knit("startUpTex.Rnw")
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Do not forget to change directory for where the data will be (see 01Probability_forClassCode.Rmd)
dataDir<-"../../finalDataSets"
# dataDir<-"."
We will start by reading in the data of college information
scorecard <- read.csv(file.path(dataDir,"college.csv"), stringsAsFactors = FALSE)
We will exclude those that are for-profit institutes:
scorecard<-scorecard[-which(scorecard$CONTROL==3),]
xlab="Out-of-state tuition fee"
ylab="Full time student retention rate"
Let's plot tuition costs and retention rate of students:
plot(scorecard[,c("TUITIONFEE_OUT","RET_FT4")],xlab=xlab,ylab=ylab)
Let's print out those observations that have 0% retention rate
scorecard[scorecard[,"RET_FT4"]==0,]
## X INSTNM STABBR ADM_RATE_ALL SATMTMID
## 1238 5930 Pennsylvania College of Health Sciences PA 398 488
## SATVRMID SAT_AVG_ALL AVGFACSAL TUITFTE TUITIONFEE_IN TUITIONFEE_OUT
## 1238 468 955 5728 13823 21502 21502
## CONTROL UGDS UGDS_WHITE UGDS_BLACK UGDS_HISP UGDS_ASIAN UGDS_AIAN
## 1238 2 1394 0.8364 0.0445 0.0509 0.0294 7e-04
## UGDS_NHPI UGDS_2MOR UGDS_NRA UGDS_UNKN INC_PCT_LO INC_PCT_M1 INC_PCT_M2
## 1238 0.0029 0.0014 0 0.0337 0.367788462 0.146634615 0.227163462
## INC_PCT_H1 INC_PCT_H2 RET_FT4 PCTFLOAN C150_4 mn_earn_wne_p10
## 1238 0.175480769 0.082932692 0 0.6735 0.6338 53500
## md_earn_wne_p10 PFTFAC
## 1238 53100 0.7564
This seems odd. For now we will drop this value, though if we were going to go forward more seriously, we'd need to investigate this.
scorecard<-scorecard[-which(scorecard[,"RET_FT4"]==0),]
plot(scorecard[,c("TUITIONFEE_OUT","RET_FT4")],xlab=xlab,ylab=ylab)
Here is code to color the private colleges separately from the public. The variable CONTROL indicates whether it is private (=2) or public (=1). Notice how I identify the colors using the numeric index of 1 or 2 (so =1 is red).
plot(scorecard[,c("TUITIONFEE_OUT","RET_FT4")],col=c("red","black")[scorecard[,"CONTROL"]],xlab=xlab,ylab=ylab)
legend("bottomright",c("public","private"),fill=c("red","black"))
To make it simple going forward, we are going to make datasets that are just private or just public
private<-subset(scorecard,CONTROL==2)
# equivalent to
# private<-scorecard[scorecard[,"CONTROL"]==2,]
public<-subset(scorecard,CONTROL==1)
# equivalent to
# public<-scorecard[scorecard[,"CONTROL"]==1,]
Here we plot the scatter plot, with lines drawn. The choices of which lines isn't important here – basically I picked them arbitrary – but notice the use of abline. If the argument is v= or h= then you get a vertical/horizontal line. But if the arguments are a= and b=, then you are giving the intercept and slope of the line.
plot(private[,c("TUITIONFEE_OUT","RET_FT4")],col="black")
m<-colMeans(private[,c("TUITIONFEE_OUT","RET_FT4")])
abline(h=m[2],col="red")
abline(a=.4,b=(m[2]-.4)/m[1],col="red")
abline(a=.9,b=(m[2]-.9)/m[1],col="red")
points(m[1],m[2],pch=19,cex=2)
Further notice the use of colMeans to get the mean of each column. There is similarly colSums, rowMeans, rowSums, etc.
Now we draw a scatter plot with the line fit by least-squares regression. Note now I can give the output from a least-squares fit (given by lm). We will look at the lm function more later. But for now notice how if I give it to abline it will draw the regression line on top of our scatter plot.
plot(private[,c("TUITIONFEE_OUT","RET_FT4")],col="black",main="Minimize Least Squares")
abline(lm(RET_FT4~TUITIONFEE_OUT,data=private),col="red",lwd=3)
lm is the function that will find the least squares fit. Just typing it at the command line will give the coefficients calculated.
