| Title: | Nonparametric Estimation of the Derivatives of a Regression Function |
|---|---|
| Description: | Estimating the first and second derivatives of a regression function by the method of Wang and Lin (2015) <http://www.jmlr.org/papers/v16/wang15b.html>. |
| Authors: | Olga Savchuk [aut, cre], Alex Volinsky [aut] |
| Maintainer: | Olga Savchuk <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0 |
| Built: | 2025-02-11 04:26:19 UTC |
| Source: | https://github.com/cran/npregderiv |
Estimating the first and second derivatives by the method of Wang and Lin (2015) requires choosing the soothing parameter that shows how many observations from each side of
an estimation point are used. Notice that the method of Wang and Lin (2015) requires an estimation point to be one of the design points. If a single value of is used for all
estimation points, it is impossible to compute a derivative estimate for the first and the last data points that constitute the left and right boundary regions, respectively.
Generally, an adaptive value of may be used (see Wang and Lin (2015)). Even though the article of Wang and Lin (2015) contains the expressions for the asymptotically optimal ,
those results are of no practical importance since they involve the higher order derivatives of the regression function. Wang and Lin (2015) implement their method for
the values for ranging from to , where is the sample size. The larger the value of , the smoother the resulting estimate. However, too large may lead to
smoothing away certain important features of the function's derivatives. We found that the value of worked reasonably well for estimating the first and second derivatives
of the regression functions in many practical cases.
The function reg_1derivWL is developed to estimate the first derivative of the underlying regression function from the original data. The function reg_2derivWL is
intended to estimate the function's second derivative. The latter function was used for estimating the second derivatives of the deflection functions in the experiment
described and analyzed in the article of Savchuk and Volinsky (2020). The deflection's second derivative is used to estimate the substrate's curvature and, ultimately, the
residual stress of the thin film deposited on the substrate.
The data set wafer40 analyzed in the article of Savchuk and Volinsky (2020) is included into the package. It contains the deflection measurements for one of the
wafers (wafer 40) used in the experiment. The measurements are taken before and after the thin film deposition in two radial directions on the round substrate and are marked as
"0 degrees" and "90 degrees" depending on the scan orientation. The design points are separated by 0.000005 m. The resulting sample size in each case is =7755. All
deflection and position measurements are recorded in meters.
Wang, W.W., Lin, L. (2015) http://www.jmlr.org/papers/v16/wang15b.html “Derivative Estimation Based on Difference Sequence via Locally Weighted Least Squares Regression”, Journal of Machine Learning Research, 16, 2617-2641.
Savchuk, O., Volinsky, A.A. (2020). “Nonparametric estimation of SiC film residual stress from the wafer profile”.
The design data must be increasing and evenly spaced. The first derivative estimate is computed at that must be one of the design points from the interior region. The interior region excludes the first points with the smallest values (left boundary) and the last points with the largest values (right boundary).
reg_1derivWL(xdat, ydat, k, u)reg_1derivWL(xdat, ydat, k, u)
xdat |
numerical vector of the increasing and evenly spaced design points |
ydat |
numerical vector of the corresponding data points |
k |
integer value of the smoothing parameter |
u |
an estimation point (scalar) that must be one of the |
The method's smoothing parameter shows how many data points are used from each side of an estimation point to compute the first derivative estimate at that point. Choose , where is the sample size. The value of can be taken as a starting point.
The computed first derivative estimate (scalar) at the point .
Wang, W.W., Lin, L. (2015) http://www.jmlr.org/papers/v16/wang15b.html “Derivative Estimation Based on Difference Sequence via Locally Weighted Least Squares Regression”, Journal of Machine Learning Research, 16, 2617-2641.
