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 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.
Let
US=rpois(150, 1.7) #simulate 150 years of
hurricane counts
a=c()
for(i in 1:1000)
a[i]=mean(sample(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%
1.2 2.8
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".
3. 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 some tweaking the default
settings. This is true of all plotting software.
Several
graphics device drivers are available
including:
Onscreen graphics: Windows, X11, or
quartz.
Print graphics: postscript, pdf, png, jpeg, and wmf.
The
onscreen devices are more commonly used. For publicationquality
graphics, the postscript, pdf, or wmf devices are preferred because they
produce scalable
images.
Use bitmap devices for drafts.
The preferred sequence is to specify a graphics device, then
call graphics functions. If you do not specify a device first, the
onscreen device is started.
Back to our
hurricane
data.
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
Error: object
"US" not found
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 of the data. First read the data from
our website.
AMO=read.table("http://myweb.fsu.edu/jelsner/extspace/AMO18562006.txt",T)
SOI=read.table("http://myweb.fsu.edu/jelsner/extspace/SOI18662006.txt",T)
NAO=read.table("http://myweb.fsu.edu/jelsner/extspace/NAO18512006.txt",T)
Next create the covariate data.
amoAO=(AMO$Aug+AMO$Sep+AMO$Oct)/3
soiAO=(SOI$Aug+SOI$Sep+SOI$Oct)/3
naoMJ=(NAO$May+NAO$Jun)/2
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.
Let's start with the simplest thing to do.
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 righthand 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="MayJun 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 "QQ 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 "boxandwhiskers 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$US>3,ylab="NAO MayJun
(sd)")
The parallel box plots help us visualize the relationship between hurricanes
and covariates.
label=factor(c(rep("On or Before
1928",78),rep("After 1928",78)))
boxplot(ushur$US~label,notch=T,xlab="",ylab="US Hurricanes")
Box plots can also be used to examine seasonality. The AMO covariate
contains monthly values from 18562006. 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.
4. 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 n1 parameters to a model with n parameters

A model with k1 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 covary.
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 leastsquares 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 AugOct AMO in units of degrees C equals 22.7 minus 0.02687 times
the MayJun 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 <2e16
***
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 RSquared: 0.01297, Adjusted Rsquared: 0.006343
Fstatistic: 1.957 on 1 and 149 DF, pvalue: 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 <2e16
***
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 sizes.
Multiple
RSquared: 0.01297, Adjusted Rsquared:
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 =
(SSYSSE)/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.
Fstatistic:
1.957 on 1 and 149 DF, pvalue: 0.1639
This is an F test for the hypothesis that the slope is zero. The
Fstatistic 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 pvalue 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 Rsquared value can be manipulated simply by adding more explanatory
variables. The adjusted Rsquared value is an attempt to correct this
shortcoming 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 Rsquared, the adjusted Rsquared 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 Rsquared.
Note: while the Rsquared is a percent, the adjusted Rsquared 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 pointwise confidence intervals on the predictions
predict(model,level=0.95,interval="confidence")
To get prediction intervals use
interval="prediction".
To ﬁnd the conﬁdence bands The conﬁdence bands would be a chore to compute by
hand. Unfortunately, it is a bit of a chore to get with the lowlevel commands
as well. The predict method also has an ability to ﬁnd the conﬁdence bands if
we learn how to ask. Generally speaking, for each value of x we want a point
to plot.
This is done as before with a data frame containing all the x values we want.
In addition, we need to ask for the interval. There are two types: conﬁdence,
or prediction. The conﬁdence will be for the mean, and the prediction for the
individual. Let’s see the output, and then go from there. This is for a 90%
conﬁdence level.
predict(lm.result,data.frame(x=sort(x)), level=.9,
interval="confidence")
fit lwr upr
1 195.6894 192.5083
198.8705
2 195.6894 192.5083
198.8705
3 194.8917 191.8028
197.9805
... skipped ...
We see we get 3 numbers back for each value of x. (note we sorted x ﬁrst to
get the proper order for plotting.)
To plot the lower band, we just need the second column which is accessed with
[,2]. So the following will plot just the lower. Notice, we make a scatterplot
with the plot command, but add the conﬁdence band with points.
plot(x,y)
abline(lm.result)
ci.lwr = predict(lm.result,data.frame(x=sort(x)),
level=.9,interval="confidence")[,2]
points(sort(x), ci.lwr,type="l") # or use
lines
Alternatively, we could plot this with the curve function as follows
curve(predict(lm.result,data.frame(x=x),
interval="confidence")[,3],add=T)
This is conceptually easier, but harder to break up, as the curve function
requires a function of x to plot.
Model adequacy
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 straightline
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.
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 oneparameter 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 10018
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:
1ppois(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 longterm 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
3.04e10 ***
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 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(207.16195.26,1,lower.tail=F)
Model adequacy is
examined with the residual deviance.
