Scenario (a): Oil excavation
Question 1:
This is supervised learning model. It is example of regression. We are interested in prediction. We predict the most promising spot to dig. Here number of observations n = 80 and number of predictors p = 24.
1: (b)
This is supervised learning model. It is example of classification. Here we are interested in prediction. We predict the whether to display advertisement A or advertisement B to each customer. Here number of observations n = 300 and number of predictors p = 3 (age, zip code, and gender).
1: (c)
This is supervised learning model. It is example of regression. Here we are interested in inference. We are interested in discovering factors that are associated with the unemployment rate across different U.S. cities. Here number of observations n = 400 and number of predictors p = 6 (the population, state, average income, crime rate, percentage of students who graduate high school and unemployment level.).
1: (d)
This is unsupervised learning model. For the each students we don’t have responses (different subtypes) of students in the application pool.
1 (e):
This is supervised learning model. It is example of classification. Here we are interested in prediction. We predict the type of cells based on a few measurements. Here number of observations n = 68 and number of predictors p = 3 (the number of branch points, the number of active processes, and the average process length).
Question 2:
2 (a):
We preferred inflexible regression model as number of predictors (number of genes) p is extremely large, and the number of observations (number of patients) n is small.
2 (b):
We preferred flexible regression model as number of predictors (math, science and history grades in the 7th grade) p is small, and the number of observations (number of students) n is extremely large.
2 (c):
We preferred inflexible regression model as we variation in data is more.
2 (d):
We preferred flexible regression model as we variation in data is less. Flexible model will perform better to find non-linear effect also.
Question 3:
3 (a):
Flexible model performs better. A flexible models fits the data well with larger sample size and performs better than inflexible model.
3 (b):
Flexible model performs worse. A flexible models overfit small number of observation.
3 (c):
Flexible model performs worse. A flexible method would fit to the noise in the error terms and increase variance.
Scenario (b): Online retailer advertisement
3 (d):
Flexible model performs better. A flexible models gets more degrees of freedom fits the data well.
3 (e):
Flexible model performs worse as it would fit to the noise in the error terms and increase variance.
Question 4:
Solution is obtained from R studio. Here we have given the some output of the program.
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> summary(College) Private Apps Accept Enroll Top10perc No :212 Min. : 81 Min. : 72 Min. : 35 Min. : 1.00 Yes:565 1st Qu.: 776 1st Qu.: 604 1st Qu.: 242 1st Qu.:15.00 Median : 1558 Median : 1110 Median : 434 Median :23.00 Mean : 3002 Mean : 2019 Mean : 780 Mean :27.56 3rd Qu.: 3624 3rd Qu.: 2424 3rd Qu.: 902 3rd Qu.:35.00 Max. :48094 Max. :26330 Max. :6392 Max. :96.00 Top25perc F.Undergrad P.Undergrad Outstate Room.Board Min. : 9.0 Min. : 139 Min. : 1.0 Min. : 2340 Min. :1780 1st Qu.: 41.0 1st Qu.: 992 1st Qu.: 95.0 1st Qu.: 7320 1st Qu.:3597 Median : 54.0 Median : 1707 Median : 353.0 Median : 9990 Median :4200 Mean : 55.8 Mean : 3700 Mean : 855.3 Mean :10441 Mean :4358 3rd Qu.: 69.0 3rd Qu.: 4005 3rd Qu.: 967.0 3rd Qu.:12925 3rd Qu.:5050 Max. :100.0 Max. :31643 Max. :21836.0 Max. :21700 Max. :8124 Books Personal PhD Terminal S.F.Ratio Min. : 96.0 Min. : 250 Min. : 8.00 Min. : 24.0 Min. : 2.50 1st Qu.: 470.0 1st Qu.: 850 1st Qu.: 62.00 1st Qu.: 71.0 1st Qu.:11.50 Median : 500.0 Median :1200 Median : 75.00 Median : 82.0 Median :13.60 Mean : 549.4 Mean :1341 Mean : 72.66 Mean : 79.7 Mean :14.09 3rd Qu.: 600.0 3rd Qu.:1700 3rd Qu.: 85.00 3rd Qu.: 92.0 3rd Qu.:16.50 Max. :2340.0 Max. :6800 Max. :103.00 Max. :100.0 Max. :39.80 perc.alumni Expend Grad.Rate Min. : 0.00 Min. : 3186 Min. : 10.00 1st Qu.:13.00 1st Qu.: 6751 1st Qu.: 53.00 Median :21.00 Median : 8377 Median : 65.00 Mean :22.74 Mean : 9660 Mean : 65.46 3rd Qu.:31.00 3rd Qu.:10830 3rd Qu.: 78.00 Max. :64.00 Max. :56233 Max. :118.00 |
> length(which(College$Private==”Yes”))
[1] 565
Out of 777, 565 colleges are private whereas 212 are non-private.
> length(which(College$Elite==”Yes”))
[1] 78
There are 78 elite universities.
