R/WREG.WLS.R
In USGS-R/WREG: USGS WREG v. 2.02
#'Weighted-Multiple-Linear Regression Program (WREG)
#'
#'@description The \code{WREG.WLS} function executes the multiple linear
#' regression analysis using weighted least-squares regression.
#'
#'@param Y A numeric vector of the dependent variable of interest, with any
#' transformations already applied.
#'@param X A numeric matrix of the independent variables in the regression, with
#' any transformations already applied. Each row represents a site and each
#' column represents a particular independe variable. (If a leading constant
#' is used, it should be included here as a leading column of ones.) The rows
#' must be in the same order as the dependent variables in \code{Y}.
#'@param recordLengths A numeric vector whose rows are in the same order as
#' \code{Y} and represent the at-site record length.
#'@param LP3 A numeric matrix containing the fitted Log-Pearson Type III
#' standard deviate, standard deviation and skew for each site. The columns of
#' the matrix represent S, K, G, and an option regional skew value \code{GR}
#' required by WREG.GLS with regSkew = TRUE. The order of the rows must be the
#' same as \code{Y}.
#'@param transY A required character string indicating if the the
#' dependentvariable was transformed by the common logarithm ('log10'),
#' transformed by the natural logarithm ('ln') or untransformed ('none').
#'@param x0 A vector containing the independent variables (as above) for a
#' particular target site. This variable is only used for ROI analysis.
#'
#'@details In this implementation, the weights for weighted least-squares
#' regression are defined by record lengths. See manual for details.
#'
#'@return All outputs are returned as part of a list. The elements of the list
#' depend on the type of regression performed. The elements of the list may
#' include: \item{Coefs}{A data frame composed of four variables: (1)
#' \code{Coefficient} contains the regression coefficeints estimated for the
#' model, (2) \code{Standard Error} contains the standard errors of
#' each regression coefficient, (3) \code{tStatistic} contains the Student's
#' T-statistic of each regression coefficient and (4) \code{pValue} contains
#' the significance probability of each regression coefficient.}
#' \item{ResLevInf}{A data frame composed of three variables for each site in
#' the regression. \code{Residual} contains the model residuals.
#' \code{Leverage} contains the leverage of each site. \code{Influence}
#' contains the influence of each site.} \item{LevLim}{The critical value of
#' leverage. See \code{\link{Leverage}}} \item{InflLim}{The critical value of
#' influence. See \code{\link{Influence}}} \item{LevInf.Sig}{A logical matrix
#' indicating if the leverage (column 1) is significant and the influence
#' (column 2) is significant for each site in the regression.}
#' \item{PerformanceMetrics}{A list of not more than ten elements. All
#' regression types return the mean squared error of residuals (\code{MSE}),
#' the coefficient of determination (\code{R2}), the adjusted coefficient of
#' determination (\code{R2_adj}) and the root mean squared error (\code{RMSE},
#' in percent). The pseudo coefficient of regression (\code{R2_pseudo}), the
#' average variance of prediction (\code{AVP}), the standard error of
#' prediction (\code{Sp}, in percent), a vector of the individual variances of
#' prediction for each site (\code{VP.PredVar}), the model-error variance
#' (\code{ModErrVar}) and the standardized model error variance
#' (\code{StanModErr}, in percent) are also returned. Details on the
#' appropriateness and applicability of performance metrics can be found in the
