## Exploratory Data Analysis (EDA) and Regression

This tutorial demonstrates some of the capabilities of R for exploring relationships among two (or more) quantitative variables.

### Bivariate exploratory data analysis

We begin by loading the Hipparcos dataset used in the descriptive statistics tutorial, found at http://astrostatistics.psu.edu/datasets/HIP_star.html. Type
```   hip  <-  read.table("http://astrostatistics.psu.edu/datasets/HIP_star.dat",
names(hip)
attach(hip)
```
In the descriptive statistics tutorial, we considered boxplots, a one-dimensional plotting technique. We may perform a slightly more sophisticated analysis using boxplots to get a glimpse at some bivariate structure. Let us examine the values of Vmag, with objects broken into categories according to the B minus V variable:
```   boxplot(Vmag~cut(B.V,breaks=(-1:6)/2),
notch=T, varwidth=T, las=1, tcl=.5,
xlab=expression("B minus V"),
ylab=expression("V magnitude"),
main="Can you find the red giants?",
cex=1, cex.lab=1.4, cex.axis=.8, cex.main=1)
axis(2, labels=F, at=0:12, tcl=-.25)
axis(4, at=3*(0:4))
```
The notches in the boxes, produced using "notch=T", can be used to test for differences in the medians (see boxplot.stats for details). With "varwidth=T", the box widths are proportional to the square roots of the sample sizes. The "cex" options all give scaling factors, relative to default: "cex" is for plotting text and symbols, "cex.axis" is for axis annotation, "cex.lab" is for the x and y labels, and "cex.main" is for main titles. The two axis commands are used to add an axis to the current plot. The first such command above adds smaller tick marks at all integers, whereas the second one adds the axis on the right.

### Scatterplots

The boxplots in the plot above are telling us something about the bivariate relationship between the two variables. Yet it is probably easier to grasp this relationship by producing a scatter plot.
```   plot(Vmag,B.V)
```
The above plot looks too busy because of the default plotting character, set let's use a different one:
```   plot(Vmag,B.V,pch=".")
```
Let's now use exploratory scatterplots to locate the Hyades stars. This open cluster should be concentrated both in the sky coordinates RA and DE, and also in the proper motion variables pm_RA and pm_DE. We start by noticing a concentration of stars in the RA distribution:
```   plot(RA,DE,pch=".")
```
See the cluster of stars with RA between 50 and 100 and with DE between 0 and 25?
```   rect(50,0,100,25,border=2)
```
Let's construct a logical (TRUE/FALSE) variable that will select only those stars in the appropriate rectangle:
```   filter1 <-  (RA>50 & RA<100 & DE>0 & DE<25)
```
Next, we select in the proper motions. (As our cuts through the data are parallel to the axes, this variable-by-variable classification approach is sometimes called Classification and Regression Trees or CART, a very common multivariate classification procedure.)
```   plot(pmRA[filter1],pmDE[filter1],pch=20)
rect(0,-150,200,50,border=2)
```
Let's replot after zooming in on the rectangle shown in red.
```   plot(pmRA[filter1],pmDE[filter1],pch=20, xlim=c(0,200),ylim=c(-150,50))
rect(90,-60,130,-10,border=2)
filter2 <- (pmRA>90 & pmRA<130 & pmDE>-60 & pmDE< -10) # Space in 'pmDE< -10' is necessary!
filter  <-  filter1 & filter2
```
Let's have a final look at the stars we have identified using the pairs command to produce all bivariate plots for pairs of variables. We'll exclude the first and fifth columns (the HIP identifying number and the parallax, which is known to lie in a narrow band by construction).
```   pairs(hip[filter,-c(1,5)],pch=20)
```
Notice that indexing a matrix or vector using negative integers has the effect of excluding the corresponding entries.