# EXAMPLE 1: Toy example xdata=1:100 ydata=xdata^2 reg_1derivWL(xdata,ydata,10,30) # EXAMPLE 2 (simulated data). m=function(x) sin(2*pi*x)+log(6+5*x) m1=function(x) # first derivative of m 2*pi*cos(2*pi*x)+5/(6+5*x) N=300 # sample size xi=2*(1:N/N)-1 # equidistant design points sigma=0.25 # noise level used in the article yi=m(xi)+rnorm(N,sd=sigma) # generating data K=0.1*N # selected value of the smoothing parameter k x_inter=xi[(K+1):(N-K)] # interior estimation region N_inter=length(x_inter) # number of points where the second derivative estimate is computed M1=array(0,N_inter) # Vector of estimated second derivatives at the points x_inter for (i in 1:N_inter) M1[i]=reg_1derivWL(xi,yi,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xi,yi,pch=20,cex=1.7,main="Regression function and generated data",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(xi,m(xi),'l',col="red",lwd=2) title(xlab="argument",ylab="regression function",line=2.5,cex.lab=1.8) legend(0.65,-0.4,lty="solid",col="red",lwd=2,legend="true regression function",bty="n", cex=1.45,seg.len=0.5) plot(x_inter,M1,'l',lwd=2,main="Actual and estimated first derivative",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(x_inter,m1(x_inter),'l',lwd=2,col="red") title(xlab="argument",ylab="second derivative",line=2.5,cex.lab=1.8) legend(0.6,-10,lty=c("solid","solid"),col=c("red","black"),lwd=c(2,2), legend=c("true first derivative", "estimate"),bty="n",cex=1.45,seg.len=0.5) par(op) # EXAMPLE 3: Estimating the first derivative of the deflection function for the data on # deflection after film deposition scanned at 0 degrees. # See Savchuk O., and Volinsky, A. (2020) for the experiment's description. xdesign=wafer40$x_dat ydata=wafer40$y_after_0 n_data=length(xdesign) # sample size (original) n_new=1034 # reducing the number of points where the second # derivative is estimated. K=ceiling(0.1*n_data) # value of the smoothing parameter index=(K+1)+0:(n_new-1)*6 # cutting about 10% of points from each side and # reducing the number of points where the second # derivative is estimated x_inter=xdesign[index] # the values of argument where the second # derivative is to be estimated y_inter=ydata[index] Der1=array(0,n_new) for (i in 1:n_new) Der1[i]=reg_1derivWL(xdesign,ydata,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xdesign*1000,ydata*10^6,pch=20,main="Deflection AFTER film deposition. Angle: 0 degrees.", xlab="",ylab="",cex.axis=1.8,cex.main=1.5) title(ylab=expression(paste("deflection, ",mu,"m")),xlab="position, mm",line=2.25,cex.lab=1.8) plot(x_inter*1000,Der1,'l',lwd=2, main="1-st derivative AFTER film deposition. Angle: 0 degrees.", xlab="",ylab="",cex.axis=1.8, cex.main=1.5) title(ylab="first derivative of deflection", xlab="position, mm", line=2.5, cex.lab=1.8) par(op)# EXAMPLE 1: Toy example xdata=1:100 ydata=xdata^2 reg_1derivWL(xdata,ydata,10,30) # EXAMPLE 2 (simulated data). m=function(x) sin(2*pi*x)+log(6+5*x) m1=function(x) # first derivative of m 2*pi*cos(2*pi*x)+5/(6+5*x) N=300 # sample size xi=2*(1:N/N)-1 # equidistant design points sigma=0.25 # noise level used in the article yi=m(xi)+rnorm(N,sd=sigma) # generating data K=0.1*N # selected value of the smoothing parameter k x_inter=xi[(K+1):(N-K)] # interior estimation region N_inter=length(x_inter) # number of points where the second derivative estimate is computed M1=array(0,N_inter) # Vector of estimated second derivatives at