4. Logistic Regression

Exercise 13: Classifying high crime rate using the `Boston` dataset

Using the Boston data set, fit classification models in order to predict whether a given suburb has a crime rate above or below the median. Explore logistic regression, LDA, and KNN models using various subsets of the predictors. Describe your findings.

Data decription

This data contains the following columns:

crim: per capita crime rate by town.
zn: proportion of residential land zoned for lots over 25,000 sq.ft.
indus: proportion of non-retail business acres per town.
chas: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox: nitrogen oxides concentration (parts per 10 million).
rm: average number of rooms per dwelling.
age: proportion of owner-occupied units built prior to 1940.
dis: weighted mean of distances to five Boston employment centres.
rad: index of accessibility to radial highways.
tax: full-value property-tax rate per $10,000.
ptratio: pupil-teacher ratio by town.
black: $1000(Bk - 0.63)^2$ where $Bk$ is the proportion of blacks by town.
lstat: lower status of the population (percent).
medv: median value of owner-occupied homes in $1000s.

So chas and rad are qualitative variables, while the rest are quantitative

Prepare the data

import pandas as pd

boston = pd.read_csv('../../datasets/boston.csv', index_col=0).reset_index(drop=True)

boston.head()

	crim	zn	indus	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv
0	0.00632	18.0	2.31	0.538	6.575	65.2	4.0900	1	296	15.3	396.90	4.98	24.0
1	0.02731	0.0	7.07	0.469	6.421	78.9	4.9671	2	242	17.8	396.90	9.14	21.6
2	0.02729	0.0	7.07	0.469	7.185	61.1	4.9671	2	242	17.8	392.83	4.03	34.7
3	0.03237	0.0	2.18	0.458	6.998	45.8	6.0622	3	222	18.7	394.63	2.94	33.4
4	0.06905	0.0	2.18	0.458	7.147	54.2	6.0622	3	222	18.7	396.90	5.33	36.2

boston.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 506 entries, 0 to 505
Data columns (total 14 columns):
crim       506 non-null float64
zn         506 non-null float64
indus      506 non-null float64
chas       506 non-null int64
nox        506 non-null float64
rm         506 non-null float64
age        506 non-null float64
dis        506 non-null float64
rad        506 non-null int64
tax        506 non-null int64
ptratio    506 non-null float64
black      506 non-null float64
lstat      506 non-null float64
medv       506 non-null float64
dtypes: float64(11), int64(3)
memory usage: 55.4 KB

Add high/low crime class variable and drop `crim` variable

We’ll let 1 denote high crime (above the median) and 0 denote below. We also drop crim variable ¹

boston['high_crim'] = (boston['crim'] > boston['crim'].median()).astype('int32')
boston = boston.drop(['crim'], axis=1)
boston.head()

	zn	indus	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv
0	18.0	2.31	0.538	6.575	65.2	4.0900	1	296	15.3	396.90	4.98	24.0
1	0.0	7.07	0.469	6.421	78.9	4.9671	2	242	17.8	396.90	9.14	21.6
2	0.0	7.07	0.469	7.185	61.1	4.9671	2	242	17.8	392.83	4.03	34.7
3	0.0	2.18	0.458	6.998	45.8	6.0622	3	222	18.7	394.63	2.94	33.4
4	0.0	2.18	0.458	7.147	54.2	6.0622	3	222	18.7	396.90	5.33	36.2

Train/test split

from sklearn.model_selection import train_test_split
import statsmodels.api as sm

X_train, X_test, y_train, y_test = train_test_split(sm.add_constant(boston.drop(['high_crim'], axis=1)), boston['high_crim'])

X_train.shape

(379, 14)

X_test.shape

(127, 14)

Dummy classifier

As a baseline, we’ll train a dummy model

from sklearn.dummy import DummyClassifier
from sklearn.metrics import accuracy_score

dummy_model = DummyClassifier()

print("The dummy model has train accuracy {}".format(
      accuracy_score(y_train, dummy_model.fit(X_train, y_train).predict(X_train))))
print("The dummy model has test accuracy {}".format(
      accuracy_score(y_test, dummy_model.fit(X_train, y_train).predict(X_test))))

The dummy model has train accuracy 0.49076517150395776
The dummy model has test accuracy 0.5433070866141733

Naive Logit, LDA and KNN classifiers

As a next step, we’ll train logistic regression, LDA and KNN models ( $K = 1, 3, 5$ ) on the full set of predictors, with no adjustments ²

from sklearn.linear_model import LogisticRegression as Logit
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA
from sklearn.neighbors import KNeighborsClassifier as KNN

import warnings  
warnings.filterwarnings('ignore')

# dict for models
models = {}

# create model instances
models['Logit'] = Logit()
models['LDA'] = LDA()
for i in range(1, 6):
    models['KNN' + str(i)] = KNN(n_neighbors=i)

for model in models:
    print("The naive {} model has train accuracy {}".format(model, 
          round(accuracy_score(y_train, 
                               models[model].fit(X_train, y_train).predict(X_train)), 4)))

/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)


The naive Logit model has train accuracy 0.8707
The naive LDA model has train accuracy 0.8602
The naive KNN1 model has train accuracy 1.0
The naive KNN2 model has train accuracy 0.9631
The naive KNN3 model has train accuracy 0.9551
The naive KNN4 model has train accuracy 0.9578
The naive KNN5 model has train accuracy 0.9393


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

for model in models:
    print("The naive {} model has test accuracy {}".format(
          model, round(accuracy_score(y_test, models[model].predict(X_test)), 4)))

The naive Logit model has test accuracy 0.811
The naive LDA model has test accuracy 0.8031
The naive KNN1 model has test accuracy 0.9213
The naive KNN2 model has test accuracy 0.9291
The naive KNN3 model has test accuracy 0.9055
The naive KNN4 model has test accuracy 0.9134
The naive KNN5 model has test accuracy 0.8898

EDA

Inspection and Visualization

import seaborn as sns
import matplotlib.pyplot as plt

%matplotlib inline
plt.style.use('seaborn-white')
sns.set_style('white')

Pairplot

sns.pairplot(boston, hue='high_crim')