#' WREG manual.} \item{X}{The input predictors.} \item{Y}{The input
#' observations.} \item{fitted.values}{A vector of model estimates from the
#' regression model.} \item{residuals}{A vector of model residuals.}
#' \item{Weighting}{The weighting matrix used to develop regression estimates.}
#' \item{Input}{A list of input parameters for error searching. Currently
#' empty.}
#'@import stats
#'
#' @examples
#' # Import some example data
# peakFQdir <- paste0(
# file.path(system.file("exampleDirectory", package = "WREG"),
# "pfqImport"))
# gisFilePath <- file.path(peakFQdir, "pfqSiteInfo.txt")
# importedData <- importPeakFQ(pfqPath = peakFQdir, gisFile = gisFilePath)
# Organizing input data
# lp3Data <- importedData$LP3f
# lp3Data$K <- importedData$LP3k$AEP_0.5
# Y <- importedData$Y$AEP_0.5
# X <- importedData$X[c("Sand", "OutletElev", "Slope")]
# recordLengths <- importedData$recLen
# transY <- "none"
#
# # Run WLS regression
# result <- WREG.WLS(Y, X, recordLengths, LP3 = lp3Data, transY)
#'
#'@export
WREG.WLS <- function(Y,X,recordLengths,LP3,transY,x0=NA) {
# 11/9/16 Greg Petrochenkov: Changed validation scheme
warn("clear")
# Some upfront error handling
wregValidation((!missing(X)&!missing(Y))&&(length(Y)!=nrow(X)), "eq", FALSE,
paste0("The length of Y must be the same as ",
"the number of rows in X."), warnFlag = TRUE)
if (!wregValidation((!missing(X)&!missing(Y))&&(length(Y)!=nrow(X)), "eq", FALSE,
"Dependent variable (Y) must be provided", warnFlag = TRUE)) {
if (!wregValidation(Y, "numeric", message =
"Dependent variable (Y) must be provided as class numeric",
warnFlag = TRUE)) {
wregValidation(sum(is.na(Y)), "eq", 0 ,
paste0("The depedent variable (Y) contains missing ",
"values. These must be removed."),
TRUE)
wregValidation(sum(is.infinite(Y)), "eq", 0 ,
paste0("The depedent variable (Y) contains infinite ",
"values. These must be removed."),
warnFlag = TRUE)
}
}
if (!wregValidation(missing(X), "eq", FALSE,
"Independent variables (X) must be provided.", warnFlag = TRUE)) {
if (!wregValidation((length(unique(apply(X,FUN=class,MARGIN=2)))!=1)|
(unique(apply(X,FUN=class,MARGIN=2))!="numeric"), "eq", FALSE,
"Independent variables (X) must be provided as class numeric.", warnFlag = TRUE)){
wregValidation(sum(is.na(as.matrix(X))), "eq", 0,
paste0("Some independent variables (X) contain missing ",
"values. These must be removed."), warnFlag = TRUE)
wregValidation(sum(is.infinite(as.matrix(X))), "eq", 0,
paste0("Some independent variables (X) contain infinite ",
"values. These must be removed."), warnFlag = TRUE)
}
}
if(!wregValidation(missing(transY)|!is.character(transY), "eq", FALSE,
"transY must be included as a character string", warnFlag = TRUE)) {
wregValidation(!is.element(transY,c("none","log10","ln")), "eq", FALSE,
"transY must be either 'none', 'log10' or 'ln'", warnFlag = TRUE)
}
## Determine if ROI is being applied
if (is.na(sum(x0))) { # ROI regression is not used.
ROI <- F
} else { # ROI regression is used.
ROI <- T
}
## Just initial values for control.
var.modelerror.k <- NA
### Correct recordLengths to be only at-site record lengths.
if(is.matrix(recordLengths)) {
recordLengths<-diag(recordLengths)
}
if (!wregValidation(missing(recordLengths), "eq", FALSE,
"Record lengths must be provided.", warnFlag = TRUE)) {
if (!wregValidation(recordLengths, "numeric",
message="Record lengths must be provided as class numeric.", warnFlag = TRUE)) {
wregValidation(sum(is.na(c(recordLengths))), "eq", 0,
paste0("Some record lengths contain missing ",
"values. These must be removed."), warnFlag = TRUE)
wregValidation(sum(is.infinite(c(recordLengths))), "eq", 0,
paste0("Some record lengths contain infinite ",
"values. These must be removed."), warnFlag = TRUE)
}
}