the points x_inter for (i in 1:N_inter) M1[i]=reg_1derivWL(xi,yi,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xi,yi,pch=20,cex=1.7,main="Regression function and generated data",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(xi,m(xi),'l',col="red",lwd=2) title(xlab="argument",ylab="regression function",line=2.5,cex.lab=1.8) legend(0.65,-0.4,lty="solid",col="red",lwd=2,legend="true regression function",bty="n", cex=1.45,seg.len=0.5) plot(x_inter,M1,'l',lwd=2,main="Actual and estimated first derivative",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(x_inter,m1(x_inter),'l',lwd=2,col="red") title(xlab="argument",ylab="second derivative",line=2.5,cex.lab=1.8) legend(0.6,-10,lty=c("solid","solid"),col=c("red","black"),lwd=c(2,2), legend=c("true first derivative", "estimate"),bty="n",cex=1.45,seg.len=0.5) par(op) # EXAMPLE 3: Estimating the first derivative of the deflection function for the data on # deflection after film deposition scanned at 0 degrees. # See Savchuk O., and Volinsky, A. (2020) for the experiment's description. xdesign=wafer40$x_dat ydata=wafer40$y_after_0 n_data=length(xdesign) # sample size (original) n_new=1034 # reducing the number of points where the second # derivative is estimated. K=ceiling(0.1*n_data) # value of the smoothing parameter index=(K+1)+0:(n_new-1)*6 # cutting about 10% of points from each side and # reducing the number of points where the second # derivative is estimated x_inter=xdesign[index] # the values of argument where the second # derivative is to be estimated y_inter=ydata[index] Der1=array(0,n_new) for (i in 1:n_new) Der1[i]=reg_1derivWL(xdesign,ydata,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xdesign*1000,ydata*10^6,pch=20,main="Deflection AFTER film deposition. Angle: 0 degrees.", xlab="",ylab="",cex.axis=1.8,cex.main=1.5) title(ylab=expression(paste("deflection, ",mu,"m")),xlab="position, mm",line=2.25,cex.lab=1.8) plot(x_inter*1000,Der1,'l',lwd=2, main="1-st derivative AFTER film deposition. Angle: 0 degrees.", xlab="",ylab="",cex.axis=1.8, cex.main=1.5) title(ylab="first derivative of deflection", xlab="position, mm", line=2.5, cex.lab=1.8) par(op)
The design data must be increasing and evenly spaced. The second derivative estimate is computed at that must be one of the design points from the interior region. The interior region excludes the first points with the smallest values (left boundary) and the last points with the largest values (right boundary).
reg_2derivWL(xdat, ydat, k, u)reg_2derivWL(xdat, ydat, k, u)
xdat |
numerical vector of the increasing and evenly spaced design points |
ydat |
numerical vector of the corresponding data points |
k |
integer value of the smoothing parameter |
u |
an estimation point (scalar) that must be one of the |
The method's smoothing parameter shows how many data points are used from each side of an estimation point to compute the second derivative estimate at that point. Choose , where is the sample size. The value of can be taken as a starting point.
The computed second derivative estimate (scalar) at the point .
Wang, W.W., Lin, L. (2015) http://www.jmlr.org/papers/v16/wang15b.html “Derivative Estimation Based on Difference Sequence via Locally Weighted Least Squares Regression”, Journal of Machine Learning Research, 16, 2617-2641.
Savchuk, O., Volinsky, A.A. (2020). “Nonparametric estimation of SiC film residual stress from the wafer profile”.
