Author: Ashish Sharma, Raj Mehrotra, Sanjeev Jha, Jingwan Li and Ze Jiang

# Predictor Identifier: Nonparametric PREDiction

#### Ashish Sharma, Raj Mehrotra, Sanjeev Jha, Jingwan Li and Ze Jiang

#### 10:30:56 25 July, 2024

    library(NPRED)
    require(zoo)
    #> Loading required package: zoo
    #>
    #> Attaching package: 'zoo'
    #> The following objects are masked from 'package:base':
    #>
    #>     as.Date, as.Date.numeric
    require(ggplot2)
    #> Loading required package: ggplot2

# 1 Synthetic data generation

    # Other statistical models for generating synthetic data see Package - synthesis
    require(synthesis)
    #> Loading required package: synthesis
    #>
    #> Attaching package: 'synthesis'
    #> The following objects are masked from 'package:NPRED':
    #>
    #>     data.gen.ar1, data.gen.ar4, data.gen.ar9

    set.seed(2020) # set seed for reproducible results
    # AR1 model from paper with 9 dummy variables
    data.ar1 <- synthesis::data.gen.ar1(500)

    # AR4 model from paper with total 9 dimensions
    data.ar4 <- synthesis::data.gen.ar4(500)

    # AR9 model from paper with total 9 dimensions
    data.ar9 <- synthesis::data.gen.ar9(500)

    # plot.zoo(cbind(data.ar1$x, data.ar4$x, data.ar9$x),
    #   xlab = NA, main = "Example of AR models",
    #   ylab = c("AR1", "AR4", "AR9"))

# 2 Partial informational correlation (PIC)

    # calculate PIC
    pic.calc(data.ar1$x, data.ar1$dp)
    #> $pmi
    #> [1] 0.69319346 0.41783003 0.26170451 0.17923658 0.12102203 0.09080493 0.05591261
    #> [8] 0.02951112 0.01654911
    #>
    #> $pic
    #> [1] 0.8660388 0.7526034 0.6383594 0.5488694 0.4636577 0.4075210 0.3252683
    #> [8] 0.2394038 0.1804341

    pic.calc(data.ar4$x, data.ar4$dp)
    #> $pmi
    #> [1]  0.273343774  0.031705618  0.001377018  0.151620432  0.235788221
    #> [6]  0.126943971  0.021324429 -0.008302517  0.030173933
    #>
    #> $pic
    #> [1] 0.6489498 0.2478761 0.0524428 0.5114477 0.6131739 0.4735201 0.2043335
    #> [8] 0.0000000 0.2419980

    pic.calc(data.ar9$x, data.ar9$dp)
    #> $pmi
    #> [1]  0.034701149 -0.008256610  0.073552041  0.353385264 -0.004119804
    #> [6]  0.027551717  0.088435930  0.091603445  0.049201990
    #>
    #> $pic
    #> [1] 0.2589377 0.0000000 0.3698593 0.7118746 0.0000000 0.2315443 0.4026324
    #> [8] 0.4091505 0.3061328

# 3 Predictor identifier: stepwise.PIC

    pic1 <- stepwise.PIC(data.ar1$x, data.ar1$dp)

    pic4 <- stepwise.PIC(data.ar4$x, data.ar4$dp)

    pic9 <- stepwise.PIC(data.ar9$x, data.ar9$dp)

# 4 Partial weights (PW)

    # calculate partial weights
    pw.calc(data.ar1$x, data.ar1$dp, pic1$cpy, pic1$cpyPIC)
    #> $pw
    #> [1] 1

    pw.calc(data.ar4$x, data.ar4$dp, pic4$cpy, pic4$cpyPIC)
    #> $pw
    #> [1] 0.5405296 0.4594704

    pw.calc(data.ar9$x, data.ar9$dp, pic9$cpy, pic9$cpyPIC)
    #> $pw
    #> [1] 0.46352053 0.32817626 0.16792648 0.04037673

