45 1.435838e-41 4.261364e-45
## 18 7.168131e-04 5.784062e-03 1.113981e-55 4.261364e-45 1.435838e-41
## 19 7.107376e-07 1.021269e-07 5.784062e-03 7.168131e-04 1.364348e-06
## 20 1.356956e-06 7.107376e-07 7.168131e-04 5.784062e-03 7.168131e-04
## 21 7.107376e-07 1.356956e-06 1.364348e-06 7.168131e-04 5.784062e-03
## 22 9.382886e-04 1.348240e-04 4.857107e-13 6.019365e-14 1.145697e-16
## 23 1.791401e-03 9.382886e-04 6.019365e-14 4.857107e-13 6.019365e-14
## 24 9.382886e-04 1.791401e-03 1.145697e-16 6.019365e-14 4.857107e-13
## 25 5.041442e-16 8.000712e-19 4.302596e-06 1.240942e-04 4.302596e-06
##              16           17           18           19           20
## 1  1.356956e-06 7.107376e-07 1.021269e-07 8.330248e-04 6.166746e-04
## 2  7.107376e-07 1.356956e-06 7.107376e-07 6.166746e-04 8.330248e-04
## 3  1.021269e-07 7.107376e-07 1.356956e-06 2.501787e-04 6.166746e-04
## 4  1.791401e-03 9.382886e-04 1.348240e-04 2.947989e-05 2.182348e-05
## 5  9.382886e-04 1.791401e-03 9.382886e-04 2.182348e-05 2.947989e-05
## 6  1.348240e-04 9.382886e-04 1.791401e-03 8.853568e-06 2.182348e-05
## 7  4.857107e-13 6.019365e-14 1.145697e-16 1.791401e-03 9.382886e-04
## 8  6.019365e-14 4.857107e-13 6.019365e-14 9.382886e-04 1.791401e-03
## 9  1.145697e-16 6.019365e-14 4.857107e-13 1.348240e-04 9.382886e-04
## 10 5.784062e-03 7.168131e-04 1.364348e-06 1.356956e-06 7.107376e-07
## 11 7.168131e-04 5.784062e-03 7.168131e-04 7.107376e-07 1.356956e-06
## 12 1.364348e-06 7.168131e-04 5.784062e-03 1.021269e-07 7.107376e-07
## 13 1.435838e-41 4.261364e-45 1.113981e-55 5.784062e-03 7.168131e-04
## 14 4.261364e-45 1.435838e-41 4.261364e-45 7.168131e-04 5.784062e-03
## 15 1.113981e-55 4.261364e-45 1.435838e-41 1.364348e-06 7.168131e-04
## 16 2.250000e-02 6.677680e-06 1.745640e-16 4.857107e-13 6.019365e-14
## 17 6.677680e-06 2.250000e-02 6.677680e-06 6.019365e-14 4.857107e-13
## 18 1.745640e-16 6.677680e-06 2.250000e-02 1.145697e-16 6.019365e-14
## 19 4.857107e-13 6.019365e-14 1.145697e-16 2.250000e-02 6.677680e-06
## 20 6.019365e-14 4.857107e-13 6.019365e-14 6.677680e-06 2.250000e-02
## 21 1.145697e-16 6.019365e-14 4.857107e-13 1.745640e-16 6.677680e-06
## 22 5.784062e-03 7.168131e-04 1.364348e-06 1.435838e-41 4.261364e-45
## 23 7.168131e-04 5.784062e-03 7.168131e-04 4.261364e-45 1.435838e-41
## 24 1.364348e-06 7.168131e-04 5.784062e-03 1.113981e-55 4.261364e-45
## 25 2.770413e-11 7.990346e-10 2.770413e-11 3.308220e-04 7.884761e-04
##              21           22           23           24           25
## 1  2.501787e-04 2.947989e-05 2.182348e-05 8.853568e-06 1.632912e-04
## 2  6.166746e-04 2.182348e-05 2.947989e-05 2.182348e-05 6.493361e-04
## 3  8.330248e-04 8.853568e-06 2.182348e-05 2.947989e-05 1.632912e-04
## 4  8.853568e-06 8.330248e-04 6.166746e-04 2.501787e-04 1.206101e-06
## 5  2.182348e-05 6.166746e-04 8.330248e-04 6.166746e-04 4.796124e-06
## 6  2.947989e-05 2.501787e-04 6.166746e-04 8.330248e-04 1.206101e-06
## 7  1.348240e-04 1.356956e-06 7.107376e-07 1.021269e-07 7.188593e-09
## 8  9.382886e-04 7.107376e-07 1.356956e-06 7.107376e-07 4.529707e-06
## 9  1.791401e-03 1.021269e-07 7.107376e-07 1.356956e-06 7.188593e-09
## 10 1.021269e-07 1.791401e-03 9.382886e-04 1.348240e-04 8.000712e-19
## 11 7.107376e-07 9.382886e-04 1.791401e-03 9.382886e-04 5.041442e-16
## 12 1.356956e-06 1.348240e-04 9.382886e-04 1.791401e-03 8.000712e-19
## 13 1.364348e-06 4.857107e-13 6.019365e-14 1.145697e-16 4.302596e-06
## 14 7.168131e-04 6.019365e-14 4.857107e-13 6.019365e-14 1.240942e-04
## 15 5.784062e-03 1.145697e-16 6.019365e-14 4.857107e-13 4.302596e-06
## 16 1.145697e-16 5.784062e-03 7.168131e-04 1.364348e-06 2.770413e-11
## 17 6.019365e-14 7.168131e-04 5.784062e-03 7.168131e-04 7.990346e-10
## 18 4.857107e-13 1.364348e-06 7.168131e-04 5.784062e-03 2.770413e-11
## 19 1.745640e-16 1.435838e-41 4.261364e-45 1.113981e-55 3.308220e-04
## 20 6.677680e-06 4.261364e-45 1.435838e-41 4.261364e-45 7.884761e-04
## 21 2.250000e-02 1.113981e-55 4.261364e-45 1.435838e-41 3.308220e-04
## 22 1.113981e-55 2.250000e-02 6.677680e-06 1.745640e-16 1.508203e-05
## 23 4.261364e-45 6.677680e-06 2.250000e-02 6.677680e-06 3.594628e-05
## 24 1.435838e-41 1.745640e-16 6.677680e-06 2.250000e-02 1.508203e-05
## 25 3.308220e-04 1.508203e-05 3.594628e-05 1.508203e-05 2.250000e-02
lambda=solve(sigma)%*%sigma0[25,-25]
lambda
##             [,1]
## 1   7.588729e-03
## 2   3.051466e-02
## 3   7.588729e-03
## 4  -3.276782e-05
## 5   1.166265e-04
## 6  -3.276782e-05
## 7  -4.212887e-03
## 8  -1.073644e-02
## 9  -4.212887e-03
## 10 -5.666790e-05
## 11 -9.187561e-05
## 12 -5.666790e-05
## 13 -5.543596e-03
## 14 -4.857869e-03
## 15 -5.543596e-03
## 16 -2.212264e-04
## 17 -4.597041e-04
## 18 -2.212264e-04
## 19  1.587928e-02
## 20  3.629635e-02
## 21  1.587928e-02
## 22  7.064917e-04
## 23  1.685196e-03
## 24  7.064917e-04
z_pred0=t(lambda)%*%datos1[,4]
z_pred0
##              [,1]
## [1,] -0.001056342
VarErropred0=sigma[1,1]-t(sigma0[25,-25])%*%solve(sigma)%*%sigma0[25,-25]
VarErropred0
##           [,1]
## [1,] 0.0224392

