Use R for Climate Research

James B. Elsner & Thomas H. Jagger

This tutorial gives you an introduction to the R programming language for statistical analysis and modeling.  Examples are taken from our research into
hurricane climate.  It was developed with support from the Florida State University, the U.S. National Science Foundation, and the Risk Prediction Initiative of the Bermuda Institute of Ocean Sciences. You should follow along with an open R session.

The key strokes are given in red font.  If you're not a fast typer, they can be copied from the pdf (not from DVD) and pasted into your R session.  Output from an R command is given in blue Courier New font.  Questions for which answers are sought and practice problems are given in green font.

The tutorial is divided into 7 sections.  We begin with some motivations for giving R a try and some information on how to download and install R on your computer.  We show you how to use R as a calculator.  In section 2 we show you how to enter data into R, how to query your data, and how to obtain some statistical information about your data.   In section 3 we provide some examples of using R for making inferences from correlations and differences in sample means.  In section 4 we provide an introduction to using R for graphing your data.  In section 5 we show you how R can be used to simulate various probability models and how to do bootstrap analysis.  In section 6 we provide more advanced material on using R for statistical modeling.  We consider linear regression, logistic regression, Poisson regression, quantile regression and regression trees.   In section 7 we list additional resource material.

1. Introduction

Benefits and drawbacks of R
Free, open-source, runs on unix, linux, windows, and macs. Built-in help system. Excellent graphing capabilities. Powerful, extensible, and easy to learn syntax. Thousands of built-in functions.

Limited graphical user interface meaning it is harder to learn at the outset. No commercial support. It's a programming language so you need to appreciate syntax issues. Common metaphors for working with computers are: browsers, iTunes, and perhaps Excel. R is nothing like these. Irregular users forget commands. There are no visual cues: A blank screen is intimidating.

Why switch to R?
If you've already mastered a statistical package and if you only carry out a limited range of statistical tests with no intention of doing something different, then don't switch.  The main reason for switching to R is to take advantage of its unrivaled coverage and availability of new, cutting edge applications in emerging statistical fields such as generalized mixed effects and generalized additive models.  Another reason is to be able to understand the literature.  More people are reporting results in the context of R, and it is important to know what they're talking about.  Many of the world's leading statisticians use and contribute to R.

"...the massive collective international effort represented by the R project exceeds the most idealistic Marxist imagination."   --- Roger Koenker

What is R?
R is an open-source statistical environment modeled after S and S-Plus (http://www.insightful.com). The S language was developed in the late 1980s at AT&T labs. The R project was started by Robert Gentleman and Ross Ihaka of the Statistics Department of the University of Auckland in 1995. It now has a large audience. It's currently maintained by the R core-development team; an international group of volunteer developers.

To get to the R project web site, open a browser and in a search window, type the key words "R project".

The R project web page is http://www.r-project.org/

Directions for obtaining the software, accompanying packages and other sources of documentation are provided at the site.

Click on the above url (universal resource locater) and browse to the site.  Click on CRAN (Comprehensive R Archive Network) to download the latest version of R.  Scroll to the nearest mirror site.  Click on the appropriate pre-compiled binary distribution.   Click on the base distribution to download.  Only the base directory is needed at this time.

Once installed, click on the icon to start R.  To open R in Linux, from a command window, type R.

R is most easily used in an interactive manner. You ask it a question and it gives you an answer. Questions are asked and answered on the command line.

The > (greater than symbol) is used as the prompt.  In what follows, it is the character that is NOT typed.

If a command is too long, a + (plus symbol) is used for the continuation prompt.  If you get completely lost on a command, you can use Ctrl c or Esc to get the prompt back and to start over.

To find out how to cite R in your publications, type:

citation()

What appears on the screen is a reference citation for R and a BibTeX entry for LaTeX users.

Most commands are functions and most functions require parentheses.  If the function has all default options, then the function is applied by typing two side by side parentheses.

Calculating

R evaluates commands typed at the prompt.  It returns the result of the computation on the screen or saves it to an object.  For example, to find the sum of the square root of 25 and 2, type:

sqrt(25)+2
[1] 7
>

The [1] says "first requested element will follow".  Here there is only one element the answer 7.  The > prompt that follows indicates R is ready for another command.

12/3-5

How would you calculate the 5th power of 2?

How about the amount of interest on \$1000, compounded annually at 4.5% a.p. for 5 years.

1000*(1+0.045)^5-1000

Getting help
The help manuals are available as html.  To access them, type:

help.start()

To quit R, type q(save="no").

2. Data

Entering data
Statistics is the analysis and modeling of data.  The most useful R command for quickly inputing small amounts of data is the c function.  This function combines (concatenates) items together.   As an example, consider a set of hypothetical annual land falling hurricane counts over a ten-year period.

2  3  0  3  1  0  0  1  2  1

To enter these into an R session, type (note: do not type the prompt only the red text).

counts = c(2,3,0,3,1,0,0,1,2,1)
counts

[1] 2 3 0 3 1 0 0 1 2 1

Notice a few things. You assigned the values to an object called counts. The assignment operator is an equal sign (=).  Another valid assignment operator is the arrow (<-).  I think it is easier to stick with the equal sign. Values do not automatically print.  They are assigned to an object name.  They can be printed by typing the object name as we did on the second line.  Finally, the values when printed are prefaced with a [1].  This indicates that the object is a vector and the first entry in the vector is a value of 2 (The number immediately to the right of [1]).  More on this later.

You can save yourself a lot of typing if you learn that the arrow keys can be used to retrieve previous commands.  Each command is stored in a history file and the up arrow key will move backwards through the history file and the down arrow forwards.  The left and right arrow keys will work as expected.

You can also enter small amounts of data with the scan function.  You enter data values one at a time line by line.  When finished type enter.

counts2 = scan()
1: 2
2: 3

>

Applying a function
R comes with many built in functions that you can apply to your counts data. One of them is the mean function for finding the average.  To use it, type:

mean(counts)
[1] 1.3

As well, you could apply the functions median and var to find the median and variance, respectively. The syntax is the same – the function name followed by parentheses to contain the argument(s).  Most functions have more than one argument.  Arguments not specified are given default values.

median(counts)
[1] 1
var(counts)
[1] 1.344444

Data vectors
The data is stored in R as a vector.  This means that R keeps track of the order that the data were entered.  There is a first element, a second element, and so on.  This is good for several reasons.
1. Our simple data vector of counts has a natural order - year 1, year 2, etc.  We do not want to mix these up.
2. We would like to be able to make changes to the data item by item instead of entering the entire data again.
3. Vectors are math objects so that math operations can be performed on them.

Let's see how these concepts apply to our landfall counts example.  First, suppose counts contain the annual landfall count from the first decade of a longer record.  We want to keep track of counts over other decades.  This could be done by the following, rather contrived, example.

counts.decade1 = counts      # make a copy
counts.decade2 = c(0,5,4,2,3,0,3,3,2,1)   # enter a new set of counts

The pound symbol (#) is used as a comment character.  Anything after the # is ignored.  Adding comments to the history file is a good way of recalling what you did and why.

Note the two different object names.  Unlike many programming languages, the period in R is only used as punctuation.  You can't use an _ (underscore) to punctuate names as the underscore is actually another assignment operator although its use is fading.

Now suppose the U.S. National Hurricane Center (NHC) reanalyzes a storm, and that the 6th year of the 2nd decade is a 1 rather than a 0 for the number of landfalls.  In this case we can type:

counts.decade2[6] = 1    # assign the 6 year of the decade a value of 1 landfall

The assignment to the 6th entry in the vector counts.decade2 is done by referencing the 6th entry of the vector with square brackets [].  It is very important to keep this in mind: Parentheses () are used for functions and square brackets [] are used for vectors (and arrays, lists, etc).

[1] 0 5 4 2 3 1 3 3 2 1
counts.decade2[2]  # print the number of landfalls during year 2 of the second decade
[1] 5
[1] 2
counts.decade2[-4]  # all but the 4th year
[1] 0 5 4 3 1 3 3 2 1
counts.decade2[c(1,3,5,7,9)]   # print the counts from the odd years
[1] 0 4 3 3 2

R lets us systematically examine our data.  For example, to find the maximum number of landfalls in the first decade, type:

[1] 3

[1] FALSE  TRUE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE

Notice the usage of double equals signs (==). This tests all the values of counts.decade1 to see if they are equal to 3. The 2nd and 4th answer yes (TRUE) the others no. Think of this as asking R a question. Is the value equal to 3? R answers all at once with a long vector of TRUE's and FALSE's. Now the question is – how can we get the years corresponding to the TRUE values? Let’s rephrase, which years have 3 landfalls?  If you guessed that the command which will work, you're on your way to mastering R.

[1] 2 4

The function which.max can be used to get the first maximum.

We might also want to know the total number of landfalls in each decade or the number of years in a decade without a landfall.  Or how about the ratio of the mean number of landfalls over the two decades.

[1] 13
[1] 24

Here we apply two functions on a single line by using the semicolon ;.

[1] 3
[1] 1

[1] 1.846154

So there is 85% more landfalls during the second decade.  Is this statistically significant?

Before moving on, it is good practice to clean up your working directory.  This is done using the rm (remove) function.

Most of your analyzes will be done on data that are read into R.  First, determine what directory you are working in by typing:

getwd()

You can change your working directory by using the setwd() function and specifying the path.

