#!/usr/bin/env python
# coding: utf-8
#
# # Logistic Regression
# ## Logistic Regression
#
# In linear regression our main interest was centered on learning the
# coefficients of a functional fit (say a polynomial) in order to be
# able to predict the response of a continuous variable on some unseen
# data. The fit to the continuous variable $y_i$ is based on some
# independent variables $x_i$. Linear regression resulted in
# analytical expressions for standard ordinary Least Squares or Ridge
# regression (in terms of matrices to invert) for several quantities,
# ranging from the variance and thereby the confidence intervals of the
# optimal parameters $\hat{\beta}$ to the mean squared error. If we can invert
# the product of the design matrices, linear regression gives then a
# simple recipe for fitting our data.
#
# Classification problems, however, are concerned with outcomes taking
# the form of discrete variables (i.e. categories). We may for example,
# on the basis of DNA sequencing for a number of patients, like to find
# out which mutations are important for a certain disease; or based on
# scans of various patients' brains, figure out if there is a tumor or
# not; or given a specific physical system, we'd like to identify its
# state, say whether it is an ordered or disordered system (typical
# situation in solid state physics); or classify the status of a
# patient, whether she/he has a stroke or not and many other similar
# situations.
#
# The most common situation we encounter when we apply logistic
# regression is that of two possible outcomes, normally denoted as a
# binary outcome, true or false, positive or negative, success or
# failure etc.
#
# Logistic regression will also serve as our stepping stone towards
# neural network algorithms and supervised deep learning. For logistic
# learning, the minimization of the cost function leads to a non-linear
# equation in the parameters $\hat{\beta}$. The optimization of the
# problem calls therefore for minimization algorithms. This forms the
# bottle neck of all machine learning algorithms, namely how to find
# reliable minima of a multi-variable function. This leads us to the
# family of gradient descent methods. The latter are the working horses
# of basically all modern machine learning algorithms.
#
# We note also that many of the topics discussed here on logistic
# regression are also commonly used in modern supervised Deep Learning
# models, as we will see later.
# ## Basics
#
# We consider the case where the dependent variables, also called the
# responses or the outcomes, $y_i$ are discrete and only take values
# from $k=0,\dots,K-1$ (i.e. $K$ classes).
#
# The goal is to predict the
# output classes from the design matrix $\boldsymbol{X}\in\mathbb{R}^{n\times p}$
# made of $n$ samples, each of which carries $p$ features or predictors. The
# primary goal is to identify the classes to which new unseen samples
# belong.
#
# Let us specialize to the case of two classes only, with outputs
# $y_i=0$ and $y_i=1$. Our outcomes could represent the status of a
# credit card user that could default or not on her/his credit card
# debt. That is
# $$
# y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
# $$
# Before moving to the logistic model, let us try to use our linear
# regression model to classify these two outcomes. We could for example
# fit a linear model to the default case if $y_i > 0.5$ and the no
# default case $y_i \leq 0.5$.
#
# We would then have our
# weighted linear combination, namely
#
#
#
# $$
# \begin{equation}
# \boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} + \boldsymbol{\epsilon},
# \label{_auto1} \tag{1}
# \end{equation}
# $$
# where $\boldsymbol{y}$ is a vector representing the possible outcomes, $\boldsymbol{X}$ is our
# $n\times p$ design matrix and $\boldsymbol{\beta}$ represents our estimators/predictors.
#
# The main problem with our function is that it takes values on the
# entire real axis. In the case of logistic regression, however, the
# labels $y_i$ are discrete variables. A typical example is the credit
# card data discussed below here, where we can set the state of
# defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons
# in the data set (see the full example below).
#
# One simple way to get a discrete output is to have sign
# functions that map the output of a linear regressor to values $\{0,1\}$,
# $f(s_i)=sign(s_i)=1$ if $s_i\ge 0$ and 0 if otherwise.
# We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
# literature. This model is extremely simple. However, in many cases it is more
# favorable to use a ``soft" classifier that outputs
# the probability of a given category. This leads us to the logistic function.
#
# The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
# In[1]:
get_ipython().run_line_magic('matplotlib', 'inline')
# Common imports
import os
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression, Ridge, Lasso
from sklearn.model_selection import train_test_split
from sklearn.utils import resample
from sklearn.metrics import mean_squared_error
from IPython.display import display
from pylab import plt, mpl
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
infile = open(data_path("chddata.csv"),'r')
# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
output = chd['CHD']
age = chd['Age']
agegroup = chd['Agegroup']
numberID = chd['ID']
display(chd)
plt.scatter(age, output, marker='o')
plt.axis([18,70.0,-0.1, 1.2])
plt.xlabel(r'Age')
plt.ylabel(r'CHD')
plt.title(r'Age distribution and Coronary heart disease')
plt.show()