# Error checking LP3
if (!wregValidation(missing(LP3), "eq", FALSE,
"The data frame LP3 must be provided.", warnFlag = TRUE)) {
if (!wregValidation(!is.data.frame(LP3), "eq", FALSE,
paste("LP3 must be provided as a data frame with elements named",
"'S', 'K' and 'G' for standard deivation, deviate and skew,",
"respectively."), warnFlag = TRUE)) {
if (!wregValidation(sum(is.element(c("S","K","G"),names(LP3)))!=3, "eq", FALSE,
paste("In valid elements: The names of the elements in LP3 are",
names(LP3),". LP3 must be provided as a data frame with elements named",
"'S', 'K' and 'G' for standard deivation, deviate and skew,",
"respectively."), warnFlag = TRUE)) {
if(!wregValidation((length(unique(apply(cbind(LP3$S,LP3$K,LP3$G),FUN=class,MARGIN=2)))!=1)|
(unique(apply(cbind(LP3$S,LP3$K,LP3$G),FUN=class,MARGIN=2))!="numeric"), "eq", FALSE,
"The data frame LP3 must be provided in a numeric class", warnFlag = TRUE)){
wregValidation(sum(is.infinite(LP3$S),is.infinite(LP3$K),is.infinite(LP3$G))>0, "eq", FALSE,
paste0("Some elements of LP3$S, LP3$K, and LP3$G contain infinite ",
"values. These must be removed."), warnFlag = TRUE)
wregValidation(sum(is.na(LP3$S),is.na(LP3$K),is.na(LP3$G))>0, "eq", FALSE,
paste0("Some elements of LP3$S, LP3$K, and LP3$G contain missing ",
"values. These must be removed."), warnFlag = TRUE)
}
}
}
}
wregValidation((!missing(X)&!missing(Y)&!missing(recordLengths)&!missing(LP3))&&
(length(unique(length(Y),nrow(X),nrow(LP3),length(recordLengths)))!=1), "eq", FALSE,
paste0("length(Y), nrow(X), nrow(LP3) and ",
"length(recordLengths) must all be equal"), warnFlag = TRUE)
if (warn("check")){
stop('Invalid inputs were provided. See warnings().', warn("get"))
}
#Convert X and Y from dataframes to matrices to work with matrix operations below
X <- as.matrix(X)
Y <- as.matrix(Y)
### Initial OLS (basis for others)
Omega <- diag(nrow(X)) # temporary weighting matrix (identity)
B_hat <- solve(t(X)%*%solve(Omega)%*%X)%*%t(X)%*%solve(Omega)%*%Y # OLS estimated coefficients
Y_hat <- X%*%B_hat # OLS model estimates
e <- Y-Y_hat # OLS residuals
MSE.OLS <- sum(e^2)/(nrow(X)-ncol(X)) # Mean square-error (k variable OLS)
MSE.OLS.0 <- sum((Y-matrix(1,ncol=1,nrow=length(Y))%*%solve(t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%matrix(1,ncol=1,nrow=length(Y)))%*%t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%Y)^2)/(nrow(X)-1) # Mean square-error (0 variable, constant OLS)
### Estimate model-error variance
B.SigReg <- solve(t(X)%*%solve(Omega)%*%X)%*%t(X)%*%solve(Omega)%*%LP3$S # OLS estimated coefficients for k-variable model of LP3 standard deviation. See Eq 15.
Yhat.SigReg <- X%*%B.SigReg # Estimates of sigma regression
S_bar <- mean(Yhat.SigReg) # average sigma of LP3
G_bar <- mean(LP3$G) # average skew of LP3
K_bar <- mean(LP3$K) # average standard deviate of LP3
c1 <- max(0,(1+K_bar^2*(1+0.75*G_bar^2)/2+K_bar*G_bar)*S_bar^2) # Leading coefficient. Eq 13
var.modelerror.k <- max(0,MSE.OLS-c1*mean(1/recordLengths)) # k-variable model-error variance. Eq 14.
### Estimate 0-order model-error variance
B.SigReg <- solve(t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%matrix(1,ncol=1,nrow=length(Y)))%*%t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%LP3$S # OLS estimated coefficients for constant model of LP3 standard deviation. See Eq 15.
Yhat.SigReg <- matrix(1,ncol=1,nrow=length(Y))%*%B.SigReg # Estimates of sigma regression
S_bar <- mean(Yhat.SigReg) # average sigma of LP3
c1.0 <- max(0,(1+K_bar^2*(1+0.75*G_bar^2)/2+K_bar*G_bar)*S_bar^2) # Leading coefficient for constant-model model-error variance. See Eq 13
var.modelerror.0 <- max(0,MSE.OLS.0-c1.0*mean(1/recordLengths)) # Constant-model model-error variance. See Eq. 14.
