Ch04-classification-lab.Rmd

# Logistic Regression, LDA, QDA, and KNN

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## The Stock Market Data

In this lab we will examine the `Smarket` 
data, which is part of the `ISLP`
library. This data set consists of percentage returns for the S&P 500
stock index over 1,250 days, from the beginning of 2001 until the end
of 2005. For each date, we have recorded the percentage returns for
each of the five previous trading days,  `Lag1`  through
 `Lag5`. We have also recorded  `Volume`  (the number of
shares traded on the previous day, in billions),  `Today`  (the
percentage return on the date in question) and  `Direction`
(whether the market was  `Up`  or  `Down`  on this date).

We start by importing  our libraries at this top level; these are all imports we have seen in previous labs.

```{python}
import numpy as np
import pandas as pd
from matplotlib.pyplot import subplots
import statsmodels.api as sm
from ISLP import load_data
from ISLP.models import (ModelSpec as MS,
                         summarize)
```

We also collect together the new imports needed for this lab.

```{python}
from ISLP import confusion_table
from ISLP.models import contrast
from sklearn.discriminant_analysis import \
     (LinearDiscriminantAnalysis as LDA,
      QuadraticDiscriminantAnalysis as QDA)
from sklearn.naive_bayes import GaussianNB
from sklearn.neighbors import KNeighborsClassifier
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression

```


Now we are ready to load the `Smarket` data.

```{python}
Smarket = load_data('Smarket')
Smarket
```
This gives a truncated listing of the data.
We can see what the variable names are.

```{python}
Smarket.columns
```
We compute the correlation matrix using the `corr()` method
for data frames, which produces a matrix that contains all of
the pairwise correlations among the variables.
 
By instructing `pandas` to use only numeric variables, the `corr()` method does not report a correlation for the `Direction`  variable because it is
 qualitative.

```{python}
Smarket.corr(numeric_only=True)

```
As one would expect, the correlations between the lagged return  variables and
today’s return are close to zero.  The only substantial correlation is between  `Year`  and
 `Volume`. By plotting the data we see that  `Volume`
is increasing over time. In other words, the average number of shares traded
daily increased from 2001 to 2005.


```{python}
Smarket.plot(y='Volume');

```


## Logistic Regression
Next, we will fit a logistic regression model in order to predict
 `Direction`  using  `Lag1`  through  `Lag5`  and
 `Volume`. The `sm.GLM()`  function fits *generalized linear models*, a class of
models that includes logistic regression.  Alternatively,
the function `sm.Logit()` fits a logistic regression
model directly. The syntax of
`sm.GLM()` is similar to that of `sm.OLS()`, except
that we must pass in the argument `family=sm.families.Binomial()`
in order to tell `statsmodels` to run a logistic regression rather than some other
type of generalized linear model.

```{python}
allvars = Smarket.columns.drop(['Today', 'Direction', 'Year'])
design = MS(allvars)
X = design.fit_transform(Smarket)
y = Smarket.Direction == 'Up'
glm = sm.GLM(y,
             X,
             family=sm.families.Binomial())
results = glm.fit()
summarize(results)

```
The smallest *p*-value here is associated with  `Lag1`. The
negative coefficient for this predictor suggests that if the market
had a positive return yesterday, then it is less likely to go up
today. However, at a value of 0.15, the *p*-value is still
relatively large, and so there is no clear evidence of a real
association between  `Lag1`  and  `Direction`.

We use the `params`  attribute of `results`
in order to access just the
coefficients for this fitted model.

```{python}
results.params

```
Likewise we can use the
`pvalues`  attribute to access the *p*-values for the coefficients.

```{python}
results.pvalues

```

The `predict()`  method of `results` can be used to predict the
probability that the market will go up, given values of the
predictors. This method returns predictions
on the probability scale. If no data set is supplied to the `predict()`
function, then the probabilities are computed for the training data
that was used to fit the logistic regression model.
As with linear regression, one can pass an optional `exog` argument consistent
with a design matrix if desired. Here we have
printed only the first ten probabilities.

```{python}
probs = results.predict()
probs[:10]

```
In order to make a prediction as to whether the market will go up or
down on a particular day, we must convert these predicted
probabilities into class labels,  `Up`  or  `Down`.  The
following two commands create a vector of class predictions based on
whether the predicted probability of a market increase is greater than
or less than 0.5.

```{python}
labels = np.array(['Down']*1250)
labels[probs>0.5] = "Up"

