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# Statistical Learning

## Conceptual

### Question 1

> For each of parts (a) through (d), indicate whether we would generally expect
> the performance of a flexible statistical learning method to be better or
> worse than an inflexible method. Justify your answer.
>
> a. The sample size n is extremely large, and the number of predictors p is
> small.
> b. The number of predictors p is extremely large, and the number of
> observations n is small.
> c. The relationship between the predictors and response is highly
> non-linear.
> d. The variance of the error terms, i.e. $\sigma^2 = Var(\epsilon)$, is
> extremely high.
### Question 2

> Explain whether each scenario is a classification or regression problem, and
> indicate whether we are most interested in inference or prediction. Finally,
> provide n and p.
>
> a. We collect a set of data on the top 500 firms in the US. For each firm
> we record profit, number of employees, industry and the CEO salary. We are
> interested in understanding which factors affect CEO salary.
>
> b. We are considering launching a new product and wish to know whether
> it will be a success or a failure. We collect data on 20 similar products
> that were previously launched. For each product we have recorded whether it
> was a success or failure, price charged for the product, marketing budget,
> competition price, and ten other variables.
>
> c. We are interested in predicting the % change in the USD/Euro exchange
> rate in relation to the weekly changes in the world stock markets. Hence we
> collect weekly data for all of 2012. For each week we record the % change
> in the USD/Euro, the % change in the US market, the % change in the British
> market, and the % change in the German market.
### Question 3

> We now revisit the bias-variance decomposition.
>
> a. Provide a sketch of typical (squared) bias, variance, training error,
> test error, and Bayes (or irreducible) error curves, on a single plot, as
> we go from less flexible statistical learning methods towards more flexible
> approaches. The x-axis should represent the amount of flexibility in the
> method, and the y-axis should represent the values for each curve. There
> should be five curves. Make sure to label each one.
>
> b. Explain why each of the five curves has the shape displayed in
> part (a).
### Question 4

> You will now think of some real-life applications for statistical learning.
>
> a. Describe three real-life applications in which classification might
> be useful. Describe the response, as well as the predictors. Is the goal of
> each application inference or prediction? Explain your answer.
>
> b. Describe three real-life applications in which regression might be
> useful. Describe the response, as well as the predictors. Is the goal of
> each application inference or prediction? Explain your answer.
>
> c. Describe three real-life applications in which cluster analysis might be
> useful.
### Question 5

> What are the advantages and disadvantages of a very flexible (versus a less
> flexible) approach for regression or classification? Under what circumstances
> might a more flexible approach be preferred to a less flexible approach? When
> might a less flexible approach be preferred?
### Question 6

> Describe the differences between a parametric and a non-parametric statistical
> learning approach. What are the advantages of a para- metric approach to
> regression or classification (as opposed to a non- parametric approach)? What
> are its disadvantages?
### Question 7

> The table below provides a training data set containing six observations,
> three predictors, and one qualitative response variable.
>
> | Obs. | $X_1$ | $X_2$ | $X_3$ | $Y$ |
> |------|-------|-------|-------|-------|
> | 1 | 0 | 3 | 0 | Red |
> | 2 | 2 | 0 | 0 | Red |
> | 3 | 0 | 1 | 3 | Red |
> | 4 | 0 | 1 | 2 | Green |
> | 5 | -1 | 0 | 1 | Green |
> | 6 | 1 | 1 | 1 | Red |
>
> Suppose we wish to use this data set to make a prediction for $Y$ when
> $X_1 = X_2 = X_3 = 0$ using $K$-nearest neighbors.
>
> a. Compute the Euclidean distance between each observation and the test
> point, $X_1 = X_2 = X_3 = 0$.
>
> b. What is our prediction with $K = 1$? Why?
>
> c. What is our prediction with $K = 3$? Why?
>
> d. If the Bayes decision boundary in this problem is highly non-linear, then
> would we expect the best value for $K$ to be large or small? Why?
```{r}
dat <- data.frame(
"x1" = c(0, 2, 0, 0, -1, 1),
"x2" = c(3, 0, 1, 1, 0, 1),
"x3" = c(0, 0, 3, 2, 1, 1),
"y" = c("Red", "Red", "Red", "Green", "Green", "Red")
)
```