We see that there is one outlying star in the e_Plx variable, indicating that its measurements are not reliable. We exclude this point:
```   filter <- filter & (e_Plx<5)
pairs(hip[filter,-c(1,5)],pch=20)
```
How many stars have we identified? The filter variable, a vector of TRUE and FALSE, may be summed to reveal the number of TRUE's (summation causes R to coerce the logical values to 0's and 1's).
```   sum(filter)
```
As a final look at these data, let's consider the HR plot of Vmag versus B.V but make the 92 Hyades stars we just identified look bigger (pch=20 instead of 46) and color them red (col=2 instead of 1). This shows the Zero Age Main Sequence, plus four red giants, with great precision.
```   plot(Vmag,B.V,pch=c(46,20)[1+filter], col=1+filter,
xlim=range(Vmag[filter]), ylim=range(B.V[filter]))
```

### Linear and polynomial regression

Here is how one may reproduce the output seen in the regression lecture, i.e., a linear regression relating BminusV to logL, where logL is the luminosity, defined to be (15 - Vmag - 5 log(Plx)) / 2.5. However, we'll use a different subset of the data than the one seen in the lecture, namely, the main-sequence Hyades:
```   mainseqhyades <- filter & (Vmag>4 | B.V<0.2)
logL <- (15 - Vmag - 5 * log10(Plx)) / 2.5
plot(x, y)
regline <- lm(y~x)
abline(regline, lwd=2, col=2)
summary(regline)
```
Note that the regression line passes exactly through the point (xbar, ybar):
```   points(mean(x), mean(y), col=3, pch=20, cex=3)
```
Here is a regression of y on exp(-x/4):
```   newx  <-  exp(-x/4)
regline2 <- lm(y~newx)
xseq  <-  seq(min(x), max(x), len=250)
lines(xseq, regline2\$coef %*% rbind(1, exp(-xseq/4)), lwd=2, col=3)
```
For an implementation of the ridge regression technique mentioned in lecture, see the lm.ridge function in the MASS package. To use this function, you must first type library(MASS). For a package that implements LASSO (which uses an L1-penalty instead of the L2-penalty of ridge regression), check out, e.g., the lasso2 package on CRAN.

Let's consider a new dataset, one that comes from NASA's Swift satellite. The statistical problem at hand is modeling the X-ray afterglow of gamma-ray bursts. First, read in the dataset:
```   grb <- read.table ("http://astrostatistics.psu.edu/datasets/GRB_afterglow.dat",
```
The skip=1 option in the previous statement tells R to ignore the first row in the data file. You will see why this is necessary if you look at the file. Let's focus on the first two columns, which are times and X-ray fluxes:
```   plot(grb[,1:2],xlab="time",ylab="flux")
```
This plot is very hard to interpret because of the scales, so let's take the log of each variable:
```   x <- log(grb[,1])
y <- log(grb[,2])
plot(x,y,xlab="log time",ylab="log flux")
```
The relationship looks roughly linear, so let's try a linear model (lm in R):
```   model1 <- lm(y~x)
abline(model1, col=2, lwd=2)
```
The "response ~ predictor(s)" format seen above is used for model formulas in functions like lm .

The model1 object just created is an object of class "lm". The class of an object in R can help to determine how it is treated by functions such as print and summary.
```   model1 # same as print(model1)
summary(model1)
```
Notice the sigma-hat, the R-squared and adjusted R-squared, and the standard errors of the beta-hats in the output from the summary function.

There is a lot of information contained in model1 that is not displayed by print or summary:
```   names(model1)
```
For instance, we will use the model1\$fitted.values and model1\$residuals information later when we look at some residuals plots.

Notice that the coefficient estimates are listed in a regression table, which is standard regression output for any software package. This table gives not only the estimates but their standard errors as well, which enables us to determine whether the estimates are very different from zero. It is possible to give individual confidence intervals for both the intercept parameter and the slope parameter based on this information, but remember that a line really requires both a slope and an intercept. Since our goal is really to estimate a line here, maybe it would be better if we could somehow obtain a confidence "interval" for the lines themselves.

This may in fact be accomplished. By viewing a line as a single two-dimensional point in (intercept, slope) space, we set up a one-to-one correspondence between all (nonvertical) lines and all points in two-dimensional space. It is possible to obtain a two-dimensional confidence ellipse for the (intercept,slope) points, which may then be mapped back into the set of lines to see what it looks like.