```
The `confusion_table()`
function from the `ISLP` package summarizes these predictions, showing   how
many observations were correctly or incorrectly classified. Our function, which is adapted from a similar function
in the module `sklearn.metrics`,  transposes the resulting
matrix and includes row and column labels.
The `confusion_table()` function takes as first argument the
predicted labels, and second argument the true labels.

```{python}
confusion_table(labels, Smarket.Direction)

```

The diagonal elements of the confusion matrix indicate correct
predictions, while the off-diagonals represent incorrect
predictions. Hence our model correctly predicted that the market would
go up on 507 days and that it would go down on 145 days, for a
total of 507 + 145 = 652 correct predictions. The `np.mean()`
function can be used to compute the fraction of days for which the
prediction was correct. In this case, logistic regression correctly
predicted the movement of the market 52.2% of the time.


```{python}
(507+145)/1250, np.mean(labels == Smarket.Direction)

```




At first glance, it appears that the logistic regression model is
working a little better than random guessing. However, this result is
misleading because we trained and tested the model on the same set of
1,250 observations. In other words, $100-52.2=47.8%$ is the
*training* error  rate. As we have seen
previously, the training error rate is often overly optimistic --- it
tends to underestimate the test error rate.  In
order to better assess the accuracy of the logistic regression model
in this setting, we can fit the model using part of the data, and
then examine how well it predicts the *held out* data.  This
will yield a more realistic error rate, in the sense that in practice
we will be interested in our model’s performance not on the data that
we used to fit the model, but rather on days in the future for which
the market’s movements are unknown.

To implement this strategy, we first create a Boolean vector
corresponding to the observations from 2001 through 2004. We  then
use this vector to create a held out data set of observations from
2005.

```{python}
train = (Smarket.Year < 2005)
Smarket_train = Smarket.loc[train]
Smarket_test = Smarket.loc[~train]
Smarket_test.shape

```


The object `train` is a vector of 1,250 elements, corresponding
to the observations in our data set. The elements of the vector that
correspond to observations that occurred before 2005 are set to
`True`, whereas those that correspond to observations in 2005 are
set to `False`.  Hence `train` is a
*boolean*   array, since its
elements are `True` and `False`.  Boolean arrays can be used
to obtain a subset of the rows or columns of a data frame
using the `loc` method. For instance,
the command `Smarket.loc[train]` would pick out a submatrix of the
stock market data set, corresponding only to the dates before 2005,
since those are the ones for which the elements of `train` are
`True`.  The `~` symbol can be used to negate all of the
elements of a Boolean vector. That is, `~train` is a vector
similar to `train`, except that the elements that are `True`
in `train` get swapped to `False` in `~train`, and vice versa.
Therefore, `Smarket.loc[~train]` yields a
subset of the rows of the data frame
of the stock market data containing only the observations for which
`train` is `False`.
The output above indicates that there are 252 such
observations.

We now fit a logistic regression model using only the subset of the
observations that correspond to dates before 2005. We then obtain predicted probabilities of the
stock market going up for each of the days in our test set --- that is,
for the days in 2005.

```{python}
X_train, X_test = X.loc[train], X.loc[~train]
y_train, y_test = y.loc[train], y.loc[~train]
glm_train = sm.GLM(y_train,
                   X_train,
                   family=sm.families.Binomial())
results = glm_train.fit()
probs = results.predict(exog=X_test)

```

Notice that we have trained and tested our model on two completely
separate data sets: training was performed using only the dates before
2005, and testing was performed using only the dates in 2005.

Finally, we compare the predictions for 2005 to the
actual movements of the market over that time period.
We will first store the test and training labels (recall `y_test` is binary).

```{python}
D = Smarket.Direction
L_train, L_test = D.loc[train], D.loc[~train]

```
Now we threshold the
fitted probability at 50% to form
our predicted labels.

```{python}
labels = np.array(['Down']*252)
labels[probs>0.5] = 'Up'
confusion_table(labels, L_test)

```
The test accuracy is about 48% while the error rate is about 52%

```{python}
np.mean(labels == L_test), np.mean(labels != L_test)

```


The `!=` notation means *not equal to*, and so the last command
computes the test set error rate. The results are rather
disappointing: the test error rate is 52%, which is worse than
random guessing! Of course this result is not all that surprising,
given that one would not generally expect to be able to use previous
days’ returns to predict future market performance. (After all, if it
were possible to do so, then the authors of this book would be out
striking it rich rather than writing a statistics textbook.)

