本文由兩部分組成

• 邏輯回歸 Logistic Regression
  問題背景：預測學生是否可以入學

• 正則化邏輯回歸 regularized logistic regression
  問題背景：預測芯片是否可以通過質量檢測。

Logistic Regression

問題背景：預測學生是否可以入學
數據集：歷屆申請者的歷史數據：兩次考試的成績，以及是否入學的結果（0表示未入學，1表示入學）
目標：建立一個分類模型，基于申請者的兩次考試成績來評估申請者可以上大學的概率

兩部分的成績

初始化和加載數據
數據集:ex2data1.txt 位于文章最后

%% Initialization clear ; close all; clc%% Load Data % The first two columns contains the exam scores and the third column % contains the label.data = load('ex2data1.txt'); X = data(:, [1, 2]); y = data(:, 3);

Part 1:Plotting

%% ==================== Part 1: Plotting ==================== % We start the exercise by first plotting the data to understand the % the problem we are working with.fprintf(['Plotting data with + indicating (y = 1) examples and o ' ...'indicating (y = 0) examples.\n']);plotData(X, y);%自己編寫該函數% Put some labels hold on; % Labels and Legend xlabel('Exam 1 score') ylabel('Exam 2 score')% Specified in plot order legend('Admitted', 'Not admitted') hold off;fprintf('\nProgram paused. Press enter to continue.\n'); pause;

需要完成如下效果:兩個坐標軸分別是兩次得分，繪制出來的點的顏色代表是否錄取。黑色代表錄取，黃色代表未錄取

plotData.m

function plotData(X, y) %PLOTDATA Plots the data points X and y into a new figure % PLOTDATA(x,y) plots the data points with + for the positive examples % and o for the negative examples. X is assumed to be a Mx2 matrix.% Create New Figure figure; hold on;% ====================== YOUR CODE HERE ====================== % Instructions: Plot the positive and negative examples on a % 2D plot, using the option 'k+' for the positive % examples and 'ko' for the negative examples. %pos=find(y==1); neg=find(y==0); plot(X(pos,1),X(pos,2),'k+','LineWidth',2,...'MarkerSize',7); plot(X(neg,1),X(neg,2),'ko','MarkerFaceColor','y',...'MarkerSize',7);% =========================================================================hold off;end

繪制結果

資料查詢
find函數
功能：找到非零元素的索引（也就是它在array中的位置）

find - Find indices and values of nonzero elementsThis MATLAB function returns a vector containing the linear indices of eachnonzero element in array X.k = find(X)k = find(X,n)k = find(X,n,direction)[row,col] = find(___)[row,col,v] = find(___)

本例中使用的是 pos=find(y==1)這里y是一個列向量，find(y== 1)得到的就是 y==1時的位置，通過pos找到矩陣X中對應的一行，進行繪制。

繪制的代碼

‘MarkerFaceColor’ - 標記填充顏色
本例中'MarkerFaceColor','y'使用的是黃色
圖片來源：matlab 幫助中心
如果沒有加入'MarkerFaceColor','y'得到的是黑白圖形

Part2 :Compute Cost and Gradient

%% ============ Part 2: Compute Cost and Gradient ============ % In this part of the exercise, you will implement the cost and gradient % for logistic regression. You neeed to complete the code in % costFunction.m% Setup the data matrix appropriately, and add ones for the intercept term [m, n] = size(X);% Add intercept term to x and X_test X = [ones(m, 1) X];% Initialize fitting parameters initial_theta = zeros(n + 1, 1);% Compute and display initial cost and gradient [cost, grad] = costFunction(initial_theta, X, y);fprintf('Cost at initial theta (zeros): %f\n', cost); fprintf('Expected cost (approx): 0.693\n'); fprintf('Gradient at initial theta (zeros): \n'); fprintf(' %f \n', grad); fprintf('Expected gradients (approx):\n -0.1000\n -12.0092\n -11.2628\n');% Compute and display cost and gradient with non-zero theta test_theta = [-24; 0.2; 0.2]; [cost, grad] = costFunction(test_theta, X, y);fprintf('\nCost at test theta: %f\n', cost); fprintf('Expected cost (approx): 0.218\n'); fprintf('Gradient at test theta: \n'); fprintf(' %f \n', grad); fprintf('Expected gradients (approx):\n 0.043\n 2.566\n 2.647\n');fprintf('\nProgram paused. Press enter to continue.\n'); pause;