Algunas funciones de covarianza espacio temporal no separables

##Funciones de covarianza espacio temporal p vector de parámetros para cada modelo
exp_esp_temp=function(h,u,p){((p[1])^2)*exp(-h/p[2]-u/p[3])}
gauss_esp_temp=function(h,u,p){(p[1]^2)*exp(-(h/p[2])^2-(u/p[3])^2)}
cressie1=function(h,u,p){(p[1]^2/((p[2]^2*u^2+1)))*exp(-(p[3]^2*h^2)/(p[2]^2*u^2+1))}
Gneiting1=function(h,u,p){p[1]^2/((p[2]*u^(2*p[3])+1)^(p[4]))*exp(-(p[6]*h^(2*p[5]))/((p[2]*u^(2*p[3])+1)^(p[4]*p[5])))}
Gneiting2=function(h,u,sigma,p)
{p[1]^2/((2^(p[3]-1))*p[7](p[3])*(p[2]*u^(2*p[3])+1)^(p[4]+p[5]))*
(((p[6]*h)/((p[2]*u^(2*p[3])+1)^(p[5]/2)))^p[3])*
besselK(((p[6]*h)/((p[2]*u^(2*p[3])+1)^(p[5]/2))),p[3])}
Iaco_Cesare=function(h,u,a,b,c){(1+h^p[1]+u^p[2])^(-p[3])}

##C R E S S I E - H U A N G (1999)