/Users/home/anaconda3/lib/python3.6/site-packages/statsmodels/nonparametric/kde.py:488: RuntimeWarning: invalid value encountered in true_divide
  binned = fast_linbin(X, a, b, gridsize) / (delta * nobs)
/Users/home/anaconda3/lib/python3.6/site-packages/statsmodels/nonparametric/kdetools.py:34: RuntimeWarning: invalid value encountered in double_scalars
  FAC1 = 2*(np.pi*bw/RANGE)**2

<seaborn.axisgrid.PairGrid at 0x110a057f0>

png

Observations about distributions

zn has very low variance in the high crime class, but much higher variance in the low crime class. It is unimodal in both classes.
indus appears bimodal in both classes and may perhaps be modeled as a mixture of normals. The high crime class appears to have a higher mean. The distribution appears to be nearly identical for both classes, so this likely will not be a helpful predictor.
chas is a binary variable heavily concentrated in the value 1 (hence a large majority of observations “bound” the Charles river).
nox appears approximately normal in the classes. The high crime class has a higher mean and a much higher variance
rm appears normal in the classes, but the distributions are very similar across classes. This will likely not be a helpful predictor.
age appears approximately normal in the low crime classes, or at least symmetric. It is unimodal in the classes. The high crime class is heavily skewed to the right with a higher mean and a much lower variance.
dis appears approximately normal in the low crime class. The high crime class is heavily skewed to the left, with a lower mean and much lower variance.
rad is approximately symmetric in both classes, unimodal in the low crime class, and bimodal in the high crime class. The high crime class has a higher mean and much higher variance.
tax is similar to rad but the high crime class is more concentrated around each mode.
ptratio appears approximately normal in the low crime class. The means for both classes are similar. The high crime class is heavily skewed to the right, with high variance.
black has very low variance in both classes. Both classes are heavily skewed to the right, with long tails. The high crime class is less concentrated about its mode.
lstat appears approximately normal in the high crime class which has a higher mean and variance. The low crime class is skewed a bit to the left.
medv appears approximately normal in both classes. The high crime class has a lower mean and higher variance.

Observations about correlations

# strong correlation 
boston.corr()[abs(boston.corr()) > 0.70]

	zn	indus	chas	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv	high_crim
zn	1.0	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
indus	NaN	1.000000	NaN	0.763651	NaN	NaN	-0.708027	NaN	0.720760	NaN	NaN	NaN	NaN	NaN
chas	NaN	NaN	1.0	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
nox	NaN	0.763651	NaN	1.000000	NaN	0.731470	-0.769230	NaN	NaN	NaN	NaN	NaN	NaN	0.723235
rm	NaN	NaN	NaN	NaN	1.0	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
age	NaN	NaN	NaN	0.731470	NaN	1.000000	-0.747881	NaN	NaN	NaN	NaN	NaN	NaN	NaN
dis	NaN	-0.708027	NaN	-0.769230	NaN	-0.747881	1.000000	NaN	NaN	NaN	NaN	NaN	NaN	NaN
rad	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.000000	0.910228	NaN	NaN	NaN	NaN	NaN
tax	NaN	0.720760	NaN	NaN	NaN	NaN	NaN	0.910228	1.000000	NaN	NaN	NaN	NaN	NaN
ptratio	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.0	NaN	NaN	NaN	NaN
black	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.0	NaN	NaN	NaN
lstat	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.000000	-0.737663	NaN
medv	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	-0.737663	1.000000	NaN
high_crim	NaN	NaN	NaN	0.723235	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.000000

With the (somewhat arbitrary) threshold of $0.70$ , we observe the following (distinct) strong correlation pairs

indus is strongly positively correlated with nox and tax and strongly negatively correlated with dis
nox is strongly positively correlated with age and high_crim and strongly negatively correlated with dis
rad is strongly positively correlated with tax
lstat is strongly positively correlated with medv

Violin, strip and count plots

sns.catplot(x='high_crim', y='zn', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='zn', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c22a915c0>

png

sns.catplot(x='high_crim', y='indus', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='indus', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c25f89c50>

png

sns.catplot(x='chas', hue='high_crim', data=boston, kind='count')

<seaborn.axisgrid.FacetGrid at 0x1c22ceb0f0>

png

sns.catplot(x='high_crim', y='nox', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='nox', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c23766240>

png

sns.catplot(x='high_crim', y='rm', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='rm', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c27e2fdd8>

png

sns.catplot(x='high_crim', y='age', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='age', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c27f96470>

png

sns.catplot(x='high_crim', y='dis', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='dis', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c28068550>

png

sns.catplot(x='high_crim', y='tax', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='tax', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c282475f8>

png

sns.catplot(x='high_crim', y='ptratio', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='ptratio', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c283a70f0>

png

sns.catplot(x='high_crim', y='black', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='black', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c2848ba90>

png

sns.catplot(x='high_crim', y='lstat', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='lstat', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c2866e400>

png

sns.catplot(x='high_crim', y='medv', data=boston, kind='violin')
sns.stripplot(x='high_crim', y='medv', data=boston, color='k', alpha=0.3)

<matplotlib.axes._subplots.AxesSubplot at 0x1c287c4a90>

png

Observations

As suggested by the pairplot, the distributions of chas and rm are very similar across classes, so we’ll drop these features.
dis appears to have two outliers in the high crime class (dis > 8)
black appears to have an outlier in the low crime class (black < 100)
tax appears to have 3-4 outliers in the low crime class (tx > 700).³

Feature Selection

Given the 0.91 correlation with tax, let’s drop rad variable. We’ll also drop chas and rm given the similarity in their distribution across classes.

boston_2 = boston.drop(['rad', 'chas', 'rm'], axis=1)

Now we’ll drop the outliers we noticed above

boston_2 = boston_2[(boston_2['dis'] < 8) | (boston_2['high_crim'] == 1)]
boston_2 = boston_2[(boston_2['black'] > 100) | (boston_2['high_crim'] == 1)]
boston_2 = boston_2[(boston_2['tax'] < 700) | (boston_2['high_crim'] == 1)]

and retest

X_train, X_test, y_train, y_test = train_test_split(sm.add_constant(boston_2.drop(['high_crim'], axis=1)), boston_2['high_crim'])

print("The dummy model has train accuracy {}".format(
      accuracy_score(y_train, dummy_model.fit(X_train, y_train).predict(X_train))))
print("The dummy model has test accuracy {}".format(
      accuracy_score(y_test, dummy_model.fit(X_train, y_train).predict(X_test))))