### Final weighting matrix
Omega <- diag((var.modelerror.k+c1/recordLengths)) # WLS weighting matrix. Eq 12
## Basic regression calculations
B_hat <- solve(t(X)%*%solve(Omega)%*%X)%*%t(X)%*%solve(Omega)%*%Y # Estimated regression coefficients. Eq 7 (9, 11, and 18)
Y_hat <- X%*%B_hat # Model estimates. Eq 8
e <- Y-Y_hat # Model residuals. Eq 30
## Performance metrics
MSE <- sum(e^2)/(nrow(X)-ncol(X)) # Mean square-error. Eq 31
SSR <- sum(e^2) # Residual sum of squares. Eq 36
SST <- sum((Y-mean(Y))^2) # Total sum of squares. Eq 37
R2 <- 1 - SSR/SST # Coefficient of determination. Eq 35
R2_adj <- 1 - SSR*(nrow(X)-1)/SST/(nrow(X)-ncol(X)) # Adjusted coefficient of determination. Eq 38
RMSE <- NA
if (transY=='log10') {
RMSE <-100*sqrt(exp(log(10)*log(10)*MSE)-1) # Root-mean-squared error, in percent. Eq 34
} else if (transY=='ln') {
RMSE <-100*sqrt(exp(MSE)-1) # Root-mean-squared error, in percent. transformed for natural logs.
}
PerfMet <- list(MSE=MSE,R2=R2,R2_adj=R2_adj,RMSE=RMSE) # Performance metrics for output (basic, for OLS)
R2_pseudo <- 1 - var.modelerror.k/var.modelerror.0 # Pseudo coefficient of determination. Eq 39
AVP <- var.modelerror.k + mean(diag(X%*%solve(t(X)%*%solve(Omega)%*%X)%*%t(X))) # Average varaince of prediction. Eq 32
VP <- vector(length=length(Y)) # Empty vector for individual variances of prediction
for (i in 1:length(VP)) { # Individual variances of prediction
VP[i] <- var.modelerror.k + X[i,]%*%solve(t(X)%*%solve(Omega)%*%X)%*%X[i,] # Individual variance of prediction. Based on Eq 32.
}
VP <- data.frame(VP); names(VP) <- 'PredVar' # Formating the VP vector for output
Sp <- Se <- NA
if (transY=='log10') {
Sp <- 100*sqrt(exp(log(10)*log(10)*AVP)-1) # Standard error of predictions. Eq 33.
Se <-100*sqrt(exp(log(10)*log(10)*var.modelerror.k)-1) # Standard model error. Not noted in the manual, but included as output in WREG 1.05. Based on Eq 33.
} else if (transY=='ln') {
# corrected for natural logs
Sp <- 100*sqrt(exp(AVP)-1)
Se <-100*sqrt(exp(var.modelerror.k)-1)
}
PerfMet <- c(PerfMet,R2_pseudo=R2_pseudo,AVP=AVP,Sp=Sp,VP=VP,ModErrVar=var.modelerror.k,StanModErr=Se) # Performance metrics for output
## Leverage and influence statistics
Lev <- Leverage(X=X,Omega=Omega,x0=x0,ROI=ROI) # Leverage subroutine
Infl <- Influence(e=e,X=X,Omega=Omega,Beta=B_hat,ROI=ROI,Lev=Lev$Leverage) # Influence subroutine
## Significance of regression parameters
B_var <- diag(solve(t(X)%*%solve(Omega)%*%X)) # Covariances of regression coefficients. Eq 46
B_tval <- B_hat/sqrt(B_var) # T-value statistics of regression coefficients. Eq 45.
B_pval <- 2*stats::pt(-abs(B_tval),df=(nrow(X)-ncol(X))) # Significnace of regression coefficients
## Create summary tables
Coefs <- data.frame(cbind(B_hat,sqrt(B_var),B_tval,B_pval)) # Regression coefficient table for output
names(Coefs) <- c('Coefficient','Standard Error','tStatistic','pValue')
ResLevInf <- data.frame(cbind(e,Lev$Leverage,Infl$Influence)) # Residuals, leverage and influence of each varaible for output
names(ResLevInf) <- c('Residual','Leverage','Influence')
LevInf.Sig<-data.frame(cbind(Lev$Significant,Infl$Significant)) # Indication of significance for leverage and Influence for output
names(LevInf.Sig) <- c('SignificantLeverage','SignificantInfluence')
## Handling output
Output <- list(Coefs=Coefs,ResLevInf=ResLevInf,LevLim=Lev$Limit,
InflLim=Infl$Limit,LevInf.Sig=LevInf.Sig,
PerformanceMetrics=PerfMet,X=X,Y=Y,fitted.values=Y_hat,residuals=e,
Weighting=Omega,Inputs=list(transY=transY))
if (ROI) { # Appended at-site estimates for ROI calculations
Y_est <- as.matrix(x0)%*%B_hat # ROI site estimate
Output <- c(Output,Y.ROI=Y_est,x0.ROI=x0)
}
class(Output) <- 'WREG.WLS'
return(Output)
}
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#'Weighted-Multiple-Linear Regression Program (WREG)
#'
#'@description The \code{WREG.WLS} function executes the multiple linear
#' regression analysis using weighted least-squares regression.