#sigma:desviacion estandar, a es el parámetros de escala del tiempo, b es el parámetros de escala del espacio, d es la dimensión espacial; a,b positivos
CH_1=function(h,u,p,d){(p[1]^2/((p[2]^2*u^2+1)^(d/2)))*exp(-(p[3]^2*h^2)/(p[2]^2*u^2+1))}
CH_2=function(h,u,p,d){(p[1]^2/((p[2]*abs(u)+1)^(d/2)))*exp(-(p[3]^2*h^2)/(p[2]*abs(u)+1))}
CH_3=function(h,u,p,d){p[1]^2*((p[2]^2)*(u^2)+1)/(((p[2]^2)*(u^2)+1)^2+(p[3]^2)*h^2)^((d+1)/2)}
CH_4=function(h,u,p,d){p[1]^2*(p[2]*abs(u)+1)/((p[2]*abs(u)+1)^2+(p[3]^2)*h^2)^((d+1)/2)}

#el caso mas general de C R E S S I E - H U A N G (1999) es cuando d=2, entonces queda
CH_1=function(h,u,p){(p[1]^2/((p[2]^2*u^2+1)))*exp(-(p[3]^2*h^2)/(p[2]^2*u^2+1))}
CH_2=function(h,u,p){(p[1]^2/((p[2]*abs(u)+1)))*exp(-(p[3]^2*h^2)/(p[2]*abs(u)+1))}
CH_3=function(h,u,p){p[1]^2*((p[2]^2)*(u^2)+1)/(((p[2]^2)*(u^2)+1)^2+(p[3]^2)*h^2)^((3)/2)}
CH_4=function(h,u,p){p[1]^2*(p[2]*abs(u)+1)/((p[2]*abs(u)+1)^2+(p[3]^2)*h^2)^((3)/2)}

Gneiting (2002), combina fun1, fun2 y psi en Gneiting

#fun1
phi1=function(r,c,gama,v){v*exp(-c*r^gama)}                                            #c>0, 0<gama<=1, siempre v=1
phi2=function(r,c,gama,v){((2^(v-1))*gamma(v))^(-1)*(c*r^0.5)^v*besselK(c*r^0.5,v)}    #c>0, v>0
phi3=function(r,c,gama,v){(1+c*r^gama)^(-v)}                                           #c>0, 0<gama<=1, v>0
phi4=function(r,c,gama,v){gama*(2^v)*(exp(c*r^0.5)+exp(-c*r^0.5))^(-v)}                #c>0, v>0, siempre gama=1

#fun2
psi1=function(r,a,alpha,beta){(a*r^alpha+1)^beta}                                      #a>0, 0<alpha<=1, 0<=beta<=1
psi2=function(r,a,alpha,beta){log(a*r^alpha+beta)/log(beta)}                           #a>0, beta>1,  0<alpha<=1
psi3=function(r,a,alpha,beta){(a*r^alpha+beta)/(beta*(a*r^alpha+1))}                   #a>0, 0<beta<=1   0<alpha<=1  

#Cualquier combinación genera una función de covarianza válida
Gneiting=function(h,u,sigma,d,a,alpha,beta,c,gama,v,psi,phi){(sigma^2/(psi((abs(u)^2),a,alpha,beta))^(d/2))*phi(h^2/(psi(abs(u)^2,a,alpha,beta)),c,gama,v)}

#el caso mas general de Gneiting (2002) es cuando d=2, entonces queda
Gneiting=function(h,u,sigma,a,alpha,beta,c,gama,v,psi,phi){(sigma^2/(psi((abs(u)^2),a,alpha,beta)))*phi(h^2/(psi(abs(u)^2,a,alpha,beta)),c,gama,v)}
####IACO_CESSARE
C_IACO_CESSARE=function(h,u,sigma,a,b,alpha,beta,gama){(1 + (h/a)^alpha + (u/b)^beta)^(-gama)}
#(Porcu, 2007) Basado en la función de supervivencia de Dagum 
#función de Dagum
Dagum=function(r,lambda,theta,epsilon){1-1/(1+lambda*r^(-theta))^epsilon}                                                                                     #lamdba, theta in (0,7), epsilon in (0,7)
Dagumm=function(r,lambda,theta,epsilon){ifelse(r==0,1,Dagum(r,lambda,theta,epsilon))}

Porcu_sep=function(h,u,lambda_h,theta_h,epsilon_h,lambda_u,theta_u,epsilon_u){Dagumm(h,lambda_h,theta_h,epsilon_h)*Dagumm(u,lambda_u,theta_u,epsilon_u)}      
Porcu_Nsep=function(h,u,lambda_h,theta_h,epsilon_h,lambda_u,theta_u,epsilon_u,vartheta){vartheta*Dagumm(h,lambda_h,theta_h,epsilon_h)+(1-vartheta)*Dagumm(u,lambda_u,theta_u,epsilon_u)}