# 5 Nonparameteric prediction: knn

    data("data3")
    x <- ts(data3[, 1]) # response
    z <- ts(data3[, -1]) # possible predictors
    zout <- ts(data.gen.ar1(500, ndim = 15)$dp) # new input

    xhat1 <- xhat2 <- x
    # xhat1 <- NPRED::knn(x,z,zout,k=5,reg=T,extrap=F)
    # xhat2 <- NPRED::knn(x,z,zout,k=5,reg=T,extrap=T)

    for (i in 1:500) {
      xhat1[i] <- NPRED::knn(x[-i], z[-i, ], z[i, ], extrap = F)
      xhat2[i] <- NPRED::knn(x[-i], z[-i, ], z[i, ], extrap = T)
    }

    if (TRUE) {
      par(mfrow = c(1, 1),
          mar=c(3,2,1,1), mgp=c(1.5,0.5,0),
          pty = c("m"))

      ts.plot(x, xhat1, xhat2, col = c("black","red","blue"), ylim = c(-10, 10), lwd = c(1, 1, 1))
      legend("topleft",
        bty = "n", lwd = 3, cex = 1, lty = 1,
        # inset = c(-0.5, 0),
        legend = c("OBS", "Pred", "Pred(extrap=T)"),
        x.intersp = 0, xjust = 0, yjust = 0, text.width = c(0, 50, 50), horiz = T,
        col = c("black","red","blue"))

      par(mfrow = c(1, 1), pty = c("s"))
      plot(xhat1, xhat2, xlim = c(-10, 10), ylim = c(-10, 10))
      abline(coef = c(0, 1), lwd = 1, col = 2)

    }

Figure 5.1: Example of KNN implemented in NPRED

Figure 5.2: Example of KNN implemented in NPRED

# 6 Illustration of the usage of partial weights

    sample <- 500
    k <- 0
    u <- runif(sample, 0, 5 * pi)
    z <- sin(u) + rnorm(sample, sd = 0.2)

    u1 <- cbind(u, runif(sample, 0, 5 * pi), runif(sample, 0, 5 * pi), runif(sample, 0, 5 * pi))
    # zhat1 <- knnregl1cv(x=z, z=u1, k=k)
    zhat1 <- sapply(1:sample, function(i) knn(x = z[-i], z = u1[-i, ], zout = u1[i, ], k = k))

    sel <- stepwise.PIC(x = z, py = u1)
    sel
    #> $cpy
    #> [1] 1
    #>
    #> $cpyPIC
    #> [1] 0.1606175
    #>
    #> $wt
    #> [1] 1
    #>
    #> $lstwet
    #>   Intercept           X
    #> 0.141643996 0.002144423
    #>
    #> $icpy
    #> [1] 1
    # zhat2 <- knnregl1cv(x=z, z=u1[,sel$cpy], k=k)
    zhat2 <- sapply(1:sample, function(i) knn(x = z[-i], z = u1[-i, sel$cpy], zout = u1[i, sel$cpy], k = k))

    if (TRUE) {
      par(mfrow = c(1, 1), pty = c("m"))

      plot(u, z, pch = 16)
      lines(sort(u), zhat1[order(u)], col = "green")
      lines(sort(u), zhat2[order(u)], col = "red")
      abline(a = 0, b = 0)
    }

Figure 6.1: Illustration of the usage of partial weights

# 7 Application to predicting drought anomalies

    # An demo example used in Jiang, Z., Rashid, M. M., Johnson, F., & Sharma, A. (2020)
    # Jiang, Z., Rashid, M. M., Johnson, F., & Sharma, A. (2021). A wavelet-based tool to modulate variance in predictors: An application to predicting drought anomalies. Environmental Modelling and Software, 135, 104907.

