机器学习编程作业（exe5）总结

偏执的太偏执、 2022-05-18 10:17 176阅读 0赞

参考博文[CheeseZH: Stanford University: Machine Learning Ex5:Regularized Linear Regression and Bias v.s. Variance][CheeseZH_ Stanford University_ Machine Learning Ex5_Regularized Linear Regression and Bias v.s. Variance]

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(1) plot the training data:

%% 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 将训练集描述出来
    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;

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function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
    %LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear 
    %regression with multiple variables
    %   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the 
    %   cost of using theta as the parameter for linear regression to fit the 
    %   data points in X and y. Returns the cost in J and the gradient in grad
    
    % Initialize some useful values
    m = length(y); % number of training examples
    
    % You need to return the following variables correctly 
    J = 0;
    grad = zeros(size(theta));
    
    % ====================== YOUR CODE HERE ======================
    % Instructions: Compute the cost and gradient of regularized linear 
    %               regression for a particular choice of theta.
    %
    %               You should set J to the cost and grad to the gradient.
    %
    
    %cost function
    % 正则化线性回归的代价函数 303.9932
    J=(1/2/m)*sum(power(X*theta-y,2))+lambda/2/m*sum(theta(2:end,:) .^ 2);
    
    %gradient
    % 正则化的线性回归代价函数的梯度
    grad = (X' * (X*theta-y)) / m;
    % -15.3030 598.1674  j =0 不需要正则化 j大于等于1时 才需要正则化参数来修正
    grad(2:end) =  grad(2:end) + lambda * theta(2:end) / m;% -15.3030 598.2507
    % =========================================================================
    
    grad = grad(:);
    
    end

%% =========== 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);
    %  [-15.303016; 598.250744] 
    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');

function [theta] = trainLinearReg(X, y, lambda)
    %TRAINLINEARREG Trains linear regression given a dataset (X, y) and a
    %regularization parameter lambda
    %   [theta] = TRAINLINEARREG (X, y, lambda) trains linear regression using
    %   the dataset (X, y) and regularization parameter lambda. Returns the
    %   trained parameters theta.
    %
    
    % Initialize Theta
    initial_theta = zeros(size(X, 2), 1); 
    
    % Create "short hand" for the cost function to be minimized
    costFunction = @(t) linearRegCostFunction(X, y, t, lambda);
    
    % Now, costFunction is a function that takes in only one argument
    options = optimset('MaxIter', 200, 'GradObj', 'on');
    
    % Minimize using fmincg
    theta = fmincg(costFunction, initial_theta, options);
    
    end

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

function [error_train, error_val] = ...
        learningCurve(X, y, Xval, yval, lambda)
    %LEARNINGCURVE Generates the train and cross validation set errors needed 
    %to plot a learning curve
    %   [error_train, error_val] = ...
    %       LEARNINGCURVE(X, y, Xval, yval, lambda) returns the train and
    %       cross validation set errors for a learning curve. In particular, 
    %       it returns two vectors of the same length - error_train and 
    %       error_val. Then, error_train(i) contains the training error for
    %       i examples (and similarly for error_val(i)).
    %
    %   In this function, you will compute the train and test errors for
    %   dataset sizes from 1 up to m. In practice, when working with larger
    %   datasets, you might want to do this in larger intervals.
    %
    
    % Number of training examples
    m = size(X, 1);
    
    % You need to return these values correctly
    error_train = zeros(m, 1);
    error_val   = zeros(m, 1);
    