# What we could attempt however is to plot the mean value for each group.
# In[2]:
agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
plt.plot(group, agegroupmean, "r-")
plt.axis([0,9,0, 1.0])
plt.xlabel(r'Age group')
plt.ylabel(r'CHD mean values')
plt.title(r'Mean values for each age group')
plt.show()
# We are now trying to find a function $f(y\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.
# In standard linear regression with a linear dependence on $x$, we would write this in terms of our model
# $$
# f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
# $$
# This expression implies however that $f(y_i\vert x_i)$ could take any
# value from minus infinity to plus infinity. If we however let
# $f(y\vert y)$ be represented by the mean value, the above example
# shows us that we can constrain the function to take values between
# zero and one, that is we have $0 \le f(y_i\vert x_i) \le 1$. Looking
# at our last curve we see also that it has an S-shaped form. This leads
# us to a very popular model for the function $f$, namely the so-called
# Sigmoid function or logistic model. We will consider this function as
# representing the probability for finding a value of $y_i$ with a given
# $x_i$.
# ## The logistic function
#
# Another widely studied model, is the so-called
# perceptron model, which is an example of a "hard classification" model. We
# will encounter this model when we discuss neural networks as
# well. Each datapoint is deterministically assigned to a category (i.e
# $y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
# classifier that outputs the probability of a given category rather
# than a single value. For example, given $x_i$, the classifier
# outputs the probability of being in a category $k$. Logistic regression
# is the most common example of a so-called soft classifier. In logistic
# regression, the probability that a data point $x_i$
# belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
# $$
# p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
# $$
# Note that $1-p(t)= p(-t)$.
# ## Examples of likelihood functions used in logistic regression and neural networks
#
# The following code plots the logistic function, the step function and other functions we will encounter from here and on.
# In[3]:
"""The sigmoid function (or the logistic curve) is a
function that takes any real number, z, and outputs a number (0,1).
It is useful in neural networks for assigning weights on a relative scale.
The value z is the weighted sum of parameters involved in the learning algorithm."""
import numpy
import matplotlib.pyplot as plt
import math as mt
z = numpy.arange(-5, 5, .1)
sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
sigma = sigma_fn(z)
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, sigma)
ax.set_ylim([-0.1, 1.1])
ax.set_xlim([-5,5])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('sigmoid function')
plt.show()
"""Step Function"""
z = numpy.arange(-5, 5, .02)
step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
step = step_fn(z)
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, step)
ax.set_ylim([-0.5, 1.5])
ax.set_xlim([-5,5])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('step function')
plt.show()
"""tanh Function"""
z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
t = numpy.tanh(z)
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, t)
ax.set_ylim([-1.0, 1.0])
ax.set_xlim([-2*mt.pi,2*mt.pi])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('tanh function')
plt.show()
# We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\beta$ in our fitting of the Sigmoid function, that is we define probabilities
# $$
# \begin{align*}
# p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
# p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
# \end{align*}
# $$
# where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$.
#
# Note that we used
# $$
# p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
# $$
# In order to define the total likelihood for all possible outcomes from a
# dataset $\mathcal{D}=\{(y_i,x_i)\}$, with the binary labels
# $y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle.
# We aim thus at maximizing
# the probability of seeing the observed data. We can then approximate the
# likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is
# $$
# \begin{align*}
# P(\mathcal{D}|\boldsymbol{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\boldsymbol{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]^{1-y_i}\nonumber \\
# \end{align*}
# $$
# from which we obtain the log-likelihood and our **cost/loss** function
# $$
# \mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\boldsymbol{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]\right).
# $$
# Reordering the logarithms, we can rewrite the **cost/loss** function as
# $$
# \mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
# $$
# The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\beta$.
# Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
# $$
# \mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
# $$
# This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression,
# in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.
#
# The cross entropy is a convex function of the weights $\boldsymbol{\beta}$ and,
# therefore, any local minimizer is a global minimizer.
#
# Minimizing this
# cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain
# $$
# \frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
# $$
# and
# $$
# \frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
# $$
# Let us now define a vector $\boldsymbol{y}$ with $n$ elements $y_i$, an
# $n\times p$ matrix $\boldsymbol{X}$ which contains the $x_i$ values and a
# vector $\boldsymbol{p}$ of fitted probabilities $p(y_i\vert x_i,\boldsymbol{\beta})$. We can rewrite in a more compact form the first
# derivative of cost function as
# $$
# \frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
# $$
# If we in addition define a diagonal matrix $\boldsymbol{W}$ with elements
# $p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})$, we can obtain a compact expression of the second derivative as
# $$
# \frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
# $$
# Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors
# $$
# \log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
# $$
# Here we defined $\boldsymbol{x}=[1,x_1,x_2,\dots,x_p]$ and $\boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p]$ leading to
