plot_binary_classification.py

# -*- coding: utf-8 -*-
"""
Binary classification
=====================

A binary classification problem is to find
a frontier to separate two clouds of points
(see `binary classification <https://en.wikipedia.org/wiki/Binary_classification>`_).

.. contents::
    :local:

.. index:: classification, binary, ROC
"""

###########################
#
# What does it solve?
# -------------------
#
# We start by generating an artificial cloud of points.

from sklearn.datasets import make_classification
X, Y = make_classification(n_samples=500, n_features=2, n_classes=2,
                           n_repeated=0, n_redundant=0)

###########################
# We plot the data.

import matplotlib.pyplot as plt
fig = plt.figure(figsize=(5, 5))
ax = plt.subplot()
ax.plot(X[Y == 0, 0], X[Y == 0, 1], "ob")
ax.plot(X[Y == 1, 0], X[Y == 1, 1], "or")

########################
# We represent the data in dataframe.

import pandas
data = pandas.DataFrame(data=X, columns=["X1", "X2"])
data["Label"] = Y.astype(float)

##########################
# 
# From a geometrical point of view, a binary classification 
# problem consists in finding the best boundary between 
# two clouds of points. The simplest is to assume that it is 
# a straight line. In this case, a logistic regression model 
# will help us.

from microsoftml import rx_logistic_regression, rx_predict
logreg = rx_logistic_regression("Label ~ X1 + X2", data=data)

##############################
# The model produces a line boundary
# whose coefficients are:
print(logreg.coef_)

###############################
# We could trace this line but this graph
# would only be valid for a linear model.
# Instead we color the background of the graph with
# the color of the class predicted by the model.

import numpy

def colorie(X, model, ax, fig):
    if isinstance(X, pandas.DataFrame):
        X = X.as_matrix()
    xmin, xmax = numpy.min(X[:, 0]), numpy.max(X[:, 0])
    ymin, ymax = numpy.min(X[:, 1]), numpy.max(X[:, 1])
    hx = (xmax - xmin) / 100
    hy = (ymax - ymin) / 100
    xx, yy = numpy.mgrid[xmin:xmax:hx, ymin:ymax:hy]
    grid = numpy.c_[xx.ravel(), yy.ravel()]
    dfgrid = pandas.DataFrame(data=grid, columns=["X1", "X2"])
    probs = rx_predict(model, dfgrid).as_matrix()[:, 1].reshape(xx.shape)

    contour = ax.contourf(xx, yy, probs, 25, cmap="RdBu", vmin=0, vmax=1)
    ax_c = fig.colorbar(contour)
    ax_c.set_label("$P(y = 1)$")
    ax_c.set_ticks([0, .25, .5, .75, 1])
    ax.set_xlim([xmin, xmax])
    ax.set_ylim([ymin, ymax])


fig = plt.figure(figsize=(7, 5))
ax = plt.subplot()
colorie(X, logreg, ax, fig)
ax.scatter(X[:, 0], X[:, 1], c=Y, s=50,
           cmap="RdBu", vmin=-.2, vmax=1.2,
           edgecolor="white", linewidth=1)
ax.set(aspect="equal", xlabel="$X_1$", ylabel="$X_2$")

##############################
#
# Evaluation
# ----------
#
# There are two tests that are carried out in a quasi-systematic way
# for this type of problem: one :epkg:`confusion matrix`
# And a curve :epkg:`ROC`.
#
# The first step is to divide the dataset into
# training set (train) and test set (test).
# This is necessary because some models can simply
# learn by heart reach a null error on the data
# used to learn. This is the case with the
# `nearest neighbors <https://en.wikipedia.org/wiki/Search_of_more_neighbourne neighbors>` _.
# The division must be random. Moreover,
# this can be repeated several times 
# to ensure the robustness of the results.

try:
    from sklearn.model_selection import train_test_split
except ImportError:
    from sklearn.cross_validation import train_test_split
data_train, data_test = train_test_split(data)

#####################
# We train on the training set.

logreg = rx_logistic_regression("Label ~ X1 + X2", data=data_train)

#####################
# We predict on the test set.
y_pred = rx_predict(logreg, data=data_test)

##########################
# We compute the :epkg:`confusion matrix`.
from sklearn.metrics import confusion_matrix
conf = confusion_matrix(data_test["Label"], y_pred["PredictedLabel"])
print(conf)

##########################
# The numbers on the diagonal indicate the observations
# ranked in the right class. Elsewhere, these are the miclassified observations
# These are blue dots on a red background for example.