#'
#'@param Y A numeric vector of the dependent variable of interest, with any
#' transformations already applied.
#'@param X A numeric matrix of the independent variables in the regression, with
#' any transformations already applied. Each row represents a site and each
#' column represents a particular independe variable. (If a leading constant
#' is used, it should be included here as a leading column of ones.) The rows
#' must be in the same order as the dependent variables in \code{Y}.
#'@param recordLengths A numeric vector whose rows are in the same order as
#' \code{Y} and represent the at-site record length.
#'@param LP3 A numeric matrix containing the fitted Log-Pearson Type III
#' standard deviate, standard deviation and skew for each site. The columns of
#' the matrix represent S, K, G, and an option regional skew value \code{GR}
#' required by WREG.GLS with regSkew = TRUE. The order of the rows must be the
#' same as \code{Y}.
#'@param transY A required character string indicating if the the
#' dependentvariable was transformed by the common logarithm ('log10'),
#' transformed by the natural logarithm ('ln') or untransformed ('none').
#'@param x0 A vector containing the independent variables (as above) for a
#' particular target site. This variable is only used for ROI analysis.
#'
#'@details In this implementation, the weights for weighted least-squares
#' regression are defined by record lengths. See manual for details.
#'
#'@return All outputs are returned as part of a list. The elements of the list
#' depend on the type of regression performed. The elements of the list may
#' include: \item{Coefs}{A data frame composed of four variables: (1)
#' \code{Coefficient} contains the regression coefficeints estimated for the
#' model, (2) \code{Standard Error} contains the standard errors of
#' each regression coefficient, (3) \code{tStatistic} contains the Student's
#' T-statistic of each regression coefficient and (4) \code{pValue} contains
#' the significance probability of each regression coefficient.}
#' \item{ResLevInf}{A data frame composed of three variables for each site in
#' the regression. \code{Residual} contains the model residuals.
#' \code{Leverage} contains the leverage of each site. \code{Influence}
#' contains the influence of each site.} \item{LevLim}{The critical value of
#' leverage. See \code{\link{Leverage}}} \item{InflLim}{The critical value of
#' influence. See \code{\link{Influence}}} \item{LevInf.Sig}{A logical matrix
#' indicating if the leverage (column 1) is significant and the influence
#' (column 2) is significant for each site in the regression.}
#' \item{PerformanceMetrics}{A list of not more than ten elements. All
#' regression types return the mean squared error of residuals (\code{MSE}),
#' the coefficient of determination (\code{R2}), the adjusted coefficient of
#' determination (\code{R2_adj}) and the root mean squared error (\code{RMSE},
#' in percent). The pseudo coefficient of regression (\code{R2_pseudo}), the
#' average variance of prediction (\code{AVP}), the standard error of
#' prediction (\code{Sp}, in percent), a vector of the individual variances of
#' prediction for each site (\code{VP.PredVar}), the model-error variance
#' (\code{ModErrVar}) and the standardized model error variance
#' (\code{StanModErr}, in percent) are also returned. Details on the
#' appropriateness and applicability of performance metrics can be found in the
#' WREG manual.} \item{X}{The input predictors.} \item{Y}{The input
#' observations.} \item{fitted.values}{A vector of model estimates from the
#' regression model.} \item{residuals}{A vector of model residuals.}