Second, download the data (Save Link or Target As) into your working directory.  The data are available here (http://garnet.fsu.edu/~jelsner/www/data.html) under hurricane landfall counts.

Third, a simple way to read data is to use the read.table function.

If the file is read in, you should get the > prompt.  Note the quotes around the file name as you are referencing a file outside of R.  If you can use full path names for files.  The header=T argument means there is a header record in the data (T stands for true).  Data read in with the read.table function are of class data.frame.  To check, type:

class(ushur)

To see if the data are what you expect, try the following commands.

names(ushur)   # list the column names, US: US hurr, MUS: major US hur, G: Gulf coast, FL: Florida, E: East coast
head(ushur)  # list the first several rows
tail(ushur)  # list the last several rows

The function read.table has options for specifying the separator character or characters between columns used in the native file.  Alternatives to the default use of a space are sep="," and sep="\t" for tab.  You can also control the choice of a missing value character, which by default is NA.  If the missing value character in the native file is a 999, specify na.strings="999".

There are several variants of read.table that differ only in having different default parameter settings.  Note in particular read.csv, which has settings that are suitable for comma delimited (csv) files that have been exported from an Excel spreadsheet.

If you want to change the names of the data frame, type:

names(ushur)[4]="GC"

This changes the 4 column name from G to GC.

Summarizing
Frequently all that's needed is a summary of your data.

mean(ushur\$US)
[1] 1.711538

To access the columns of a data frame, you give the name of the data frame followed by the dollar sign (\$) followed by the column name.

What is the median number of U.S. hurricanes over this time period?  What is the variance?

The output carries too many digits.  Since there are only about 150 or so years, the number of significant digits is 2 or 3.  This can be changed using the signif function.

signif(mean(ushur\$US),3)
[1] 1.71

Note that the signif function wraps around the mean function.

The mean or average number of counts over a time period is called a rate.  How many years are in the data set?

dim(ushur)  # dimensions (rows by columns) of the data frame, which here is the number of years by the number of variables
[1] 156  6

So the annual hurricane rate is 1.71 hurricanes per year over the 156-year period.

To find out which year(s) have the maximum, type

ushur\$Year[ushur\$US==max(ushur\$US)]
[1] 1886

Note how the command structure allows embedding.  It is often best to read the expression from right to left.  The first function on the right is max, so this returns a number giving the maximum annual count.  Then we want the index(es) for that count, finally we want the year(s) corresponding to the index(es).

Similarly, to find the years with no hurricanes, type:

ushur\$Year[ushur\$US==min(ushur\$US)]
[1] 1853 1862 1863 1864 1868 1872 1884 1889 1890 1892 1895 1902 1905 1907 1914
[16] 1922 1927 1930 1931 1937 1951 1958 1962 1973 1978 1981 1982 1990 1994 2000
[31] 2001 2006

Note the output is indexed by the values inside the brackets on the left-hand side.

sum(ushur\$US==0)  # How many years have zero hurricanes?
table(ushur\$US)  # How many years have H hurricanes?

Is there a trend in the U.S. landfall counts?  One way to get an answer is to correlate the counts with the year.

cor(ushur\$Year,ushur\$US)
[1] -0.003594048

What if someone explains that since you are using count data, you should use the rank correlation.  By default, R provides the Pearson product-moment correlation coefficient.  To change the default, type:

cor(ushur\$Year,ushur\$US,method="s")  # method="s" refers to the Spearman rank correlation

3. Simple Inferences

"The modern world, despite a surfeit of obfuscation, complication, and downright deceit, is not impenetrable, is not unknowable, and---if the right questions are asked---is even more intriguing than we think.  All it takes is a new way of looking."---Levitt and Dubner (Freakonomics)

Correlation
The correlation is small and not likely significant.  To test for significance, type:

cor.test(ushur\$Year,ushur\$US)
Pearson's product-moment correlation

data:  ushur\$Year and ushur\$US
t = -0.0446, df = 154, p-value = 0.9645
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.1606441  0.1536335
sample estimates:
cor
-0.003594048

More information is returned when inference is sought.  Here we examine the significance of the computed correlation as a test on whether there is a trend over time in U.S. hurricane counts.  Thus the null hypothesis is no trend (zero correlation).

The t statistic (value) is calculated as the ratio of the correlation to the standard error of the correlation and this t value is used as the quantile of a t distribution with N-1 dof (pt(-0.0446,154)*2).  The corresponding p-value of 0.96 is evidence in support of the null hypothesis on a scale between 0 and 1. Thus we confidently state that the small negative correlation is not significant.

This is exactly the answer we get if we perform a linear regression of U.S. counts on year and test for a significant trend (see Section 5).  If you learned statistics from a textbook, you will recall distribution tables appeared in the Appendix.  With R these values are accessible through calls to the appropriate functions (e.g., probabilities from a t distribution using pt).

Let's get some corresponding climate data.  The data are available here (http://garnet.fsu.edu/~jelsner/www/data.html) under hurricane landfall counts.  The data consist of monthly values for three covariates that have been identified as important in modulating Atlantic hurricanes; the North Atlantic Oscillation (NAO), the Southern Oscillation (SOI), and the Atlantic Multidecadal Oscillation (AMO).

amoAO=(amo\$Aug+amo\$Sep+amo\$Oct)/3    # compute the hurricane season (Aug-Oct) average
cor(amoAO,ushur\$US)    # this gives an error, the AMO record begins in 1856
cor(amoAO,ushur\$US,use="complete.obs")   # compute correlations over all complete obs

Is the correlation between the AMO and U.S. hurricanes significant?

cor.test(amoAO,ushur\$US)

Suggestive, but inconclusive.

A p value is an estimate of the probability that a particular result, or a result more extreme than the result observed, could have occurred by chance if the null hypothesis were true.
In short, the p value is a measure of the credibility of the null hypothesis.

However, the p value is often interpreted as evidence against the null hypothesis, thus a small p value indicates evidence to reject the null. The interpretation of the p value as evidence against the null hypothesis is:

 p value range interpreted as evidence against the null hypothesis 0-0.01 convincing 0.01-0.05 moderate 0.05-0.15 suggestive, but inconclusive > 0.15 none

Similar to what we did with the amoAO, create two new objects soiAO and naoMJ for the August through October SOI and the May through June NAO respectively. Using these objects, which covariate appears to be most strongly correlated with U.S. hurricane counts?  Note that the relationship with the NAO is a lead relationship (May-June is before the hurricane season starts) so that it can be used directly for prediction.  Is there any significant correlation between the covariates?

Two sample tests
Is there evidence of more U.S. hurricanes recently?  One way to examine this question is to divide the record into two samples (e.g., first half versus second half) and compare the means from both samples.  The null hypothesis in this case is that the sample means are not significantly different.

Before we can carry out a test to compare two sample means, we need to test whether the sample variances are significantly different.  Here we use Fisher's F test.  The test is based on the ratio of variances, which under the null hypothesis is 1 and has an F distribution.

Since there are 156 years in the record (length(ushur\$US)), we take the first 78 years for the first sample (s1) and the next 78 years for the second sample (s2).

s1=ushur\$US[1:78]
s2=ushur\$US[79:156]

The ratio of variances is:

var(s1)/var(s2)
[1] 1.081287

The value is not equal to one, but is it significantly different?  To test, type:

var.test(s1,s2)
data:  s1 and s2
F = 1.0813, num df = 77, denom df = 77, p-value = 0.7326
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.6894304 1.6958656
sample estimates:
ratio of variances
1.081287

The F value is the ratio of the variance in sample 1 to the variance in sample 2.   Each sample contributes (n-1) degrees of freedom (here, n=78 years).  The p-value exceeds 0.15 by a wide margin, thus we have no evidence that the ratio of variances is different than one.

The 95% confidence interval refers to the ratio of the variances and we see that the interval includes the value of 1.

Having discounted unequal variance, there are a couple of tests for comparing two sample means:  (1) Student's t test when the samples are independent, the variances constant, and the values are normally distributed, and (2) Wilcoxon's rank sum test when the samples are independent, but the values are not normally distributed.

In the case of count data, the values are non normally distributed, but this violation is usually not critical, so let's try both tests. The null hypothesis is that the difference in means is zero.

t.test(s1,s2,var.equal=T)
Two Sample t-test

data:  s1 and s2
t = 0.1651, df = 154, p-value = 0.869
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-0.4217579  0.4986809
sample estimates:
mean of x mean of y
1.730769  1.692308

Here the t value of 0.1651 is computed as the difference between the two means divided by the standard error of the difference.  In R code

(mean(s1)-mean(s2))/sqrt(var(s1)/length(s1)+var(s2)/length(s2))

The degrees of freedom (df) are the sum of the sample sizes minus 2.  The
p value exceeds 0.15 by a wide margin, thus we have no evidence that the difference in means is other than zero.

Note that this is a 2-sided p value with the alternative that the difference in means in not equal to zero.  If you are testing whether the mean of one sample is greater than the mean of the other sample, then you would include the argument alternative="greater".

Note the default is var.equal=F, so that if you type t.test(s1,s2) you get an approximation (Welch) to the degrees of freedom.

The 95% confidence interval is around the difference in means.  Since the interval contains zero, there is no evidence against the null hypothesis.

Sometimes two sample data come from paired observations.  In this case you should always use the paired=T argument.