# just print output
lm(RET_FT4~TUITIONFEE_OUT,data=private)
##
## Call:
## lm(formula = RET_FT4 ~ TUITIONFEE_OUT, data = private)
##
## Coefficients:
## (Intercept) TUITIONFEE_OUT
## 4.863e-01 9.458e-06
We can also save this output as an object
# save as object to have results:
lmPrivate<-lm(RET_FT4~TUITIONFEE_OUT,data=private)
But there's actually a lot more created by lm that doesn't show up when you just print out the object.
names(lmPrivate)
## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "xlevels" "call" "terms" "model"
The actual values of the least squares line are saved as the coefficients element of the list. You can access them via standard list usage:
# first way to get coefficients: list value
lmPrivate$coefficients
## (Intercept) TUITIONFEE_OUT
## 4.863443e-01 9.458235e-06
Or you can use a built-in function coef to grab them.
# second way: built in function
coef(lmPrivate)
## (Intercept) TUITIONFEE_OUT
## 4.863443e-01 9.458235e-06
Absolute error Here we use the rq function in the quantreg package to find the coefficients for minimizing absolute error. The argument tau is set to 0.5 to signify we want average absolute error (other choices of tau could choose different error).
library(quantreg)
## Loading required package: SparseM
##
## Attaching package: 'SparseM'
## The following object is masked from 'package:base':
##
## backsolve
rqPrivate<-rq(RET_FT4~TUITIONFEE_OUT,data=private,tau=0.5)
We plot the results of both least squares and absolute error below.
plot(private[,c("TUITIONFEE_OUT","RET_FT4")],col="black",main="Minimize Least Squares",xlab=xlab,ylab=ylab)
abline(lmPrivate,col="red",lwd=3)
abline(rqPrivate,col="blue",lwd=3)
legend("bottomright",c("Squared error","Absolute error"),fill=c("red","blue"))
The following function creates bootstrap confidence intervals comparing two groups.
The input is like the permutation.test I wrote above: the two vectors of data, the function FUN that defines your statistic, and the number of repetitions. It returns a list with the confidence interval and the vector of bootstrapped statistics.
bootstrapLM <- function(y,x, repetitions, confidence.level=0.95){
# calculate the observed statistics
stat.obs <- coef(lm(y~x))
# calculate the bootstrapped statistics
bootFun<-function(){
sampled <- sample(1:length(y), size=length(y),replace = TRUE)
coef(lm(y[sampled]~x[sampled]))
}
stat.boot<-replicate(repetitions,bootFun())
nm <-deparse(substitute(x))
row.names(stat.boot)[2]<-nm
level<-1-confidence.level
confidence.interval <- apply(stat.boot,1,quantile,probs=c(level/2,1-level/2))
return(list(confidence.interval = cbind("lower"=confidence.interval[1,],"estimate"=stat.obs,"upper"=confidence.interval[2,]), bootStats=stat.boot))
}
We run this function with our college data. Notice that I use 'with', rather than just calling bootstrapLM with private$RET_FT4 and private$TUITIONFEE_OUT as the input. This is for the sole purpose so that my output is prettier and says TUITIONFEE_OUT as the header for my confidence interval, rather than private$TUITIONFEE_OUT.
privateBoot<-with(private,bootstrapLM(y=RET_FT4,x=TUITIONFEE_OUT,repetitions=10000))
privateBoot$conf
## lower estimate upper
## (Intercept) 4.627906e-01 4.863443e-01 5.089126e-01
## TUITIONFEE_OUT 8.789113e-06 9.458235e-06 1.014601e-05
Notice, if you haven't already, that when I use the $ to access an element of a list or data.frame, that I don't have to put down the whole name (e.g. confidence.interval), as long as it uniquely identifies one of the names (and since my other name for this output is bootStats, I can just use conf). However, if I use the bracket version to access it, I have to write out the whole name.
# this works
privateBoot[["confidence.interval"]]
## lower estimate upper
## (Intercept) 4.627906e-01 4.863443e-01 5.089126e-01
## TUITIONFEE_OUT 8.789113e-06 9.458235e-06 1.014601e-05
# this doesn't (just returns NULL)
privateBoot[["conf"]]
## NULL
Using the wrong name doesn't give an error however, it just returns NULL, meaning the element named conf was empty/undefined.
# We will again print out the confidence interval for only the slope, only we will multiply by 10,000 to get the change for a 10,000 dollar increase, rather than 1 dollar.