The dummy model has train accuracy 0.49444444444444446
The dummy model has test accuracy 0.49166666666666664

for model in models:
    print("The {} model has train accuracy {}".format(model, 
          round(accuracy_score(y_train, 
                               models[model].fit(X_train, y_train).predict(X_train)), 4)))

The Logit model has train accuracy 0.875
The LDA model has train accuracy 0.8944
The KNN1 model has train accuracy 1.0
The KNN2 model has train accuracy 0.9639
The KNN3 model has train accuracy 0.9583
The KNN4 model has train accuracy 0.9528
The KNN5 model has train accuracy 0.9333


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

for model in models:
    print("The {} model has test accuracy {}".format(model, 
          round(accuracy_score(y_test, 
                               models[model].fit(X_train, y_train).predict(X_test)), 4)))

The Logit model has test accuracy 0.8083
The LDA model has test accuracy 0.8
The KNN1 model has test accuracy 0.9
The KNN2 model has test accuracy 0.925
The KNN3 model has test accuracy 0.9083
The KNN4 model has test accuracy 0.9333
The KNN5 model has test accuracy 0.9167


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

Collinearity

We’re still getting a collinearity warning from the LDA model. Let’s try dropping any more correlated variables

boston_3 = boston_2.drop(['lstat', 'nox', 'indus'], axis=1)

X_train, X_test, y_train, y_test = train_test_split(sm.add_constant(boston_3.drop(['high_crim'], axis=1)), boston_3['high_crim'])

print("The dummy model has train accuracy {}".format(
      accuracy_score(y_train, dummy_model.fit(X_train, y_train).predict(X_train))))
print("The dummy model has test accuracy {}".format(
      accuracy_score(y_test, dummy_model.fit(X_train, y_train).predict(X_test))))

The dummy model has train accuracy 0.5305555555555556
The dummy model has test accuracy 0.5333333333333333

for model in models:
    print("The {} model has train accuracy {}".format(model, 
          round(accuracy_score(y_train, 
                               models[model].fit(X_train, y_train).predict(X_train)), 4)))

The Logit model has train accuracy 0.8667
The LDA model has train accuracy 0.8333
The KNN1 model has train accuracy 1.0
The KNN2 model has train accuracy 0.9583
The KNN3 model has train accuracy 0.95
The KNN4 model has train accuracy 0.9444
The KNN5 model has train accuracy 0.9361


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

for model in models:
    print("The {} model has test accuracy {}".format(model, 
          round(accuracy_score(y_test, 
                               models[model].fit(X_train, y_train).predict(X_test)), 4)))

/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)


The Logit model has test accuracy 0.8417
The LDA model has test accuracy 0.825
The KNN1 model has test accuracy 0.95
The KNN2 model has test accuracy 0.9333
The KNN3 model has test accuracy 0.8833
The KNN4 model has test accuracy 0.9083
The KNN5 model has test accuracy 0.8917


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

Interactions

Our set of features is now small enough

boston_3.head()

	zn	age	dis	tax	ptratio	black	medv
0	18.0	65.2	4.0900	296	15.3	396.90	24.0
1	0.0	78.9	4.9671	242	17.8	396.90	21.6
2	0.0	61.1	4.9671	242	17.8	392.83	34.7
3	0.0	45.8	6.0622	222	18.7	394.63	33.4
4	0.0	54.2	6.0622	222	18.7	396.90	36.2

that we can include all interaction terms and see if we can get improvement

from itertools import combinations

# all pairs of columns except high_crim
col_pairs = combinations(boston_3.drop(['high_crim'], axis=1), 2)

# assemble interactions in dataframe
interactions = pd.DataFrame({col1 + ':' + col2: boston_3[col1]*boston_3[col2] 
                               for (col1, col2) in col_pairs})
interactions.head()

	zn:age	zn:dis	zn:tax	zn:ptratio	zn:black	zn:medv	age:dis	age:tax	age:ptratio	age:black	...	dis:tax	dis:ptratio	dis:black	dis:medv	tax:ptratio	tax:black	tax:medv	ptratio:black	ptratio:medv	black:medv
0	1173.6	73.62	5328.0	275.4	7144.2	432.0	266.66800	19299.2	997.56	25877.880	...	1210.6400	62.57700	1623.321000	98.16000	4528.8	117482.40	7104.0	6072.570	367.20	9525.600
1	0.0	0.00	0.0	0.0	0.0	0.0	391.90419	19093.8	1404.42	31315.410	...	1202.0382	88.41438	1971.441990	107.28936	4307.6	96049.80	5227.2	7064.820	384.48	8573.040
2	0.0	0.00	0.0	0.0	0.0	0.0	303.48981	14786.2	1087.58	24001.913	...	1202.0382	88.41438	1951.225893	172.35837	4307.6	95064.86	8397.4	6992.374	617.66	13631.201
3	0.0	0.00	0.0	0.0	0.0	0.0	277.64876	10167.6	856.46	18074.054	...	1345.8084	113.36314	2392.325986	202.47748	4151.4	87607.86	7414.8	7379.581	624.58	13180.642
4	0.0	0.00	0.0	0.0	0.0	0.0	328.57124	12032.4	1013.54	21511.980	...	1345.8084	113.36314	2406.087180	219.45164	4151.4	88111.80	8036.4	7422.030	676.94	14367.780

5 rows × 21 columns

sns.pairplot(pd.concat([interactions, boston_3['high_crim']], axis=1), hue='high_crim')

/Users/home/anaconda3/lib/python3.6/site-packages/statsmodels/nonparametric/kde.py:488: RuntimeWarning: invalid value encountered in true_divide
  binned = fast_linbin(X, a, b, gridsize) / (delta * nobs)
/Users/home/anaconda3/lib/python3.6/site-packages/statsmodels/nonparametric/kdetools.py:34: RuntimeWarning: invalid value encountered in double_scalars
  FAC1 = 2*(np.pi*bw/RANGE)**2

