tensorflow基本算法(1):线性回归 linear regression

参考维基百科:

在统计学中,线性回归(Linear regression)是利用称为线性回归方程的最小二乘函数对一个或多个自变量和因变量之间关系进行建模的一种回归分析。这种函数是一个或多个称为回归系数的模型参数的线性组合。只有一个自变量的情况称为简单回归,大于一个自变量情况的叫做多元回归。(这反过来又应当由多个相关的因变量预测的多元线性回归区别,而不是一个单一的标量变量。)

在线性回归中,数据使用线性预测函数来建模,并且未知的模型参数也是通过数据来估计。这些模型被叫做线性模型。最常用的线性回归建模是给定X值的y的条件均值是X的仿射函数。不太一般的情况,线性回归模型可以是一个中位数或一些其他的给定X的条件下y的条件分布的分位数作为X的线性函数表示。像所有形式的回归分析一样,线性回归也把焦点放在给定X值的y的条件概率分布,而不是X和y的联合概率分布(多元分析领域)。

线性回归是回归分析中第一种经过严格研究并在实际应用中广泛使用的类型。这是因为线性依赖于其未知参数的模型比非线性依赖于其未知参数的模型更容易拟合,而且产生的估计的统计特性也更容易确定。

线性回归有很多实际用途。分为以下两大类:

  1. 如果目标是预测或者映射,线性回归可以用来对观测数据集的和X的值拟合出一个预测模型。当完成这样一个模型以后,对于一个新增的X值,在没有给定与它相配对的y的情况下,可以用这个拟合过的模型预测出一个y值。
  2. 给定一个变量y和一些变量X1...Xp,这些变量有可能与y相关,线性回归分析可以用来量化y与Xj之间相关性的强度,评估出与y不相关的Xj,并识别出哪些Xj的子集包含了关于y的冗余信息。

线性回归模型经常用最小二乘逼近来拟合,但他们也可能用别的方法来拟合,比如用最小化“拟合缺陷”在一些其他规范里(比如最小绝对误差回归),或者在桥回归中最小化最小二乘损失函数的惩罚。相反,最小二乘逼近可以用来拟合那些非线性的模型。因此,尽管“最小二乘法”和“线性模型”是紧密相连的,但他们是不能划等号的。

理论模型及估计:

tensorflow基本算法(1):线性回归 linear regression_第1张图片

总结

线性回归的主要思想就是通过历史数据拟合出一条直线,用这条直线对新的数据进行预测

代价函数为: 

                 

训练方法:梯度下降算法

tensorflow实现:

1.不使用eager

#!/usr/bin/env python
# filename: linear regression

import tensorflow as tf
import numpy
import matplotlib.pyplot as plt
rng = numpy.random

# Parameters
learning_rate = 0.01
training_epochs = 1000
display_step = 50

# Training Data
train_X = numpy.asarray([3.3,4.4,5.5,6.71,6.93,4.168,9.779,6.182,7.59,2.167,
                         7.042,10.791,5.313,7.997,5.654,9.27,3.1])
train_Y = numpy.asarray([1.7,2.76,2.09,3.19,1.694,1.573,3.366,2.596,2.53,1.221,
                         2.827,3.465,1.65,2.904,2.42,2.94,1.3])
n_samples = train_X.shape[0]

# tf Graph Input
X = tf.placeholder("float")
Y = tf.placeholder("float")
​
# Set model weights
W = tf.Variable(rng.randn(), name="weight")
b = tf.Variable(rng.randn(), name="bias")

# Construct a linear model
pred = tf.add(tf.multiply(X, W), b)

# Mean squared error
cost = tf.reduce_sum(tf.pow(pred-Y, 2))/(2*n_samples)
# Gradient descent
optimizer = tf.train.GradientDescentOptimizer(learning_rate).minimize(cost)

# Initialize the variables (i.e. assign their default value)
init = tf.global_variables_initializer()

# Start training
with tf.Session() as sess:
    sess.run(init)
​
    # Fit all training data
    for epoch in range(training_epochs):
        for (x, y) in zip(train_X, train_Y):
            sess.run(optimizer, feed_dict={X: x, Y: y})
​
        #Display logs per epoch step
        if (epoch+1) % display_step == 0:
            c = sess.run(cost, feed_dict={X: train_X, Y:train_Y})
            print("Epoch:", '%04d' % (epoch+1), "cost=", "{:.9f}".format(c), \
                "W=", sess.run(W), "b=", sess.run(b))
​
    print("Optimization Finished!")
    training_cost = sess.run(cost, feed_dict={X: train_X, Y: train_Y})
    print("Training cost=", training_cost, "W=", sess.run(W), "b=", sess.run(b), '\n')
​
    #Graphic display
    plt.plot(train_X, train_Y, 'ro', label='Original data')
    plt.plot(train_X, sess.run(W) * train_X + sess.run(b), label='Fitted line')
    plt.legend()
    plt.show()

结果输出:

Epoch: 0050 cost= 0.290731698 W= -0.00836047 b= 2.6571696
Epoch: 0100 cost= 0.266084820 W= 0.006981368 b= 2.5468016
Epoch: 0150 cost= 0.244280651 W= 0.02141102 b= 2.4429953
Epoch: 0200 cost= 0.224992946 W= 0.03498184 b= 2.3453684
Epoch: 0250 cost= 0.207929820 W= 0.047745574 b= 2.2535467
Epoch: 0300 cost= 0.192835465 W= 0.05974988 b= 2.167189
Epoch: 0350 cost= 0.179481432 W= 0.07104068 b= 2.085963
Epoch: 0400 cost= 0.167667642 W= 0.081660114 b= 2.0095677
Epoch: 0450 cost= 0.157216385 W= 0.091648005 b= 1.9377158
Epoch: 0500 cost= 0.147970721 W= 0.10104162 b= 1.8701389
Epoch: 0550 cost= 0.139791384 W= 0.109876454 b= 1.8065817
Epoch: 0600 cost= 0.132555142 W= 0.11818599 b= 1.7468032
Epoch: 0650 cost= 0.126153350 W= 0.12600137 b= 1.69058
Epoch: 0700 cost= 0.120489597 W= 0.13335216 b= 1.637699
Epoch: 0750 cost= 0.115478866 W= 0.14026573 b= 1.5879631
Epoch: 0800 cost= 0.111046173 W= 0.1467677 b= 1.5411884
Epoch: 0850 cost= 0.107124634 W= 0.15288284 b= 1.497197
Epoch: 0900 cost= 0.103655063 W= 0.15863435 b= 1.4558206
Epoch: 0950 cost= 0.100585289 W= 0.1640441 b= 1.4169033
Epoch: 1000 cost= 0.097869352 W= 0.16913211 b= 1.3803006
Optimization Finished!
Training cost= 0.09786935 W= 0.16913211 b= 1.3803006 

tensorflow基本算法(1):线性回归 linear regression_第2张图片

2. 使用eager:

#!/usr/bin/env python
# filename: linear regression eager api

from __future__ import absolute_import, division, print_function
​
import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
import tensorflow.contrib.eager as tfe

# Set Eager API
tfe.enable_eager_execution()

# Training Data
train_X = [3.3, 4.4, 5.5, 6.71, 6.93, 4.168, 9.779, 6.182, 7.59, 2.167,
           7.042, 10.791, 5.313, 7.997, 5.654, 9.27, 3.1]
train_Y = [1.7, 2.76, 2.09, 3.19, 1.694, 1.573, 3.366, 2.596, 2.53, 1.221,
           2.827, 3.465, 1.65, 2.904, 2.42, 2.94, 1.3]
n_samples = len(train_X)
​
# Parameters
learning_rate = 0.01
display_step = 100
num_steps = 1000

# Weight and Bias
W = tfe.Variable(np.random.randn())
b = tfe.Variable(np.random.randn())
​
# Linear regression (Wx + b)
def linear_regression(inputs):
    return inputs * W + b
​
# Mean square error
def mean_square_fn(model_fn, inputs, labels):
    return tf.reduce_sum(tf.pow(model_fn(inputs) - labels, 2)) / (2 * n_samples)

# SGD Optimizer
optimizer = tf.train.GradientDescentOptimizer(learning_rate=learning_rate)
​
# Compute gradients
grad = tfe.implicit_gradients(mean_square_fn)

# Initial cost, before optimizing
print("Initial cost= {:.9f}".format(
    mean_square_fn(linear_regression, train_X, train_Y)),
    "W=", W.numpy(), "b=", b.numpy())
​
# Training
for step in range(num_steps):
​
    optimizer.apply_gradients(grad(linear_regression, train_X, train_Y))
​
    if (step + 1) % display_step == 0 or step == 0:
        print("Epoch:", '%04d' % (step + 1), "cost=",
              "{:.9f}".format(mean_square_fn(linear_regression, train_X, train_Y)),
              "W=", W.numpy(), "b=", b.numpy())
​
# Graphic display
plt.plot(train_X, train_Y, 'ro', label='Original data')
plt.plot(train_X, np.array(W * train_X + b), label='Fitted line')
plt.legend()
plt.show()

结果输出:

Initial cost= 0.351724952 W= 0.120354496 b= 2.2911706
Epoch: 0001 cost= 0.254229784 W= 0.08552867 b= 2.2844243
Epoch: 0100 cost= 0.183475479 W= 0.0665762 b= 2.1107843
Epoch: 0200 cost= 0.160497516 W= 0.08774154 b= 1.9607316
Epoch: 0300 cost= 0.142475009 W= 0.10648614 b= 1.8278408
Epoch: 0400 cost= 0.128339261 W= 0.12308694 b= 1.7101487
Epoch: 0500 cost= 0.117252141 W= 0.13778897 b= 1.605918
Epoch: 0600 cost= 0.108556032 W= 0.15080954 b= 1.513608
Epoch: 0700 cost= 0.101735294 W= 0.16234097 b= 1.4318552
Epoch: 0800 cost= 0.096385524 W= 0.1725536 b= 1.3594522
Epoch: 0900 cost= 0.092189491 W= 0.18159817 b= 1.2953304
Epoch: 1000 cost= 0.088898353 W= 0.18960832 b= 1.2385421

tensorflow基本算法(1):线性回归 linear regression_第3张图片

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