We recall that the logistic regression model had very underwhelming
*p*-values associated with all of the predictors, and that the
smallest *p*-value, though not very small, corresponded to
 `Lag1`. Perhaps by removing the variables that appear not to be
helpful in predicting  `Direction`, we can obtain a more
effective model. After all, using predictors that have no relationship
with the response tends to cause a deterioration in the test error
rate (since such predictors cause an increase in variance without a
corresponding decrease in bias), and so removing such predictors may
in turn yield an improvement.  Below we refit the logistic
regression using just  `Lag1`  and  `Lag2`, which seemed to
have the highest predictive power in the original logistic regression
model.

```{python}
model = MS(['Lag1', 'Lag2']).fit(Smarket)
X = model.transform(Smarket)
X_train, X_test = X.loc[train], X.loc[~train]
glm_train = sm.GLM(y_train,
                   X_train,
                   family=sm.families.Binomial())
results = glm_train.fit()
probs = results.predict(exog=X_test)
labels = np.array(['Down']*252)
labels[probs>0.5] = 'Up'
confusion_table(labels, L_test)

```


Let’s evaluate the overall accuracy as well as the accuracy within the days when
logistic regression predicts an increase.

```{python}
(35+106)/252,106/(106+76)

```

Now the results appear to be a little better: 56% of the daily
movements have been correctly predicted. It is worth noting that in
this case, a much simpler strategy of predicting that the market will
increase every day will also be correct 56% of the time! Hence, in
terms of overall error rate, the logistic regression method is no
better than the naive approach. However, the confusion matrix
shows that on days when logistic regression predicts an increase in
the market, it has a 58% accuracy rate. This suggests a possible
trading strategy of buying on days when the model predicts an
increasing market, and avoiding trades on days when a decrease is
predicted. Of course one would need to investigate more carefully
whether this small improvement was real or just due to random chance.

Suppose that we want to predict the returns associated with particular
values of  `Lag1`  and  `Lag2`. In particular, we want to
predict  `Direction`  on a day when  `Lag1`  and
 `Lag2`  equal $1.2$ and $1.1$, respectively, and on a day when they
equal $1.5$ and $-0.8$.  We do this using the `predict()`
function.

```{python}
newdata = pd.DataFrame({'Lag1':[1.2, 1.5],
                        'Lag2':[1.1, -0.8]});
newX = model.transform(newdata)
results.predict(newX)

```


## Linear Discriminant Analysis


We begin by performing LDA on the  `Smarket`  data, using the function
`LinearDiscriminantAnalysis()`, which we have abbreviated `LDA()`. We 
fit the model using only the observations before 2005.

```{python}
lda = LDA(store_covariance=True)

```

Since the `LDA` estimator automatically 
adds an intercept, we should remove the column corresponding to the
intercept in both `X_train` and `X_test`. We can also directly
use the labels rather than the Boolean vectors `y_train`.

```{python}
X_train, X_test = [M.drop(columns=['intercept'])
                   for M in [X_train, X_test]]
lda.fit(X_train, L_train)

```
Here we have used the list comprehensions introduced
in Section~\ref{Ch3-linreg-lab:multivariate-goodness-of-fit}. Looking at our first line above, we see that the right-hand side is a list
of length two. This is because the code `for M in [X_train, X_test]` iterates over a list
of length two. While here we loop over a list,
the list comprehension method works when looping over any iterable object.
We then apply the `drop()` method to each element in the iteration, collecting
the result in a list. The left-hand side tells `Python` to unpack this list
of length two, assigning its elements to the variables `X_train` and `X_test`. Of course,
this overwrites the previous values of `X_train` and `X_test`.

Having fit the model, we can extract the means in the two classes with the `means_` attribute. These are the average of each predictor within each class, and
are used by LDA as estimates of $\mu_k$.  These suggest that there is
a tendency for the previous 2 days’ returns to be negative on days
when the market increases, and a tendency for the previous days’
returns to be positive on days when the market declines.

```{python}
lda.means_

```


The estimated prior probabilities are stored in the `priors_` attribute.
The package `sklearn` typically uses this trailing `_` to denote
a quantity estimated when using the `fit()` method. We can be sure of which
entry corresponds to which label by looking at the `classes_` attribute.

```{python}
lda.classes_

```


The LDA output indicates that $\hat\pi_{Down}=0.492$ and
$\hat\pi_{Up}=0.508$.


```{python}
lda.priors_

```


The linear discriminant vectors can be found in the `scalings_` attribute:

```{python}
lda.scalings_

```

These values provide the linear combination of `Lag1`  and `Lag2`  that are used to form the LDA decision rule. In other words, these are the multipliers of the elements of $X=x$ in (\ref{Ch4:bayes.multi}).
  If $-0.64\times `Lag1`  - 0.51 \times `Lag2` $ is large, then the LDA classifier will predict a market increase, and if it is small, then the LDA classifier will predict a market decline.

```{python}
lda_pred = lda.predict(X_test)