<seaborn.axisgrid.PairGrid at 0x1c287fa0b8>

png

interactions.corr()[abs(interactions.corr()) > 0.70]

	zn:age	zn:dis	zn:tax	zn:ptratio	zn:black	zn:medv	age:dis	age:tax	age:ptratio	age:black	...	dis:tax	dis:ptratio	dis:black	dis:medv	tax:ptratio	tax:black	tax:medv	ptratio:black	ptratio:medv	black:medv
zn:age	1.000000	0.809044	0.836926	0.862228	0.872434	0.811637	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
zn:dis	0.809044	1.000000	0.977948	0.963280	0.973811	0.912649	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	0.712078	NaN	NaN	NaN	NaN	NaN	NaN
zn:tax	0.836926	0.977948	1.000000	0.965968	0.972944	0.900585	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
zn:ptratio	0.862228	0.963280	0.965968	1.000000	0.991143	0.929864	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
zn:black	0.872434	0.973811	0.972944	0.991143	1.000000	0.956084	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	0.701793	NaN	NaN	NaN	NaN	NaN	NaN
zn:medv	0.811637	0.912649	0.900585	0.929864	0.956084	1.000000	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	0.701289	NaN	NaN	NaN	NaN	NaN	NaN
age:dis	NaN	NaN	NaN	NaN	NaN	NaN	1.0	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
age:tax	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.000000	0.842898	NaN	...	NaN	NaN	-0.763407	-0.745526	0.898734	NaN	NaN	NaN	NaN	NaN
age:ptratio	NaN	NaN	NaN	NaN	NaN	NaN	NaN	0.842898	1.000000	NaN	...	NaN	NaN	-0.731611	-0.786899	NaN	NaN	NaN	NaN	NaN	NaN
age:black	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.0	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
age:medv	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
dis:tax	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	1.000000	0.709582	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
dis:ptratio	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	0.709582	1.000000	0.935323	0.764576	NaN	NaN	NaN	NaN	NaN	NaN
dis:black	NaN	NaN	NaN	NaN	NaN	NaN	NaN	-0.763407	-0.731611	NaN	...	NaN	0.935323	1.000000	0.852583	NaN	NaN	NaN	NaN	NaN	NaN
dis:medv	NaN	0.712078	NaN	NaN	0.701793	0.701289	NaN	-0.745526	-0.786899	NaN	...	NaN	0.764576	0.852583	1.000000	NaN	NaN	NaN	NaN	NaN	NaN
tax:ptratio	NaN	NaN	NaN	NaN	NaN	NaN	NaN	0.898734	NaN	NaN	...	NaN	NaN	NaN	NaN	1.000000	NaN	NaN	NaN	NaN	NaN
tax:black	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	1.0	NaN	NaN	NaN	NaN
tax:medv	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	1.0	NaN	NaN	NaN
ptratio:black	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.0	NaN	NaN
ptratio:medv	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	1.00000	0.90046
black:medv	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	0.90046	1.00000

21 rows × 21 columns

drop_cols = [col_name for col_name in interactions.columns if 'zn' in col_name]
drop_cols += ['dis:medv', 'age:ptratio', 'dis:black', 'tax:ptratio', 'dis:ptratio', 'dis:black', 'ptratio:medv',
              'age:dis', 'age:medv', 'dis:tax', 'tax:medv']
interactions = interactions.drop(drop_cols, axis=1)

boston_4 = pd.concat([boston_3.drop(['high_crim'], axis=1), interactions, boston_3['high_crim']], axis=1)
boston_4.head()

	zn	age	dis	tax	ptratio	black	medv	age:tax	age:black	tax:black	ptratio:black	black:medv
0	18.0	65.2	4.0900	296	15.3	396.90	24.0	19299.2	25877.880	117482.40	6072.570	9525.600
1	0.0	78.9	4.9671	242	17.8	396.90	21.6	19093.8	31315.410	96049.80	7064.820	8573.040
2	0.0	61.1	4.9671	242	17.8	392.83	34.7	14786.2	24001.913	95064.86	6992.374	13631.201
3	0.0	45.8	6.0622	222	18.7	394.63	33.4	10167.6	18074.054	87607.86	7379.581	13180.642
4	0.0	54.2	6.0622	222	18.7	396.90	36.2	12032.4	21511.980	88111.80	7422.030	14367.780

# standardize everything except high_crim since interaction terms are so much bigger
boston_4 = boston_4.drop(['high_crim'], axis=1)
boston_4 = (boston_4 - boston_4.mean())/boston_4.std()
boston_4 = pd.concat([boston_4, boston_3['high_crim']], axis=1)

X_train, X_test, y_train, y_test = train_test_split(sm.add_constant(boston_4.drop(['high_crim'], axis=1)), boston_4['high_crim'])

print("The dummy model has train accuracy {}".format(
      accuracy_score(y_train, dummy_model.fit(X_train, y_train).predict(X_train))))
print("The dummy model has test accuracy {}".format(
      accuracy_score(y_test, dummy_model.fit(X_train, y_train).predict(X_test))))

The dummy model has train accuracy 0.5055555555555555
The dummy model has test accuracy 0.55

for model in models:
    print("The {} model has train accuracy {}".format(model, 
          round(accuracy_score(y_train, 
                               models[model].fit(X_train, y_train).predict(X_train)), 4)))

The Logit model has train accuracy 0.8472
The LDA model has train accuracy 0.8444
The KNN1 model has train accuracy 1.0
The KNN2 model has train accuracy 0.95
The KNN3 model has train accuracy 0.9444
The KNN4 model has train accuracy 0.9194
The KNN5 model has train accuracy 0.9222


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

for model in models:
    print("The {} model has test accuracy {}".format(model, 
          round(accuracy_score(y_test, 
                               models[model].fit(X_train, y_train).predict(X_test)), 4)))

The Logit model has test accuracy 0.8333
The LDA model has test accuracy 0.825
The KNN1 model has test accuracy 0.8833
The KNN2 model has test accuracy 0.8917
The KNN3 model has test accuracy 0.9083
The KNN4 model has test accuracy 0.8667
The KNN5 model has test accuracy 0.8833


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

The KNN3 and KNN4 models are better with these interactions, but still tied with KNN2 for first with 93.3% accuracy

Nonlinearity and transformations

Some naive transformations

We weren’t able to beat the accuracy of KNN2 by including interactions. Let’s try some transformations on the original features

from numpy import log

df = boston_3.drop(['high_crim'], axis=1)

# log df
log_df = log(df.drop(['zn'], axis=1)).rename({col: 'log(' + col + ')' for col in boston_3.columns}, axis='columns')

# sq df
sq_df = df.drop(['zn'], axis=1)**2
sq_df = sq_df.rename({col: col + '^2' for col in boston_3.columns}, axis='columns')
sq_df.head()