A non-parametric alternative to the Student's t test when the values are not normally distributed is the Wilcoxon rank-sum test.  The test statistic (W) is calculated by combining samples and arranging them in order (e.g., largest to smallest).  A rank is assigned to each value and the ranks are added for each of the two samples.  The significance is assessed based on the size difference between the cumulative rankings.

wilcox.test(s1,s2)
Wilcoxon rank sum test with continuity correction

data:  s1 and s2
W = 3084.5, p-value = 0.8783
alternative hypothesis: true location shift is not equal to 0

The function wilcox.test uses a normal approximation to work out a standard normal value from the rank sum value (W).  From this a p value is used to assess the hypothesis that the two means are the same.

Here the p value, as with the t test, is much greater than 0.15, thus we have no evidence that the difference in means is other than zero.

Typically the t test will give the lower p value, so the rank sum test is said to be conservative.  The rank sum test is more appropriate than the t test when the values are not normal.

4. Graphs and Plots

"It is well and good to opine or theorize about a subject, as humankind is wont to do, but when moral posturing is replaced by an honest assessment of the data, the result is often a new, surprising insight."---Levitt and Dubner (Freakonomics)

If you do nothing else with your data you will likely at least make a graph.  R has a wide range of plotting capabilities.  It takes time to become a master, but a little effort goes a long way.

Although the commands are simple and somewhat intuitive, to get a publication quality figure usually requires tweaking the default settings.  This is true of all plotting software.

Since we need access to the various columns in our data frame, it simplifies typing if we use the attach and detach functions.  The attach function makes the columns available to use in commands without the need to preface the data frame name.  For example, if you type:

US

The error tells you that there is no object with that name in your working directory.  To list the objects, type ls().  Now type:

attach(ushur)
US

This makes things easier.  It’s good practice to use the detach function when you finish with the data frame to avoid masking objects with the same name from other data frames or objects.

Scatter plots
To start, let's look at the U.S. hurricane counts over time as a scatter plot.

plot(Year, US)

The default settings on plot give you a first look, but the plot needs to be modified before serious study.

plot(Year, US, type="l")    # Note the letter in quotes is a small "l" for line, not the number 1

This is often how you see count data plotted in the hurricane literature.  Ignoring for now the axis labels, what's wrong with it?  To help you with the answer, grab the lower right corner of the plot window and stretch it to the right.  See it?

To avoid this problem, use the type="h" option.

plot(Year,US,type="h")
plot(Year,US,type="h",lwd=3)   # This increases the line width from a default of 1 pixel unit to 3

To make it prettier, type:

plot(Year,US,ylim=c(0,8),ylab="U.S. Hurricanes",type="h",lwd=3,las=1)

Is there any correlation between the covariates?  One way to examine this is to graph a scatterplot.

plot(amoAO,naoMJ)
plot(amoAO,soiAO)

What do you conclude?

Distribution plots
Another useful data display for discrete data is the histogram.  This provides information on the type of distribution.  Knowing what kind of distribution your data conform to is important for modeling.

hist(US)

Not bad, but the location of the abscissa labels makes it difficult to understand what bar belongs to what hurricane count.  Moreover, since the values are discrete, the years with 0 landfalls are added to the years with 1 landfall.  You can see this by looking at the output from table(US) as we did previously.

There are many options for improving the histogram.  You can see these options by typing a question mark before the function name and leaving off the parentheses.

?hist  # brings up a help menu on the topic of histograms

Here is one that works by specifying the break points with the breaks argument.  Note also the use of the seq function.

hist(US,breaks=seq(-0.5,7.5,1))

That's more like it.  Now we can see that the mode is 1 (not 0) and that 2s are more frequent than 0s.  The distribution is said to be "skew to the right" (or positively skew) because the long tail is on the right-hand side of the histogram.

To overlay a theoretical (model) distribution, type:

xs=0:7
ys=dpois(xs,1.71)
lines(xs,ys*156,col="red")

The data fits the theoretical distribution quite well.  We will look at these theoretical distributions a bit later.

barplot(table(US),ylab="Number of Years",xlab="Number of Hurricanes")
barplot(table(US)/length(US),ylab="Proportion of Years",xlab="Number of Hurricanes")

Note that the command table(US)/length(US) produces proportions with the result handed off to barplot to make a graph.  The graph has the same shape as the previous one, but the height axis is between 0 and 1 as it measures the proportion of years.

To examine the distribution of the covariates together with the distribution of US hurricanes, type the following sequence of commands:

par(mfrow=c(2,2))
hist(US,breaks=seq(-0.5,7.5,1))
hist(amoAO)
hist(soiAO)
hist(naoMJ)

The par function sets the graphical parameters.  Here they are set by the argument mfrow, which should be read as "multiframe, rowwise, 2 x 2 layout".  Thus each plot is added to the graph in lexicographical ordering (top to bottom, left to right).

For continuous variables like the covariates considered here, it is sometimes better to plot an estimate of the probability density function.  The probability density function (pdf) can be seen as a smoothed version of the histogram.   The probability of observing a value between any two locations [a, b] is given by the integral of the density function evaluated between a and b.

The plot of a probability density function first requires a call to the density function.  By default an appropriate smoothing value (bandwidth) will be chosen as the standard deviation of the smoothing kernel (window).  To examine the pdf for the default and other bandwidths, type:

par(mfrow=c(2,2))
plot(density(naoMJ))
plot(density(naoMJ,bw=0.15))
plot(density(naoMJ,bw=0.45))
plot(density(naoMJ,bw=0.7))

To add the locations of the values, use the rug function.

par(mfrow=c(1,1))
plot(density(naoMJ),main="",xlab="May-Jun NAO (sd)")
rug(naoMJ)

One purpose of plotting the pdf is to assess whether the data can be assumed normally distributed.  Clearly in the case of the NAO, the data appears to be normal.

For a better assessment, you can plot the kth smallest NAO value against the expected value of the kth smallest value out of n in a standard normal distribution.  In this way you would expect the points on the plot to form a straight line if the data come from a normal distribution with any mean and standard deviation.  The function for doing this is called qqnorm.

qqnorm(naoMJ)

As the default title of the plot indicates, plots of this kind are also called "Q-Q plots" (quantile versus quantile). The sample of values has heavy tails if the outer parts of the curve are steeper than the middle part.

A "box plot", or more descriptively a "box-and-whiskers plot", is a graphical summary of a set of values.  It is used to examine the distribution of your data.  To get a box plot of the NAO values, type:

boxplot(naoMJ)

The box in the middle indicates that the middle 50% of the data (between the first and third quartiles) lies between about -1 and +0.5 s.d.  The median value is the thick horizontal line inside the box.  The lines ("whiskers") show the largest/smallest value that falls within a distance of 1.5 times the box size from the nearest quartile.  If any value is farther away, it is considered "extreme" and is shown separately.

Boxplots can also be used to look at conditional distributions.   For instance, what is the distribution of the NAO values conditional on the upcoming season having few or many U.S. hurricanes?  This is question that is important as we move toward modeling our data.  In section 2 we noted a negative correlation between the number of U.S. hurricanes and the NAO thus we would expect to see lower values of NAO when the rate of U.S. hurricanes is above normal.

boxplot(naoMJ~ushur\$US>3,ylab="NAO May-Jun (sd)")

The parallel box plots help us visualize the relationship between hurricanes and covariates.

label=factor(c(rep("On or Before 1928",78),rep("After 1928",78)))
boxplot(ushur\$US~label,notch=T,xlab="",ylab="US Hurricanes")

Box plots can also be used to examine seasonality.  The AMO covariate contains monthly values from 1856-2006.  Suppose we want to look at a box plot of the AMO values by month.  Note: Don't worry about memorizing this code.

amo2=na.omit(amo)   # omit rows with no data
amo2=stack(amo2[,2:13])   # stack the columns
ind=factor(as.character(amo2\$ind),levels=month.abb)   # create an ordered factor (months)
plot(ind,amo2\$values)

Note that the examples as described will plot the ﬁgures on the screen.  In order to create hard copy output, we need to specify an output device.
R supports several devices, but by far the most commonly used are the postscript and pdf devices. We ﬁrst open the device explicitly. For
example, postscript(file="filename") or pdf(file="filename").

All plotting commands that follow will be written to that device, until you close it with a dev.off() command. For example,

postscript(file="AMO.pdf") # file name for pdf file that is dropped outside of the R session in your working directory
plot(ind,amo2\$values)  # plotting command(s)
dev.off() # close postscript device

The width and height of the postscript plot is specified with the height= and width= arguments in the postscript function.  Units on the values are inches.

5. Probability and Distributions

Sampling
The concepts of randomness and probability are central to statistics and data modeling.  Here we look at the basic ideas of probability with the help of R functions.

Much of the earliest work in probability theory was about games and gambling.  The basic notion is that of random sampling from well-shuffled card decks or picking colored balls from an urn.

In R you can simulate these situations with the sample function.  If you want to pick five numbers at random from the set of number 1 through 40, then you type:

sample(1:40,5)

The first argument is a vector of values to be sampled and the second is the sample size.  Your sample likely will be different than mine.  To sample 5 years of U.S. hurricanes, you would type:

sample(ushur\$US, 5)

Note that the default behavior of sample is sampling without replacement.  In the case of the set of numbers 1 through 40, each number will be different.  In the case of the set of hurricane counts, each count will correspond to a different year.  If you want sampling with replacement, you need to add the argument replace=TRUE.