```

As we observed in our comparison of classification methods
 (Section~\ref{Ch4:comparison.sec}),  the LDA and logistic
regression predictions are almost identical.

```{python}
confusion_table(lda_pred, L_test)

```


We can also estimate the
probability of each class for
each point in a training set. Applying a 50% threshold to the posterior probabilities of
being in class one allows us to
recreate the predictions contained in `lda_pred`.

```{python}
lda_prob = lda.predict_proba(X_test)
np.all(
       np.where(lda_prob[:,1] >= 0.5, 'Up','Down') == lda_pred
       )

```


Above, we used the `np.where()`  function that
creates an array with value `'Up'` for indices where
the second column of `lda_prob` (the estimated
posterior probability of `'Up'`) is greater than 0.5.
For problems with more than two classes the labels are chosen as the class whose posterior probability is highest:

```{python}
np.all(
       [lda.classes_[i] for i in np.argmax(lda_prob, 1)] == lda_pred
       )

```


If we wanted to use a posterior probability threshold other than
50% in order to make predictions, then we could easily do so. For
instance, suppose that we wish to predict a market decrease only if we
are very certain that the market will indeed decrease on that
day --- say, if the posterior probability is at least 90%.
We know that the first column of `lda_prob` corresponds to the
label `Down` after having checked the `classes_` attribute, hence we use
the column index 0 rather than 1 as we did above.

```{python}
np.sum(lda_prob[:,0] > 0.9)

```

No days in 2005 meet that threshold! In fact, the greatest posterior
probability of decrease in all of 2005 was 52.02%.

The LDA classifier above is the first classifier from the
`sklearn` library. We will use several other objects
from this library. The objects
follow a common structure that simplifies tasks such as cross-validation,
which we will see in Chapter~\ref{Ch5:resample}. Specifically,
the methods first create a generic classifier without
referring to any data. This classifier is then fit
to data with the `fit()`  method and predictions are
always produced with the `predict()` method. This pattern
of first instantiating the classifier, followed by fitting it, and
then producing predictions is an explicit design choice of `sklearn`. This uniformity
makes it possible to cleanly copy the classifier so that it can be fit
on different data; e.g. different training sets arising in cross-validation.
This standard pattern also allows for a predictable formation of workflows.



## Quadratic Discriminant Analysis
We will now fit a QDA model to the  `Smarket`  data. QDA is
implemented via
`QuadraticDiscriminantAnalysis()`
in the `sklearn` package, which we abbreviate to `QDA()`.
The syntax is very similar to `LDA()`.

```{python}
qda = QDA(store_covariance=True)
qda.fit(X_train, L_train)

```

The `QDA()` function will again compute `means_` and `priors_`.

```{python}
qda.means_, qda.priors_

```

The `QDA()` classifier will estimate one covariance per class. Here is the
estimated covariance in the first class:

```{python}
qda.covariance_[0]

```
The output contains the group means. But it does not contain the
coefficients of the linear discriminants, because the QDA classifier
involves a quadratic, rather than a linear, function of the
predictors. The `predict()`  function works in exactly the
same fashion as for LDA.

```{python}
qda_pred = qda.predict(X_test)
confusion_table(qda_pred, L_test)

```
Interestingly, the QDA predictions are accurate almost 60% of the
time, even though the 2005 data was not used to fit the model.

```{python}
np.mean(qda_pred == L_test)

```


This level of accuracy is quite impressive for stock market data, which is
known to be quite hard to model accurately.  This suggests that the
quadratic form assumed by QDA may capture the true relationship more
accurately than the linear forms assumed by LDA and logistic
regression.  However, we recommend evaluating this method’s
performance on a larger test set before betting that this approach
will consistently beat the market!


## Naive Bayes
Next, we fit a naive Bayes model to the `Smarket` data. The syntax is
similar to that of `LDA()` and `QDA()`. By
default, this implementation `GaussianNB()` of the naive Bayes classifier models each
quantitative feature using a Gaussian distribution. However, a kernel
density method can also be used to estimate the distributions.

```{python}
NB = GaussianNB()
NB.fit(X_train, L_train)

```


The classes are stored as `classes_`.

```{python}
NB.classes_

```


The class prior probabilities are stored in the `class_prior_` attribute.

```{python}
NB.class_prior_

```


The parameters of the features can be found in the `theta_` and `var_` attributes. The number of rows
is equal to the number of classes, while the number of columns is equal to the number of features.
We see below that the mean for feature `Lag1` in the `Down` class is 0.043.

```{python}
NB.theta_

```

Its variance is 1.503.

```{python}
NB.var_

```
How do we know the names of these attributes? We use `NB?` (or `?NB`).