# increments of $10,000 for the slope
privateBoot$conf[2,]*10000
## lower estimate upper
## 0.08789113 0.09458235 0.10146007
Here we plot different lines that correspond to the limits of the confidence intervals (both intercept and slope).
plot(private[,c("TUITIONFEE_OUT","RET_FT4")],col="black")
abline(a=privateBoot$conf[1,1],b=privateBoot$conf[2,1],col="red",lwd=3)
abline(a=privateBoot$conf[1,3],b=privateBoot$conf[2,3],col="blue",lwd=3)
abline(a=privateBoot$conf[1,1],b=privateBoot$conf[2,3],col="green",lwd=3)
abline(a=privateBoot$conf[1,3],b=privateBoot$conf[2,1],col="green",lwd=3)
abline(lmPrivate,lwd=3)
Now we can use the summary of our lm result to get a more informative summary of our least squares fit.
summary(lmPrivate)
##
## Call:
## lm(formula = RET_FT4 ~ TUITIONFEE_OUT, data = private)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44411 -0.04531 0.00525 0.05413 0.31388
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.863e-01 1.020e-02 47.66 <2e-16 ***
## TUITIONFEE_OUT 9.458e-06 3.339e-07 28.32 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.08538 on 783 degrees of freedom
## Multiple R-squared: 0.5061, Adjusted R-squared: 0.5055
## F-statistic: 802.3 on 1 and 783 DF, p-value: < 2.2e-16
The confint will give the confidence intervals for the coefficients of the lm fit (i.e. based on the parametric model)
confint(lmPrivate)
## 2.5 % 97.5 %
## (Intercept) 4.663136e-01 5.063750e-01
## TUITIONFEE_OUT 8.802757e-06 1.011371e-05
We will combine the bootstrap and parametric confidence intervals of \(\beta_1\) into one matrix
b1CI<-rbind(c(lower=confint(lmPrivate)[2,1],estimate=unname(coef(lmPrivate)[2]),upper=confint(lmPrivate)[2,2]),privateBoot$conf[2,])
print(b1CI)
## lower estimate upper
## [1,] 8.802757e-06 9.458235e-06 1.011371e-05
## [2,] 8.789113e-06 9.458235e-06 1.014601e-05
We can plot these confidence intervals side-by-side. Note that use par(mar=...) to define how much space is on each side of my plot, in the form of a vector c(bottom, left, top, right), i.e. the amount of space around the plot for these four sides. The default (see help of par) is c(5, 4, 4, 2) + 0.1. I am going to increase the amount for the x-axis so that there is space for my labels to be turned sideways.
library(gplots)
##
## Attaching package: 'gplots'
## The following object is masked from 'package:stats':
##
## lowess
par(mar=c(8,4,4,1))
plotCI(b1CI[,2],ui=b1CI[,3] ,li=b1CI[,1],main="compare confidence intervals",col=c("blue","red"),pch=19,xaxt="n",xlim=c(0,3))
axis(1,at=c(1,2),labels=c("Parameteric","Bootstrap"),las=2)
We can predict a value using our coefficients. Here we consider prediction at the value 20,000
coef(lmPrivate)[1]+coef(lmPrivate)[2]*20000
## (Intercept)
## 0.675509
There is also a function predict that does this for you that is useful, particularly if you have more complicated models. Its input is rather fussy, however. You have to give the argument newdata, to give the values you want to predict (so you can do multiple predictions at one time), and the fussy part is that newdata has to be a data.frame of the same format as your original data that you ran lm on. In particular, it has to have the same column names.
predict(lmPrivate, newdata=data.frame(TUITIONFEE_OUT=20000))
## 1
## 0.675509
We can use the predict function to give both confidence and prediction intervals for our predictions.
predict(lmPrivate, newdata=data.frame(TUITIONFEE_OUT=20000),interval="confidence")
## fit lwr upr
## 1 0.675509 0.6670314 0.6839866
predict(lmPrivate, newdata=data.frame(TUITIONFEE_OUT=20000),interval="prediction")
## fit lwr upr
## 1 0.675509 0.5076899 0.843328
Now we plot prediction intervals using the predict function for a wide range of \(x_i\). First we calculate the confidence and prediction intervals for a range of tuition values
tuit<-seq(2000,60000,by=1000)
cint<-predict(lmPrivate,newdata=data.frame(TUITIONFEE_OUT=tuit),interval="confidence")
pint<-predict(lmPrivate,newdata=data.frame(TUITIONFEE_OUT=tuit),interval="prediction")
This returns a matrix with three columns: the estimate, and the lower/upper values of the interval. We will plot the data and the prediction line. Then we will plot the intervals as lines as well. We could use lines separately for the upper and lower. But we will use a handy function called matlines that plots (as lines) each column of the matrix against an x. (there's also a matplot and matpoints that correspond to plot and points).
