Stats Homework (R)
Part 1 of 2
• Read section 11 of handbook_statsV13
• What is the difference between these two models?
– g1 = glm(admit ~ ., data=mydata)
– g2 = glm(admit ~ ., data=mydata, family=“binomial”)
• Compute MSE (mean squared error)
– mean((g1$fitted.values – mydata$admit)^2)
– mean((g2$fitted.values – mydata$admit)^2)
• Which model is better in terms of MSE
Stats Homework (R)
Part 2 of 2
• Consider two loss functions
– MSE.loss = function(y, yhat) mean((y - yhat)^2)
– log.loss = function(y, yhat) -mean(y * log(yhat) + (1-y) * log(1-yhat))
• Which model (g1 or g2) is better in terms of loss2?
– Hint: log.loss assumes yhat is a probability between 0 and 1
– What are range(g1$fitted.values) and range(g2$fitted.values)?
• Let s = g1$fitted.values > 0 & g1$fitted.values < 1
– What is log.loss(mydata$admit[s], g1$fitted.values[s])?
• Let baseline.fit be rep(mean(admit), length(admit))
– How do the two models above compare to this baseline
– Compare both g1 and g2 to this baseline using both loss functions
Functions
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•
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fact = function(x) if(x < 2) 1 else x*fact(x-1)
fact(5)
fact2 = function(x) gamma(x+1)
fact(5)
fact(1:5)
fact(seq(1,5,0.5))
help(swiss)
What are statistics packages good for?
• Plotting
– Scatter plots, histograms, boxplots
• Visualization
– Dendrograms, heatmaps
• Modeling
– Linear Regression
– Logistic Regression
• Hypothesis Testing
Objects
Data Types
•
•
•
Numbers: 1, 1.5
Strings
Functions:
– sqrt, help, summary
•
3:30
c(1:5, 25:30)
rnorm(100, 0, 1), rbinom(100, 10, .5)
rep(0,10)
scan( filename, 0)
matrix(rep(0,100), ncol=10)
diag(rep(1,10))
cbind(1:10, 21:30)
rbind(1:10, 21:30)
Data Frames
– swiss
– read.table (filename, header=T)
All: summary, help, print, cat
Numbers: +,*,sqrt
Strings: grep, substr, paste
Vectors: x = (3:30)
–
–
–
–
–
•
x+5
plot(x)
mean(x), var(x), median(x)
mean((x – mean(x))^2)
x[3], x[3:6], x[-1], x[x>20]
Matrices: m = matrix(1:100,ncol=10)
– m+3, m* 3, m %*% m
– m[3,3], m[1:3,1:3], m[3,], m[m < 10]
– diag(m), t(m)
Matrices
–
–
–
–
•
•
•
•
•
Vectors:
–
–
–
–
–
•
Operations
•
Data Frames
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–
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names(swiss)
plot(swiss$Education, swiss$Fertility)
plot(swiss[,1], swiss[,2])
pairs(swiss)
plot(swiss$Education, swiss$Fertility)
plot(swiss$Education, swiss$Fertility,
col = 3 + (swiss$Catholic > 50))
boxplot(split(swiss$Fertility, swiss$Catholic > 50))
boxplot(split(swiss$Fertility, swiss$Education > median(swiss$Education)))
pairs(swiss)
Correlations
> cor(swiss$Fertility, swiss$Catholic)
[1] 0.4636847
> cor(swiss$Fertility, swiss$Education)
[1] -0.6637889
> round(100*cor(swiss))
Fertility
Agriculture
Examination
Education
Catholic
Infant.Mortality
Fertility Agriculture Examination Education Catholic Infant.Mortality
100
35
-65
-66
46
42
35
100
-69
-64
40
-6
-65
-69
100
70
-57
-11
-66
-64
70
100
-15
-10
46
40
-57
-15
100
18
42
-6
-11
-10
18
100
• Many functions in R are polymorphic:
– cor works on vectors as well as matrices
Regression
> plot(swiss$Education, swiss$Fertility)
> g = glm(swiss$Fertility ~ swiss$Education)
> abline(g)
> summary(g)
Call:
glm(formula = swiss$Fertility ~ swiss$Education)
Deviance Residuals:
Min
1Q
Median
-17.036
-6.711
-1.011
3Q
9.526
Max
19.689
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
79.6101
2.1041 37.836 < 2e-16 ***
swiss$Education -0.8624
0.1448 -5.954 3.66e-07 ***
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
plot(swiss$Education, swiss$Fertility, type='n’); abline(g)
