Linearregression 1 Model representation2 Cost function3 Gradient descent4 Gradient descent for linear regression1 Mul2ple features2 Feature Scaling3 Learningrate4 Features and polynomial regression5 Normal equa2on编程作业

Linear’regression

发现这个教程是最入门的一个教程了，老师讲的很好，也很通俗，每堂课后面还有编程作业，全程用matlab编程，只需要填写核心代码，很适合自学。

1.1 Model representation

起始给出了预测房价的例子。

这个问题属于监督问题，每个样本都给出了准确的答案。

同时属于回归问题，对给定值预测实际输出。

定义(x(i),y(i))(x(i),y(i))为第i个样本，x表示输入值，y表示输出值，上标表示样本。

以下是机器学习运行模型

对于假设h我们可以用一条直线描述，用线性函数预测房价值。

hθ(x)=θ0+θ1∗xhθ(x)=θ0+θ1∗x

1.2 Cost function

我们取怎样的θθ值可以使预测值更加准确呢？

想想看，我们应使得每一个预测值和真实值差别不大，可以定义代价函数如下

J(θ0,θ1)=12m∑mi=1(hθ(x(i))−y(i))2J(θ0,θ1)=12m∑i=1m(hθ(x(i))−y(i))2

通过使J值取最小来满足需求

下面通过图形方式感受一下代价函数

1.3 Gradient descent

怎样使我们的代价函数取得最小值呢

下面我们采取梯度下降法。

好比我们下山，每次在一点环顾四周，往最陡峭的路向下走，用图形的方式更形象的表示

Gradient descent algorithm

repeat until convergence{

θj=θj−α∂∂θjJ(θ0,θ1)θj=θj−α∂∂θjJ(θ0,θ1)(forj=0andj=1)(forj=0andj=1)

}

注意更新theta值应同时更新，matlab中向量更新即为同时更新，所以应使上式向量化（之后会讲解向量化含义），也可采取下面方式

1.4 Gradient descent for linear regression

repeat until convergence{

θj=θj−α∂∂θjJ(θ0,θ1)θj=θj−α∂∂θjJ(θ0,θ1)(forj=0andj=1)(forj=0andj=1)

}

∂∂θjJ(θ0,θ1)==∂∂θj12m∑i=1m(hθ(x(i)−y(i)))2∂∂θj12m∑i=1m(hθ(θ0+θ1x)−y(i))2∂∂θjJ(θ0,θ1)=∂∂θj12m∑i=1m(hθ(x(i)−y(i)))2=∂∂θj12m∑i=1m(hθ(θ0+θ1x)−y(i))2

j=0:∂∂θjJ(θ0,θ1)=1m∑mi=1(hθ(x(i)−y(i)))j=0:∂∂θjJ(θ0,θ1)=1m∑i=1m(hθ(x(i)−y(i)))

j=1:∂∂θjJ(θ0,θ1)=1m∑mi=1(hθ(x(i)−y(i)))∗x(i)j=1:∂∂θjJ(θ0,θ1)=1m∑i=1m(hθ(x(i)−y(i)))∗x(i)

2.1 Mul2ple features

如果输入值不止一个，我们的假设函数应修改为

hθ(x)=θ0+θ1x1+θ2x2+⋯+θnxnhθ(x)=θ0+θ1x1+θ2x2+⋯+θnxn

为了结构统一，我们设x0=1x0=1

hθ(x)=θ0+θ1x1+θ2x2+⋯+θnxn=θTxhθ(x)=θ0+θ1x1+θ2x2+⋯+θnxn=θTx

如此一来，便将变量向量化了

New algorithm

repeat until convergence{

θj=θj−α∂∂θjJ(θ)=θj−α1m∑mi=1(hθ(x(i)−y(i)))∗x(i)jθj=θj−α∂∂θjJ(θ)=θj−α1m∑i=1m(hθ(x(i)−y(i)))∗xj(i)(forj=0,1,2⋯n)(forj=0,1,2⋯n)

}

2.2 Feature Scaling

面对输入数据各个特征值范围差距过大的问题，我们可以对输入数据进行标准化。

x(j)i=x(j)i−avg(xi)Sixi(j)=xi(j)−avg(xi)Si

其中SiSi可以为标准差，也可以为max(xi)−min(xi)max(xi)−min(xi)

2.3 Learning’rate

如果αα太小，则梯度下降法会收敛缓慢如果αα太大，则梯度下降法每次迭代可能不下降，最终导致不收敛。

2.4 Features and polynomial regression

除了线性回归外，我们也能采用多项式回归

举例如下假设函数

hθ(x)=θ0+θ1x+θ2x2+θ3x3hθ(x)=θ0+θ1x+θ2x2+θ3x3

我们可以定义为

hθ(x)=θ0+θ1x1+θ2x2+θ3x3=θ0+θ1x1+θ2x21+θ3x31hθ(x)=θ0+θ1x1+θ2x2+θ3x3=θ0+θ1x1+θ2x12+θ3x13

对于多项式回归，标准化更加重要。

2.5 Normal equa2on

除了梯度下降法，另一种求最小值的方式则是让代价函数导数为0，求θθ值

J(θ)=12m∑mi=1(hθ(x(i))−y(i))2J(θ)=12m∑i=1m(hθ(x(i))−y(i))2

∂∂θjJ(θ)=0∂∂θjJ(θ)=0for every j

求得:θ=(XTX)−1XTyθ=(XTX)−1XTy

下面这个图比较了两个算法之间的区别

特殊情况：

由于用标准方程法时，涉及到要计算矩阵XTX的逆矩阵。但是XTX的结果有可能不可逆。 当使用python的numpy计算时，其会返回广义的逆结果。 主要原因： 出现这种情况的主要原因，主要有特征值数量多于训练集个数、特征值之间线性相关（如表示面积采用平方米和平方公里同时出现在特征值中）。 因此，首先需要考虑特征值是否冗余，并且清除不常用、区分度不大的特征值。对于(XTX)(XTX)不可逆的情况下，我们可以采取减少特征量和使用正规化方式来改善。