# EXAMPLE 1 (simulated data). # The regression function is taken from p. 2631 of the article of Wang and Lin (2015). m=function(x) # Regression function from the article of Wang and Lin (2015) 32*exp(-8*(1-2*x)^2)*(1-2*x) m2=function(x) # second derivative of m -2048*exp(-8*(1-2*x)^2)*(128*x^3-192*x^2+90*x-13) N=100 # sample size xi=1:N/N # equidistant design points sigma=0.2 # noise level used in the article yi=m(xi)+rnorm(N,sd=sigma) # generating data K=0.1*N # selected value of the smoothing parameter k x_inter=xi[(K+1):(N-K)] # interior estimation region N_inter=length(x_inter) # number of points where the second derivative estimate is computed M2=array(0,N_inter) # Vector of estimated second derivatives at the points x_inter for (i in 1:N_inter) M2[i]=reg_2derivWL(xi,yi,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xi,yi,pch=20,cex=1.7,main="Regression function and generated data",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(xi,m(xi),'l',col="red",lwd=2) title(xlab="argument",ylab="regression function",line=2.5,cex.lab=1.8) legend(0.35,5,lty="solid",col="red",lwd=2,legend="true regression function",bty="n", cex=1.45,seg.len=0.5) plot(x_inter,M2,'l',lwd=2,main="Actual and estimated second derivative",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(x_inter,m2(x_inter),'l',lwd=2,col="red") title(xlab="argument",ylab="second derivative",line=2.5,cex.lab=1.8) legend(0.01,800,lty=c("solid","solid"),col=c("red","black"),lwd=c(2,2), legend=c("true second derivative", "estimate"),bty="n",cex=1.45,seg.len=0.5) par(op) # EXAMPLE 2: Estimating the second derivative of the deflection function for the data on # deflection after film deposition scanned at 90 degrees. # See Savchuk O., and Volinsky, A. (2020) for the experiment's description. xdesign=wafer40$x_dat ydata=wafer40$y_after_90 n_data=length(xdesign) # sample size (original) n_new=1034 # reducing the number of points where the second # derivative is estimated. K=ceiling(0.1*n_data) # value of the smoothing parameter index=(K+1)+0:(n_new-1)*6 # cutting about 10% of points from each side and reducing the # number of points where the second derivative is estimated x_inter=xdesign[index] # the values of argument where the second derivative is to be # estimated y_inter=ydata[index] Der2=array(0,n_new) for (i in 1:n_new) Der2[i]=reg_2derivWL(xdesign,ydata,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xdesign*1000,ydata*10^6,pch=20,main="Deflection AFTER film deposition. Angle: 90 degrees.", xlab="",ylab="",cex.axis=1.8,cex.main=1.5) title(ylab=expression(paste("deflection, ", mu, "m")), xlab="position, mm", line=2.25, cex.lab=1.8) plot(x_inter*1000,Der2,'l',lwd=2, main="2-nd derivative AFTER film deposition. Angle: 90 degrees.",xlab="",ylab="", cex.axis=1.8, cex.main=1.5) title(ylab="second derivative of deflection, 1/m", xlab="position, mm", line=2.5, cex.lab=1.8) par(op)# EXAMPLE 1 (simulated data). # The regression function is taken from p. 2631 of the article of Wang and Lin (2015). m=function(x) # Regression function from the article of Wang and Lin (2015) 32*exp(-8*(1-2*x)^2)*(1-2*x) m2=function(x) # second derivative of m -2048*exp(-8*(1-2*x)^2)*(128*x^3-192*x^2+90*x-13) N=100 # sample size xi=1:N/N # equidistant design points sigma=0.2 # noise level used in the article yi=m(xi)+rnorm(N,sd=sigma) # generating data K=0.1*N # selected value of the smoothing parameter k x_inter=xi[(K+1):(N-K)] # interior estimation region N_inter=length(x_inter) # number of points where the second derivative estimate is computed M2=array(0,N_inter) # Vector of estimated second derivatives at the points x_inter for (i in 1:N_inter) M2[i]=reg_2derivWL(xi,yi,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xi,yi,pch=20,cex=1.7,main="Regression