To simulate 10 coin tosses we could write:

sample(c("H","T"), 10, replace=T)

Distributions
The standard distributions that turn up in connection with data modeling and statistical tests are built into R.  Here we look only at a few of these distributions, but all distributions follow the same pattern.

Four fundamental items can be calculated for a statistical distribution:

• Density or point probability
• Cumulative probability, distribution function
• Quantiles
• Random numbers

Densities
For all distributions implemented in R, there is a function for each of the four items.  For instance, for the normal distribution, these are named dnorm, pnorm, qnorm, and rnorm, respectively (density, probability, quantile, and random).

The density for a continuous distribution is a measure of the probability of "getting a value close to x".  The probability of getting a value in a particular interval is the area under the corresponding part of the curve.   For discrete distributions, the term "density" is used for the point probability--the probability of getting exactly the value x.  Technically, this is correct: It is a density with respect to a counting measure.

The density item of the distribution functions can be used to "see" the distribution.  For  instance, to see the well-known bell curve of the normal distribution, type:

x=seq(-4,4,0.1)
plot(x,dnorm(x),type="l")

An alternative is to type:

curve(dnorm(x),from=-4, to=4)

This is often a convenient way of making graphs, but it requires that the y-values can be expressed as a simple functional expression in x.

For discrete distributions, where variables can take on only distinct values, it is preferable to draw a pin diagram.  For instance, what is the probability of getting exactly x successes in 50 trials with the probability of success for each trial at 33%?

x=0:50
plot(x,dbinom(x,size=50,prob=0.33),type="h")

Notice that there are three arguments to the dbinom function.  In addition to x, you need to specify the number of trials (size) and the probability (prob).  The distribution drawn corresponds to, for example the number of 5s or 6s in 50 throws of a fair die.  Actually, dnorm also takes more than one argument, namely the mean and standard deviation, which by default are set to 0 and 1, respectively since most often it is the standard normal distribution that is requested.  It is type="h" (as in histogram-like) that causes the pins to be drawn.

Animation
To see how the graph varies with respect to the parameters, we can animate the evolution by drawing a series of graphs (Baclawski, 2008).  We need to create and display a series of graphs in quick succession.  We do this by defining a function then applying it to a sequence of parameters.

There are several R utilities available for repeatedly applying a function.  The  most common is sapply.  The sapply function applies every element of the first argument to the function defined in the second argument and returns a vector with the values computed by the function.

The function we want to apply in this case is the plot function.  However, because the graphs are drawn quickly it is necessary to add a pause between them.  This is done with the Sys.sleep function.  It causes the program to stop running for a specified number of seconds.  In animation this is called the dwell rate.

Our first animation shows how the binomial distribution varies with the probability of success.

binom=function(p)
{
plot(0:10,dbinom(0:10,10,p),type="h",xlab="No. Successes",ylab="Probability of Success")
Sys.sleep(0.2)
}

To run your function binom on a series of probabilities from 0 to 1 in steps of .1, type:

sapply(seq(0,1,.1),binom)

Notes: (1) To suppress the output to the screen, we can assign it to a dummy variable. (2) To make the y-axis scale the same for each plot, we can add ylim=c(0,0.5) to the plot function.

binom=function(p)
{
plot(0:10,dbinom(0:10,10,p),ylim=c(0,0.5),type="h",xlab="No. Successes",
ylab="Probability of Success")
Sys.sleep(0.1)
}
ignore=
sapply(seq(0,1,.01),binom)

Next we look at what happens when the number of trials varies but the probability of success is fixed at p=1/2.

binom=function(size)
{
plot(0:size,dbinom(0:size,size,1/2),ylim=c(0,0.2),type="h",xlab="No. Successes",
ylab="Probability of Success")
Sys.sleep(0.1)
}
ignore=
sapply(1:100,binom)

The result is quite remarkable.  The graph quickly converges to a shape that changes little except that it becomes narrower.  This is the graphical manifestation of the Central Limit Theorem (the most important theorem in probability).  To see the convergence more dramatically, we need to control the x-axis.

binom=function(size)
{
plot(0:size,dbinom(0:size,size,1/2),ylim=c(0,0.15),
xlim=c((size/2)-2*sqrt(size),(size/2)+2*sqrt(size)),type="h",
xlab="No. Successes",ylab="Probability of Success")
Sys.sleep(0.1)
}
ignore=
sapply(1:200,binom)

Repeat using a probability of success = 1/4.

Cumulative distribution functions
The cumulative probability function describes the probability of "hitting" x or less in a given distribution.  The corresponding R functions begin with a 'p' (for probability) by convention.

Just as you can plot densities, you can also plot cumulative distribution functions, but that is usually not very informative.  More often, actual numbers are desired.  For example say you are interested in knowing the probability of more than 3 U.S. hurricanes next year, given that the mean number is 1.71.  In this case you would type:

1-ppois(3,1.71)
[1] 0.09469102

Thus knowing the rate of U.S. hurricanes is 1.71 per year, the probability of observing more than 3 U.S. hurricanes next year is 9.5% assuming all else the same.

Question: What is the probability that next year's NAO value averaged over May and June will be less than or equal to -1 sd?

pnorm(-1,mean=mean(naoMJ),sd=sd(naoMJ))
[1] 0.2508814

Similar to dnorm, we can also plot the cumulative distribution:

curve(pnorm(x),from=-4, to=4)

Quantiles
The quantile function is the inverse of the cumulative distribution function.   The p-quantile is the value with the property that there is a probability p of getting a value less than or equal to it.  The median is by definition the 50% quantile.  Thus the 25% quantile of the NAO data should be about -1.  To verify, type:

qnorm(.25,mean=mean(naoMJ),sd=sd(naoMJ))
[1] -1.002741

To find the 95% confidence interval about the population mean NAO value, we can type:

xbar=mean(naoMJ)  #sample mean
sigma= sd(naoMJ)   # sample standard deviation
n=length(naoMJ)     # sample size
sem=sigma/sqrt(n)  # standard error of the mean
xbar+sem*qnorm(0.025); xbar+sem*qnorm(0.975)   # lower and upper end points of the 95% confidence interval about the mean
[1] -0.4907953
[1] -0.1803586

Note that qnorm(0.025) = -qnorm(0.975).

We can also plot the  quantile function with suitable limits:

curve(qnorm(x), from=0, to=1)

Random numbers
To many people it sounds like a contradiction to generate random numbers on a computer since computer results are predictable and reproducible.  What is in fact possible is to generate sequences of "pseudo-random" numbers, which for practical purposes behave as if they were drawn randomly.

The use of the functions in R that generate random numbers is straightforward.  The first argument specifies the number of random values to generate and the subsequent arguments refer to characteristics of the distributions similar to those for the other distributional functions.

To generate 10 random numbers from a standard normal distribution, type:

rnorm(10)

Or

rnorm(10,mean=-0.336,sd=0.99)  # 10 random normal values with the same mean and variance as the NAO May-June data
rbinom(10, size=20,prob=0.5)  # 10 random binomial values where the number represents the number of successes in 20 trials
plot(ushur\$Year,rpois(length(ushur\$US),mean(ushur\$US)),type="h",lwd=3)

This last one simulates and plots the U.S. hurricane record.  Note that these simulations often contain periods of high and low activity by chance.

Finally, let's generate a histogram of random Poisson counts and compare it with our histogram of U.S. hurricanes from section 3.

hist(rpois(length(ushur\$US),mean(ushur\$US)),breaks=seq(-0.5,7.5,1),main="", xlab="Random Counts")

Bootstrap samples
You've probably heard the old phrase about pulling yourself up by your own bootstraps.  That is where the term 'bootstrap' in statistics comes from.  You might only have a single sample of hurricane observations over time some time period, but you can sample (bootstrap) from the original sample in many ways so long as you allow some values to appear more than once and other not to appear at all (sampling with replacement).  You calculate the mean of the values in each bootstrap sample and obtain a confidence interval by looking at the extreme highs and lows using the quantile function.

a=numeric(1000)
for(i in 1:1000) a[i]=mean(sample(ushur\$US,10,replace=T))
hist(a,main="", xlab="Historical Annual U.S. Hurricane Rate")
quantile(a,c(0.025,0.975))
2.5% 97.5%
0.9   2.7

Note the large range of bootstrapped hurricane rates.  A 95% confidence interval spans a factor of 3.  This is characteristic of small count data and should be kept in mind when describing periods as "active" and "inactive".

6. Data Models

Introduction
Theoretical distributions like the normal and the Poisson, are examples of data models.  If your data conform to a theoretical distribution then the analysis is simplified, because the properties of the distributions are known.

Fitting models to data is a central function of R.  The object is to determine a minimal adequate model from the large set of potential models that might be used to describe the given set of data.

Parsimony says that, other things being equal, we prefer:
• A model with n-1 parameters to a model with n parameters
• A model with k-1 explanatory variable to a model with k explanatory variables
• A linear model to a model that is curved
• A model without interactions to a model with interactions

We also prefer models containing explanatory variables that are easy to measure over variables that are difficult to measure or are derived (e.g., EOFs).  And we prefer models that are based on a sound mechanistic understanding of the process over purely empirical functions.

All of the above are subject to the caveats that the simplifications make good scientific sense and do not lead to significant reductions in explanatory power.  There is no perfect model.