The residual deviance
is
195.26
on 154 degrees of freedom. A pvalue from a chisquared distribution with this
quantile value and this dof is evidence in favor of the null hypothesis that
the model is adequate. A small pvalue 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 overdispersed relative to the Poisson
distribution; underdispersion occurs when variance is smaller than the
mean. If a Poisson model seems appropriate but doesn’t fit well,
overdispersion or underdispersion 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 semiparametric technique because it relies on the nonparametric
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 equalsized 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:
StormMax=read.csv("extremedatasince1899.csv",header=T)
head(StormMax)
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 3.33e06
***
Yr
0.0003521 0.0003880 0.907 0.364
The answer is no.
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$stats[3,]~xx),col="green")
The graph verifies the lack of trend in the central tendency of
maximum cyclone intensities. It also shows a tendency for the strongest
storms to get stronger over time. To quantify this trend and determine
significance we turn to a quantile regression model.
Quantile regression models
The functionality is available in the
quantreg package thanks
to Roger Koenker.
install.packages("quantreg")
library(quantreg)
citation("quantreg")
We begin by running a median regression on year
and checking if the parameter (slope) is significantly different from
zero.
summary(rq(Wmax~Yr,tau=0.5),se="iid")
Call: rq(formula =
Wmax ~ Yr, tau = 0.5)
tau: [1] 0.5
Coefficients:
Value Std. Error t value Pr(>t)
(Intercept) 118.58813 437.70671 0.27093 0.78661
Yr
0.02639 0.21955 0.12022 0.90438
As expected we find no significant trend in the median storm intensity over
time. How about trends in the upper quantiles?
summary(rq(Wmax~Yr,tau=c(0.75,0.9,0.95)),se="iid")
At the 75th and 90th percentiles, there is
suggestive, but inconclusive evidence of a trend, but at the 95th percentile
there is convincing evidence of an increase in the intensity of the strongest
hurricanes in accord with MPI theory.
To show the results on a plot, type:
model=rq(Wmax~Yr,tau=seq(0.7,0.95,0.01))
plot(summary(model,alpha=.05,se="iid"),parm=2,pch=19,cex=1.2,mar=c(5,5,4,2)+0.1,ylab="Trend
(kt/yr)",xlab="Quantile",main="North Atlantic
Hurricanes")
The dots with lines indicate the trend for quantiles in the range between 0.7
and 0.95 by increments of 0.01. The gray shading indicates the 95%
confidence interval on the slope estimate. The red solid line is the
least squares regression slope and the dotted lines are the 95% confidence
interval about that estimate.
Quantile regression is not linear regression on the quantiles
globalTCmax4=read.table("http://myweb.fsu.edu/jelsner/extspace/globalTCmax4.txt",T,stringsAsFactors=F,na.strings="NaN")
#load the data fix North Atlantic
globalTCmax4$Wmax=globalTCmax4$WmaxST*0.514444444
IO=subset(globalTCmax4,Basin=="IO")
IO1=tapply(IO$Wmax,IO$Year,quantile, p=.9)
IO1.df=data.frame(Year=unique(IO$Year),Wmax=IO1,weights=tapply(IO$Year,IO$Year,length))
lm90=lm(Wmax~Year,data=IO1.df)
qr90=rq(tau=.9,Wmax~Year,data=IO)
qr90
lm90
plot(Wmax~Year,data=IO,pch=16,cex=.7)
abline(qr90)
points(Wmax~Year,data=IO1.df,pch=15,col="green",cex=.7)
abline(lm90,col="green")
title("Quantile regression fit (black) \n versus linear regression of
quantiles (green) \n for the 90% quantile")
text(x=c(1992,1991),y=c(46,41.2),c("slope=0.42
m/s/yr","m=.36 m/s/yr"))
rq90=round(resid(qr90),10)
cat("Quantile Regression Residuals\n")
r1=c(sum(rq90<0),sum(rq90<=0))/length(resid(qr90))
#.90 is between these observations we check out o.k. here with
quantile regression
cat("Percentage of residuals less than or less than or equal to 0\n")
round(100*r1,2)
quantile(prob=.90, rq90)
rl90=round(IO$Wmaxpredict(lm90,newdata=IO),10)
r2=c(sum(rl90<0),sum(rl90<=0))/length(resid(qr90))
#This is too low we only have 84% of observations below or equal
fitted line.
cat("Linear Model Residuals using all data points for prediction\n")
cat("Percentage of residuals less than or less than or equal to 0\n")
round(100*r2,2)
quantile(prob=.90, rl90)
Summary
Quantile regression provides a more complete picture of how the distribution
of the response variable is conditioned on the values of the covariates.
5. Additional Reading and
References
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, Chapman
& Hall/CRC.
R
Reference Card
UCLA Academic Technology Services
http://www.ats.ucla.edu/stat/r