## Applied

### Question 8

> This exercise relates to the `College` data set, which can be found in
> the file `College.csv`. It contains a number of variables for 777 different
> universities and colleges in the US. The variables are
>
> * `Private` : Public/private indicator
> * `Apps` : Number of applications received
> * `Accept` : Number of applicants accepted
> * `Enroll` : Number of new students enrolled
> * `Top10perc` : New students from top 10% of high school class
> * `Top25perc` : New students from top 25% of high school class
> * `F.Undergrad` : Number of full-time undergraduates
> * `P.Undergrad` : Number of part-time undergraduates
> * `Outstate` : Out-of-state tuition
> * `Room.Board` : Room and board costs
> * `Books` : Estimated book costs
> * `Personal` : Estimated personal spending
> * `PhD` : Percent of faculty with Ph.D.'s
> * `Terminal` : Percent of faculty with terminal degree
> * `S.F.Ratio` : Student/faculty ratio
> * `perc.alumni` : Percent of alumni who donate
> * `Expend` : Instructional expenditure per student
> * `Grad.Rate` : Graduation rate
>
> Before reading the data into `R`, it can be viewed in Excel or a text
> editor.
>
> a. Use the `read.csv()` function to read the data into `R`. Call the loaded
> data `college`. Make sure that you have the directory set to the correct
> location for the data.
>
> b. Look at the data using the `View()` function. You should notice that the
> first column is just the name of each university. We don't really want `R`
> to treat this as data. However, it may be handy to have these names for
> later. Try the following commands:
>
> ```r
> rownames(college) <- college[, 1]
> View(college)
> ```
>
> You should see that there is now a `row.names` column with the name of
> each university recorded. This means that R has given each row a name
> corresponding to the appropriate university. `R` will not try to perform
> calculations on the row names. However, we still need to eliminate the
> first column in the data where the names are stored. Try
>
> ```r
> college <- college [, -1]
> View(college)
> ```
>
> Now you should see that the first data column is `Private`. Note that
> another column labeled `row.names` now appears before the `Private` column.
> However, this is not a data column but rather the name that R is giving to
> each row.
>
> c.
> i. Use the `summary()` function to produce a numerical summary of the
> variables in the data set.
> ii. Use the `pairs()` function to produce a scatterplot matrix of the
> first ten columns or variables of the data. Recall that you can
> reference the first ten columns of a matrix A using `A[,1:10]`.
> iii. Use the `plot()` function to produce side-by-side boxplots of
> `Outstate` versus `Private`.
> iv. Create a new qualitative variable, called `Elite`, by _binning_ the
> `Top10perc` variable. We are going to divide universities into two
> groups based on whether or not the proportion of students coming from
> the top 10% of their high school classes exceeds 50%.
>
> ```r
> > Elite <- rep("No", nrow(college))
> > Elite[college$Top10perc > 50] <- "Yes"
> > Elite <- as.factor(Elite)
> > college <- data.frame(college, Elite)
> ```
>
> Use the `summary()` function to see how many elite universities there
> are. Now use the `plot()` function to produce side-by-side boxplots of
> `Outstate` versus `Elite`.
> v. Use the `hist()` function to produce some histograms with differing
> numbers of bins for a few of the quantitative variables. You may find
> the command `par(mfrow=c(2,2))` useful: it will divide the print
> window into four regions so that four plots can be made
> simultaneously. Modifying the arguments to this function will divide
> the screen in other ways.
> vi. Continue exploring the data, and provide a brief summary of what you
> discover.
### Question 9

> This exercise involves the Auto data set studied in the lab. Make sure
> that the missing values have been removed from the data.
>
> a. Which of the predictors are quantitative, and which are qualitative?
>
> b. What is the range of each quantitative predictor? You can answer this using
> the `range()` function.
>
> c. What is the mean and standard deviation of each quantitative predictor?
>
> d. Now remove the 10th through 85th observations. What is the range, mean, and
> standard deviation of each predictor in the subset of the data that
> remains?
>
> e. Using the full data set, investigate the predictors graphically, using
> scatterplots or other tools of your choice. Create some plots highlighting
> the relationships among the predictors. Comment on your findings.
>
> f. Suppose that we wish to predict gas mileage (`mpg`) on the basis of the
> other variables. Do your plots suggest that any of the other variables
> might be useful in predicting `mpg`? Justify your answer.
### Question 10

> This exercise involves the `Boston` housing data set.
>
> a. To begin, load in the `Boston` data set. The `Boston` data set is part of
> the `ISLR2` library in R.
>
> ```r
> > library(MASS)
> ```
>
> Now the data set is contained in the object `Boston`.
>
> ```r
> > Boston
> ```
>
> Read about the data set:
>
> ```r
> > ?Boston
> ```
>
> How many rows are in this data set? How many columns? What do the rows and
> columns represent?
>
> b. Make some pairwise scatterplots of the predictors (columns) in this data
> set. Describe your findings.
>
> c. Are any of the predictors associated with per capita crime rate? If so,
> explain the relationship.
>
> d. Do any of the census tracts of Boston appear to have particularly high
> crime rates? Tax rates? Pupil-teacher ratios? Comment on the range of each
> predictor.
>
> e. How many of the census tracts in this data set bound the Charles river?
>
> f. What is the median pupil-teacher ratio among the towns in this data set?
>
> g. Which census tract of Boston has lowest median value of owner-occupied
> homes? What are the values of the other predictors for that census tract,
> and how do those values compare to the overall ranges for those predictors?
> Comment on your findings.
>
> h. In this data set, how many of the census tract average more than seven
> rooms per dwelling? More than eight rooms per dwelling? Comment on the
> census tracts that average more than eight rooms per dwelling.
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