Models in R follow the following formula:   response variable ~ explanatory variable(s), where the ~ reads "is modeled as a function of".

There is a temptation to become personally attached to a particular model.  You do well to remember the following truths about models:
• All models are wrong.
• Some models are better than others.
• Some models are useful.
• The correct model can never be known with certainty.
• The simpler the model, the better it is all else the same.

After fitting a model to your data it is essential that you investigate how well the model describes the data.  In particular, are there any systematic trends in the goodness of fit?  This is done by examining the model residuals which are computed as the difference between the observed response and the predicted response.

Linear regression
In section 2 we saw how to get the correlation between two variables. Correlation is nothing more than a statistical term that indicates whether two variables move together.  It tends to be cold outside when it snows; these two factors are positively correlated.  Sunshine and rain, meanwhile, are negatively correlated.

Easy enough so long as there are only a couple of variables.  With many variables things get harder.  Regression analysis is the tool that enables a geographer to sort out piles of data with many variables.  It does so by artificially holding constant every variable except the two he wishes to focus on, then showing how these two co-vary.

It has been said that economics is a science with excellent tools for gaining answers but a serious shortage of interesting questions.  Geography is in the other boat.  It has no shortage of interesting questions, but it tends to lack tools for gaining answers.  Of all the tools in an economists toolbox, the most useful is regression.

Mathematically, regression attributes to a response variable y a normal distribution whose expected value depends on a covariate x, typically in the following way:  E[y] =  a + b x.  The parameters a and b are estimated by a least-squares minimization procedure.  It is called "linear" regression because the variables multiplied by their parameters are added together.

For linear regression, the function lm is used (linear model).  To obtain a linear regression model of the August through October AMO on the May through June NAO, type:

lm(amoAO~naoMJ)

Call:
lm(formula = amoAO ~ naoMJ)

Coefficients:
(Intercept)        naoMJ
22.79375     -0.02687

The argument to the lm function is a model formula.  The variable to the left of the tilde is the dependent variable and the variable to the right is the independent variable.

Regression is a model of the response (dependent) variable (usually denoted by "y") ON the explanatory (independent) variable, (usually denoted by "x").  You regress y onto x.  Unlike correlation, which is not a model for the data, it matters which variable is which and the choice is made by the investigator, not the software.

Continuous explanatory variables are called "covariates".

The statement "I regressed variable A and variable B" is vague.

The coefficients are the values of the intercept and slope of the straight line through the scatter of points when the NAO is on the horizontal axis.  The equation for the line is read as:

The mean of Aug-Oct AMO in units of degrees C equals 22.7 minus 0.02687 times the May-Jun NAO in units of s.d.  Thus the units on the intercept are the same as the units of the response variable, in this case degrees C, and the units on the slope are degrees C/s.d.

To see the line, type:

plot(naoMJ,amoAO)
abline(lm(amoAO~naoMJ))

The regression model is different if the NAO is regressed on the AMO.  Try it.  Also consider a regression of hurricane counts on year.

In its raw form, the output of lm is very brief.  All you see is the estimated intercept and estimated slope values, but there are no tests of significance.  The result of lm is a model object.  This is a distinctive concept in R.  Where other statistical programs focus on generating printed output that can be controlled by setting options, you get instead the result of a model fit encapsulated in an object from which the desired quantities can be obtained using other functions.

More output is obtained if you type:

summary(lm(amoAO~naoMJ))
Call:
lm(formula = amoAO ~ naoMJ)

Residuals:
Min       1Q   Median       3Q      Max
-0.52648 -0.14007 -0.02344  0.12599  0.65141

Coefficients:
Estimate Std. Error  t value Pr(>|t|)
(Intercept) 22.79375    0.02008 1135.221   <2e-16 ***
naoMJ       -0.02687    0.01920   -1.399    0.164
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2345 on 149 degrees of freedom
(5 observations deleted due to missingness)
Multiple R-Squared: 0.01297,    Adjusted R-squared: 0.006343
F-statistic: 1.957 on 1 and 149 DF,  p-value: 0.1639

Model description
Let's dissect the output.

Call:
lm(formula = amoAO ~ naoMJ)

The output starts with a repeat of the function call.  This is not very interesting, but it is useful if the result is saved in an object that is examined later.

Residuals:
Min       1Q   Median       3Q      Max
-0.52648 -0.14007 -0.02344  0.12599  0.65141

This gives us a quick look at the distribution of the residuals.  The residuals are the key for understanding how well the model fits the data and whether there are problems.  The average of the residuals is zero by definition, so the median should not be far from zero and the minimum and maximum should roughly be equal in absolute magnitude.  Similar with the 1st (1Q) and 3rd (3Q) quartile values.

Coefficients:
Estimate Std. Error  t valu  Pr(>|t|)
(Intercept) 22.79375    0.02008 1135.221   <2e-16 ***
naoMJ       -0.02687    0.01920   -1.399    0.164
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here we again see the regression coefficients including the intercept and slope, but this time with accompanying standard errors, t values and p values.  Note that the slope value and its statistics are given to the right of the covariate, but they refer to the coefficient that is multiplied by the covariate values, not the covariate values themselves.  To the right of the p values is an indicator of the level of significance.  Here we note that the NAO is not a very good predictor of the AMO.

Residual standard error: 0.2345 on 149 degrees of freedom

This is the residual variation, an expression of the variation of the observations around the regression line, estimating the model parameter sigma.  The degrees of freedom is the number of observations minus 2 (since there are two model parameters).

(5 observations deleted due to missingness)

This indicates that there are 5 missing values in either the response or covariate or both.  Rows with missing values are not included in the model (deleted).  Care must be exercised when comparing models built with differing data set sizes.

Multiple R-Squared: 0.01297,    Adjusted R-squared: 0.006343

The first item is the R squared value.  When multiplied by 100%, it is interpreted as the variance explained by the covariate.

SSY=deviance(lm(amoAO~naoMJ))
SSE=deviance(lm(amoAO~1))
R squared = (SSY-SSE)/SSY

The adjusted R squared is an index that accounts for the loss in the degrees of freedom when adding a covariate to the model.  It is important in the context of multiple regression.

F-statistic: 1.957 on 1 and 149 DF,  p-value: 0.1639

This is an F test for the hypothesis that the slope is zero.  The F-statistic has a value of 1.957 on 1 and 149 degrees of freedom (DF). For the case of a one covariate model, it gives the same p-value as the one on the slope coefficient.  It is more interesting in the case of more than one covariate.

Another example: airquality data set.

?airquality

Regress ozone concentration (Ozone) on solar radiation (Solar.R), wind speed (Wind), and temperature (Temp).

model=lm(Ozone~Solar.R+Wind+Temp, data=airquality)
summary(model)

The R-squared value can be manipulated simply by adding more explanatory variables. The adjusted R-squared value is an attempt to correct this short-coming by adjusting both the numerator and the denominator by their respective degrees of freedom.

adjusted R squared = 1- (1 - R2 )[(n - 1)/(n - p - 1)]

R2 = R squared
n = number of observations
p = number of explanatory variables

Unlike the R-squared, the adjusted R-squared will decline in value if the contribution to the explained deviation by the additional variable is less than the impact due to the loss in degrees of freedom. Thus it is a better way to compare models than using R-squared.

Note: while the R-squared is a percent, the adjusted R-squared is not and should be referred to as an index value.

To obtain confidence intervals on the model coefficients type:

confint(model)

Model predictions
To get predictions from the model for each set of explanatory variables type:

predict(model)

To get point-wise confidence intervals on the predictions

predict(model,level=0.95,interval="confidence")

To get prediction intervals use interval="prediction".

By fitting a linear regression model, you are making several implicit assumptions including linearity, normally distributed residuals, constant variance, and  independence.

• Linearity assumption: The means of the subpopulations fall on a straight-line function of the explanatory variables.
• Normality assumption: There is a normally distributed subpopulation of responses for each value of the explanatory variable.
• Constant variance assumption: The subpopulation standard deviation are all equal.
• Independence assumption: The selection of an observation from any of the subpopulation is independent of the selection of any other observation.

The first three assumptions are best checked using plots.

plot(model)

The fourth assumption can be checked by plotting the autocorrelation function of the model residuals.

acf(residuals(model))

It is also a good idea to check for collinearity among the explanatory variables.  This can be done with the cor function as we saw earlier.  If the correlation between two explanatory variables is between -0.6 and +0.6 then collinearity is usually not a big problem.

Logistic regression
Modeling the relationship between explanatory and response variables is a fundamental activity in applied statistics. Often the response is not a numerical value.  Instead, the response is one of two possible outcomes (a binary response), e.g., landfall or no landfall.

Here we look at binary response data and its analysis through logistic regression.  Concepts from simple and multiple regression carry over to logistic regression.  The idea of maximum likelihood (ML) estimation is central to the modeling of binary data.

All applications of logistic regression have two things in common, (1) Binary response, (2) Involve the idea of prediction of a chance, probability, proportion or percentage.  The response variable is bounded from below by 0 and from above by 1.

The mean of a binary response is a probability.  For example, generate a set of random binary responses.

mean(rbinom(30,1,0.5))

Repeat several times, then change 30 to 300 and repeat several times.

Thus a logistic regression model specifies that a probability---such as the probability of a hurricane---is related to the explanatory variables, such as SST and ENSO.