function and generated data",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(xi,m(xi),'l',col="red",lwd=2) title(xlab="argument",ylab="regression function",line=2.5,cex.lab=1.8) legend(0.35,5,lty="solid",col="red",lwd=2,legend="true regression function",bty="n", cex=1.45,seg.len=0.5) plot(x_inter,M2,'l',lwd=2,main="Actual and estimated second derivative",xlab="",ylab="", cex.axis=1.8,cex.main=1.5) lines(x_inter,m2(x_inter),'l',lwd=2,col="red") title(xlab="argument",ylab="second derivative",line=2.5,cex.lab=1.8) legend(0.01,800,lty=c("solid","solid"),col=c("red","black"),lwd=c(2,2), legend=c("true second derivative", "estimate"),bty="n",cex=1.45,seg.len=0.5) par(op) # EXAMPLE 2: Estimating the second derivative of the deflection function for the data on # deflection after film deposition scanned at 90 degrees. # See Savchuk O., and Volinsky, A. (2020) for the experiment's description. xdesign=wafer40$x_dat ydata=wafer40$y_after_90 n_data=length(xdesign) # sample size (original) n_new=1034 # reducing the number of points where the second # derivative is estimated. K=ceiling(0.1*n_data) # value of the smoothing parameter index=(K+1)+0:(n_new-1)*6 # cutting about 10% of points from each side and reducing the # number of points where the second derivative is estimated x_inter=xdesign[index] # the values of argument where the second derivative is to be # estimated y_inter=ydata[index] Der2=array(0,n_new) for (i in 1:n_new) Der2[i]=reg_2derivWL(xdesign,ydata,K,x_inter[i]) op=par(no.readonly=TRUE) par(mfrow=c(1,2)) plot(xdesign*1000,ydata*10^6,pch=20,main="Deflection AFTER film deposition. Angle: 90 degrees.", xlab="",ylab="",cex.axis=1.8,cex.main=1.5) title(ylab=expression(paste("deflection, ", mu, "m")), xlab="position, mm", line=2.25, cex.lab=1.8) plot(x_inter*1000,Der2,'l',lwd=2, main="2-nd derivative AFTER film deposition. Angle: 90 degrees.",xlab="",ylab="", cex.axis=1.8, cex.main=1.5) title(ylab="second derivative of deflection, 1/m", xlab="position, mm", line=2.5, cex.lab=1.8) par(op)
The data frame on the deflection measurements is described and analyzed in the article of Savchuk and Volinsky (2020).
wafer40wafer40
The data frame has five variables:
design points (separated by 0.000005 m);
deflections measurements (in m) before film deposition at 0 degrees;
deflections measurements (in m) after film deposition at 0 degrees;
deflections measurements (in m) before film deposition at 90 degrees;
deflections measurements (in m) after film deposition at 90 degrees.
Savchuk, O., Volinsky, A.A. (2020). “Nonparametric estimation of SiC film residual stress from the wafer profile”.
# EXAMPLE: Plotting scatter plot of deflection in the case of 0 degrees before the film # deposition. Note: position is displayed in mm, whereas the deflection is displayed in # micrometers. No smoothing is used. The original n=7755 data points are plotted. dev.new() plot(wafer40$x_dat*1000,wafer40$y_before_0*1000000,pch=20, main="Deflection BEFORE film deposition. Angle: 0 degrees.",xlab="",ylab="", cex.axis=2,cex.main=2) title(ylab=expression(paste("deflection, ", mu, "m")), xlab="position, mm", line=2.25, cex.lab=2)# EXAMPLE: Plotting scatter plot of deflection in the case of 0 degrees before the film # deposition. Note: position is displayed in mm, whereas the deflection is displayed in # micrometers. No smoothing is used. The original n=7755 data points are plotted. dev.new() plot(wafer40$x_dat*1000,wafer40$y_before_0*1000000,pch=20, main="Deflection BEFORE film deposition. Angle: 0 degrees.",xlab="",ylab="", cex.axis=2,cex.main=2) title(ylab=expression(paste("deflection, ", mu, "m")), xlab="position, mm", line=2.25, cex.lab=2)