#' \item{Weighting}{The weighting matrix used to develop regression estimates.}
#' \item{Input}{A list of input parameters for error searching. Currently
#' empty.}
#'@import stats
#'
#' @examples
#' # Import some example data
# peakFQdir <- paste0(
# file.path(system.file("exampleDirectory", package = "WREG"),
# "pfqImport"))
# gisFilePath <- file.path(peakFQdir, "pfqSiteInfo.txt")
# importedData <- importPeakFQ(pfqPath = peakFQdir, gisFile = gisFilePath)
# Organizing input data
# lp3Data <- importedData$LP3f
# lp3Data$K <- importedData$LP3k$AEP_0.5
# Y <- importedData$Y$AEP_0.5
# X <- importedData$X[c("Sand", "OutletElev", "Slope")]
# recordLengths <- importedData$recLen
# transY <- "none"
#
# # Run WLS regression
# result <- WREG.WLS(Y, X, recordLengths, LP3 = lp3Data, transY)
#'
#'@export
WREG.WLS <- function(Y,X,recordLengths,LP3,transY,x0=NA) {
# 11/9/16 Greg Petrochenkov: Changed validation scheme
warn("clear")
# Some upfront error handling
wregValidation((!missing(X)&!missing(Y))&&(length(Y)!=nrow(X)), "eq", FALSE,
paste0("The length of Y must be the same as ",
"the number of rows in X."), warnFlag = TRUE)
if (!wregValidation((!missing(X)&!missing(Y))&&(length(Y)!=nrow(X)), "eq", FALSE,
"Dependent variable (Y) must be provided", warnFlag = TRUE)) {
if (!wregValidation(Y, "numeric", message =
"Dependent variable (Y) must be provided as class numeric",
warnFlag = TRUE)) {
wregValidation(sum(is.na(Y)), "eq", 0 ,
paste0("The depedent variable (Y) contains missing ",
"values. These must be removed."),
TRUE)
wregValidation(sum(is.infinite(Y)), "eq", 0 ,
paste0("The depedent variable (Y) contains infinite ",
"values. These must be removed."),
warnFlag = TRUE)
}
}
if (!wregValidation(missing(X), "eq", FALSE,
"Independent variables (X) must be provided.", warnFlag = TRUE)) {
if (!wregValidation((length(unique(apply(X,FUN=class,MARGIN=2)))!=1)|
(unique(apply(X,FUN=class,MARGIN=2))!="numeric"), "eq", FALSE,
"Independent variables (X) must be provided as class numeric.", warnFlag = TRUE)){
wregValidation(sum(is.na(as.matrix(X))), "eq", 0,
paste0("Some independent variables (X) contain missing ",
"values. These must be removed."), warnFlag = TRUE)
wregValidation(sum(is.infinite(as.matrix(X))), "eq", 0,
paste0("Some independent variables (X) contain infinite ",
"values. These must be removed."), warnFlag = TRUE)
}
}
if(!wregValidation(missing(transY)|!is.character(transY), "eq", FALSE,
"transY must be included as a character string", warnFlag = TRUE)) {
wregValidation(!is.element(transY,c("none","log10","ln")), "eq", FALSE,
"transY must be either 'none', 'log10' or 'ln'", warnFlag = TRUE)
}
## Determine if ROI is being applied
if (is.na(sum(x0))) { # ROI regression is not used.
ROI <- F
} else { # ROI regression is used.
ROI <- T
}
## Just initial values for control.
var.modelerror.k <- NA
### Correct recordLengths to be only at-site record lengths.
if(is.matrix(recordLengths)) {
recordLengths<-diag(recordLengths)
}
if (!wregValidation(missing(recordLengths), "eq", FALSE,
"Record lengths must be provided.", warnFlag = TRUE)) {
if (!wregValidation(recordLengths, "numeric",
message="Record lengths must be provided as class numeric.", warnFlag = TRUE)) {
wregValidation(sum(is.na(c(recordLengths))), "eq", 0,
paste0("Some record lengths contain missing ",
"values. These must be removed."), warnFlag = TRUE)
wregValidation(sum(is.infinite(c(recordLengths))), "eq", 0,
paste0("Some record lengths contain infinite ",
"values. These must be removed."), warnFlag = TRUE)
}
}
# Error checking LP3
if (!wregValidation(missing(LP3), "eq", FALSE,