We can easily verify the mean computation:

```{python}
X_train[L_train == 'Down'].mean()

```


Similarly for the variance:

```{python}
X_train[L_train == 'Down'].var(ddof=0)

```
Since `NB()` is a classifier in the `sklearn` library, making predictions
uses the same syntax as for `LDA()` and `QDA()` above.

```{python}
nb_labels = NB.predict(X_test)
confusion_table(nb_labels, L_test)

```

Naive Bayes performs well on these data, with accurate predictions over 59% of the time. This is slightly worse than QDA, but much better than LDA.

As for `LDA`, the `predict_proba()` method estimates the probability that each observation belongs to a particular class.

```{python}
NB.predict_proba(X_test)[:5]

```


## K-Nearest Neighbors
We will now perform KNN using the `KNeighborsClassifier()` function. This function works similarly
to the other model-fitting functions that we have
encountered thus far.

As is the
case for LDA and QDA, we fit the classifier
using the `fit` method. New
predictions are formed using the `predict` method
of the object returned by `fit()`.

```{python}
knn1 = KNeighborsClassifier(n_neighbors=1)
X_train, X_test = [np.asarray(X) for X in [X_train, X_test]]
knn1.fit(X_train, L_train)
knn1_pred = knn1.predict(X_test)
confusion_table(knn1_pred, L_test)

```

The results using $K=1$ are not very good, since only $50%$ of the
observations are correctly predicted. Of course, it may be that $K=1$
results in an overly-flexible fit to the data.

```{python}
(83+43)/252, np.mean(knn1_pred == L_test)

```

We repeat the
analysis below using $K=3$.

```{python}
knn3 = KNeighborsClassifier(n_neighbors=3)
knn3_pred = knn3.fit(X_train, L_train).predict(X_test)
np.mean(knn3_pred == L_test)
```
The results have improved slightly. But increasing *K* further
provides no further improvements. It appears that for these data, and this train/test split,
QDA gives the best results of the methods that we have examined so
far.


KNN does not perform well on the `Smarket`  data, but it often does provide impressive results. As an example we will apply the KNN approach to the `Caravan`  data set, which is part of the `ISLP` library.  This data set includes 85
predictors that measure demographic characteristics for 5,822
individuals. The response variable is  `Purchase`, which
indicates whether or not a given individual purchases a caravan
insurance policy. In this data set, only 6% of people purchased
caravan insurance.

```{python}
Caravan = load_data('Caravan')
Purchase = Caravan.Purchase
Purchase.value_counts()

```


The method `value_counts()` takes a `pd.Series` or `pd.DataFrame` and returns
a `pd.Series` with the corresponding counts
for each unique element. In this case `Purchase` has only `Yes` and `No` values
and the method returns how many values of each there are.

```{python}
348 / 5822

```


Our features will include all columns except `Purchase`.

```{python}
feature_df = Caravan.drop(columns=['Purchase'])

```

Because the KNN classifier predicts the class of a given test
observation by identifying the observations that are nearest to it,
the scale of the variables matters. Any variables that are on a large
scale will have a much larger effect on the *distance* between
the observations, and hence on the KNN classifier, than variables that
are on a small scale. For instance, imagine a data set that contains
two variables,  `salary`  and  `age`  (measured in dollars
and years, respectively). As far as KNN is concerned, a difference of
1,000 USD in salary is enormous compared to a difference of 50 years in
age. Consequently,  `salary`  will drive the KNN classification
results, and  `age`  will have almost no effect. This is contrary
to our intuition that a salary difference of 1,000 USD is quite small
compared to an age difference of 50 years.  Furthermore, the
importance of scale to the KNN classifier leads to another issue: if
we measured  `salary`  in Japanese yen, or if we measured
 `age`  in minutes, then we’d get quite different classification
results from what we get if these two variables are measured in
dollars and years.

A good way to handle this problem is to *standardize*  the data so that all variables are
given a mean of zero and a standard deviation of one. Then all
variables will be on a comparable scale. This is accomplished
using
the `StandardScaler()`
transformation.

```{python}
scaler = StandardScaler(with_mean=True,
                        with_std=True,
                        copy=True)

```
The argument `with_mean` indicates whether or not
we should subtract the mean, while `with_std` indicates
whether or not we should scale the columns to have standard
deviation of 1 or not. Finally, the argument `copy=True`
indicates that we will always copy data, rather than
trying to do calculations in place where possible.