    require(WASP)
    #> Loading required package: WASP
    #>
    #> Attaching package: 'WASP'
    #> The following objects are masked from 'package:synthesis':
    #>
    #>     data.gen.HL, data.gen.Rossler, data.gen.SW, data.gen.ar1,
    #>     data.gen.ar4, data.gen.ar9, data.gen.tar1, data.gen.tar2
    #> The following objects are masked from 'package:NPRED':
    #>
    #>     data.gen.ar1, data.gen.ar4, data.gen.ar9, knn, knnregl1cv, pic.calc
    #load response and predictor variables
    data("SPI.12", package="WASP");
    data("data.CI", package="WASP")
    data("Ind_AWAP.2.5", package="WASP")

    #study grids and period
    Grid <- sample(Ind_AWAP.2.5,1)
    Grid = 149 #Grid used in the SI
    SPI.12.ts <- window(SPI.12, start=c(1910,1),end=c(2009,12))
    data.CI.ts <- window(data.CI, start=c(1910,1),end=c(2009,12))
    #partition into two folds
    folds <- cut(seq(1,nrow(SPI.12.ts)),breaks=2,labels=FALSE)
    sub.cali <- which(folds==1, arr.ind=TRUE); sub.vali <- which(folds==2, arr.ind=TRUE)
    #-------------------------------------------------------------------------------------
    ###calibration and selection
    data <- list(x=SPI.12.ts[sub.cali,Grid],dp=data.CI.ts[sub.cali,])

    #variance transformation - calibration
    modwt <- modwt.vt(data, wf="haar", J=8, boundary="periodic")

    #stepwise PIC selection
    sel <- NPRED::stepwise.PIC(modwt$x, modwt$dp.n)
    #-------------------------------------------------------------------------------------
    ###validation and prediction
    data.val <- list(x=SPI.12.ts[sub.vali,Grid],dp=data.CI.ts[sub.vali,])

    #variance transformation - validation
    modwt.val <- modwt.vt.val(data.val, J=8, modwt)

    #knn prediction
    cpy <- sel$cpy; wt <- sel$wt
    x=data$x; z=modwt$dp.n[,cpy]; zout=modwt.val$dp.n[,cpy]
    mod <- knn(x, z, zout, k=5, pw=wt, extrap=T)

    #-------------------------------------------------------------------------------------
    ###plot
    start.cal <- c(1910,1); start.val <- c(1960,1)
    ndim = ncol(data.CI.ts); CI.names = colnames(data.CI.ts)
    par(mfcol=c(ndim+1,2),mar=c(2,4,2,2),mgp=c(1.5,0.5,0),
        bg = "white",pty="m",cex.lab=1.5,ps=8)
    #----------------------------------------------
    #plot before and after vt - calibration
    if(TRUE){

      x <- ts(modwt$x, start=start.cal, frequency = 12)
      dp <- ts(modwt$dp, start=start.cal, frequency = 12)
      dp.n <- ts(modwt$dp.n, start=start.cal, frequency = 12)

      ts.plot(x,xlab=NA, main=paste0("Sampled Grid: ", Grid), ylab=paste0("SPI",12), col=c("black"),lwd=c(1))
      for(nc in 1:ndim)
        ts.plot(dp[,nc],dp.n[,nc],xlab=NA,ylab=paste0(CI.names[nc]),
                col=c("red","blue"),lwd=c(1,1),lty=c(1,2))

    }
    #----------------------------------------------
    #plot before and after vt - validation
    if(TRUE){

      x <- ts(modwt.val$x, start=start.val, frequency = 12)
      dp <- ts(modwt.val$dp, start=start.val, frequency = 12)
      dp.n <- ts(modwt.val$dp.n, start=start.val, frequency = 12)

      ts.plot(x, xlab=NA, main=paste0("Sampled Grid: ",Grid), ylab=paste0("SPI",12), col=c("black"),lwd=c(1))
      for(nc in 1:ndim)
        ts.plot(dp[,nc],dp.n[,nc],xlab=NA,ylab=paste0(CI.names[nc]),
                col=c("red","blue"),lwd=c(1,1),lty=c(1,2))

    }

Figure 7.1: Application to predicting drought anomalies