# normalize
log_df = (log_df - log_df.mean())/log_df.std()
sq_df = (sq_df - sq_df.mean())/sq_df.std()

# concat
boston_5 = pd.concat([df, log_df, sq_df, boston_3['high_crim']], axis=1)
boston_5.head()

	zn	age	dis	tax	ptratio	black	medv	log(age)	log(dis)	log(tax)	log(ptratio)	log(black)	log(medv)	age^2	dis^2	tax^2	ptratio^2	black^2	medv^2
0	18.0	65.2	4.0900	296	15.3	396.90	24.0	0.107596	0.507131	-0.612061	-1.415498	0.304822	0.347164	-0.401921	0.035014	-0.687747	-1.436963	0.526018	-0.038104
1	0.0	78.9	4.9671	242	17.8	396.90	21.6	0.425009	0.890189	-1.119060	-0.217665	0.304822	0.090660	0.176982	0.538601	-0.872406	-0.361721	0.526018	-0.243573
2	0.0	61.1	4.9671	242	17.8	392.83	34.7	-0.000495	0.890189	-1.119060	-0.217665	0.291773	1.244740	-0.553769	0.538601	-0.872406	-0.361721	0.450105	1.141108
3	0.0	45.8	6.0622	222	18.7	394.63	33.4	-0.480184	1.282996	-1.336186	0.172721	0.297561	1.151781	-1.033382	1.304266	-0.931391	0.065127	0.483582	0.974897
4	0.0	54.2	6.0622	222	18.7	396.90	36.2	-0.199926	1.282996	-1.336186	0.172721	0.304822	1.347768	-0.787061	1.304266	-0.931391	0.065127	0.526018	1.340776

boston_5.info()

<class 'pandas.core.frame.DataFrame'>
Int64Index: 480 entries, 0 to 505
Data columns (total 20 columns):
zn              480 non-null float64
age             480 non-null float64
dis             480 non-null float64
tax             480 non-null int64
ptratio         480 non-null float64
black           480 non-null float64
medv            480 non-null float64
log(age)        480 non-null float64
log(dis)        480 non-null float64
log(tax)        480 non-null float64
log(ptratio)    480 non-null float64
log(black)      480 non-null float64
log(medv)       480 non-null float64
age^2           480 non-null float64
dis^2           480 non-null float64
tax^2           480 non-null float64
ptratio^2       480 non-null float64
black^2         480 non-null float64
medv^2          480 non-null float64
high_crim       480 non-null int32
dtypes: float64(18), int32(1), int64(1)
memory usage: 96.9 KB

X_train, X_test, y_train, y_test = train_test_split(sm.add_constant(boston_5.drop(['high_crim'], axis=1)), boston_5['high_crim'])

print("The dummy model has train accuracy {}".format(
      accuracy_score(y_train, dummy_model.fit(X_train, y_train).predict(X_train))))
print("The dummy model has test accuracy {}".format(
      accuracy_score(y_test, dummy_model.fit(X_train, y_train).predict(X_test))))

The dummy model has train accuracy 0.49166666666666664
The dummy model has test accuracy 0.5416666666666666

for model in models:
    print("The {} model has train accuracy {}".format(model, 
          round(accuracy_score(y_train, 
                               models[model].fit(X_train, y_train).predict(X_train)), 4)))

The Logit model has train accuracy 0.8694
The LDA model has train accuracy 0.9
The KNN1 model has train accuracy 1.0
The KNN2 model has train accuracy 0.9639
The KNN3 model has train accuracy 0.9583
The KNN4 model has train accuracy 0.9528
The KNN5 model has train accuracy 0.9472


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

for model in models:
    print("The {} model has test accuracy {}".format(model, 
          round(accuracy_score(y_test, 
                               models[model].fit(X_train, y_train).predict(X_test)), 4)))

The Logit model has test accuracy 0.8583
The LDA model has test accuracy 0.875
The KNN1 model has test accuracy 0.925
The KNN2 model has test accuracy 0.95
The KNN3 model has test accuracy 0.95
The KNN4 model has test accuracy 0.9417
The KNN5 model has test accuracy 0.9583


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")

Recursive feature elimination

As an alternative to the visual methods of feature elimination used above, let’s use sklearn’s recursive feature selection, which works for the parametric models.

The parametric models have been underperforming the KNN models throughout, so this will perhaps give them an advantage.

On the original data set

boston['const'] = 1
boston = pd.concat([boston['const'], boston.drop(['high_crim', 'const'], axis=1), boston['high_crim']], axis=1)
boston.head()

	const	zn	indus	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv
0	1	18.0	2.31	0.538	6.575	65.2	4.0900	1	296	15.3	396.90	4.98	24.0
1	1	0.0	7.07	0.469	6.421	78.9	4.9671	2	242	17.8	396.90	9.14	21.6
2	1	0.0	7.07	0.469	7.185	61.1	4.9671	2	242	17.8	392.83	4.03	34.7
3	1	0.0	2.18	0.458	6.998	45.8	6.0622	3	222	18.7	394.63	2.94	33.4
4	1	0.0	2.18	0.458	7.147	54.2	6.0622	3	222	18.7	396.90	5.33	36.2

# standardize appropriate columns
cols = [col for col in boston.columns if col not in {'const', 'chas', 'high_crim'}]
df = boston[cols]
df = (df - df.mean()) / df.std()
df = pd.concat([boston.const, boston.chas, df, boston.high_crim], axis=1)
df.head()

	const	zn	indus	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv
0	1	0.284548	-1.286636	-0.144075	0.413263	-0.119895	0.140075	-0.981871	-0.665949	-1.457558	0.440616	-1.074499	0.159528
1	1	-0.487240	-0.592794	-0.739530	0.194082	0.366803	0.556609	-0.867024	-0.986353	-0.302794	0.440616	-0.491953	-0.101424
2	1	-0.487240	-0.592794	-0.739530	1.281446	-0.265549	0.556609	-0.867024	-0.986353	-0.302794	0.396035	-1.207532	1.322937
3	1	-0.487240	-1.305586	-0.834458	1.015298	-0.809088	1.076671	-0.752178	-1.105022	0.112920	0.415751	-1.360171	1.181589
4	1	-0.487240	-1.305586	-0.834458	1.227362	-0.510674	1.076671	-0.752178	-1.105022	0.112920	0.440616	-1.025487	1.486032

