Logistic Regression

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• Author: Diego Inácio
• GitHub: github.com/diegoinacio
• Notebook: regression_logistic.ipynb

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Overview and implementation of Logistic Regression analysis.

%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt
import numpy as np

from regression__utils import *

$$ \large h_{\theta}(x)=g(\theta^Tx)=\frac{e^{\theta^Tx}}{1+e^{\theta^Tx}}=\frac{1}{1+e^{-\theta^Tx}} $$

where:

$$ \large \theta^Tx= \begin{bmatrix} \theta_0 \\ \theta_1 \\ \vdots \\ \theta_i \end{bmatrix} \begin{bmatrix} 1 & x_{11} & \cdots & x_{1i} \\ 1 & x_{21} & \cdots & x_{2i} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & x_{n1} & \cdots & x_{ni} \end{bmatrix} $$

where:

• $\large h_\theta(x)$ is the hypothesis;
• $\large g(z)$ is the logistic function or sigmoid;
• $\large \theta_i$ is the parameters (or weights).

def arraycast(f):
    '''
    Decorator for vectors and matrices cast
    '''
    def wrap(self, *X, y=[]):
        X = np.array(X)
        X = np.insert(X.T, 0, 1, 1)
        if list(y):
            y = np.array(y)[np.newaxis]
            return f(self, X, y)
        return f(self, X)
    return wrap

class logisticRegression(object):
    def __init__(self, rate=0.001, iters=1024):
        self._rate = rate
        self._iters = iters
        self._theta = None
    @property
    def theta(self):
        return self._theta
    def _sigmoid(self, Z):
        return 1/(1 + np.exp(-Z))
    def _dsigmoid(self, Z):
        return self._sigmoid(Z)*(1 - self._sigmoid(Z))
    @arraycast
    def fit(self, X, y=[]):
        self._theta = np.ones((1, X.shape[-1]))
        for i in range(self._iters):
            thetaTx = np.dot(X, self._theta.T)
            h = self._sigmoid(thetaTx)
            delta = h - y.T
            grad = np.dot(X.T, delta).T
            self._theta -= grad*self._rate
    @arraycast
    def pred(self, x):
        return self._sigmoid(np.dot(x, self._theta.T)) > 0.5

# Synthetic data 5
x1, x2, y = synthData5()

%%time
# Training
rlogb = logisticRegression(rate=0.001, iters=512)
rlogb.fit(x1, x2, y=y)
rlogb.pred(x1, x2)

Wall time: 52.9 ms

To find the boundary line components:

$$ \large \theta_0+\theta_1 x_1+\theta_2 x_2=0 $$

Considering $\large x_2$ as the dependent variable:

$$ \large x_2=-\frac{\theta_0+\theta_1 x_1}{\theta_2} $$

# Prediction
w0, w1, w2 = rlogb.theta[0]
f = lambda x: - (w0 + w1*x)/w2