This transformation can be fit
and then applied to arbitrary data. In the first line
below, the parameters for the scaling are computed and
stored in `scaler`, while the second line actually
constructs the standardized set of features.

```{python}
scaler.fit(feature_df)
X_std = scaler.transform(feature_df)

```
Now every column of `feature_std` below has a standard deviation of
one and a mean of zero.

```{python}
feature_std = pd.DataFrame(
                 X_std,
                 columns=feature_df.columns);
feature_std.std()

```

Notice that the standard deviations are not quite $1$ here; this is again due to some procedures using the $1/n$ convention for variances (in this case `scaler()`), while others use $1/(n-1)$ (the `std()` method). See the footnote on page~\pageref{Ch4-varformula}.
In this case it does not matter, as long as the variables are all on the same scale.

Using the function `train_test_split()`  we now split the observations into a test set,
containing 1000 observations, and a training set containing the remaining
observations. The argument `random_state=0` ensures that we get
the same split each time we rerun the code.

```{python}
(X_train,
 X_test,
 y_train,
 y_test) = train_test_split(np.asarray(feature_std),
                            Purchase,
                            test_size=1000,
                            random_state=0)

```
`?train_test_split` reveals that the non-keyword arguments can be `lists`, `arrays`, `pandas dataframes` etc that all have the same length (`shape[0]`) and hence are *indexable*. In this case they are the dataframe `feature_std` and the response variable `Purchase`.
 {Note that we have converted `feature_std` to an `ndarray` to address a bug in `sklearn`.}
We fit a KNN model on the training data using $K=1$,
and evaluate its performance on the test data.

```{python}
knn1 = KNeighborsClassifier(n_neighbors=1)
knn1_pred = knn1.fit(X_train, y_train).predict(X_test)
np.mean(y_test != knn1_pred), np.mean(y_test != "No")

```


The KNN error rate on the 1,000 test observations is about $11%$.
At first glance, this may appear to be fairly good. However, since
just over 6% of customers purchased insurance, we could get the error
rate down to almost 6% by always predicting `No` regardless of the
values of the predictors! This is known as the *null rate*.}

Suppose that there is some non-trivial cost to trying to sell
insurance to a given individual. For instance, perhaps a salesperson
must visit each potential customer.  If the company tries to sell
insurance to a random selection of customers, then the success rate
will be only 6%, which may be far too low given the costs
involved.  Instead, the company would like to try to sell insurance
only to customers who are likely to buy it. So the overall error rate
is not of interest. Instead, the fraction of individuals that are
correctly predicted to buy insurance is of interest.

```{python}
confusion_table(knn1_pred, y_test)

```

It turns out that KNN with $K=1$ does far better than random guessing
among the customers that are predicted to buy insurance. Among 62
such customers, 9, or 14.5%, actually do purchase insurance.
This is double the rate that one would obtain from random guessing.

```{python}
9/(53+9)
```


### Tuning Parameters

The number of neighbors in KNN is referred to as a *tuning parameter*, also referred to as a *hyperparameter*.
We do not know *a priori* what value to use. It is therefore of interest
to see how the classifier performs on test data as we vary these
parameters. This can be achieved with a `for` loop, described in Section~\ref{Ch2-statlearn-lab:for-loops}.
Here we use a for loop to look at the accuracy of our classifier in the group predicted to purchase
insurance as we vary the number of neighbors from 1 to 5:

```{python}
for K in range(1,6):
    knn = KNeighborsClassifier(n_neighbors=K)
    knn_pred = knn.fit(X_train, y_train).predict(X_test)
    C = confusion_table(knn_pred, y_test)
    templ = ('K={0:d}: # predicted to rent: {1:>2},' +
            '  # who did rent {2:d}, accuracy {3:.1%}')
    pred = C.loc['Yes'].sum()
    did_rent = C.loc['Yes','Yes']
    print(templ.format(
          K,
          pred,
          did_rent,
          did_rent / pred))

```
We see some variability ---  the numbers for `K=4` are very different from the rest.

### Comparison to Logistic Regression
As a comparison, we can also fit a logistic regression model to the
data. This can also be done
with `sklearn`, though by default it fits
something like the *ridge regression* version
of logistic regression, which we introduce in Chapter~\ref{Ch6:varselect}. This can
be modified by appropriately setting the argument `C` below. Its default
value is 1 but by setting it to a very large number, the algorithm converges to the same solution as the usual (unregularized)
logistic regression estimator discussed above.

