机器学习——正则化线性回归和偏差/方差（Matlab代码实现）

目录

💥1 概述

📚2 运行结果

🎉3 参考文献

👨‍💻4 Matlab代码

💥1 概述

本文使用水库水位的变化来实现正则化线性回归，以预测大坝流出的水量。后续本文将对调试学习算法进行一些诊断，并检查偏差与方差的影响。

第 1 部分：正则化线性回归

我们将实现正则化线性回归，以利用水库水位的变化来预测从大坝流出的水量。

第 1.1 部分：加载和可视化数据

我们将首先可视化包含有关水位变化 x 和从大坝流出的水量 y 的历史记录的数据集。

此数据集分为三个部分：

模型将学习的训练集：X， y用于确定正则化参数的交叉验证集：Xval、yval用于评估性能的测试集。这些是你的模型在训练过程中没有看到的“看不见的”例子：Xtest，ytest。

📚2 运行结果

🎉3 参考文献

[1]刘建伟,崔立鹏,刘泽宇,罗雄麟.正则化稀疏模型[J].计算机学报,,38(07):1307-1325.

👨‍💻4 Matlab代码

主函数部分代码：

%% Initialization

clear ; close all; clc

%% =========== Part 1: Loading and Visualizing Data =============

% We start the exercise by first loading and visualizing the dataset.

% The following code will load the dataset into your environment and plot

% the data.

%

% Load Training Data

fprintf('Loading and Visualizing Data ...\n')

% Load from ex5data1:

% You will have X, y, Xval, yval, Xtest, ytest in your environment

load ('ex5data1.mat');

% m = Number of examples

m = size(X, 1);

% Plot training data

plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);

xlabel('Change in water level (x)');

ylabel('Water flowing out of the dam (y)');

fprintf('Program paused. Press enter to continue.\n');

pause;

%% =========== Part 2: Regularized Linear Regression Cost =============

% You should now implement the cost function for regularized linear

% regression.

%

theta = [1 ; 1];

J = linearRegCostFunction([ones(m, 1) X], y, theta, 1);

fprintf(['Cost at theta = [1 ; 1]: %f '...

'\n(this value should be about 303.993192)\n'], J);

fprintf('Program paused. Press enter to continue.\n');

pause;

%% =========== Part 3: Regularized Linear Regression Gradient =============

% You should now implement the gradient for regularized linear

% regression.

%

theta = [1 ; 1];

[J, grad] = linearRegCostFunction([ones(m, 1) X], y, theta, 1);

fprintf(['Gradient at theta = [1 ; 1]: [%f; %f] '...

'\n(this value should be about [-15.303016; 598.250744])\n'], ...

grad(1), grad(2));

fprintf('Program paused. Press enter to continue.\n');

pause;

%% =========== Part 4: Train Linear Regression =============

% Once you have implemented the cost and gradient correctly, the

% trainLinearReg function will use your cost function to train

% regularized linear regression.

%

% Write Up Note: The data is non-linear, so this will not give a great

% fit.

%

% Train linear regression with lambda = 0

lambda = 0;

[theta] = trainLinearReg([ones(m, 1) X], y, lambda);

% Plot fit over the data

plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);

xlabel('Change in water level (x)');

ylabel('Water flowing out of the dam (y)');

hold on;

plot(X, [ones(m, 1) X]*theta, '--', 'LineWidth', 2)

hold off;

fprintf('Program paused. Press enter to continue.\n');

pause;

%% =========== Part 5: Learning Curve for Linear Regression =============

% Next, you should implement the learningCurve function.

%

% Write Up Note: Since the model is underfitting the data, we expect to

% see a graph with "high bias" -- Figure 3 in ex5.pdf

%

lambda = 0;

[error_train, error_val] = ...

learningCurve([ones(m, 1) X], y, ...

[ones(size(Xval, 1), 1) Xval], yval, ...

lambda);

plot(1:m, error_train, 1:m, error_val);

title('Learning curve for linear regression')

legend('Train', 'Cross Validation')

xlabel('Number of training examples')

ylabel('Error')

axis([0 13 0 150])

fprintf('# Training Examples\tTrain Error\tCross Validation Error\n');

for i = 1:m

fprintf(' \t%d\t\t%f\t%f\n', i, error_train(i), error_val(i));

end

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