    % ====================== YOUR CODE HERE ======================
    % Instructions: Fill in this function to return training errors in 
    %               error_train and the cross validation errors in error_val. 
    %               i.e., error_train(i) and 
    %               error_val(i) should give you the errors
    %               obtained after training on i examples.
    %
    % Note: You should evaluate the training error on the first i training
    %       examples (i.e., X(1:i, :) and y(1:i)).
    %
    %       For the cross-validation error, you should instead evaluate on
    %       the _entire_ cross validation set (Xval and yval).
    %
    % Note: If you are using your cost function (linearRegCostFunction)
    %       to compute the training and cross validation error, you should 
    %       call the function with the lambda argument set to 0. 
    %       Do note that you will still need to use lambda when running
    %       the training to obtain the theta parameters.
    %
    % Hint: You can loop over the examples with the following:
    %
    %       for i = 1:m
    %           % Compute train/cross validation errors using training examples 
    %           % X(1:i, :) and y(1:i), storing the result in 
    %           % error_train(i) and error_val(i)
    %           ....
    %           
    %       end
    %
    
    % ---------------------- Sample Solution ----------------------
    for i=1:m
        % 将训练集数目逐个增加  进行训练得出参数 
        % 这样就能画出随着训练数据集个数增加,训练集与测试集 线性回归代价函数的大小变化
        theta = trainLinearReg(X(1:i,:), y(1:i), lambda);
        % 用得出来的参数 分别放到训练集和验证集中计算代价函数
        error_train(i) = linearRegCostFunction(X(1:i,:), y(1:i), theta, 0);
        error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
    end
    % -------------------------------------------------------------
    
    % =========================================================================
    
    end

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在你完成了polyFeatures.m。ex5 m脚本将运用线性回归代价函数进行多项式回归。记住，即使我们在特征向量中有多项式项，我们仍然在解线性回归优化问题。多项式项已经变成了我们可以用于线性回归的特征。我们使用的是相同的代价函数和梯度，这是你为这个练习的早期所写的。

对于这部分的练习，你将使用一个8次方的多项式。事实证明，如果我们直接在训练数据集上运行预测数据，不能很好发挥作用，因为这些特性会被严重地伸缩（例如，一个x=40的例子现在有一个特性s=40^8=6.5 x 1012）。因此，您需要使用特性规范化。在学习多项式回归的参数0之前，ex5.m将首先调用featureNormalize函数，使训练集的特征规范化，并分别存储mu和sigma参数。我们已经为你实现了这个featureNormalize.m，它和第一次练习的功能是一样的。在学习了参数![70 8][]之后，您应该会看到两个图（图4,5），用于多项式回归的lamda=0。

从图4中，您应该看到，多项式能够很好地拟合数据集，从而获得一个较低的训练数据集错误。然而，多项式的拟合是非常复杂的，甚至在极端情况下也会下降。这是多项式回归模型对训练数据的过度拟合的一个暗示，并不能拟合所有数据集点。为了更好地理解非正则化（lamda=0）模型的问题，您可以看到学习曲线（图5）显示了低训练数据集错误低的相同效果，但是交叉验证错误很高。在培训和交叉验证错误之间存在差距，这表明存在一个高方差问题。

function [X_poly] = polyFeatures(X, p)
    %POLYFEATURES Maps X (1D vector) into the p-th power
    %   [X_poly] = POLYFEATURES(X, p) takes a data matrix X (size m x 1) and
    %   maps each example into its polynomial features where
    %   X_poly(i, :) = [X(i) X(i).^2 X(i).^3 ...  X(i).^p];
    %
    
    
    % You need to return the following variables correctly.
    X_poly = zeros(numel(X), p);
    
    % ====================== YOUR CODE HERE ======================
    % Instructions: Given a vector X, return a matrix X_poly where the p-th 
    %               column of X contains the values of X to the p-th power.
    %
    % 增加多项式特征向量 x^2
    for i=1:p
        X_poly(:,i)=X .^ i;
    end
    
    % =========================================================================
    
    end

%% =========== Part 6: Feature Mapping for Polynomial Regression =============
    %  One solution to this is to use polynomial regression. You should now
    %  complete polyFeatures to map each example into its powers
    %
    
    p = 8;
    