"The data frame LP3 must be provided.", warnFlag = TRUE)) {
if (!wregValidation(!is.data.frame(LP3), "eq", FALSE,
paste("LP3 must be provided as a data frame with elements named",
"'S', 'K' and 'G' for standard deivation, deviate and skew,",
"respectively."), warnFlag = TRUE)) {
if (!wregValidation(sum(is.element(c("S","K","G"),names(LP3)))!=3, "eq", FALSE,
paste("In valid elements: The names of the elements in LP3 are",
names(LP3),". LP3 must be provided as a data frame with elements named",
"'S', 'K' and 'G' for standard deivation, deviate and skew,",
"respectively."), warnFlag = TRUE)) {
if(!wregValidation((length(unique(apply(cbind(LP3$S,LP3$K,LP3$G),FUN=class,MARGIN=2)))!=1)|
(unique(apply(cbind(LP3$S,LP3$K,LP3$G),FUN=class,MARGIN=2))!="numeric"), "eq", FALSE,
"The data frame LP3 must be provided in a numeric class", warnFlag = TRUE)){
wregValidation(sum(is.infinite(LP3$S),is.infinite(LP3$K),is.infinite(LP3$G))>0, "eq", FALSE,
paste0("Some elements of LP3$S, LP3$K, and LP3$G contain infinite ",
"values. These must be removed."), warnFlag = TRUE)
wregValidation(sum(is.na(LP3$S),is.na(LP3$K),is.na(LP3$G))>0, "eq", FALSE,
paste0("Some elements of LP3$S, LP3$K, and LP3$G contain missing ",
"values. These must be removed."), warnFlag = TRUE)
}
}
}
}
wregValidation((!missing(X)&!missing(Y)&!missing(recordLengths)&!missing(LP3))&&
(length(unique(length(Y),nrow(X),nrow(LP3),length(recordLengths)))!=1), "eq", FALSE,
paste0("length(Y), nrow(X), nrow(LP3) and ",
"length(recordLengths) must all be equal"), warnFlag = TRUE)
if (warn("check")){
stop('Invalid inputs were provided. See warnings().', warn("get"))
}
#Convert X and Y from dataframes to matrices to work with matrix operations below
X <- as.matrix(X)
Y <- as.matrix(Y)
### Initial OLS (basis for others)
Omega <- diag(nrow(X)) # temporary weighting matrix (identity)
B_hat <- solve(t(X)%*%solve(Omega)%*%X)%*%t(X)%*%solve(Omega)%*%Y # OLS estimated coefficients
Y_hat <- X%*%B_hat # OLS model estimates
e <- Y-Y_hat # OLS residuals
MSE.OLS <- sum(e^2)/(nrow(X)-ncol(X)) # Mean square-error (k variable OLS)
MSE.OLS.0 <- sum((Y-matrix(1,ncol=1,nrow=length(Y))%*%solve(t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%matrix(1,ncol=1,nrow=length(Y)))%*%t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%Y)^2)/(nrow(X)-1) # Mean square-error (0 variable, constant OLS)
### Estimate model-error variance
B.SigReg <- solve(t(X)%*%solve(Omega)%*%X)%*%t(X)%*%solve(Omega)%*%LP3$S # OLS estimated coefficients for k-variable model of LP3 standard deviation. See Eq 15.
Yhat.SigReg <- X%*%B.SigReg # Estimates of sigma regression
S_bar <- mean(Yhat.SigReg) # average sigma of LP3
G_bar <- mean(LP3$G) # average skew of LP3
K_bar <- mean(LP3$K) # average standard deviate of LP3
c1 <- max(0,(1+K_bar^2*(1+0.75*G_bar^2)/2+K_bar*G_bar)*S_bar^2) # Leading coefficient. Eq 13
var.modelerror.k <- max(0,MSE.OLS-c1*mean(1/recordLengths)) # k-variable model-error variance. Eq 14.
### Estimate 0-order model-error variance
B.SigReg <- solve(t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%matrix(1,ncol=1,nrow=length(Y)))%*%t(matrix(1,ncol=1,nrow=length(Y)))%*%solve(Omega)%*%LP3$S # OLS estimated coefficients for constant model of LP3 standard deviation. See Eq 15.
Yhat.SigReg <- matrix(1,ncol=1,nrow=length(Y))%*%B.SigReg # Estimates of sigma regression
S_bar <- mean(Yhat.SigReg) # average sigma of LP3
c1.0 <- max(0,(1+K_bar^2*(1+0.75*G_bar^2)/2+K_bar*G_bar)*S_bar^2) # Leading coefficient for constant-model model-error variance. See Eq 13
var.modelerror.0 <- max(0,MSE.OLS.0-c1.0*mean(1/recordLengths)) # Constant-model model-error variance. See Eq. 14.