plot(private[,c("TUITIONFEE_OUT","RET_FT4")],col="black",main="Minimize Least Squares",xlab=xlab,ylab=ylab)
abline(lm(RET_FT4~TUITIONFEE_OUT,data=private),col="black",lwd=3)
matlines(tuit,cint[,-1],lty=2,lwd=3,col="red")
matlines(tuit,pint[,-1],lty=3,lwd=3,col="blue")
legend("bottomright",c("Prediction","Conf. Int","Pred Int"),lty=c(1,2,3),col=c("black","red","blue"),lwd=3)
By default, predict just returns the predictions for the x-values of the observed data used in fitting the line, if you don't provide a newdata argument
Notice that predict is a general function that will work for many curve fitting functions. If we type help of predict it is not helpful because that's the generic function. If we want help to see what we can do for the predict of a lm response, we need to look at help of predict.lm. Uncomment the code below to see the difference.
# help(predict)
# help(predict.lm)
Now we consider the relationship of retention and tuition to average SAT score SAT_AVG_ALL, for both public and private institutes. Notice I can quickly draw the regression line by putting the call to lm inside the abline call.
par(mfrow=c(2,2))
plot(RET_FT4~SAT_AVG_ALL,data=public,main="Public")
abline(lm(RET_FT4~SAT_AVG_ALL,data=public))
plot(TUITIONFEE_OUT~SAT_AVG_ALL,data=public,main="Public")
abline(lm(TUITIONFEE_OUT~SAT_AVG_ALL,data=public))
plot(RET_FT4~SAT_AVG_ALL,data=private,main="Private")
abline(lm(RET_FT4~SAT_AVG_ALL,data=private))
plot(TUITIONFEE_OUT~SAT_AVG_ALL,data=private,main="Private")
abline(lm(TUITIONFEE_OUT~SAT_AVG_ALL,data=private))
We will fit a quadratic model, i.e. where SAT_AVG_ALL is both linear and quadratic term in predicting our variables. We use the same function (lm) and now our formula has both SAT_AVG_ALL as well as SAT_AVE_ALL^2 for predicting our y (either RET_FT4 or TUITIONFEE_OUT). We put I() around the quadratic term to make sure that lm recognizes the quadratic correctly. .
modelRET2<-lm(RET_FT4~SAT_AVG_ALL+I(SAT_AVG_ALL^2),data=private)
modelTUT2<-lm(TUITIONFEE_OUT~SAT_AVG_ALL+I(SAT_AVG_ALL^2),data=private)
Now we will plot the fitted quadratic curve. I define a function quadCurve that will plot a quadratic function from a vector of coefficients.
quadCurve<-function(x,cf){cf[1]+cf[2]*x+cf[3]*x^2}
Then we can find the coefficients from lm fitting the model using the function coef.
coefRET2<-coef(modelRET2)
coefTUT2<-coef(modelTUT2)
Now we can plot the quadratic curves using those coefficients.
par(mfrow=c(1,2))
plot(RET_FT4~SAT_AVG_ALL,data=private,main="Private schools, Retention")
curve(quadCurve(x,cf=coefRET2),add=TRUE,col="red",lwd=2)
plot(TUITIONFEE_OUT~SAT_AVG_ALL,data=private,main="Private schools, Tuition")
curve(quadCurve(x,cf=coefTUT2),add=TRUE,col="red",lwd=2)
We repeat the same code, only now fitting a cubic curve.
modelRET3<-lm(RET_FT4~SAT_AVG_ALL+I(SAT_AVG_ALL^2)+I(SAT_AVG_ALL^3),data=private)
modelTUT3<-lm(TUITIONFEE_OUT~SAT_AVG_ALL+I(SAT_AVG_ALL^2)+I(SAT_AVG_ALL^3),data=private)
cubicCurve<-function(x,cf){cf[1]+cf[2]*x+cf[3]*x^2+cf[4]*x^3}
par(mfrow=c(1,2))
plot(RET_FT4~SAT_AVG_ALL,data=private,main="Private schools")
curve(cubicCurve(x,cf=coef(modelRET3)),add=TRUE,col="red",lwd=2)
plot(TUITIONFEE_OUT~SAT_AVG_ALL,data=private,main="Private schools")
curve(cubicCurve(x,cf=coef(modelTUT3)),add=TRUE,col="red",lwd=2)
The following is code to calculate the running mean/median (using rollmedian and rollmean in zoo package) to describe the relationship between variables x and y. This code is not particularly important and mainly for instruction purposes, since the LOESS fit is a much better way to do this and doesn't use these functions.