sigmoid.m函數
預測函數
$$h_{\theta }(x)=g({\theta }^{T}x)$$
其中 g為sigmoid function
$$g(z)=\frac{1}{1+e^{?z}}$$

function g = sigmoid(z) %SIGMOID Compute sigmoid function % g = SIGMOID(z) computes the sigmoid of z.% You need to return the following variables correctly g = zeros(size(z));% ====================== YOUR CODE HERE ====================== % Instructions: Compute the sigmoid of each value of z (z can be a matrix, % vector or scalar).g=1./(1+exp(-z));%這里需要點除，否則報錯% =============================================================end

函數的性質

sigmoid(100000)
ans = 1
sigmoid(0)
ans = 0.5000
sigmoid(-100000)
ans =0

代價函數
$$J(\theta )=?\frac{1}{m}\sum _{i=1}^m\left[y^{(i)}\times log\left(h_{\theta }(x^{(i)})\right)+(1?y^{(i)})\times log\left(1?h_{\theta }(x^{(i)})\right)\right]$$

代價函數的梯度是一個向量，第$j^{th}$個元素被定義為
$$\frac{?J(\theta )}{?{\theta }_j}=\frac{1}{m}\sum _{i=1}^m(h_{\theta }(x^{(i)})?y^{(i)})x_j^{(i)}$$

costFunction.m

function [J, grad] = costFunction(theta, X, y) %COSTFUNCTION Compute cost and gradient for logistic regression % J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the % parameter for logistic regression and the gradient of the cost % w.r.t. to the parameters.% 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 of a particular choice of theta. % You should set J to the cost. % Compute the partial derivatives and set grad to the partial % derivatives of the cost w.r.t. each parameter in theta % % Note: grad should have the same dimensions as theta % h=sigmoid(X*theta); first=y.*log(h);%第一項，點乘 second=(1-y).*log(1-h);%第二項，同樣是點乘 J=-1/m*sum(first+second);%求和，代價函數grad=1/m*X'*(h-y);% =============================================================end

Part 3: Optimizing using fminunc

下面是調用內置函數 fminunc

首先是定義fminunc的選項設置GradObj為on，是告訴函數返回值有兩個：cost 和 gradient
設置MaxIter為400，是告訴函數 最多迭代400次就要結束運行。

options = optimset('GradObj', 'on', 'MaxIter', 400);

@(t)(costFunction(t, X, y))創造一個函數，參數為t的函數，調用costFunction

[theta, cost] = ...fminunc(@(t)(costFunction(t, X, y)), initial_theta, options); %% ============= Part 3: Optimizing using fminunc ============= % In this exercise, you will use a built-in function (fminunc) to find the % optimal parameters theta.% Set options for fminunc options = optimset('GradObj', 'on', 'MaxIter', 400);% Run fminunc to obtain the optimal theta % This function will return theta and the cost [theta, cost] = ...fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);% Print theta to screen fprintf('Cost at theta found by fminunc: %f\n', cost); fprintf('Expected cost (approx): 0.203\n'); fprintf('theta: \n'); fprintf(' %f \n', theta); fprintf('Expected theta (approx):\n'); fprintf(' -25.161\n 0.206\n 0.201\n');% Plot Boundary plotDecisionBoundary(theta, X, y);% Put some labels hold on; % Labels and Legend xlabel('Exam 1 score') ylabel('Exam 2 score')% Specified in plot order legend('Admitted', 'Not admitted') hold off;fprintf('\nProgram paused. Press enter to continue.\n'); pause;

函數fminunc不需要自己手動寫循環，不需要手動為梯度下降法設置學習率，只需要提供計算代價和梯度的函數costFunction.它會收斂到正確的最優參數，并且返回cost和$\theta$