X, y = df.drop(['high_crim'], axis=1), df['high_crim']

from sklearn.feature_selection import RFECV

Logit_rfecv = RFECV(models['Logit'], cv=10).fit(X, y)
Logit_rfecv.ranking_

/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
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  FutureWarning)





array([ 2,  6,  1,  8,  1, 10,  5,  3,  1,  1,  7,  1,  9,  4])

LDA_rfecv = RFECV(models['LDA'], cv=10).fit(X, y)
LDA_rfecv.ranking_

/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")





array([14, 13,  8, 11,  1, 12,  3, 10,  2,  5,  6,  7,  9,  4])

Logit_df = boston[boston.columns[list(Logit_rfecv.support_) + [True]]]

X_train, X_test, y_train, y_test = train_test_split(Logit_df.drop(['high_crim'], axis=1), Logit_df['high_crim'])

print("The Logit model has train accuracy {}".format(
      round(accuracy_score(y_train, 
                               models['Logit'].fit(X_train, y_train).predict(X_train)), 4)))
print("The Logit model has test accuracy {}".format(
      round(accuracy_score(y_test, 
                               models['Logit'].fit(X_train, y_train).predict(X_test)), 4)))

The Logit model has train accuracy 0.8496
The Logit model has test accuracy 0.874


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)

LDA_df = boston[boston.columns[list(LDA_rfecv.support_) + [True]]]

X_train, X_test, y_train, y_test = train_test_split(LDA_df.drop(['high_crim'], axis=1), LDA_df['high_crim'])

print("The LDA model has train accuracy {}".format(
      round(accuracy_score(y_train, 
                               models['LDA'].fit(X_train, y_train).predict(X_train)), 4)))
print("The LDA model has test accuracy {}".format(
      round(accuracy_score(y_test, 
                               models['LDA'].fit(X_train, y_train).predict(X_test)), 4)))

The LDA model has train accuracy 0.8232
The LDA model has test accuracy 0.8346

On the original data + interactions

# all pairs of columns in except high_crim
col_pairs = combinations(boston.drop(['high_crim', 'const'], axis=1), 2)

# assemble interactions in dataframe
interactions = pd.DataFrame({col1 + ':' + col2: boston[col1]*boston[col2] for (col1, col2) in col_pairs})

# total dataframe
boston_interact = pd.concat([boston.drop(['high_crim'], axis=1), interactions, boston['high_crim']], axis=1)
boston_interact.head()

	const	zn	indus	nox	rm	age	dis	rad	tax	...	tax:black	tax:lstat	tax:medv	ptratio:black	ptratio:lstat	ptratio:medv	black:lstat	black:medv	lstat:medv
0	1	18.0	2.31	0.538	6.575	65.2	4.0900	1	296	...	117482.40	1474.08	7104.0	6072.570	76.194	367.20	1976.5620	9525.600	119.520
1	1	0.0	7.07	0.469	6.421	78.9	4.9671	2	242	...	96049.80	2211.88	5227.2	7064.820	162.692	384.48	3627.6660	8573.040	197.424
2	1	0.0	7.07	0.469	7.185	61.1	4.9671	2	242	...	95064.86	975.26	8397.4	6992.374	71.734	617.66	1583.1049	13631.201	139.841
3	1	0.0	2.18	0.458	6.998	45.8	6.0622	3	222	...	87607.86	652.68	7414.8	7379.581	54.978	624.58	1160.2122	13180.642	98.196
4	1	0.0	2.18	0.458	7.147	54.2	6.0622	3	222	...	88111.80	1183.26	8036.4	7422.030	99.671	676.94	2115.4770	14367.780	192.946

5 rows × 93 columns

# standardize appropriate columns
cols = [col for col in boston_interact.columns if col not in {'const', 'chas', 'high_crim'}]
df = boston_interact[cols]
df = (df - df.mean()) / df.std()
df = pd.concat([boston_interact.const, boston_interact.chas, df, boston_interact.high_crim], axis=1)
df.head()

	const	zn	indus	nox	rm	age	dis	rad	tax	...	tax:black	tax:lstat	tax:medv	ptratio:black	ptratio:lstat	ptratio:medv	black:lstat	black:medv	lstat:medv
0	1	0.284548	-1.286636	-0.144075	0.413263	-0.119895	0.140075	-0.981871	-0.665949	...	-0.330250	-0.854748	-0.345437	-0.258522	-1.118571	-0.264575	-0.879380	0.296187	-1.366892
1	1	-0.487240	-0.592794	-0.739530	0.194082	0.366803	0.556609	-0.867024	-0.986353	...	-0.661922	-0.709601	-0.818692	0.281461	-0.525338	-0.146023	-0.247621	0.062881	-0.458594
2	1	-0.487240	-0.592794	-0.739530	1.281446	-0.265549	0.556609	-0.867024	-0.986353	...	-0.677164	-0.952880	-0.019292	0.242036	-1.149159	1.453735	-1.029928	1.301754	-1.129965
3	1	-0.487240	-1.305586	-0.834458	1.015298	-0.809088	1.076671	-0.752178	-1.105022	...	-0.792561	-1.016341	-0.267065	0.452754	-1.264078	1.501210	-1.191739	1.191400	-1.615512
4	1	-0.487240	-1.305586	-0.834458	1.227362	-0.510674	1.076671	-0.752178	-1.105022	...	-0.784763	-0.911961	-0.110322	0.475855	-0.957558	1.860432	-0.826227	1.482160	-0.510804

5 rows × 93 columns

X, y = df.drop(['high_crim'], axis=1), df['high_crim']

from sklearn.feature_selection import RFECV

Logit_rfecv = RFECV(models['Logit'], cv=5).fit(X, y)
Logit_rfecv.ranking_

/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
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/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)





array([17, 85, 90, 68,  1, 12, 84, 41, 25, 23, 27, 30, 37, 58, 89, 59, 56,
       83,  6, 19, 44, 62, 88, 87, 67, 48, 16, 69, 63, 81,  8, 53, 79, 49,
        7, 29, 82, 71, 76, 36, 35, 34, 15, 75, 51, 33, 74,  5, 11, 77, 45,
       91, 14, 31, 28, 13, 72, 57,  4, 52, 26, 43, 73, 42, 78, 47, 70, 39,
       60, 38, 40, 55, 24, 10,  9, 86, 65, 54, 61, 18, 46,  1, 20, 32,  2,
       21, 64,  3, 50, 80, 66, 22])