fig = plt.figure(figsize=(7, 5))
ax = plt.subplot()
colorie(data_test[["X1", "X2"]], logreg, ax, fig)
ax.scatter(data_test["X1"], data_test["X2"], c=data_test["Label"], s=50,
           cmap="RdBu", vmin=-.2, vmax=1.2,
           edgecolor="white", linewidth=1)
ax.set(aspect="equal", xlabel="$X_1$", ylabel="$X_2$")
ax.set_title("Résultats sur la base de test")

########################
# Some points are more or less far from the boundary
# the model found between the two classes. The more
# a point is far from the border, the more the model is sure of its class.
# This is true for logistic regression but also for most
# of the models used to make a binary classification.
# This is how the model is able to give an relevance indicator:
# the distance between the point and the boundary the model found.
# This distance is converted into a score or a probability.
# The :epkg:`confusion matrix` does not use this information,
# The curve :epkg:`ROC` does. Let's see how.
# This is inspired by the example
# `Receiver Operating Characteristic (ROC) <http://scikit-learn.org/stable/auto_examples/model_selection/plot_roc.html>` _.
# The curve :epkg:`ROC` is the same with scores or probabilities.
print(y_pred[:5])

###########################
# For a binary classification, :epkg:`microsoftml` returns 
# the predicted label, a score and the probability of each
# observation to be in class 1.
# We build the :epkg:`ROC` curve.
# The model is right if it finds the right class.
# with is translated by ``y_pred["PredictedLabel"] == data_test["Label"]``.
# The score is ``y_pred["Probability"]`` when the predicted class is 1,
# ``1 - y_pred["Probability"]`` otherwise.

from sklearn.metrics import roc_curve
prob_pred = [y_pred.loc[i, "Probability"] if y_pred.loc[i, "PredictedLabel"] \
             else (1 - y_pred.loc[i, "Probability"]) for i in range(0, y_pred.shape[0])]
good = y_pred["PredictedLabel"].as_matrix() == (data_test["Label"] == 1).as_matrix()
fpr, tpr, th = roc_curve(good.ravel(), prob_pred)

################################
# For index ``i``, ``th[i]`` is a threshold,
# ``tpr[i]`` is the ratio of correclty classified observations 
# with a probability higher than``th[i]``.
# ``fpr[i]`` is the ratio of misclassified observations
# with a probability higher than``th[i]``.

print("i=2", th[2], tpr[2], fpr[2])
print("i={0}".format(len(th) - 2), th[-2], tpr[-2], fpr[-2])

##############################
# The lower the score is, the higher the proportion is.
# When the threshold is very low, the model returns a score or a probability
# always above the threshold and the two proportions are equal to 1.
# If the threshold is high, the two proportions are null.
# We draw the curve by using many distinct thresholds.

plt.figure()
lw = 2
plt.plot(fpr, tpr, color='darkorange', lw=lw, label='ROC Curve')
plt.plot([0, 1], [0, 1], color='navy', lw=lw, linestyle='--')
plt.xlim([0.0, 1.0])
plt.ylim([0.0, 1.05])
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title('ROC')
plt.legend(loc="lower right")


#####################
# The area under the curve or :epkg:`AUC` (Area Under the Curve)
# is related to the relevance of the model.
# The higher the score is, the larger the proportion of correclty classified
# observations is compared to the proportion of misclassified examples. 
# When these proportions are always equal whatever the threshold is,
# the curve fits the first diagonal and the model
# has learned nothing. The higher the curve is, the better the model is.
# This is why the area under the curve is calculated.

from sklearn.metrics import auc
print(auc(fpr, tpr))

#####################################
# The curve :epkg:`ROC` always applies to a binary decision.
# The binary classification can be turned into three binary decisions:
#
# * The model correclty classified an example, either in class 0 or class 1.
# * The model correclty classified an example in class 0.
# * The model correclty classified an example in class 1.
#
# If the answers are linked, the model usually gives
# different performance on these three issues.
# We compute the curves :epkg:`ROC` related to these three questions.
fpr_cl = dict()
tpr_cl = dict()
fpr_cl["classe 0"], tpr_cl["classe 0"], _ = roc_curve(
    1 - data_test["Label"], 1 - y_pred["Probability"])
fpr_cl["classe 1"], tpr_cl["classe 1"], _ = roc_curve(
    data_test["Label"], y_pred["Probability"])  # y_test == 1
fpr_cl["all"], tpr_cl["all"] = fpr, tpr  # We already computed this.

##############################
# We plot.
plt.figure()
for key in fpr_cl:
    plt.plot(fpr_cl[key], tpr_cl[key], label=key)

plt.plot([0, 1], [0, 1], color='navy', lw=lw, linestyle='--')
plt.xlim([0.0, 1.0])
plt.ylim([0.0, 1.05])
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title('ROC(s)')
plt.legend(loc="lower right")