    % Map X onto Polynomial Features and Normalize
    % 对训练集的特征向量 进行扩展为8个特征向量 并进行特征向量归一化
    X_poly = polyFeatures(X, p);
    % ex5.m将首先调用featureNormalize函数，使训练集的特征规范化，并分别存储mu和sigma
    [X_poly, mu, sigma] = featureNormalize(X_poly);  % Normalize
    X_poly = [ones(m, 1), X_poly];                   % Add Ones
    
    % Map X_poly_test and normalize (using mu and sigma)
    % 对测试集的特征向量 进行扩展为8个特征向量 并进行特征向量归一化
    X_poly_test = polyFeatures(Xtest, p);
    X_poly_test = bsxfun(@minus, X_poly_test, mu);
    X_poly_test = bsxfun(@rdivide, X_poly_test, sigma);
    X_poly_test = [ones(size(X_poly_test, 1), 1), X_poly_test];         % Add Ones
    
    % Map X_poly_val and normalize (using mu and sigma)
    % 对验证测试集的特征向量 进行扩展为8个特征向量 并进行特征向量归一化
    X_poly_val = polyFeatures(Xval, p);
    X_poly_val = bsxfun(@minus, X_poly_val, mu);
    X_poly_val = bsxfun(@rdivide, X_poly_val, sigma);
    X_poly_val = [ones(size(X_poly_val, 1), 1), X_poly_val];           % Add Ones
    
    fprintf('Normalized Training Example 1:\n');
    fprintf('  %f  \n', X_poly(1, :));
    
    fprintf('\nProgram paused. Press enter to continue.\n');
    pause;
    
    
    
    %% =========== Part 7: Learning Curve for Polynomial Regression =============
    %  Now, you will get to experiment with polynomial regression with multiple
    %  values of lambda. The code below runs polynomial regression with 
    %  lambda = 0. You should try running the code with different values of
    %  lambda to see how the fit and learning curve change.
    %
    
    lambda = 0;
    [theta] = trainLinearReg(X_poly, y, lambda);
    
    % Plot training data and fit
    % 对原来的训练集增加多项式特征后 再次训练出新的参数 然后进行拟合训练集 可以发现过拟合现象
    figure(1);
    plot(X, y, 'rx', 'MarkerSize', 10, 'LineWidth', 1.5);
    plotFit(min(X), max(X), mu, sigma, theta, p);
    xlabel('Change in water level (x)');
    ylabel('Water flowing out of the dam (y)');
    title (sprintf('Polynomial Regression Fit (lambda = %f)', lambda));
    % 画出增加多项式特征后, 训练集和验证测试集的误差进行绘图 
    % 可以看出来训练集拟合的很好 然而验证测试集的误差很大
    figure(2);
    [error_train, error_val] = ...
        learningCurve(X_poly, y, X_poly_val, yval, lambda);
    plot(1:m, error_train, 1:m, error_val);
    
    title(sprintf('Polynomial Regression Learning Curve (lambda = %f)', lambda));
    xlabel('Number of training examples')
    ylabel('Error')
    axis([0 13 0 100])
    legend('Train', 'Cross Validation')
    
    fprintf('Polynomial Regression (lambda = %f)\n\n', lambda);
    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
    
    fprintf('Program paused. Press enter to continue.\n');
    pause;

function [X_norm, mu, sigma] = featureNormalize(X)
    %FEATURENORMALIZE Normalizes the features in X 
    %   FEATURENORMALIZE(X) returns a normalized version of X where
    %   the mean value of each feature is 0 and the standard deviation
    %   is 1. This is often a good preprocessing step to do when
    %   working with learning algorithms.
    
    mu = mean(X);
    X_norm = bsxfun(@minus, X, mu);
    
    sigma = std(X_norm);
    X_norm = bsxfun(@rdivide, X_norm, sigma);
    % ============================================================
    
    end

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在您完成代码之后，ex5.m的下一部分。将运行你的函数可以画出曲线交叉验证的高方差情况。lamda选择使用哪个入参数。您应该看到类似于下图的情节。在这个图中，我们可以看到A的最佳值是3。由于数据集的训练和验证分割的随机性，交叉验证错误有时可能比训练错误更低。