Unlike the
`statsmodels` package, `sklearn` focuses less on
inference and more on classification. Hence,
the `summary` methods seen in `statsmodels`
and our simplified version seen with `summarize` are not
generally available for the classifiers in `sklearn`.

```{python}
logit = LogisticRegression(C=1e10, solver='liblinear')
logit.fit(X_train, y_train)
logit_pred = logit.predict_proba(X_test)
logit_labels = np.where(logit_pred[:,1] > .5, 'Yes', 'No')
confusion_table(logit_labels, y_test)

```

We used the argument `solver='liblinear'` above to
avoid a warning with the default solver which would indicate that
the algorithm does not converge.

If we use $0.5$ as the predicted probability cut-off for the
classifier, then we have a problem: only two of the test observations
are predicted to purchase insurance.  However, we are not required to use a
cut-off of $0.5$. If we instead predict a purchase any time the
predicted probability of purchase exceeds $0.25$, we get much better
results: we predict that 29 people will purchase insurance, and we are
correct for about 31% of these people. This is almost five times
better than random guessing!

```{python}
logit_labels = np.where(logit_pred[:,1]>0.25, 'Yes', 'No')
confusion_table(logit_labels, y_test)
```

```{python}
9/(20+9)

```
## Linear and Poisson Regression on the Bikeshare Data
Here we fit linear and  Poisson regression models to the `Bikeshare` data, as described in Section~\ref{Ch4:sec:pois}.
The response `bikers` measures the number of bike rentals per hour
in Washington, DC in the period 2010--2012.

```{python}
Bike = load_data('Bikeshare')

```
Let's have a peek at the dimensions and names of the variables in this dataframe.

```{python}
Bike.shape, Bike.columns

```

### Linear Regression

We begin by fitting a linear regression model to the data.

```{python}
X = MS(['mnth',
        'hr',
        'workingday',
        'temp',
        'weathersit']).fit_transform(Bike)
Y = Bike['bikers']
M_lm = sm.OLS(Y, X).fit()
summarize(M_lm)

```

There are 24 levels in `hr` and 40 rows in all.
In `M_lm`, the first levels `hr[0]` and `mnth[Jan]`  are treated
as the baseline values, and so no coefficient estimates are provided
for them: implicitly, their coefficient estimates are zero, and all
other levels are measured relative to these baselines. For example,
the Feb coefficient of $6.845$ signifies that, holding all other
variables constant, there are on average about 7 more riders in
February than in January. Similarly there are about 16.5 more riders
in March than in January.

The results seen in Section~\ref{sec:bikeshare.linear}
used a slightly different coding of the variables `hr` and `mnth`, as follows:

```{python}
hr_encode = contrast('hr', 'sum')
mnth_encode = contrast('mnth', 'sum')

```
Refitting again:

```{python}
X2 = MS([mnth_encode,
         hr_encode,
        'workingday',
        'temp',
        'weathersit']).fit_transform(Bike)
M2_lm = sm.OLS(Y, X2).fit()
S2 = summarize(M2_lm)
S2
```

What is the difference between the two codings?  In `M2_lm`, a
coefficient estimate is reported for all but level `23` of `hr`
and level `Dec` of `mnth`. Importantly, in `M2_lm`, the (unreported) coefficient estimate
for the last level of `mnth` is not zero: instead, it equals the
negative of the sum of the coefficient estimates for all of the
other levels. Similarly, in `M2_lm`, the coefficient estimate
for the last level of `hr` is the negative of the sum of the
coefficient estimates for all of the other levels. This means that the
coefficients of `hr` and `mnth` in `M2_lm` will always sum
to zero, and can be interpreted as the difference from the mean
level. For example, the coefficient for January of $-46.087$ indicates
that, holding all other variables constant, there are typically 46
fewer riders in January relative to the yearly average.

It is important to realize that the choice of coding really does not
matter, provided that we interpret the  model output correctly in light
of the coding used. For example, we see that the predictions from the
linear model are the same regardless of coding:

```{python}
np.sum((M_lm.fittedvalues - M2_lm.fittedvalues)**2)

```

The sum of squared differences is zero. We can also see this using the
`np.allclose()` function:

```{python}
np.allclose(M_lm.fittedvalues, M2_lm.fittedvalues)

```


To reproduce the left-hand side of Figure~\ref{Ch4:bikeshare}
we must first obtain the coefficient estimates associated with
`mnth`. The coefficients for January through November can be obtained
directly from the `M2_lm` object. The coefficient for December
must be explicitly computed as the negative sum of all the other
months. We first extract all the coefficients for month from
the coefficients of `M2_lm`.

```{python}
coef_month = S2[S2.index.str.contains('mnth')]['coef']
coef_month

