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