library(zoo)
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
#put the response in order of x data
xorder<-private$SAT_AVG_ALL[order(private$SAT_AVG_ALL)]
yorder<-private$RET_FT4[order(private$SAT_AVG_ALL)]
# find the predictions for different window sizes
ypred11<-rollmedian(yorder,k=11)
xpred11<-rollmedian(xorder,k=11)
ypred51<-rollmedian(yorder,k=51)
xpred51<-rollmedian(xorder,k=51)
ypred101<-rollmedian(yorder,k=101)
xpred101<-rollmedian(xorder,k=101)
par(mfrow=c(1,1))
plot(xorder,yorder,main="Running Median Curve",xlab="SAT_AVG_ALL",ylab="RET_FT4")
lines(xpred11,ypred11,col="red",lwd=3)
lines(xpred51,ypred51,col="blue",lwd=3)
lines(xpred101,ypred101,col="green",lwd=3)
legend("bottomright",c("k=11","k=51","k=101"),fill=c("red","blue","green"),title="window size")
Now we are going to do loess, and compare it to running mean. Since loess is working with means, and before we did a rolling median, we are going to calculate the rolling mean
ypred11_mean<-rollmean(yorder,k=11)
ypred51_mean<-rollmean(yorder,k=51)
In the next plot we plot the moving average in the first plot, and the loess smoothing in the second plot. Notice that the function that does loess smoothing is loess.smooth and I can apply lines directly to it to plot the result ontop of my data. degree=0 means to take smoothed means (we'll see later we can use other options too). span controls the width of our smoothing, like bandwidth for density.
I get a lot of warnings on these calls because I am picking not very logical parameters for the span argument for illustration purposes.
par(mfrow=c(1,2))
plot(xorder,yorder,main="Moving Average",xlab="SAT_AVG_ALL",ylab="RET_FT4")
lines(xpred11,ypred11_mean,col="red",lwd=3)
lines(xpred51,ypred51_mean,col="blue",lwd=3)
legend("bottomright",c("k=11","k=51"),fill=c("red","blue"),title="window size")
plot(xorder,yorder,main="Kernel Smoothing")
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.01,degree=0),col="red",lwd=3)
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.1,degree=0),col="blue",lwd=3)
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,degree=0),col="green",lwd=3)
legend("bottomright",c("span=0.01","span=0.08","span=2/3 (default)"),fill=c("red","blue","green"),title="window size")
Now we consider using different fits in our local window. Specifically using the mean (degree=0), versus a local line (degree=1) and a local quadratic fit (degree=2).
par(mfrow=c(2,2))
plot(xorder,yorder,main="Mean")
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.01,degree=0),col="red",lwd=3)
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.1,degree=0),col="blue",lwd=3)
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,degree=0),col="green",lwd=3)
legend("bottomright",c("span=0.01","span=0.1","span=2/3"),fill=c("red","blue","green"),title="window size")
plot(xorder,yorder,main="Linear Regression")
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.01,degree=1),col="red",lwd=3)
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## pseudoinverse used at 985
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## neighborhood radius 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## reciprocal condition number -0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## There are other near singularities as well. 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## pseudoinverse used at 985
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## neighborhood radius 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## reciprocal condition number -0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## There are other near singularities as well. 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## pseudoinverse used at 985
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## neighborhood radius 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## reciprocal condition number -0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## There are other near singularities as well. 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## pseudoinverse used at 985
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## neighborhood radius 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## reciprocal condition number -0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## There are other near singularities as well. 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## pseudoinverse used at 985
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## neighborhood radius 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## reciprocal condition number -0
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## There are other near singularities as well. 1
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
## Warning in simpleLoess(y, x, w, span, degree = degree, parametric = FALSE, :
## zero-width neighborhood. make span bigger
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.1,degree=1),col="blue",lwd=3)
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,degree=1),col="green",lwd=3)
legend("bottomright",c("span=0.01","span=0.1","span=2/3"),fill=c("red","blue","green"),title="window size")
plot(xorder,yorder,main="Quadratic Regression")
#lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.01,degree=2),col="red",lwd=3) #can't run this because span is too small for quadratic
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,span=0.1,degree=2),col="blue",lwd=3)
lines(loess.smooth(x=private$SAT_AVG_ALL,y=private$RET_FT4,degree=2),col="green",lwd=3)
legend("bottomright",c("span=0.01","span=0.08","span=2/3"),fill=c("red","blue","green"),title="window size")
Now we consider the data you used in labs on craigs list rentals. We plot the price against the size of the rental.