LDA_rfecv = RFECV(models['LDA'], cv=5).fit(X, y)
LDA_rfecv.ranking_

/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/discriminant_analysis.py:388: UserWarning: Variables are collinear.
  warnings.warn("Variables are collinear.")





array([92, 15, 22, 50, 44, 29,  4,  5,  3, 41,  8, 13, 47, 31, 88, 81, 23,
       65, 72, 78, 80, 42, 38, 37, 79, 39, 18, 89, 25, 49, 51,  1, 21, 26,
       20, 59, 32, 66, 84, 52, 69, 16, 17, 91, 77, 85, 67, 27, 74, 73, 14,
       10,  9, 57, 53, 40, 35, 68, 55, 28, 60, 70, 61, 30, 75, 87, 48,  7,
       34, 71, 36, 45, 43,  6, 62, 83, 33,  2, 19, 11, 56, 76, 24, 12, 90,
       86, 54, 46, 58, 82, 63, 64])

Logit_df = df[df.columns[list(Logit_rfecv.support_) + [True]]]

X_train, X_test, y_train, y_test = train_test_split(Logit_df.drop(['high_crim'], axis=1), Logit_df['high_crim'])

print("The Logit model has train accuracy {}".format(
      round(accuracy_score(y_train, 
                               models['Logit'].fit(X_train, y_train).predict(X_train)), 4)))
print("The Logit model has test accuracy {}".format(
      round(accuracy_score(y_test, 
                               models['Logit'].fit(X_train, y_train).predict(X_test)), 4)))

The Logit model has train accuracy 0.8707
The Logit model has test accuracy 0.8346


/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)
/Users/home/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.
  FutureWarning)

LDA_df = df[df.columns[list(LDA_rfecv.support_) + [True]]]

X_train, X_test, y_train, y_test = train_test_split(LDA_df.drop(['high_crim'], axis=1), LDA_df['high_crim'])

print("The LDA model has train accuracy {}".format(
      round(accuracy_score(y_train, 
                               models['LDA'].fit(X_train, y_train).predict(X_train)), 4)))
print("The LDA model has test accuracy {}".format(
      round(accuracy_score(y_test, 
                               models['LDA'].fit(X_train, y_train).predict(X_test)), 4)))

The LDA model has train accuracy 0.7599
The LDA model has test accuracy 0.7638

On the original data + interactions + transformations

boston_interact_trans = pd.concat([boston.drop(['high_crim'], axis=1), interactions,
                                   log_df, sq_df, boston['high_crim']], axis=1)
boston_interact_trans.head()

	const	zn	indus	nox	rm	age	dis	rad	tax	...	log(ptratio)	log(black)	log(medv)	age^2	dis^2	tax^2	ptratio^2	black^2	medv^2
0	1	18.0	2.31	0.538	6.575	65.2	4.0900	1	296	...	-1.415498	0.304822	0.347164	-0.401921	0.035014	-0.687747	-1.436963	0.526018	-0.038104
1	1	0.0	7.07	0.469	6.421	78.9	4.9671	2	242	...	-0.217665	0.304822	0.090660	0.176982	0.538601	-0.872406	-0.361721	0.526018	-0.243573
2	1	0.0	7.07	0.469	7.185	61.1	4.9671	2	242	...	-0.217665	0.291773	1.244740	-0.553769	0.538601	-0.872406	-0.361721	0.450105	1.141108
3	1	0.0	2.18	0.458	6.998	45.8	6.0622	3	222	...	0.172721	0.297561	1.151781	-1.033382	1.304266	-0.931391	0.065127	0.483582	0.974897
4	1	0.0	2.18	0.458	7.147	54.2	6.0622	3	222	...	0.172721	0.304822	1.347768	-0.787061	1.304266	-0.931391	0.065127	0.526018	1.340776

5 rows × 105 columns

# standardize appropriate columns
cols = [col for col in boston_interact_trans.columns if col not in {'const', 'chas', 'high_crim'}]
df = boston_interact_trans[cols]
df = (df - df.mean()) / df.std()
df = pd.concat([boston_interact_trans.const, boston_interact_trans.chas, df, boston_interact_trans.high_crim], axis=1)
df.head()

	const	zn	indus	nox	rm	age	dis	rad	tax	...	log(ptratio)	log(black)	log(medv)	age^2	dis^2	tax^2	ptratio^2	black^2	medv^2
0	1	0.284548	-1.286636	-0.144075	0.413263	-0.119895	0.140075	-0.981871	-0.665949	...	-1.415498	0.304822	0.347164	-0.401921	0.035014	-0.687747	-1.436963	0.526018	-0.038104
1	1	-0.487240	-0.592794	-0.739530	0.194082	0.366803	0.556609	-0.867024	-0.986353	...	-0.217665	0.304822	0.090660	0.176982	0.538601	-0.872406	-0.361721	0.526018	-0.243573
2	1	-0.487240	-0.592794	-0.739530	1.281446	-0.265549	0.556609	-0.867024	-0.986353	...	-0.217665	0.291773	1.244740	-0.553769	0.538601	-0.872406	-0.361721	0.450105	1.141108
3	1	-0.487240	-1.305586	-0.834458	1.015298	-0.809088	1.076671	-0.752178	-1.105022	...	0.172721	0.297561	1.151781	-1.033382	1.304266	-0.931391	0.065127	0.483582	0.974897
4	1	-0.487240	-1.305586	-0.834458	1.227362	-0.510674	1.076671	-0.752178	-1.105022	...	0.172721	0.304822	1.347768	-0.787061	1.304266	-0.931391	0.065127	0.526018	1.340776

5 rows × 105 columns

df = df.dropna()

X, y = df.drop(['high_crim'], axis=1), df['high_crim']

from sklearn.feature_selection import RFECV

Logit_rfecv = RFECV(models['Logit'], cv=5).fit(X, y)
Logit_rfecv.ranking_

array([ 10,  91,  99,  60,   1,  41,  86,  59,  49,  35,  64,  20,  51,
        71,  84,  54,  40, 101,  32,   2,  39,  63,  85, 102,  96,  74,
        27,  88,  33,  75,  25,  90,  34,  92,  24,  95,  72,  76,  87,
        45,  44,  23,  42,  83,  68,  28,  81,   4,  21,  79,  58,  94,
        14,  29,  46,  15,  77,  78,   9,  57,  18,   3,  61,  43,  93,
        69,  82,  52,  98,  47,  62,  37,  36,   5,  38,  80,  53,  66,
        67,  16,  50,   1,  73,  31,  12,  30,  48,  11,  55,  89,  70,
        26,  22,  13,   8,  65,  97, 100,  17,   6,   7,  19,   1,  56])