### Final weighting matrix
Omega <- diag((var.modelerror.k+c1/recordLengths)) # WLS weighting matrix. Eq 12
## Basic regression calculations
B_hat <- solve(t(X)%*%solve(Omega)%*%X)%*%t(X)%*%solve(Omega)%*%Y # Estimated regression coefficients. Eq 7 (9, 11, and 18)
Y_hat <- X%*%B_hat # Model estimates. Eq 8
e <- Y-Y_hat # Model residuals. Eq 30
## Performance metrics
MSE <- sum(e^2)/(nrow(X)-ncol(X)) # Mean square-error. Eq 31
SSR <- sum(e^2) # Residual sum of squares. Eq 36
SST <- sum((Y-mean(Y))^2) # Total sum of squares. Eq 37
R2 <- 1 - SSR/SST # Coefficient of determination. Eq 35
R2_adj <- 1 - SSR*(nrow(X)-1)/SST/(nrow(X)-ncol(X)) # Adjusted coefficient of determination. Eq 38
RMSE <- NA
if (transY=='log10') {
RMSE <-100*sqrt(exp(log(10)*log(10)*MSE)-1) # Root-mean-squared error, in percent. Eq 34
} else if (transY=='ln') {
RMSE <-100*sqrt(exp(MSE)-1) # Root-mean-squared error, in percent. transformed for natural logs.
}
PerfMet <- list(MSE=MSE,R2=R2,R2_adj=R2_adj,RMSE=RMSE) # Performance metrics for output (basic, for OLS)
R2_pseudo <- 1 - var.modelerror.k/var.modelerror.0 # Pseudo coefficient of determination. Eq 39
AVP <- var.modelerror.k + mean(diag(X%*%solve(t(X)%*%solve(Omega)%*%X)%*%t(X))) # Average varaince of prediction. Eq 32
VP <- vector(length=length(Y)) # Empty vector for individual variances of prediction
for (i in 1:length(VP)) { # Individual variances of prediction
VP[i] <- var.modelerror.k + X[i,]%*%solve(t(X)%*%solve(Omega)%*%X)%*%X[i,] # Individual variance of prediction. Based on Eq 32.
}
VP <- data.frame(VP); names(VP) <- 'PredVar' # Formating the VP vector for output
Sp <- Se <- NA
if (transY=='log10') {
Sp <- 100*sqrt(exp(log(10)*log(10)*AVP)-1) # Standard error of predictions. Eq 33.
Se <-100*sqrt(exp(log(10)*log(10)*var.modelerror.k)-1) # Standard model error. Not noted in the manual, but included as output in WREG 1.05. Based on Eq 33.
} else if (transY=='ln') {
# corrected for natural logs
Sp <- 100*sqrt(exp(AVP)-1)
Se <-100*sqrt(exp(var.modelerror.k)-1)
}
PerfMet <- c(PerfMet,R2_pseudo=R2_pseudo,AVP=AVP,Sp=Sp,VP=VP,ModErrVar=var.modelerror.k,StanModErr=Se) # Performance metrics for output
## Leverage and influence statistics
Lev <- Leverage(X=X,Omega=Omega,x0=x0,ROI=ROI) # Leverage subroutine
Infl <- Influence(e=e,X=X,Omega=Omega,Beta=B_hat,ROI=ROI,Lev=Lev$Leverage) # Influence subroutine
## Significance of regression parameters
B_var <- diag(solve(t(X)%*%solve(Omega)%*%X)) # Covariances of regression coefficients. Eq 46
B_tval <- B_hat/sqrt(B_var) # T-value statistics of regression coefficients. Eq 45.
B_pval <- 2*stats::pt(-abs(B_tval),df=(nrow(X)-ncol(X))) # Significnace of regression coefficients
## Create summary tables
Coefs <- data.frame(cbind(B_hat,sqrt(B_var),B_tval,B_pval)) # Regression coefficient table for output
names(Coefs) <- c('Coefficient','Standard Error','tStatistic','pValue')
ResLevInf <- data.frame(cbind(e,Lev$Leverage,Infl$Influence)) # Residuals, leverage and influence of each varaible for output
names(ResLevInf) <- c('Residual','Leverage','Influence')
LevInf.Sig<-data.frame(cbind(Lev$Significant,Infl$Significant)) # Indication of significance for leverage and Influence for output
names(LevInf.Sig) <- c('SignificantLeverage','SignificantInfluence')
## Handling output
Output <- list(Coefs=Coefs,ResLevInf=ResLevInf,LevLim=Lev$Limit,
InflLim=Infl$Limit,LevInf.Sig=LevInf.Sig,
PerformanceMetrics=PerfMet,X=X,Y=Y,fitted.values=Y_hat,residuals=e,
Weighting=Omega,Inputs=list(transY=transY))
if (ROI) { # Appended at-site estimates for ROI calculations
Y_est <- as.matrix(x0)%*%B_hat # ROI site estimate
Output <- c(Output,Y.ROI=Y_est,x0.ROI=x0)
}
class(Output) <- 'WREG.WLS'
return(Output)
}
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