在实践中，特别是对于小型的训练集，当你绘制学习曲线来调试你的算法时，以确定训练错误和交叉验证错误。具体地说，为了诊断训练集错误和验证集中错误，你应该首先随机选择来自训练集和测试集。然后,您将学习使用随机选择训练集训练出来的参数和在随机选择的参数训练集和交叉验证集评估参数。上面的步骤应该多次重复(50)和平均误差可用于确定我的训练误差和交叉验证错误的例子,对于这个可选(劣质的)练习,您应该实现上述策略来计算学习曲线。为了参考，图10显示了我们获得的多项式回归的学习曲线lamda = 0.01。由于随机选择的例子，您的图形可能会有所不同。

%% =========== Part 8: Validation for Selecting Lambda =============
    %  You will now implement validationCurve to test various values of 
    %  lambda on a validation set. You will then use this to select the
    %  "best" lambda value.
    %
    
    [lambda_vec, error_train, error_val] = ...
        validationCurve(X_poly, y, X_poly_val, yval);
    
    close all;
    plot(lambda_vec, error_train, lambda_vec, error_val);
    legend('Train', 'Cross Validation');
    xlabel('lambda');
    ylabel('Error');
    
    fprintf('lambda\t\tTrain Error\tValidation Error\n');
    for i = 1:length(lambda_vec)
    	fprintf(' %f\t%f\t%f\n', ...
                lambda_vec(i), error_train(i), error_val(i));
    end
    
    fprintf('Program paused. Press enter to continue.\n');
    pause;

function [lambda_vec, error_train, error_val] = ...
        validationCurve(X, y, Xval, yval)
    %VALIDATIONCURVE Generate the train and validation errors needed to
    %plot a validation curve that we can use to select lambda
    %   [lambda_vec, error_train, error_val] = ...
    %       VALIDATIONCURVE(X, y, Xval, yval) returns the train
    %       and validation errors (in error_train, error_val)
    %       for different values of lambda. You are given the training set (X,
    %       y) and validation set (Xval, yval).
    %
    
    % Selected values of lambda (you should not change this)
    % 正则化参数的取值
    lambda_vec = [0 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10]';
    
    % You need to return these variables correctly.
    error_train = zeros(length(lambda_vec), 1);
    error_val = zeros(length(lambda_vec), 1);
    
    % ====================== YOUR CODE HERE ======================
    % Instructions: Fill in this function to return training errors in 
    %               error_train and the validation errors in error_val. The 
    %               vector lambda_vec contains the different lambda parameters 
    %               to use for each calculation of the errors, i.e, 
    %               error_train(i), and error_val(i) should give 
    %               you the errors obtained after training with 
    %               lambda = lambda_vec(i)
    %
    % Note: You can loop over lambda_vec with the following:
    %
    %       for i = 1:length(lambda_vec)
    %           lambda = lambda_vec(i);
    %           % Compute train / val errors when training linear 
    %           % regression with regularization parameter lambda
    %           % You should store the result in error_train(i)
    %           % and error_val(i)
    %           ....
    %           
    %       end
    %
    %
    for i=1:length(lambda_vec)
        % 将训练集数目 中正则化参数进行变化  进行训练得出参数 
        % 这样就能画出随着正则化参数增加,训练集与测试集 线性回归代价函数的大小变化
        lambda=lambda_vec(i);
        theta = trainLinearReg(X, y, lambda);
        % 用得出来的参数 分别放到训练集和验证集中计算代价函数
        error_train(i) = linearRegCostFunction(X, y, theta, 0);
        error_val(i) = linearRegCostFunction(Xval, yval, theta, 0);
    end;
    % =========================================================================
    
    end

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[CheeseZH_ Stanford University_ Machine Learning Ex5_Regularized Linear Regression and Bias v.s. Variance]: https://www.cnblogs.com/CheeseZH/p/4673488.html
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