Data Science and Machine Learning Algorithms from Scratch-Logistic Regression-04

Logistic Regression is more modeled from the probabilistic level, but in our practical application, it is more used for binary classification problems.

Case:

Liezi: As shown in the figure above: there are two types of data: square=0 and circle=1

Coordinate axis type: X1 ,X2 ,0/1

2 ,2 , 0

2 ,4 , 0

3 ,3 , 0

4 ,5 , 0

8 ,1 , 1

8.5,3.5 , 1

9 ,1 , 1

10 ,4 , 1

# prediction function
from math import exp
def prediction(row, coefficients):
    yhat = coefficients[0]
    for i in range(len(row)-1):
        yhat+=coefficients[i+1]*row[i]
    return 1/(1+exp(-yhat))

dataset = [[2,2,0],
           [2,4,0],
           [3,3,0],
           [4,5,0],
           [8,1,1],
           [8.5,3.5,1],
           [9,1,1],
           [10,4,1]]

coef = [-0.5,0.9,-1.5] # 我们随便定义一组系数

# prediction function
from math import exp
def prediction(row, coefficients):
    yhat = coefficients[0]
    for i in range(len(row)-1):
        yhat+=coefficients[i+1]*row[i]
    return 1/(1+exp(-yhat))

for row in dataset:
    yhat = prediction(row,coef)
    print("真实类别%.3f, 预测类别 %.3f [%d]" % (row[-1], yhat, round(yhat)))

Use gradient descent

from math import exp


def prediction(row, coefficients):
    yhat = coefficients[0]
    for i in range(len(row) - 1):
        yhat += coefficients[i + 1] * row[i]
    return 1 / (1 + exp(-yhat))


def using_sgd_method_to_calculate_coefficients(training_dataset, learning_rate, n_times_epoch):
    coefficients = [0.0 for i in range(len(training_dataset[0]))]
    for epoch in range(n_times_epoch):
        the_sum_of_error = 0
        for row in training_dataset:
            y_hat = prediction(row, coefficients)
            error = row[-1] - y_hat
            the_sum_of_error += error ** 2
            coefficients[0] = coefficients[0] + learning_rate * error * y_hat * (1.0 - y_hat)
            for i in range(len(row) - 1):
                coefficients[i + 1] = coefficients[i + 1] + learning_rate * error * y_hat * (1.0 - y_hat) * row[i]
        print("第 【%d】步,我们使用的学习率是 【%.3f】,误差是 【%.3f】" % (
            epoch, learning_rate, the_sum_of_error))
    return coefficients


dataset = [[2, 2, 0],
           [2, 4, 0],
           [3, 3, 0],
           [4, 5, 0],
           [8, 1, 1],
           [8.5, 3.5, 1],
           [9, 1, 1],
           [10, 4, 1]]

learning_rate = 0.1

n_times_epoch = 1000

coef = using_sgd_method_to_calculate_coefficients(dataset, learning_rate, n_times_epoch)

print(coef)

dataset = [[2,2,0],
           [2,4,0],
           [3,3,0],
           [4,5,0],
           [8,1,1],
           [8.5,3.5,1],
           [9,1,1],
           [10,4,1]]

coef = [-1.15556711935848, 1.4830491764089766, -2.307569789313787] # 将梯度下降得出的系数带入

# prediction function
from math import exp
def prediction(row, coefficients):
    yhat = coefficients[0]
    for i in range(len(row)-1):
        yhat+=coefficients[i+1]*row[i]
    return 1/(1+exp(-yhat))

for row in dataset:
    yhat = prediction(row,coef)
    print("真实类别%.3f, 预测类别 %.3f [%d]" % (row[-1], yhat, round(yhat)))