R Code:
# Set the working directory
# Read the data
College=read.csv(‘College.csv’,header=1)
#To view the data
View(College)
#For removing first column
rownames(College)= College[,1]
College=College[,-1]
View(College )
#Summary of Data
summary(College)
#scatterplot of the column PhD versus the column Grad.Rate.
plot(College$Grad.Rate,College$PhD,xlab=”Grad.Rate”,ylab=”PhD”,main=”Scatter Plot”)
#Number of Private Colleges
length(which(College$Private==”Yes”))
#(g)
Elite=rep(“No”,nrow (College))
Elite[College$Top10perc>50]=”Yes”
Elite=as.factor(Elite)
College=data.frame(College,Elite)
#How many elite universities are there?
length(which(College$Elite==”Yes”))
#Box Plot
plot(College$Elite, College$Outstate, xlab = “Elite University”, ylab =”Out of State tuition in USD”, main = “Outstate Tuition Plot”)
# (h) Histogram
par(mfrow=c(2,2))
hist(College$Top10perc, col = 4, xlab = “Top 10%”, ylab = “Count”,main=””)
hist(College$Top25perc, col = 6, xlab = “Top 25%”, ylab = “Count”,main=””)
hist(College$Books, col = 2, xlab = “Books”, ylab = “Count”,main=””)
hist(College$PhD, col = 3, xlab = “PhD”, ylab = “Count”,main=””)
Question 5:
One can see that variable is negatively skewed.
One can study the relation between different predictors by observing the above scatter plots of predictors.
(e)
> coef(summary(lm.model))[1:20,1:4]
Estimate Std. Error
(Intercept) 5.2523261623 1.005248e+01
Onset.Delta -0.0004375411 4.516149e-05
Symptom.Speech -0.0225458435 9.120369e-02
Symptom.WEAKNESS -0.1030724618 8.140383e-02
Site.of.Onset.Onset..Bulbar -0.3672390249 2.492628e-01
Site.of.Onset.Onset..Limb -0.2722652619 2.516364e-01
Race…Caucasian -0.1472308983 9.490391e-02
Age -0.0005163468 1.814937e-03
Sex.Female -0.0605416162 9.573716e-02
Sex.Male 0.0195376288 8.868793e-02
Mother -0.0462694066 7.581925e-02
Family 0.0067961369 5.629768e-02
Study.Arm.PLACEBO -3.1549354316 1.918113e+00
Study.Arm.ACTIVE -3.0519454033 1.917047e+00
max.alsfrs.score 0.0622267218 7.830124e-02
min.alsfrs.score -0.1666353650 8.465840e-02
last.alsfrs.score 0.4090457993 2.074172e-01
mean.alsfrs.score -0.1208903623 3.538618e-01
num.alsfrs.score.visits -0.0058420080 7.139877e-01
sum.alsfrs.score -0.0778763045 7.899171e-02
t value Pr(>|t|)
(Intercept) 0.522490777 6.014605e-01
Onset.Delta -9.688368476 3.700894e-21
Symptom.Speech -0.247203188 8.048088e-01
Symptom.WEAKNESS -1.266186858 2.057820e-01
Site.of.Onset.Onset..Bulbar -1.473300789 1.410284e-01
Site.of.Onset.Onset..Limb -1.081979013 2.795589e-01
Race…Caucasian -1.551368083 1.211739e-01
Age -0.284498398 7.760955e-01
Sex.Female -0.632373219 5.273077e-01
Sex.Male 0.220296371 8.256916e-01
Mother -0.610259353 5.418479e-01
Family 0.120717895 9.039421e-01
Study.Arm.PLACEBO -1.644811989 1.003665e-01
Study.Arm.ACTIVE -1.592003115 1.117439e-01
max.alsfrs.score 0.794709319 4.269974e-01
min.alsfrs.score -1.968326306 4.934477e-02
last.alsfrs.score 1.972092254 4.891257e-02
mean.alsfrs.score -0.341631529 7.327100e-01
num.alsfrs.score.visits -0.008182225 9.934735e-01
sum.alsfrs.score -0.985879479 3.244639e-01
We observed that R2 is 0.46 suggest that fitting is not too good.
We observed that RMSE is 0.4138632.
The error rate produced by using a simple linear regression on this data is much too high. It could be useful for variables selection. Bias–variance tradeoff shows that predictor with less bias has larger variance.
R Code:
#(a)
a=load(‘als.rData’)
#(b)
length(train.X)
length(train.y)
length(test.X)
length(test.y)
#(c)
summary(train.y) #Summary of train.y
hist(train.y,breaks = 40) #Histogram
#(d)
colnames(train.X)[1:20]
pairs(train.X[,21:25])
#Fitting of Regression model
lm.model=lm(train.y ~., data = data.frame(train.y,train.X))
#First 20 coefficients
coef(summary(lm.model))[1:20,1:4]
#R squared
summary(lm.model)$r.squared
pred=predict(lm.model)
#RMSE
RMSE=sqrt(mean((pred-train.y)^2))
RMSE
Reference:
James, G., Witten, D., Hastie, T. and Tibshirani, R., 2013. An introduction to statistical learning (Vol. 112). New York: springer.
Montgomery, D.C., Peck, E.A. and Vining, G.G., 2012. Introduction to linear regression analysis (Vol. 821). John Wiley & Sons.