Performing all the calculations necessary to do this is somewhat tedious, but fortunately, someone else has already done it and made it available to all R users through CRAN, the Comprehensive R Archive Network. The necessary functions are part of the "car" (companion to applied regression) package, which may installed onto the V: drive (we don't have write access to the default location where R packages are installed) as follows:
```   install.packages("car",lib="V:/") # lib=... is not always necessary!
```
You will have to choose a CRAN mirror as part of the installation process. Once the car package is installed, its contents can be loaded into the current R session using the library function:
```   library(car,lib.loc="V:/")
```
If all has gone well, there is now a new set of functions, along with relevant documentation. Here is a 95% confidence ellipse for the (intercept,slope) pairs:
```   confidence.ellipse(model1)
```
Remember that each point on the boundary or in the interior of this ellipse represents a line. If we were to plot all of these lines on the original scatterplot, the region they described would be a 95% confidence band for the true regression line. A graduate student, Derek Young, and I wrote a simple function to draw the borders of this band on a scatterplot. You can see this function at www.stat.psu.edu/~dhunter/R/confidence.band.r"; to read it into R, use the source function:
```   source("http://www.stat.psu.edu/~dhunter/R/confidence.band.r")
confidence.band(model1)
```
In this dataset, the confidence band is so narrow that it's hard to see. However, the borders of the band are not straight. You can see the curvature much better when there are fewer points or more variation, as in:
```   tmpx <- 1:10
tmpy <- 1:10+rnorm(10) # Add random Gaussian noise
confidence.band(lm(tmpy~tmpx))
```
Also note that increasing the sample size increases the precision of the estimated line, thus narrowing the confidence band. Compare the previous plot with the one obtained by replicating tmpx and tmpy 25 times each:
```   tmpx25 <- rep(tmpx,25)
tmpy25 <- rep(tmpy,25)
confidence.band(lm(tmpy25~tmpx25))
```
A related phenomenon is illustrated if we are given a value of the predictor and asked to predict the response. Two types of intervals are commonly reported in this case: A prediction interval for an individual observation with that predictor value, and a confidence interval for the mean of all individuals with that predictor value. The former is always wider than the latter because it accounts for not only the uncertainty in estimating the true line but also the individual variation around the true line. This phenomenon may be illustrated as follows. Again, we use a toy data set here because the effect is harder to observe on our astronomical dataset. As usual, 95% is the default confidence level.
```   confidence.band(lm(tmpy~tmpx))
predict(lm(tmpy~tmpx), data.frame(tmpx=7), interval="prediction")
text(c(7,7,7), .Last.value, "P",col=4)
predict(lm(tmpy~tmpx), data.frame(tmpx=7), interval="conf")
text(c(7,7,7), .Last.value, "C",col=5)
```

### Polynomial curve-fitting: Still linear regression!

Because there appears to be a bit of a bend in the scatterplot, let's try fitting a quadratic curve instead of a linear curve. Note: Fitting a quadratic curve is still considered linear regression. This may seem strange, but the reason is that the quadratic regression model assumes that the response y is a linear combination of 1, x, and x2. Notice the special form of the lm command when we implement quadratic regression. The I function means "as is" and it resolves any ambiguity in the model formula:
```   model2 <- lm(y~x+I(x^2))
summary(model2)
```
Here is how to find the estimates of beta using the closed-form solution seen in lecture:
```   X <- cbind(1, x, x^2) # Create nx3 X matrix
solve(t(X) %*% X) %*% t(X) %*% y # Compare to the coefficients above
```
Plotting the quadratic curve is not a simple matter of using the abline function. To obtain the plot, we'll first create a sequence of x values, then apply the linear combination implied by the regression model using matrix multiplication:
```   xx <- seq(min(x),max(x),len=200)
yy <- model2\$coef %*% rbind(1,xx,xx^2)
lines(xx,yy,lwd=2,col=3)
```

### Diagnostic residual plots

Comparing the (red) linear fit with the (green) quadratic fit visually, it does appear that the latter looks slightly better. However, let's check some diagnostic residual plots for these two models. To do this, we'll use the plot.lm command, which is capable of producing six different types of diagnostic plots. We will only consider two of the six: A plot of residuals versus fitted values and a normal quantile-quantile (Q-Q) plot.
```   plot.lm(model1, which=1:2)
```
It is not actually necessary to type plot.lm in the previous command; plot would have worked just as well. This is because model1 is an object of class "lm" -- a fact that can be verified by typing "class(model1)" -- and so R knows to apply the function plot.lm if we simply type "plot(model1, which=1:2)".