```
Next, we append `Dec` as the negative of the sum of all other months.

```{python}
months = Bike['mnth'].dtype.categories
coef_month = pd.concat([
                       coef_month,
                       pd.Series([-coef_month.sum()],
                                  index=['mnth[Dec]'
                                 ])
                       ])
coef_month

```
Finally, to make the plot neater, we’ll just use the first letter of each month, which is the $6$th entry of each of
the labels in the index.

```{python}
fig_month, ax_month = subplots(figsize=(8,8))
x_month = np.arange(coef_month.shape[0])
ax_month.plot(x_month, coef_month, marker='o', ms=10)
ax_month.set_xticks(x_month)
ax_month.set_xticklabels([l[5] for l in coef_month.index], fontsize=20)
ax_month.set_xlabel('Month', fontsize=20)
ax_month.set_ylabel('Coefficient', fontsize=20);

```

Reproducing the  right-hand plot in Figure~\ref{Ch4:bikeshare}  follows a similar process.

```{python}
coef_hr = S2[S2.index.str.contains('hr')]['coef']
coef_hr = coef_hr.reindex(['hr[{0}]'.format(h) for h in range(23)])
coef_hr = pd.concat([coef_hr,
                     pd.Series([-coef_hr.sum()], index=['hr[23]'])
                    ])

```

We now make the hour plot.

```{python}
fig_hr, ax_hr = subplots(figsize=(8,8))
x_hr = np.arange(coef_hr.shape[0])
ax_hr.plot(x_hr, coef_hr, marker='o', ms=10)
ax_hr.set_xticks(x_hr[::2])
ax_hr.set_xticklabels(range(24)[::2], fontsize=20)
ax_hr.set_xlabel('Hour', fontsize=20)
ax_hr.set_ylabel('Coefficient', fontsize=20);

```

### Poisson Regression

Now we fit instead a Poisson regression model to the
`Bikeshare` data. Very little changes, except that we now use the
function `sm.GLM()` with the Poisson family specified:

```{python}
M_pois = sm.GLM(Y, X2, family=sm.families.Poisson()).fit()

```

We can plot the coefficients associated with `mnth` and `hr`, in order to reproduce  Figure~\ref{Ch4:bikeshare.pois}. We first complete these coefficients as before.

```{python}
S_pois = summarize(M_pois)
coef_month = S_pois[S_pois.index.str.contains('mnth')]['coef']
coef_month = pd.concat([coef_month,
                        pd.Series([-coef_month.sum()],
                                   index=['mnth[Dec]'])])
coef_hr = S_pois[S_pois.index.str.contains('hr')]['coef']
coef_hr = pd.concat([coef_hr,
                     pd.Series([-coef_hr.sum()],
                     index=['hr[23]'])])
```
The plotting is as before.

```{python}
fig_pois, (ax_month, ax_hr) = subplots(1, 2, figsize=(16,8))
ax_month.plot(x_month, coef_month, marker='o', ms=10)
ax_month.set_xticks(x_month)
ax_month.set_xticklabels([l[5] for l in coef_month.index], fontsize=20)
ax_month.set_xlabel('Month', fontsize=20)
ax_month.set_ylabel('Coefficient', fontsize=20)
ax_hr.plot(x_hr, coef_hr, marker='o', ms=10)
ax_hr.set_xticklabels(range(24)[::2], fontsize=20)
ax_hr.set_xlabel('Hour', fontsize=20)
ax_hr.set_ylabel('Coefficient', fontsize=20);

```
We compare the fitted values of the two models.
The fitted values are stored in the `fittedvalues` attribute
returned by the `fit()` method for both the linear regression and the Poisson
fits. The linear predictors are stored as the attribute `lin_pred`.

```{python}
fig, ax = subplots(figsize=(8, 8))
ax.scatter(M2_lm.fittedvalues,
           M_pois.fittedvalues,
           s=20)
ax.set_xlabel('Linear Regression Fit', fontsize=20)
ax.set_ylabel('Poisson Regression Fit', fontsize=20)
ax.axline([0,0], c='black', linewidth=3,
          linestyle='--', slope=1);

```

The predictions from the Poisson regression model are correlated with
those from the linear model; however, the former are non-negative. As
a result the Poisson regression predictions tend to be larger than
those from the linear model for either very low or very high levels of
ridership.

In this section, we fit Poisson regression models using the `sm.GLM()` function with the argument
`family=sm.families.Poisson()`. Earlier in this lab we used the `sm.GLM()` function
with `family=sm.families.Binomial()` to perform logistic regression. Other
choices for the `family` argument can be used to fit other types
of GLMs. For instance, `family=sm.families.Gamma()` fits a Gamma regression
model.