# $$
# p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
# $$
# Till now we have mainly focused on two classes, the so-called binary
# system. Suppose we wish to extend to $K$ classes. Let us for the sake
# of simplicity assume we have only two predictors. We have then following model
# $$
# \log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
# $$
# and
# $$
# \log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
# $$
# and so on till the class $C=K-1$ class
# $$
# \log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
# $$
# and the model is specified in term of $K-1$ so-called log-odds or
# **logit** transformations.
#
# In our discussion of neural networks we will encounter the above again
# in terms of a slightly modified function, the so-called **Softmax** function.
#
# The softmax function is used in various multiclass classification
# methods, such as multinomial logistic regression (also known as
# softmax regression), multiclass linear discriminant analysis, naive
# Bayes classifiers, and artificial neural networks. Specifically, in
# multinomial logistic regression and linear discriminant analysis, the
# input to the function is the result of $K$ distinct linear functions,
# and the predicted probability for the $k$-th class given a sample
# vector $\boldsymbol{x}$ and a weighting vector $\boldsymbol{\beta}$ is (with two
# predictors):
# $$
# p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
# $$
# It is easy to extend to more predictors. The final class is
# $$
# p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
# $$
# and they sum to one. Our earlier discussions were all specialized to
# the case with two classes only. It is easy to see from the above that
# what we derived earlier is compatible with these equations.
#
# To find the optimal parameters we would typically use a gradient
# descent method. Newton's method and gradient descent methods are
# discussed in the material on [optimization
# methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).
# ## Wisconsin Cancer Data
#
# We show here how we can use a simple regression case on the breast
# cancer data using Logistic regression as our algorithm for
# classification.
# In[4]:
import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression
# Load the data
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# In addition to the above scores, we could also study the covariance (and the correlation matrix).
# We use **Pandas** to compute the correlation matrix.
# In[5]:
import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression
cancer = load_breast_cancer()
import pandas as pd
# Making a data frame
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
fig, axes = plt.subplots(15,2,figsize=(10,20))
malignant = cancer.data[cancer.target == 0]
benign = cancer.data[cancer.target == 1]
ax = axes.ravel()
for i in range(30):
_, bins = np.histogram(cancer.data[:,i], bins =50)
ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)
ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)
ax[i].set_title(cancer.feature_names[i])
ax[i].set_yticks(())
ax[0].set_xlabel("Feature magnitude")
ax[0].set_ylabel("Frequency")
ax[0].legend(["Malignant", "Benign"], loc ="best")
fig.tight_layout()
plt.show()
import seaborn as sns
correlation_matrix = cancerpd.corr().round(1)
# use the heatmap function from seaborn to plot the correlation matrix
# annot = True to print the values inside the square
plt.figure(figsize=(15,8))
sns.heatmap(data=correlation_matrix, annot=True)
plt.show()
# In the above example we note two things. In the first plot we display
# the overlap of benign and malignant tumors as functions of the various
# features in the Wisconsing breast cancer data set. We see that for
# some of the features we can distinguish clearly the benign and
# malignant cases while for other features we cannot. This can point to
# us which features may be of greater interest when we wish to classify
# a benign or not benign tumour.
#
# In the second figure we have computed the so-called correlation
# matrix, which in our case with thirty features becomes a $30\times 30$
# matrix.
#
# We constructed this matrix using **pandas** via the statements
# In[6]:
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
# and then
# In[7]:
correlation_matrix = cancerpd.corr().round(1)
# Diagonalizing this matrix we can in turn say something about which
# features are of relevance and which are not. This leads us to
# the classical Principal Component Analysis (PCA) theorem with
# applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).
#
# Here we present a further way to present our results in terms of a so-called **confusion matrix**, the cumulative gain and the **ROC** curve.
# This way of displaying our data are based upon different ways to classify our possible outcomes. Before we proceed we need some definitions.
# 1. **TP**: true positive or in other words, something equivalent with a proper classification
#
# 2. **TN**: true negative, which is equivalent with a correct rejection
#
# 3. **FP**: false positive, or in simpler words something that is equivalent with a false alarm
#
# 4. **FN**: false negative, which is mean to be equivalent with a miss.
#
# The total data set is then the sum of the true positive and true negative targets or outputs, labeled by $n$.
# Based on this we can then define the accuracy score as the sum of correctly predicted **TP** and **TN** cases divided by the sum of true positive and treue negative events in our data set, or as
# $$
# \mathrm{Accuracy} = \frac{\sum_{i=0}^{n-1}I(y_i=\tilde{y}_i)}{n}.
# $$
# In[8]:
import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression
# Load the data
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
print(X_train.shape)
print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import cross_validate
#Cross validation
accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']
print(accuracy)
print("Test set accuracy with Logistic Regression and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = logreg.predict(X_test_scaled)
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = logreg.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
plt.show()
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
plt.show()