Looking at the first plot, residuals vs. fitted, we immediately see a problem with model 1. A "nice" residual plot should have residuals both above and below the zero line, with the vertical spread around the line roughly of the same magnitude no matter what the value on the horizontal axis. Furthermore, there should be no obvious curvature pattern. The red line is a lowess smoother produced to help discern any patterns (more on lowess later), but this line is not necessary in the case of model1 to see the clear pattern of negative residuals on the left, positive in the middle, and negative on the right. There is curvature here that the model missed!

Pressing the return key to see the second plot reveals a normal quantile-quantile plot. The idea behind this plot is that it will make a random sample from a normal distribution look like a straight line. To the extent that the normal Q-Q plot does not look like a straight line, the assumption of normality of the residuals is suspicious. For model1, the clear S-shaped pattern indicates non-normality of the residuals.

How do the same plots look for the quadratic fit?
```   plot(model2, which=1:2)
```
These plots are much better-looking. There is a little bit of waviness in the residuals vs. fitted plot, but the pattern is nowhere near as obvious as it was before. And there appear to be several outliers among the residuals on the normal Q-Q plot, but the normality assumption looks much less suspect here.

The residuals we have been using in the above plots are the ordinary residuals. However, it is important to keep in mind that even if all of the assumptions of the regression model are perfectly true (including the assumption that all errors have the same variance), the variances of the residuals are not equal. For this reason, it is better to use the studentized residuals. Unfortunately, R reports the ordinary residuals by default and it is necessary to call another function to obtain the studentized residuals. The good news is that in most datasets, residual plots using the studentized residuals are essentially indistinguishable in shape from residual plots using the ordinary residuals, which means that we would come to the same conclusions regardless of which set of residuals we use.
```   rstu  <-  rstudent(model2)
plot(model2\$fit, rstu)
```
To see how similar the studentized residuals are to a scaled version of the ordinary residuals (called the standardized residuals), we can depict both on the same plot:
```   rsta  <-  rstandard(model2)
points(model2\$fit, rsta, col=2, pch=3)
```

### Collinearity and variance inflation factors

Let's check the variance inflation factors (VIFs) for the quadratic fit. The car package that we installed earlier contains a function called vif that does this automatically. Check its help page by typing "?vif" if you wish. Note that it does not make sense to look at variance inflation factors for model1, which has only one term (try it and see what happens). So we'll start by examining model2.
```   vif(model2)
```
The VIFs of more than 70 indicate a high degree of collinearity between the values of x and x^2 (the two predictors). This is not surprising, since x has a range from about 5 to 13. In fact, it is easy to visualize the collinearity in a plot:
```   plot(x,x^2) # Note highly linear-looking plot
```
To correct the collinearity, we'll replace x and x^2 by (x-m) and (x-m)^2, where m is the sample mean of x:
```   centered.x  <-  x-mean(x)
model2.2  <-  lm(y ~ centered.x + I(centered.x^2))
```
This new model has much lower VIFs, which means that we have greatly reduced the collinearity. However, the fit is exactly the same: It is still the best-fitting quadratic curve. We may demonstrate this by plotting both fits on the same set of axes:
```   plot(x,y,xlab="log time",ylab="log flux")
yy2  <-  model2.2\$coef %*% rbind(1, xx-mean(x), (xx-mean(x))^2)
lines(xx, yy, lwd=2, col=2)
lines(xx, yy2, lwd=2, col=3, lty=2)
```

### Model selection using AIC and BIC

Let's compare the AIC and BIC values for the linear and the quadratic fit. Without getting too deeply into details, the idea behind these criteria is that we know the model with more parameters (the quadratic model) should achieve a higher maximized log-likelihood than the model with fewer parameters (the linear model). However, it may be that the additional increase in the log-likelihood statistic achieved with more parameters is not worth adding the additional parameters. We may test whether it is worth adding the additional parameters by penalizing the log-likeilhood by subtracting some positive multiple of the number of parameters. In practice, for technical reasons we take -2 times the log-likelihood, add a positive multiple of the number of parameters, and look for the smallest resulting value. For AIC, the positive multiple is 2; for BIC, it is the natural log of n, the number of observations. We can obtain both the AIC and BIC results using the AIC function. Remember that R is case-sensitive, so "AIC" must be all capital letters.
```   AIC(model1)
AIC(model2)
```
The value of AIC for model2 is smaller than that for model1, which indicates that model2 provides a better fit that is worth the additional parameters. However, AIC is known to tend to overfit sometimes, meaning that it sometimes favors models with more parameters than they should have. The BIC uses a larger penalty than AIC, and it often seems to do a slightly better job; however, in this case we see there is no difference in the conclusion:
```   n <- length(x)
AIC(model1, k=log(n))
AIC(model2, k=log(n))
```
It did not make any difference in the above output that we used model2 (with the uncentered x values) instead of model2.2 (with the centered values). However, if we had looked at the AIC or BIC values for a model containing ONLY the quadratic term but no linear term, then we would see a dramatic difference. Which one of the following would you expect to be higher (i.e., indicating a worse fit), and why?
```   AIC(lm(y~I(x^2)), k=log(n))
AIC(lm(y~I(centered.x^2)), k=log(n))
```