LDA_rfecv = RFECV(models['LDA'], cv=5).fit(X, y)
LDA_rfecv.ranking_

array([102,  21,  25,  56,  10,  37,  43,   7,   3,  31,   1,   6,  24,
        48,  89,  91,  26,  27,  99,  68,  69,  32, 100,  58,  76,  83,
        22,  34,  16,  64,  85,   1,  15,  73,  57,  86,  66,  65,  81,
        55,  94,  20,  19,  41,  84, 101,  95,  38,  60,  90,  11,  75,
         9,  74,  40,  47,  42,  28,  33,  18,  53,  82,  62,  45,  61,
        63,  92,  52,  44,  93,  46,  14,  13,   8,  51,  77,  70,  36,
        17,   4,  39,  87,  35,   5,  29,  54,  30,  23,  49,  97,  72,
        78,  71,  50,  59,  12,  98,  80,  88,  67,   2,   1,  96,  79])

Logit_df = df[df.columns[list(Logit_rfecv.support_) + [True]]]

X_train, X_test, y_train, y_test = train_test_split(Logit_df.drop(['high_crim'], axis=1), Logit_df['high_crim'])

print("The Logit model has train accuracy {}".format(
      round(accuracy_score(y_train, 
                               models['Logit'].fit(X_train, y_train).predict(X_train)), 4)))
print("The Logit model has test accuracy {}".format(
      round(accuracy_score(y_test, 
                               models['Logit'].fit(X_train, y_train).predict(X_test)), 4)))

The Logit model has train accuracy 0.8806
The Logit model has test accuracy 0.8917

LDA_df = df[df.columns[list(LDA_rfecv.support_) + [True]]]

X_train, X_test, y_train, y_test = train_test_split(LDA_df.drop(['high_crim'], axis=1), LDA_df['high_crim'])

print("The LDA model has train accuracy {}".format(
      round(accuracy_score(y_train, 
                               models['LDA'].fit(X_train, y_train).predict(X_train)), 4)))
print("The LDA model has test accuracy {}".format(
      round(accuracy_score(y_test, 
                               models['LDA'].fit(X_train, y_train).predict(X_test)), 4)))

The LDA model has train accuracy 0.875
The LDA model has test accuracy 0.8583

Findings

The maximum accuracy we were able to achieve for the parametric models using recursive feature elimination was 89% for the LDA model. This is still less than 90%, a lower bound for the performance of all KNN models we considered

Best test accuracy result

The best test accuracy result we were able to achieve was 95\%.

This accuracy was achieved by KNN 1 by adding a constant, using ad-hoc feature elimination, and removing outliers from the original feature set

boston_3['const'] = 1
KNN1_df = pd.concat([boston_3['const'], boston_3.drop(['const'], axis=1)], axis=1)
KNN1_df.head()

	const	zn	age	dis	tax	ptratio	black	medv
0	1	18.0	65.2	4.0900	296	15.3	396.90	24.0
1	1	0.0	78.9	4.9671	242	17.8	396.90	21.6
2	1	0.0	61.1	4.9671	242	17.8	392.83	34.7
3	1	0.0	45.8	6.0622	222	18.7	394.63	33.4
4	1	0.0	54.2	6.0622	222	18.7	396.90	36.2

KNN2 and KNN3 also achieved 95% test accuracy on the same data and features, with log and square transformations, standardized

boston_4['const'] = 1
KNN2_df = pd.concat([boston_4['const'], boston_4.drop(['const'], axis=1)], axis=1)
KNN2_df.head()

	const	zn	age	dis	tax	ptratio	black	medv	age:tax	age:black	tax:black	ptratio:black	black:medv
0	1	0.410014	-0.161725	0.275539	-0.670974	-1.433859	0.442341	0.152644	-0.555560	0.152130	-0.326258	-0.244323	0.290271
1	1	-0.457932	0.332698	0.758621	-0.991114	-0.290962	0.442341	-0.105157	-0.565460	0.621200	-0.657951	0.288645	0.059675
2	1	-0.457932	-0.309691	0.758621	-0.991114	-0.290962	0.398359	1.302009	-0.773072	-0.009700	-0.673195	0.249732	1.284156
3	1	-0.457932	-0.861858	1.361771	-1.109685	0.120480	0.417810	1.162367	-0.995673	-0.521068	-0.788600	0.457713	1.175084
4	1	-0.457932	-0.558708	1.361771	-1.109685	0.120480	0.442341	1.463135	-0.905796	-0.224495	-0.780801	0.480514	1.462467

These results depend on random values of hyperparameters (e.g. train_test_split, cross-validation parameter in RFECV), so the results will differ each time this kernel is run.

Given an overall preference for simplicity (e.g. due to the bias-variance tradeoff) and considering the computational resource advantage, the best choice seems to be KNN1 above, i.e.

model: KNN1
features: zn, age, dis, tax, ptratio, black, medv
data: boston with features above with outliers removed (dis > 8, black < 100, tax > 700)

For fun, let’s fit it one more time on the full dataset

X, y = KNN1_df.drop(['high_crim'], axis=1), KNN1_df['high_crim']

print("The KNN1 model has total accuracy {}".format(
      round(accuracy_score(y, models['KNN1'].fit(X, y).predict(X)), 4)))

The KNN1 model has total accuracy 1.0

Very nice!

Footnotes

Since high_crim is derived from crim using it for prediction is a somewhat artificial scenario. It seems like in a real-life situation, either (1) crim data would be present, in which case we might be doing linear reg on crim instead of classification of high_crim, and we could classify post-hoc or (2) crim wouldn’t be present but high_crim would be in which case treating it as a classification problem would make sense. ↩
I’m calling these “naive” models. ↩
The pattern here seems opposite of what might be expected, namely that high crime would be associated with lower property tax rate and vice versa. One potential explanation for this is that crime are disproportionately reported or responded to in towns with higher property tax rate. Another is that towns with higher crime tend to raise more revenue taxes (as opposed to say, sales tax). ↩

islr notes and exercises from An Introduction to Statistical Learning