craigs<-read.csv(file.path(dataDir,"craigslist.csv"),header=TRUE)
plot(price~size,data=craigs,xlim=c(0,5000))
Now we do the same thing, only now we draw a loess smoothing line on top of the data so that we can have a better sense of the pattern in the data.
loessCraigs<-loess.smooth(y=craigs$price,x=craigs$size)
plot(price~size,data=craigs,xlim=c(0,3000),ylim=c(0,20000))
lines(loessCraigs,col="red",lwd=3)
The library hexbin gives a visualization of 2D histogram.
library(hexbin)
wh<-with(craigs,which(size<3000 & size>0 & price<20000))
bin<-hexbin(x=craigs$size[wh], y=craigs$price[wh], xbins=100)
plot(bin, main="Hexagonal Binning",xlab="Size",ylab="Price")
Here we represent the density of points with a smooth density estimate for 2D using the function smoothScatter
smoothScatter(y=craigs$price,x=craigs$size,xlim=c(0,3000),nrpoints = 500)
We can also add a loess line on the smoothed density estimate (unlike hexbin, which doesn't look right…) and change the colors
mycolramp<-colorRampPalette(c("white", head(blues9,3),"blue","purple","red","yellow"))
smoothScatter(y=craigs$price,x=craigs$size,xlim=c(0,3000),nrpoints = 1200,ylim=c(0,20000),colramp=mycolramp)
lines(loessCraigs,col="red",lwd=3)
Here we read in information regarding global land temperatures. We format the year and month as well (complicated code for splitting up strings that I will not explain here).
temp<-read.csv(file.path(dataDir,"GlobalLandTemperaturesByMajorCity.csv"),header=TRUE)
head(temp)
## dt AverageTemperature AverageTemperatureUncertainty City
## 1 1849-01-01 26.704 1.435 Abidjan
## 2 1849-02-01 27.434 1.362 Abidjan
## 3 1849-03-01 28.101 1.612 Abidjan
## 4 1849-04-01 26.140 1.387 Abidjan
## 5 1849-05-01 25.427 1.200 Abidjan
## 6 1849-06-01 24.844 1.402 Abidjan
## Country Latitude Longitude
## 1 Côte D'Ivoire 5.63N 3.23W
## 2 Côte D'Ivoire 5.63N 3.23W
## 3 Côte D'Ivoire 5.63N 3.23W
## 4 Côte D'Ivoire 5.63N 3.23W
## 5 Côte D'Ivoire 5.63N 3.23W
## 6 Côte D'Ivoire 5.63N 3.23W
temp$Year<-as.numeric(sapply(strsplit(as.character(temp$dt),"-"),.subset2,1))
temp$Month<-as.numeric(sapply(strsplit(as.character(temp$dt),"-"),.subset2,2))
Plot all of the temperatures by year
par(mfrow=c(1,2))
plot(temp$Year,temp$AverageTemperature,main="All data points, all months")
lines(loess.smooth(x=temp$Year,y=temp$AverageTemperature,degree=2),col="red",lwd=2)
smoothScatter(x=temp$Year,y=temp$AverageTemperature,main="All data points, all months, smoothed")
Now we plot all of the temperatures by year, but highlight some cities and months to show why this is not a useful plot.
plot(temp$Year,temp$AverageTemperature,main="Cities highlighed")
lines(loess.smooth(x=temp$Year,y=temp$AverageTemperature,degree=2),col="red",lwd=2)
tempNY_01<-subset(temp,City=="New York" & Month==1)
tempNY_08<-subset(temp,City=="New York" & Month==8)
tempLA_01<-subset(temp,City=="Los Angeles" & Month==1)
tempLA_08<-subset(temp,City=="Los Angeles" & Month==8)
with(tempNY_01,points(x=Year,y=AverageTemperature,col="skyblue1"))
with(tempNY_08,points(x=Year,y=AverageTemperature,col="blue"))
with(tempLA_01,points(x=Year,y=AverageTemperature,col="pink"))
with(tempLA_08,points(x=Year,y=AverageTemperature,col="red"))
legend("bottomleft",c("NY, January","NY, August","LA, January","LA, August"),fill=c("skyblue1","blue","pink","red"),bg="white")
For different cities and different months, we can plot the data with a smooth curve over it using scatter.smooth. This is a single function that combines the plotting of the points and the smooth curve. It is handy, though you have less control over things like the color of the line, etc. (not to be confused with smoothScatter that we used above!)
