5-1 simple linear regression
Linear regression algorithm
- Return to solve the problem
- Idea is simple and easy to implement
- The basis for many powerful nonlinear model
- The results have a good interpretability
- Machine learning contains many important ideas
Comparative classification, features are two axes.
in conclusion
Explanation
Linear Regression Problem Description
Solutions
Almost all parameters learning algorithms are such routine
5-2 the least squares method
Derivation, too
(1) b of the Derivatives
(2) a partial derivative
5-3 to achieve a simple linear regression
First, look at a simple example
import numpy as np
import matplotlib.pyplot as plt
x = np.array([1., 2., 3., 4., 5.])
y = np.array([1., 3., 2., 3., 5.])
plt.scatter(x, y)
plt.axis([0, 6, 0, 6])
plt.show()
Next, calculate the correlation values
x_mean = np.mean(x)
y_mean = np.mean(y)
num = 0.0
d = 0.0
for x_i, y_i in zip(x, y):
num += (x_i - x_mean) * (y_i - y_mean)
d += (x_i - x_mean) ** 2
a = num/d
b = y_mean - a * x_mean
print("(a, b): ", (a, b)) # (a, b): (0.8, 0.39999999999999947)
y_hat = a * x + b
plt.scatter(x, y)
plt.plot(x, y_hat, color='r')
plt.axis([0, 6, 0, 6])
plt.show()
x_predict = 6
y_predict = a * x_predict + b
y_predict
Package own linear regression
# playML\SimpleLinearRegression.py
import numpy as np
class SimpleLinearRegression1:
def __init__(self):
"""初始化Simple Linear Regression 模型"""
self.a_ = None
self.b_ = None
def fit(self, x_train, y_train):
"""根据训练数据集x_train,y_train训练Simple Linear Regression模型"""
assert x_train.ndim == 1, \
"Simple Linear Regressor can only solve single feature training data."
assert len(x_train) == len(y_train), \
"the size of x_train must be equal to the size of y_train"
x_mean = np.mean(x_train)
y_mean = np.mean(y_train)
num = 0.0
d = 0.0
for x, y in zip(x_train, y_train):
num += (x - x_mean) * (y - y_mean)
d += (x - x_mean) ** 2
self.a_ = num / d
self.b_ = y_mean - self.a_ * x_mean
return self
def predict(self, x_predict):
"""给定待预测数据集x_predict，返回表示x_predict的结果向量"""
assert x_predict.ndim == 1, \
"Simple Linear Regressor can only solve single feature training data."
assert self.a_ is not None and self.b_ is not None, \
"must fit before predict!"
return np.array([self._predict(x) for x in x_predict])
def _predict(self, x_single):
"""给定单个待预测数据x，返回x的预测结果值"""
return self.a_ * x_single + self.b_
def __repr__(self):
return "SimpleLinearRegression1()"
Call their own definition of simple linear regression
from playML.SimpleLinearRegression import SimpleLinearRegression1
reg1 = SimpleLinearRegression1()
reg1.fit(x, y)
print(reg1.predict(np.array([x_predict])))
print("(a, b): ", (reg1.a_, reg1.b_))
# [5.2]
# (a, b): (0.8, 0.39999999999999947)Draw
y_hat1 = reg1.predict(x)
plt.scatter(x, y)
plt.plot(x, y_hat1, color='r')
plt.axis([0, 6, 0, 6])
plt.show()5-4 vectorization
A front for the forms and inefficient performance.
consider 向量
Improved front of the algorithm. The main change fit () method.
Add the following class.
# playML.SimpleLinearRegression.py
class SimpleLinearRegression2:
def __init__(self):
"""初始化Simple Linear Regression模型"""
self.a_ = None
self.b_ = None
def fit(self, x_train, y_train):
"""根据训练数据集x_train,y_train训练Simple Linear Regression模型"""
assert x_train.ndim == 1, \
"Simple Linear Regressor can only solve single feature training data."
assert len(x_train) == len(y_train), \
"the size of x_train must be equal to the size of y_train"
x_mean = np.mean(x_train)
y_mean = np.mean(y_train)
self.a_ = (x_train - x_mean).dot(y_train - y_mean) / (x_train - x_mean).dot(x_train - x_mean)
self.b_ = y_mean - self.a_ * x_mean
return self
def predict(self, x_predict):
"""给定待预测数据集x_predict，返回表示x_predict的结果向量"""
assert x_predict.ndim == 1, \
"Simple Linear Regressor can only solve single feature training data."
assert self.a_ is not None and self.b_ is not None, \
"must fit before predict!"
return np.array([self._predict(x) for x in x_predict])
def _predict(self, x_single):
"""给定单个待预测数据x_single，返回x_single的预测结果值"""
return self.a_ * x_single + self.b_
def __repr__(self):
return "SimpleLinearRegression2()"
Call the class
from playML.SimpleLinearRegression import SimpleLinearRegression2
reg2 = SimpleLinearRegression2()
reg2.fit(x, y)
print("(a, b): ", (reg1.a_, reg1.b_))
# (a, b): (0.8, 0.39999999999999947)Paint
y_hat2 = reg2.predict(x)
plt.scatter(x, y)
plt.plot(x, y_hat2, color='r')
plt.axis([0, 6, 0, 6])
plt.show()
To quantify the performance test implemented
m = 1000000
big_x = np.random.random(size=m)
big_y = big_x * 2 + 3 + np.random.normal(size=m)
%timeit reg1.fit(big_x, big_y)
%timeit reg2.fit(big_x, big_y)
# 1.48 s ± 35.4 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
# 22.9 ms ± 1.57 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)