Interpretation of logistic regression coefficients are made in terms of statements about odds and odds ratios.  Although the model, estimation, and inference procedure differ from those of ordinary regression, the practical use of logistic regression analysis parallels the usual regression analysis closely.

Problems with linear regression when the response variable is binary
The problem with linear regression with a binary response variable arises from the fact that probabilities have maximum and minimum values of 1 and 0.  By definition, probabilities and proportions cannot exceed 1 or fall below 0.  Yet, the linear regression line can extend beyond these limits. Depending on the slope of the regression line and the observed values of the explanatory variable, a model can give predicted values of the response above 1 and below 0.  Values beyond the range of 0 and 1 make no sense, and have little predictive use.

Example: Consider the data collected from the recovered solid boosters of the 23 previous shuttle flights (before the Challenger disaster).  The damage variable is coded as 0 for no damage, and 1 for damage.  The temperature is the air temperature in F at the launch site.

temp=c(66,70,69,68,67,72,73,70,57,63,70,78,67,53,67,75,70,81,76,79,75,76,58)
damage=c(0,1,0,0,0,0,0,0,1,1,1,0,0,1,0,0,0,0,0,0,1,0,1)

A scatter plot of damage and temperature with the best fit line (damage is the response variable) is obtained by typing:

plot(damage~temp)
abline(lm(damage~temp))

The plot shows there is less likely to be damage when temperatures are warmer, but a line does not do justice to the relationship.  Regress
damage on temp.  Is the model significant?

model.lm=lm(damage~temp)
summary(model.lm)

Yes, temperature is significant in explaining O-ring damage.
Write an equation for the model.
Predicted damage = 2.905 –0.037 x temperature

The predicted damage can be interpreted as the predicted probability of damage for the given temperature. What is the predicted probability of damage if the temperature is 77F, 65F, 51F?

signif(predict(model.lm,list(temp=c(77,65,51))),2)
1     2     3
0.026 0.480 1.000

Although this appears to be a reasonable prediction equation, it can lead to nonsensical predictions.  For example, what is the predicted probability of damage when the temperature is 85F or 45F?

signif(predict(model.lm,list(temp=c(85,45))),2)

These are nonsense probabilities. The bounded nature of probabilities is not incorporated into the linear regression model.

Another clue to the inadequacy of the linear regression model for these data is shown in the pattern of the residuals.

plot(temp,residuals(model.lm))

Rather than exhibiting a random scatter of points, the plot of residuals versus temperature shows two slanted lines.

Because of these problems we need to consider another type of model for binary data.

A generalized linear model
A regression model has two parts.  1) the response function, 2) regression structure.  With linear regression the response function is the mean of y given x and the regression structure is a summation of the explanatory variables multiplied by their coefficients.  In equation form the two parts are separated by an equal sign

mean(y|x) = a + b * x

In R the two parts of a regression model are separated by the tweedle.

A generalization of the linear regression model is made by considering the response function to be the logit of the mean.  The logit is given by

logit(mean) = log(mean/(1-mean))

Note that the logit is the logarithm of the odds expressed as a fraction.  Odds are for:against.  If there is a 2:1 odds of something happening, then the probability it will happen is 2/(1+2) or 2/3.  (2/3 / (1-2/3)) = 2, which is the fraction for/against.

The inverse of the logit function is called the logistic function, so that if logit(mean)=a, then

mean=exp(a)/(1+exp(a))

Note that the logit is a nonlinear function, but with logistic regression there is no change to the regression structure, so we call the model a generalized linear model.  It is linear in the regression structure, but nonlinear in the response function.

Since the mean is a probability for a set of binary outcomes, it is customary to use the symbol pi, so the logit function is written as

logit(pi) = log(pi/(1-pi))

Example
: Shuttle O-ring damage
To model the probability of O-ring damage as a logistic regression using air temperature at the launch site as the explanatory variable, type

model.glm=glm(damage~temp,family=binomial)
summary(model.glm)

pchisq(28.267-20.315,1,lower.tail=F)  #Is the model significant?

To predict the probability of O-ring damage when the launch temperature is 55F, type:

predict(model.glm,list(temp=55),type="response")

To plot the model on the scatter of points, type:

plot(temp,damage)
lt=seq(50,90,.1)
lines(lt,predict(model,list(temp=lt),type="response"))

Practice
Create a binary variable of US hurricane counts based on whether or not a year had 2 or more hurricanes (e.g., US>1).  Regress this binary variable on NAOmj (North Atlantic Oscillation during May and June in units of s.d.), SOIao (Southern Oscillation Index during August through October in units of s.d.), SSTao (Atlantic sea-surface temperature during August through October in units of degrees C), and SSNao (sunspot number averaged over August through October).  Which variables are significant?  Rerun the model with the insignificant variable(s) removed.  Is the reduced model significant?  Is the reduced model adequate?  Use the model to predict the probability of a multiple hit year given an NAO value of -2 s.d., a SOI value of 2 s.d., and a SST value of 23 C.

Poisson regression
Poisson distribution
The Poisson distribution is one of the most useful and important of the discrete probability models for describing count data.  We know how many times something happened.  The number of lightning strikes over Tallahassee or the number of hurricanes affecting the state of Florida, are examples.

The Poisson distribution is a one-parameter distribution with the interesting property that its variance is equal to its mean.  A large number of processes show variance increasing with the mean, often faster than linearly.

The density function of the Poisson distribution shows the probability of obtaining a count x when the mean count per unit is lambda.  It is given by the exponential of minus lambda times lambda raised to the x power all divided by x factorial.

The probability of zero events is obtained by setting x to 0, which yields exp(-lambda). Thus the probability of zero hurricanes affecting the U.S. next year, given a mean rate of 1.7 hurricanes/year is 18%.

exp(-1.7)
[1] 0.1826835

Of course, this implies that the probability of at least 1 hurricane is 100-18 or 82%.

The dpois function evaluates the above equation for any count of interest.  To determine the probability of observing exactly 1 hurricane when the rate is 1.7 hurricanes per year, type:

dpois(1,1.7)

Or the probability of 5 hurricanes when the rate is 1.7 expressed in percent.

dpois(5,1.7)*100

To answer the question, what is the probability of 1 or fewer hurricanes we use the cumulative probability function ppois. Thus to answer the question, what is the probability of more than 2 hurricanes, we type:

1-ppois(3,1.7)
ppois(3,1.7,lower.tail=F)

If we want to simulate 100 seasons of hurricane strikes on the U.S. coast with a rate that is commensurate to the long-term rate of 1.7 hurricanes per year, we type:
rpois(100,1.7)

To see the distribution, type:

table(rpois(100,1.7))

Repeat for another 100 years.

To plot the distribution, type:

barplot(table(rpois(100,1.7)))

Poisson regression model
What if the rate of hurricanes depends on time or climate variables?  Note the way the question is worded.  We are interested in the rate of hurricanes because given the rate, the counts follow a Poisson distribution.  Thus we are not modeling counts directly.

There's a mistaken belief that Poisson regression is least squares regression on the logarithm of counts.  It's not.  It is a regression using the logarithm of the rate.  It's nonlinear in the regression function, but linear in regression structure and the model coefficients are determined by the method of maximum likelihoods rather than the method of least squares.

Our interest here is to study the relationship between the climate variables and coastal hurricane numbers.  We do this with a regression model that specifies that the logarithm of the annual rate is linearly related to the covariates.

The Poisson regression model attributes to a response variable y a Poisson distribution whose expected value depends on a covariate x, typically in the following way:  log(E[y]) = a + b x.  The parameters a and b are estimated via the maximum likelihood procedure.

summary(glm(ushur\$US~naoMJ,family=poisson))
Call:
glm(formula = ushur\$US ~ naoMJ, family = poisson)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-2.1480  -0.8054  -0.1172   0.5346   2.8388

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)   0.44195    0.07019   6.297 3.04e-10 ***
naoMJ        -0.21708    0.06336  -3.426 0.000612 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 207.16  on 155  degrees of freedom
Residual deviance: 195.26  on 154  degrees of freedom
AIC: 515.54

Number of Fisher Scoring iterations: 5

We note that the coefficient on the NAO covariate is highly significant (p value < 0.001).  The negative sign on the coefficient indicates that the there are fewer U.S. hurricanes when the NAO is high.

The value of -0.217 indicates that for a one s.d. change in the NAO, the difference in the logarithm of expected counts is -0.217.  If there are other covariates we must add "given
the other covariates in the model are held constant".  Since log(A)-log(B) = log(A/B),  we can interpret the parameter estimate as the log of the ratio of expected counts.

The residual deviance replaces the residual standard error and is used to determine model fit.

With Poisson regression, like logistic regression, the drop in deviance (deviance of the null model minus the deviance of the full model) is used to test whether the model is significant against the null hypothesis that none of the covariates are useful.  Since the model contains significant covariates the model itself is significant.  To see this, type:

pchisq(207.16-195.26,1,lower.tail=F)

Model adequacy is examined with the residual deviance.

The residual deviance is
195.26 on 154 degrees of freedom. A p-value from a chi-squared distribution with this quantile value and this dof is evidence in favor of the null hypothesis that the model is adequate.  A small p-value is evidence against model adequacy.

Type:

pchisq(195.26,154,lower.tail=F)

Here we find moderate evidence against model adequacy indicating that other covariates are missing from the model.  NAO is not the entire story.