ylab="Average Temperature"
xlab="Year"
par(mfrow=c(2,2))
tempNY_01<-subset(temp,City=="New York" & Month==1)
tempNY_08<-subset(temp,City=="New York" & Month==8)
tempLA_01<-subset(temp,City=="Los Angeles" & Month==1)
tempLA_08<-subset(temp,City=="Los Angeles" & Month==8)
scatter.smooth(tempNY_01$Year,tempNY_01$AverageTemperature,main="NY, January",lwd=2,ylab=ylab,xlab=xlab)
scatter.smooth(tempNY_08$Year,tempNY_08$AverageTemperature,main="NY, August",lwd=2,ylab=ylab,xlab=xlab)
scatter.smooth(tempLA_01$Year,tempLA_01$AverageTemperature,main="LA, January",lwd=2,ylab=ylab,xlab=xlab)
scatter.smooth(tempLA_08$Year,tempLA_08$AverageTemperature,main="LA, August",lwd=2,ylab=ylab,xlab=xlab)
Here I make a function to plot confidence intervals around the plot of the loess curves
If you notice, I'm going back and forth between different implementations of loess. predict on top of loess gives standard error estimates as well as the curve, but is rather akward to use; while loess.smooth (or scatter.smooth) is great for drawing the curve, but doesn't provide CI. However, loess.smooth and loess have different defaults, the most important being that loess.smooth by default sets degree=1 (i.e. linear fit locally) while loess is degree=2 (quadratic locally)}. To make it match the above, I will stick with degree=1 and set the span=2/3 to match the default of scatter.smooth that I did above.
loessWithCI<-function(dataset,...){
xseq<-seq(min(dataset$Year),max(dataset$Year),length=100)
loessPred<-predict(loess(AverageTemperature~Year,data=dataset,span=2/3,degree=1),newdata=data.frame(Year=xseq),se=TRUE)
plot(AverageTemperature~Year,data=dataset,...)
lines(xseq,loessPred$fit,lwd=2)
lines(xseq,loessPred$fit - qt(0.975,loessPred$df)*loessPred$se,lwd=2, lty=2)
lines(xseq,loessPred$fit + qt(0.975,loessPred$df)*loessPred$se,lwd=2, lty=2)
}
par(mfrow=c(2,2))
loessWithCI(tempNY_01,main="NY, January")
loessWithCI(tempNY_08,main="NY, August")
loessWithCI(tempLA_01,main="LA, January")
loessWithCI(tempLA_08,main="LA, August")
Now we use smoothed estimates of the data to be able to plot many cities on the same plot. I use subset to subset the data to 8 cities.
temp08<-subset(temp,Month==8 & City %in% c("Peking","Los Angeles","Toronto","Riyadh","Kabul","Mexico","Rome","New York"))
The factors in this data (like city, or country) are based on the full dataset. This can be annoying when you subset the data, because even though there are no representations of these levels in this subset, they levels still stick around.
# still have 100 cities listed as levels, even though only have 8
nlevels(temp08$City)
## [1] 100
So we can use the function droplevels to get rid of unused levels for all of the factors in the data
temp08<-droplevels(temp08) #otherwise the unused cities are still part of factor.
First we will make a vector that gives a color to each city. We will give the names of the vector those of the cities.
cityColors<-palette()[1:nlevels(temp08$City)]
names(cityColors)<-levels(temp08$City)
Now we make a blank plot. We will draw AverageTemperature as a function of Year for our points, but not actually draw it by setting type="n"
plot(temp08$Year,temp08$AverageTemperature,type="n",main="August, several Cities",xlab="Year",ylab="Temperature")#blank plot
Now, for each city, we will plot the loess curve on this blank plot. We will use the function by (basically the same as aggregate) to go over a data.frame based on a factor. This is like tapply, only tapply works on subsetting a vector based on a factor and by or aggregate work on subsetting a data.frame by a factor.