### Other methods of curve-fitting

Let's try a nonparametric fit, given by loess or lowess. First we plot the linear (red) and quadratic (green) fits, then we overlay the lowess fit in blue:
```   plot(x,y,xlab="log time",ylab="log flux")
abline(model1, lwd=2, col=2)
lines(xx, yy, lwd=3, col=3)
npmodel1 <- lowess(y~x)
lines(npmodel1, col=4, lwd=2)
```
It is hard to see the pattern of the lowess curve in the plot. Let's replot it with no other distractions. Notice that the "type=n" option to plot function causes the axes to be plotted but not the points.
```   plot(x,y,xlab="log time",ylab="log flux", type="n")
lines(npmodel1, col=4, lwd=2)
```
This appears to be a piecewise linear curve. An analysis that assumes a piecewise linear curve will be carried out on these data later in the week.

In the case of non-polynomial (but still parametric) curve-fitting, we can use nls. If we replace the response y by the original (nonlogged) flux values, we might posit a parametric model of the form flux = exp(a+b*x), where x=log(time) as before. Compare a nonlinear approach (in red) with a nonparametric approach (in green) for this case:
```   flux <- grb[,2]
nlsmodel1 <- nls(flux ~ exp(a+b*x), start=list(a=0,b=0))
npmodel2 <- lowess(flux~x)
plot(x, flux, xlab="log time", ylab="flux")
lines(xx, exp(9.4602-.9674*xx), col=2, lwd=2)
lines(npmodel2, col=3, lwd=2)
```
Interestingly, the coefficients of the nonlinear least squares fit are different than the coefficients of the original linear model fit on the logged data, even though these coefficients have exactly the same interpretation: If flux = exp(a + b*x), then shouldn't log(flux) = a + b*x? The difference arises because these two fitting methods calculate (and subsequently minimize) the residuals on different scales. Try plotting exp(a + b*xx) on the scatterplot of x vs. flux for both (a,b) solutions to see what happens. Next, try plotting a + b*xx on the scatterplot of x vs. log(flux) to see what happens.

If outliers appear to have too large an influence over the least-squares solution, we can also try resistant regression, using the lqs function in the MASS package. The basic idea behind lqs is that the largest residuals (presumably corresponding to "bad" outliers) are ignored. The results for our log(flux) vs. log(time) example look terrible but are very revealing. Can you understand why the output from lqs looks so very different from the least-squares output?
```   library(MASS)
lqsmodel1 <- lqs(y~x, method="lts")
plot(x,y,xlab="log time",ylab="log flux")
abline(model1,col=2)
abline(lqsmodel1,col=3)
```
Finally, let's consider least absolute deviation regression, which may be considered a milder form of resistant regression than lqs. In least absolute deviation regression, even large residuals have an influence on the regression line (unlike in lqs), but this influence is less than in least squares regression. To implement it, we'll use a function called rq (regression quantiles) in the "quantreg" package. Like the "car" package, this package is not part of the standard distribution of R, so we'll need to download it. In order to do this, we must tell R where to store the installed library using the install.packages function.
```   install.packages("quantreg",lib="V:/") # lib=... is not always necessary!
library(quantreg, lib.loc="V:/")
```
Assuming the quantreg package is loaded, we may now compare the least-squares fit (red) with the least absolute deviations fit (green). In this example, the two fits are nearly identical:
```   rqmodel1 <- rq(y~x)
plot(x,y,xlab="log time",ylab="log flux")
abline(model1,col=2)
abline(rqmodel1,col=3)
```