While a Poisson distribution for given x has variance of y equal to mean of y, this assumption may not be supported in the data.  If variance tends to exceed the mean, the data is over-dispersed relative to the Poisson distribution; under-dispersion occurs when variance is smaller than the mean.  If a Poisson model seems appropriate but doesn’t fit well, over-dispersion or under-dispersion could be the reason.

The AIC is the Akaike information criteria which is equal to 2 times the number of model parameters minus 2 times the logarithm of the likelihood function.   The AIC is an operational way of trading off the complexity of an estimated model against how well the model fits the data.  The smaller the AIC, the better the model.  Adding a covariate that captures the probability of hurricane genesis will likely improve the model.

Practice
Fit a Poisson regression model to the U.S. hurricane counts using all three covariates.  Interpret the results.

Quantile regression
Introduction
Quantile regression extends ordinary least squares regression model to conditional quantiles (e.g., 90th percentile) of the response variable.

Recall that ordinary regression is a model for the conditional mean (the mean of the response is conditional on the value of the explanatory variables). It is a semi-parametric technique because it relies on the non-parametric quantiles, but uses parameters to assess the relationship between the quantile and the covariates.

Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. The quantiles mark a set of ordered data into equal-sized data subsets.

Download the set of per storm maximum wind speeds (Save Link or Target As) into your working directory.  The data are available here under per storm maximum wind speeds.

Read the data into your current R session and check the values from the first six rows by typing:

Yr Region     Wmax   nao   soi        sst      sstmda sun split
1 1899  Basin 96.64138 -0.64 -0.21 0.05193367 -0.03133333 8.4     0
2 1899   East 90.19791 -0.64 -0.21 0.05193367 -0.03133333 8.4     0
3 1899  Basin 90.35300 -0.64 -0.21 0.05193367 -0.03133333 8.4     0
4 1899   East 37.51933 -0.64 -0.21 0.05193367 -0.03133333 8.4     0
5 1899  Basin 51.06743 -0.64 -0.21 0.05193367 -0.03133333 8.4     0
6 1899  Basin 40.00000 -0.64 -0.21 0.05193367 -0.03133333 8.4     0

Here Wmax is the maximum estimated wind speed for each cyclone in the North Atlantic by region.  Basin refers to the region away from the U.S. coast.

Quantiles and distributions
Attach the data frame, subset by Basin, and determine the quartiles (0.25 and 0.75 quantiles) of the wind speed distribution.

StormMaxBasin=subset(StormMax,Region=="Basin")
attach(StormMaxBasin)
quantile(Wmax,c(0.25,0.75))

25%      75%
50.43998 95.57234

We see that 25% of the storms have a maximum wind speed less than 51 kt and 75% have a maximum wind speed less than 96 kt so that 50% of all storms have a maximum wind speed between 51 and 96 kt (interquartile range).

To see if there is a trend in maximum wind speed values over time, we can use the linear regression technology from a previous section.  However, we note that the distribution of per cyclone maximum wind speed is not normal so it is best to transform the wind speeds before using linear regression.  A logarithmic transform is often a good choice.

To examine the distributions of the raw and log transformed wind speeds, type:

par(mfrow=c(2,1))
hist(Wmax)
hist(log(Wmax),breaks=12)

Or

par(mfrow=c(1,2))
boxplot(Wmax)
boxplot(log(Wmax))

To see if there is a significant trend in the mean wind speed over time, type:

summary(lm(log(Wmax)~Yr))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.5535805  0.7600147   4.676 3.33e-06 ***
Yr          0.0003521  0.0003880   0.907    0.364

Conditional quantiles
However, this is a model for the conditional mean (here, conditional on year).  What about higher wind speeds?  The theory of maximum potential intensity refers to storms at their maximum possible wind speed.  The subset of upper quartile maximum wind speeds would be on average a subset of storms closer their maximum possible wind speed.

So what we need is a model for the conditional quantiles.

Here we restrict the dataset to the years since satellite data.

detach(StormMaxBasin)
StormMaxBasin=subset(StormMaxBasin,Yr>1977)
attach(StormMaxBasin)

x=boxplot(Wmax~as.factor(Yr),plot=F)
boxplot(Wmax~as.factor(Yr),ylim=c(35,175),xlab="Year",ylab="Intensity (kt)")
xx=1:29
abline(lm(x\$stats[5,]~xx),col="red")
abline(lm(x\$stats[4,]~xx),col="blue")
abline(lm(x\$stats[3,]~xx),col="green")

The graph verifies the lack of trend in the central tendency of maximum cyclone intensities.  It also shows a tendency for the strongest storms to get stronger over time.  To quantify this trend and determine significance we turn to a quantile regression model.

Quantile regression models
The functionality is available in the quantreg package thanks to Roger Koenker.

install.packages("quantreg")
library(quantreg)
citation("quantreg")

We begin by running a median regression on year and checking if the parameter (slope) is significantly different from zero.

summary(rq(Wmax~Yr,tau=0.5),se="iid")
Call: rq(formula = Wmax ~ Yr, tau = 0.5)

tau: [1] 0.5

Coefficients:
Value     Std. Error t value   Pr(>|t|)
(Intercept) 118.58813 437.70671    0.27093   0.78661
Yr           -0.02639   0.21955   -0.12022   0.90438

As expected we find no significant trend in the median storm intensity over time.  How about trends in the upper quantiles?

summary(rq(Wmax~Yr,tau=c(0.75,0.9,0.95)),se="iid")

At the 75th and 90th percentiles, there is suggestive, but inconclusive evidence of a trend, but at the 95th percentile there is convincing evidence of an increase in the intensity of the strongest hurricanes in accord with MPI theory.

To show the results on a plot, type:

model=rq(Wmax~Yr,tau=seq(0.7,0.95,0.01))
plot(summary(model,alpha=.05,se="iid"),parm=2,pch=19,cex=1.2,mar=c(5,5,4,2)+0.1,ylab="Trend (kt/yr)",xlab="Quantile",main="North Atlantic Hurricanes")

The dots with lines indicate the trend for quantiles in the range between 0.7 and 0.95 by increments of 0.01.  The gray shading indicates the 95% confidence interval on the slope estimate.  The red solid line is the least squares regression slope and the dotted lines are the 95% confidence interval about that estimate.

Summary
Quantile regression provides a more complete picture of how the distribution of the response variable is conditioned on the covariates.

Classification and regression trees

Introduction
An inherent limitation of multiple regression models is the difficulty in providing a simple holistic interpretation of the results.  This limitation is most apparent when attempts are made to use the model for making decisions.  An alternative to multiple regression is a family of models known as classification and regression trees (C&RT).  These models are easier to interpret in the context of decision making.

It is a classification tree if the response variable is categorical (warm vs cold) and a regression tree if the response variable is continuous (temperature).

C&RT models have been used in many fields. An early application of a C&RT model involved the identification of ship structures from radar profiles. Hospitals use them to identify indicators of heart failure.  C&RT models are widely used in the finance industry.

C&RT models are nonparametric.  Explanatory variables and their interactions are selected based on their ability to parse the dependent variable. When the more stringent theoretical and distributional assumptions of more traditional methods (linear regression) are met, the traditional methods are preferable. But as an exploratory model, or as a model of last resort when traditional methods fail C&RT models are, in the opinion of many researchers, quite useful.

When there are many explanatory variables C&RT models can be quite complex, but graphs help simplify interpretation.

Example: Pollution
Before thinking about how C&RT models work, it helps to see one in action.  Download and install the tree package from CRAN and read the data.  The package is available thanks to Brian Ripley.

install.packages("tree")

Provided that your machine has a proper internet connection and you have write permission in the appropriate system directories, the installation of the package should proceed automatically.  Once the package is installed, it needs to be made accessible to the current R session.  To do this type:

library(tree)
citation("tree")

These procedures provide access to a large collection of specialized packages for statistical analysis.

The data we will use are in the file
pollute.txt.

attach(pollute)
names(pollute)
[1] "Pollution"  "Temp"       "Industry"   "Population" "Wind"
[6] "Rain"       "Wet.days"

Create a C&RT model object using the tree function.

model=tree(Pollution~Temp+Industry+Population+Wind+Rain+Wet.days)
plot(model)
text(model)

Instead of interpreting the parameter values from the table of coefficients, the model is interpreted from the upside-down tree diagram.

You follow a path from the top of the diagram (root of an upside-down tree) and proceed to one of the terminal nodes (called a leaf) by following a series of rules (called splits).  The number at the tip of the leaf is the mean value of the response variable in that subset of values (in this case SO_2 concentration).  The inequality listed on the branch is the rule.  Rules are applied to the set of explanatory variables.

In the pollution example the first split of the response variable Pollution is based on the explanatory variable Industry.  The split is a rule (logical operation) on the explanatory variable using a threshold value.  Is the value of Industry less than 748 units?  If yes, branch to the left; if no branch to the right.  The left branch proceeds to another split while the right branch proceeds to a leaf.

The model is fitted using binary recursive partitioning.  The response variable is successively split along coordinate axes of the explanatory variables so that at any node, the split which maximally distinguishes the response variable is selected.  Splitting continues until nodes cannot be split or there are too few cases (less than 6 by default).

Each explanatory variable is assessed, and the variable explaining the greatest amount of deviance in the response variable is selected.  Deviance is calculated on the basis of a threshold value which produces two mean values for the response (one mean above the threshold, the other below the threshold).
Note that the variable Rain does not enter the model.