by(temp08,temp08$City,function(x){
lines(loess.smooth(x=x$Year, y=x$AverageTemperature),col=cityColors[unique(x$City)])
})
## Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
legend("topleft",levels(temp08$City),ncol=2,fill=cityColors)
## Error in strwidth(legend, units = "user", cex = cex, font = text.font): plot.new has not been called yet
Notice that the limits of my plot are slightly bigger than they need because the individual points take up more space than necessary and I used the individual points to set my blank plot. I could instead find explicitly what I need the limit to be for the yaxis using by again:
loessByCity08<-by(temp08,temp08$City,function(x){
loess.smooth(x=x$Year, y=x$AverageTemperature)
})
xlim<-range(sapply(loessByCity08,function(x){range(x$x)}))
ylim<-range(sapply(loessByCity08,function(x){range(x$y)}))
Then I repeat the above, only now in my blank plot, I set xlim and ylim using the above values:
plot(temp08$Year,temp08$AverageTemperature,type="n",xlim=xlim,ylim=ylim,main="August, several Cities",xlab="Year",ylab="Temperature")#blank plot
by(temp08,temp08$City,function(x){
lines(loess.smooth(x=x$Year, y=x$AverageTemperature),col=cityColors[unique(x$City)])
})
## temp08$City: Kabul
## NULL
## ------------------------------------------------------------
## temp08$City: Los Angeles
## NULL
## ------------------------------------------------------------
## temp08$City: Mexico
## NULL
## ------------------------------------------------------------
## temp08$City: New York
## NULL
## ------------------------------------------------------------
## temp08$City: Peking
## NULL
## ------------------------------------------------------------
## temp08$City: Riyadh
## NULL
## ------------------------------------------------------------
## temp08$City: Rome
## NULL
## ------------------------------------------------------------
## temp08$City: Toronto
## NULL
legend("topleft",names(loessByCity08),ncol=2,fill=cityColors)
Here it doesn't make much difference, but with some datasets it can.
We are going to repeat the above, only now we are going to subtract off the temperature in a particular year, so we only plot the difference from that year.
yearCenter<-1849
loessByCity08_center1<-by(temp08,temp08$City,function(x){
tempCenter<-x$AverageTemperature[x$Year==yearCenter]
loess.smooth(x=x$Year, y=x$AverageTemperature-tempCenter)
})
xlim<-range(sapply(loessByCity08_center1,function(x){range(x$x)}))
ylim<-range(sapply(loessByCity08_center1,function(x){range(x$y)}))
plot(0,0,type="n",xlim=xlim,ylim=ylim,xlab="Year",ylab=paste("Difference from",yearCenter))#blank plot
trash<-mapply(loessByCity08_center1,cityColors,FUN=function(x,col){lines(x,col=col,lwd=3)})
legend("topleft",names(loessByCity08_center1),ncol=2,fill=cityColors)
That didn't work so well, so lets plot the individual data plots for each of the cities, and highlight the point of 1849 to see what went wrong.
par(mfrow=c(2,4))
test<-by(temp08,temp08$City,function(x){
whYear<-which(x$Year==yearCenter)
tempCenter<-x$AverageTemperature[whYear]
plot(x$Year,x$AverageTemperature-tempCenter,main=as.character(unique(x$City)))
lines(loess.smooth(x=x$Year, y=x$AverageTemperature-tempCenter),col="red")
points(x$Year[whYear],(x$AverageTemperature-tempCenter)[whYear],col="blue",pch=19,cex=2)
})
Instead I want to center by the value of the loess curve at that year, shown in the following plot.
loessCenterValue<-by(temp08,temp08$City,function(x){
predict(loess(AverageTemperature~Year,data=x,span=2/3,degree=1), newdata=data.frame(Year=yearCenter))
})
par(mfrow=c(2,4))
test<-by(temp08,temp08$City,function(x){
whYear<-which(x$Year==yearCenter)
plot(x$Year,x$AverageTemperature,main=as.character(unique(x$City)))
lines(loess.smooth(x=x$Year, y=x$AverageTemperature),col="red")
predValue<-predict(loess(AverageTemperature~Year,data=x,span = 2/3, degree = 1), newdata=data.frame(Year=yearCenter))
points(x$Year[whYear],predValue,col="green",pch=19,cex=2)
})
Now we subtract those values off to recenter the curves to all cross at the same point in that year.
ylim<-range(mapply(loessByCity08,loessCenterValue,FUN=function(x,val){range(x$y-val)}))
plot(0,0,type="n",xlim=xlim,ylim=ylim,xlab="Year",ylab=paste("Difference from",yearCenter))#blank plot
trash<-mapply(loessByCity08,cityColors,loessCenterValue,FUN=function(x,col,center){
x$y<-x$y-center
lines(x,col=col,lwd=3)})
legend("topleft",names(loessByCity08_center1),ncol=2,fill=cityColors)