比较标准方程法和梯度下降法：

这两个方法都是旨在获取使代价函数值最小的参数θ，两个方法各有优缺点：

1）梯度下降算法

优点：当训练集很大的时候（百万级），速度很快。

缺点：需要调试出合适的学习速率α、需要多次迭代、特征值数量级不一致时需要特征缩放。

2）标准方程法：

优点：不需要α、不需要迭代、不需要特征缩放，直接解出结果。

缺点：运算量大，当训练集很大时速度非常慢。

综合：因此，当训练集百万级时，考虑使用梯度下降算法；训练集在万级别时，考虑使用标准方程法。在万到百万级区间时，看情况使用，主要还是使用梯度下降算法。

编程作业

ex1.m

%% Machine Learning Online Class - Exercise 1: Linear Regression% Instructions% ------------%% This file contains code that helps you get started on the% linear exercise. You will need to complete the following functions% in this exericse:%%warmUpExercise.m%plotData.m%gradientDescent.m%computeCost.m%gradientDescentMulti.m%computeCostMulti.m%featureNormalize.m%normalEqn.m%% For this exercise, you will not need to change any code in this file,% or any other files other than those mentioned above.%% x refers to the population size in 10,000s% y refers to the profit in $10,000s%%% Initializationclear ; close all; clc%% ==================== Part 1: Basic Function ====================% Complete warmUpExercise.mfprintf('Running warmUpExercise ... \n');fprintf('5x5 Identity Matrix: \n');warmUpExercise()fprintf('Program paused. Press enter to continue.\n');pause;%% ======================= Part 2: Plotting =======================fprintf('Plotting Data ...\n')data = load('ex1data1.txt');X = data(:, 1); y = data(:, 2);m = length(y); % number of training examples% Plot Data% Note: You have to complete the code in plotData.mplotData(X, y);fprintf('Program paused. Press enter to continue.\n');pause;%% =================== Part 3: Cost and Gradient descent ===================X = [ones(m, 1), data(:,1)]; % Add a column of ones to xtheta = zeros(2, 1); % initialize fitting parameters% Some gradient descent settingsiterations = 1500;alpha = 0.01;fprintf('\nTesting the cost function ...\n')% compute and display initial costJ = computeCost(X, y, theta);fprintf('With theta = [0 ; 0]\nCost computed = %f\n', J);fprintf('Expected cost value (approx) 32.07\n');% further testing of the cost functionJ = computeCost(X, y, [-1 ; 2]);fprintf('\nWith theta = [-1 ; 2]\nCost computed = %f\n', J);fprintf('Expected cost value (approx) 54.24\n');fprintf('Program paused. Press enter to continue.\n');pause;fprintf('\nRunning Gradient Descent ...\n')% run gradient descenttheta = gradientDescent(X, y, theta, alpha, iterations);% print theta to screenfprintf('Theta found by gradient descent:\n');fprintf('%f\n', theta);fprintf('Expected theta values (approx)\n');fprintf(' -3.6303\n 1.1664\n\n');% Plot the linear fithold on; % keep previous plot visibleplot(X(:,2), X*theta, '-')legend('Training data', 'Linear regression')hold off % don't overlay any more plots on this figure% Predict values for population sizes of 35,000 and 70,000predict1 = [1, 3.5] *theta;fprintf('For population = 35,000, we predict a profit of %f\n',...predict1*10000);predict2 = [1, 7] * theta;fprintf('For population = 70,000, we predict a profit of %f\n',...predict2*10000);fprintf('Program paused. Press enter to continue.\n');pause;%% ============= Part 4: Visualizing J(theta_0, theta_1) =============fprintf('Visualizing J(theta_0, theta_1) ...\n')% Grid over which we will calculate Jtheta0_vals = linspace(-10, 10, 100);theta1_vals = linspace(-1, 4, 100);% initialize J_vals to a matrix of 0'sJ_vals = zeros(length(theta0_vals), length(theta1_vals));% Fill out J_valsfor i = 1:length(theta0_vals)for j = 1:length(theta1_vals)t = [theta0_vals(i); theta1_vals(j)];J_vals(i,j) = computeCost(X, y, t);endend% Because of the way meshgrids work in the surf command, we need to% transpose J_vals before calling surf, or else the axes will be flippedJ_vals = J_vals';% Surface plotfigure;surf(theta0_vals, theta1_vals, J_vals)xlabel('\theta_0'); ylabel('\theta_1');% Contour plotfigure;% Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20))xlabel('\theta_0'); ylabel('\theta_1');hold on;plot(theta(1), theta(2), 'rx', 'MarkerSize', 10, 'LineWidth', 2);

ComputeCost.m

gradientDescent.m

function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)%GRADIENTDESCENT Performs gradient descent to learn theta% theta = GRADIENTDESCENT(X, y, theta, alpha, num_iters) updates theta by % taking num_iters gradient steps with learning rate alpha% Initialize some useful valuesm = length(y); % number of training examplesJ_history = zeros(num_iters, 1);for iter = 1:num_iters% ====================== YOUR CODE HERE ======================% Instructions: Perform a single gradient step on the parameter vector%theta. %% Hint: While debugging, it can be useful to print out the values% of the cost function (computeCost) and gradient here.%theta = theta - alpha/m*X'*(X*theta - y);% ============================================================% Save the cost J in every iteration J_history(iter) = computeCost(X, y, theta);endend