Since the first split is on the variable Industry at a value of 748, let's look at how Pollution is partitioned on this variable.

low=(Industry<748)
tapply(Pollution,low,mean)

FALSE     TRUE
67.00000 24.91667

This tells us that when Industry is greater or equal to 748, the mean value for Pollution is 67 and when it is less than 748, the mean value of Pollution is 24.92.  To see this graphically we can use the following lines of code.

plot(Industry,Pollution,pch=16)
abline(v=748,lty=2)
lines(c(0,748),c(24.92,24.92))
lines(c(748,max(Industry)),c(67,67))

The plot shows that the split of Pollution on Industry<748 results in two means for Pollution that are quite a bit different (in fact they are maximally different over all such splits and over all explanatory variables).  The split results in less than 6 observations of Pollution for Industry>=748, so a terminal node is produced and the mean value of Pollution for these (5) observations is 67.

How the procedure works
The procedure works like this; for a given explanatory variable
• Select a threshold value (the vertical dotted line)
• Compute the mean of the response above and below this threshold (the two horizontal lines)
• Use the means to compute the deviance
• Go through all possible values of the threshold (values on the x axis)
• Look to see which value of the threshold gives the lowest deviance
• Split the entire data set into high and low subsets on the basis of this threshold for this variable
• Repeat the entire procedure on each subset of the data on either side of the threshold
• Keep going until no additional reduction in deviance is obtained, or there are too few data points to merit further subdivision

The value of a split is defined as the reduction in the deviance and is represented on the tree plot as the length of the hanging branch.  Thus for instance, the reduction in deviance is larger for the split on Population than for the split on Wet.days.

The key questions are:  Which variables to use for the split, and how best to achieve the split for the selected variable.

Tree regression is analogous to forward selection in multiple regression.

Model output is printed by typing:

model
node), split, n, deviance, yval
* denotes terminal node

1) root 41 22040 30.05
2) Industry < 748 36 11260 24.92
4) Population < 190 7  4096 43.43 *
5) Population > 190 29  4187 20.45
10) Wet.days < 108 11    96 12.00 *
11) Wet.days > 108 18  2826 25.61
22) Temp < 59.35 13  1895 29.69
44) Wind < 9.65 8  1213 33.88 *
45) Wind > 9.65 5   318 23.00 *
23) Temp > 59.35 5   152 15.00 *
3) Industry > 748 5  3002 67.00 *

The terminal nodes are denoted by an asterisk (there are 6 of them).  The node number is on the left and labeled by the variable on which the split at that node was made.  Next comes the split criterion which shows the threshold value and an inequality sign.  The number of cases going into the split (or terminal node) comes next.  The deviance value and the mean over all cases are shown as the next two figures.

At the root node the mean is 30.05 (mean(Pollution)) with a deviance of 22040 from 41 cases.  With the split on Industry at 748 we see the deviance decreases to 11260 (node 2) plus 3002 (node 3).  Since node 3 has less than 6 cases, it becomes a terminal node.  The remaining deviance from node 2 is split using Population at a value of 190. The highest mean value of Pollution according to this partition is 67 (node 3) and the lowest is 12 (node 10).

Note how the nodes are nested: within node 2, for example, node 4 is terminal but node 5 is not; within node 5 node 10 is terminal but node 11 is not; within node 11, node 23 is terminal but node 22 is not, and so on.

Interpretation
Tree regressions lend themselves to circumspect and critical analysis of complex .  In the present example, the aim is to understand the causes of variation in air pollution levels from case to case.  The interpretation of the tree regression is as follows:

• The five most extreme cases of Industry stand out and need to be considered separately.
• For the rest, Population is the most important variable, but interestingly, it is low populations that are associated with the highest levels of pollution.  Ask yourself which might be cause, and which might be effect.
• For high levels of population (> 190), the number of wet days is a key determinant of pollution; the places with the fewest wet days (less than 108 per year) have the lowest pollution levels.
• For those places with more than 108 wet days, it is temperature that is most important in explaining variation in pollution levels; the warmest places have the lowest air pollution levels.
• For the cooler places with more wet days, it is wind speed that matters: he windier places are less polluted.
This kind of complex and contingent explanation is easier to make with tree regressions.

Prediction
Like multiple regression, the tree regression can be used for making predictions.  The predicted values are easier to anticipate in this case.  For example, it is clear that if the Industry variable exceeds 748, then the model will prediction a pollution value of 67.  If Industry is less than 748 and Population is less than 190, then the predicted Pollution value is 43.43.  To see this type:

predict(model,data.frame(Industry=750,Population=0,Wet.days=0,Temp=0,Wind=0,Rain=0))
1
67

predict(model,data.frame(Industry=740,Population=180,Wet.days=0,Temp=0,Wind=0,Rain=0))
1
43.42857

Note that we can zero out the values for the other predictors since they do not enter the prediction for these values of Industry and Industry and Population.  Try

predict(model,data.frame(Industry=740,Population=200,Wet.days=120,Temp=50,Wind=8,Rain=0))

Prediction using regression trees is simply following the set of rules as organized by the tree.  We start with the value of the variable Industry and work our way up (down) the tree to we get to a terminal node.  The value of the response variable representing the subset mean is the predicted value.

The standard error on the predicted value is more difficult to work out.  We can think of using the standard error on the subset mean, but that ignores the uncertainty surrounding the other decisions that came before.

Model simplification
Model simplification in regression trees is similar to simplification in multiple regression.  There is a trade-off between bias and variance.  If the model is overly complex with many variables then the bias will be low but variance will be high on the predicted values.  If there are too few variables then variance will be low, but the predicted values will be biased against the excluded variables.

To deal with this trade-off, you should consider pruning your tree.  The function prune.tree determines a nested sequence of smaller trees from the largest tree by snipping off the least important splits.  The function returns the tree size (number of terminal nodes, \$size), the total deviance of each tree (\$dev), and the drop in deviance by going from a smaller tree to a larger tree (\$k).

prune.tree(model)

Thus it is useful to plot the deviance as a function of tree size.

plot(prune.tree(model))

We can see that the drop in deviance diminishes with each node added.  Deviance declines with increasing complexity.  In fact, we could argue that a tree with 4 terminal nodes is sufficiently complex as the drop in deviance from 4 to 5 nodes is relatively small.

Suppose we want the best tree with four nodes.

model2=prune.tree(model,best=4)
plot(model2)
text(model2)
model2

Classification
If the response variable is categorical rather than continuous, the tree model is one of classification.  A classification tree is based on a set of if-then rules.  "If petal length > x, then..., but if petal length < x, then something else".

Consider the intrinsic data frame iris.

data(iris)
attach(iris)
?iris

Use Species as the response variable and the others as explanatory variables.

ir.tr=tree(Species~.,iris)
summary(ir.tr)
Classification tree:
tree(formula = Species ~ ., data = iris)
Variables actually used in tree construction:
[1] "Petal.Length" "Petal.Width"  "Sepal.Length"
Number of terminal nodes:  6
Residual mean deviance:  0.1253 = 18.05 / 144
Misclassification error rate: 0.02667 = 4 / 150

plot(ir.tr)
text(ir.tr,cex=0.9)

We used a classification tree to help group hurricanes by genesis type in Objective classification of Atlantic basin hurricanes, Journal of Climate, v9, 2880-2889, 1996.

Baclawski, K., 2008: Introduction to Probability with R, Chapman & Hall/CRC.
Crawley, M.J., 2007: The R Book, John Wiley & Sons.
Dalgaard, P., 2002: Introductory Statistics with R, Springer.
Maindonald, J.H., 2004:
Using R for Data Analysis and Graphics: Introduction, Code and Commentary.
Murrell, P., 2006: R Graphics, Chapman & Hall/CRC.
Tufte, E.R, 2001: The Visual Display of Quantitative Information, 2nd Ed., Graphics Press.
Verzani, J., 2004: Using R for Introductory Statistics, Taylor & Francis.

R Reference Card

8. Other Material

Random walks
Consider the random variable X_n = 1, if the nth toss of a coin is heads and X_n = -1 if it is tails.  Now form a new random variable S_n = X_1 + X_2 + ... + X_n, then S_n is the position of a random walk after n steps: A step to the right is +1, a step to the left is -1, so the sum of the first n steps is the position of the walker at that time.

To plot the walk, type:

n=100
support=1:n
walk=cumsum(2*rbinom(n,1,0.5)-1)
plot(support,walk,type="l")

Repeat for n=1000.

The results of random walk analysis have been applied to computer science, physics, climatology, ecology, economics and a number of other fields as a fundamental model for random processes in time. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks.

To animate the random walk, we define a function that draws the graph up to time x.  For this we use the indexing function to perform the selection on the random walk, and then plot the graph with a suitable pause.

graph=function(k)
{
plot(1:k, walk[support<=k], xlim=c(1,n),
ylim=c(-n/10,n/10),type="l")
Sys.sleep(0.0)
}

To run it, type:

ignore=sapply(support, graph)

Try running a few times to acquire some intuition about what a random walk looks like.

Event table to time series

tbl=table(factor(c(1701,1701,1703,1714,1715,1716,1716,1716,1720),levels=1700:1720))
tbl1=as.data.frame(tbl)
tbl1\$Yr=as.numeric(as.character(tbl1[,1]))
tbl1=tbl1[,c("Yr","Freq")]