text(swiss$Education, swiss$Fertility, dimnames(swiss)[[1]])
g = glm(Fertility ~ . , data=swiss); summary(g)
Call:
glm(formula = Fertility ~ ., data = swiss)
Deviance Residuals:
Min
1Q
-15.2743
-5.2617
Median
0.5032
3Q
4.1198
Max
15.3213
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
66.91518
10.70604
6.250 1.91e-07 ***
Agriculture
-0.17211
0.07030 -2.448 0.01873 *
Examination
-0.25801
0.25388 -1.016 0.31546
Education
-0.87094
0.18303 -4.758 2.43e-05 ***
Catholic
0.10412
0.03526
2.953 0.00519 **
Infant.Mortality 1.07705
0.38172
2.822 0.00734 **
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘
’ 1
(Dispersion parameter for gaussian family taken to be
51.34251)
http://www.nytimes.com/2009/01/07/technology/business-computing/07program.html
Plot
Example
• help(ldeaths)
• plot(mdeaths, col="blue", ylab="Deaths", sub="Male (blue),
Female (pink)", ylim=range(c(mdeaths, fdeaths)))
• lines(fdeaths, lwd=3, col="pink")
• abline(v=1970:1980, lty=3)
• abline(h=seq(0,3000,1000), lty=3, col="red")
Periodicity
• plot(1:length(fdeaths), fdeaths, type='l')
• lines((1:length(fdeaths))+12, fdeaths, lty=3)
par(mfrow=c(2,2))
plot(fdeaths, type='p', main="points")
plot(fdeaths, type='l', main="lines")
plot(fdeaths, type='b', main="b")
plot(fdeaths, type='o', main="o")
Type
Argument
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•
•
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plot(as.vector(mdeaths), as.vector(fdeaths))
g=glm(fdeaths ~ mdeaths)
abline(g)
g$coef
(Intercept)
-45.2598005
mdeaths
0.4050554
Scatter
Plot
• par(mfrow=c(2,1))
• hist(fdeaths/mdeaths, nclass=30)
• plot(density(fdeaths/mdeaths))
Hist &
Density
Data Frames
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•
•
•
help(cars)
names(cars)
summary(cars)
plot(cars)
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•
•
•
•
•
•
•
cars2 = cars
cars2$speed2 = cars$speed^2
cars2$speed3 = cars$speed^3
summary(cars2)
names(cars2)
plot(cars2)
options(digits=2)
cor(cars2)
Normality
par(mfrow=c(2,1))
plot(density(cars$dist/cars$sp
eed))
lines(density(rnorm(1000000,
mean(cars$dist/cars$spe
ed), sqrt(var(cars$dist/
cars$speed)))),
col="red")
qqnorm(cars$dist/cars$speed
)
abline(mean(cars$dist/cars$s
peed),
sqrt(var(cars$dist/cars$s
peed)))
Stopping Distance Increases Quickly
With Speed
•
•
plot(cars$speed,
cars$dist/cars$speed)
boxplot(split(cars$dist/cars$sp
eed,
round(cars$speed/10)*10))
Quadratic
Model of
Stopping
Distance
plot(cars$speed, cars$dist)
cars$speed2 = cars$speed^2
g2 = glm(cars$dist ~ cars$speed2)
lines(cars$speed, g2$fitted.values)
Bowed Residuals
g1 = glm(dist ~ poly(speed, 1),
data=cars)
g2 = glm(dist ~ poly(speed, 2),
data=cars)
par(mfrow=c(1,2))
boxplot(split(g1$resid,
round(cars$speed/5)));
abline(h=0)
boxplot(split(g2$resid,
round(cars$speed/5)));
abline(h=0)
Help, Demo, Example
• demo(graphics)
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example(plot)
example(lines)
help(cars)
help(WWWusage)
example(abline)
example(text)
example(par)
• pairs
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–
–
–
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• utils::data(anorexia,
package="MASS")
• pairs(anorexia, col=c("red",
"green",
"blue")[anorexia$Treat])
• boxplots
– example(boxplot)
– help(chickwts)
• demo(plotmath)
example(pairs)
help(quakes)
help(airquality)
help(attitude)
Anorexia
• counting
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example(table)
example(quantile)
example(hist)
help(faithful)
• Randomness
– example(rnorm)
– example(rbinom)
– example(rt)
• Normality
– example(qqnorm)
• Regression
– help(cars)
– example(glm)
– demo(lm.glm)