Cost at theta found by fminunc: 0.203498 Expected cost (approx): 0.203 theta: -25.161343 0.206232 0.201472 Expected theta (approx):-25.1610.2060.201

后面是繪制決策邊界

對于函數
$$h_{\theta }(x)=g({\theta }_1+{\theta }_2{x}_2+{\theta }_3{x}_3)$$
由于sigmoid函數分界點為z=0，z>0時,輸出為1；z<0時，輸出為0；

所以分界點時，有：這里的變量是($x_2,x_3$)
$${\theta }_1+{\theta }_2{x}_2+{\theta }_3{x}_3=0$$
這里下標使用1，2，3的原因是：matlab中初始下標是從1開始，而不是0，便于計算

其中繪圖函數代碼如下
plotDecisionBoundary.m函數

function plotDecisionBoundary(theta, X, y) %PLOTDECISIONBOUNDARY Plots the data points X and y into a new figure with %the decision boundary defined by theta % PLOTDECISIONBOUNDARY(theta, X,y) plots the data points with + for the % positive examples and o for the negative examples. X is assumed to be % a either % 1) Mx3 matrix, where the first column is an all-ones column for the % intercept. % 2) MxN, N>3 matrix, where the first column is all-ones% Plot Data plotData(X(:,2:3), y);%調用前面的繪圖函數 hold onif size(X, 2) <= 3% Only need 2 points to define a line, so choose two endpointsplot_x = [min(X(:,2))-2, max(X(:,2))+2];% Calculate the decision boundary lineplot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));% Plot, and adjust axes for better viewingplot(plot_x, plot_y)% Legend, specific for the exerciselegend('Admitted', 'Not admitted', 'Decision Boundary')axis([30, 100, 30, 100]) else% Here is the grid rangeu = linspace(-1, 1.5, 50);v = linspace(-1, 1.5, 50);z = zeros(length(u), length(v));% Evaluate z = theta*x over the gridfor i = 1:length(u)for j = 1:length(v)z(i,j) = mapFeature(u(i), v(j))*theta;endendz = z'; % important to transpose z before calling contour% Plot z = 0% Notice you need to specify the range [0, 0]contour(u, v, z, [0, 0], 'LineWidth', 2) end hold offend

繪制直線需要知道 點對(x,y)的值，
舉例子

plot_x=[0,10]; plot_y=[3,13]; plot(plot_x,plot_y);

繪制的圖形為：

對于本題，對應點對是什么呢？
這里的變量是($x_2,x_3$),其中$x_2=$plot_x, 而$x_3=$plot_y可由${\theta }_1+{\theta }_2{x}_2+{\theta }_3{x}_3=0$解出來。

% Only need 2 points to define a line, so choose two endpointsplot_x = [min(X(:,2))-2, max(X(:,2))+2];% Calculate the decision boundary line，這實際上是x3plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));

Part 4: Predict and Accuracies

%% ============== Part 4: Predict and Accuracies ============== % After learning the parameters, you'll like to use it to predict the outcomes % on unseen data. In this part, you will use the logistic regression model % to predict the probability that a student with score 45 on exam 1 and % score 85 on exam 2 will be admitted. % % Furthermore, you will compute the training and test set accuracies of % our model. % % Your task is to complete the code in predict.m% Predict probability for a student with score 45 on exam 1 % and score 85 on exam 2 prob = sigmoid([1 45 85] * theta); fprintf(['For a student with scores 45 and 85, we predict an admission ' ...'probability of %f\n'], prob); fprintf('Expected value: 0.775 +/- 0.002\n\n');% Compute accuracy on our training set p = predict(theta, X);fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100); fprintf('Expected accuracy (approx): 89.0\n');fprintf('\n');

下面是predict.m函數的內容

返回值p是預測值，結果只能是0或者1，0代表不能上大學，1代表能夠上大學。
X的維度是m*(n+1)，m表示訓練樣本的數量。theta長度是(n+1)*1

index=find(sigmoid(X*theta)>=0.5);%找到＞=0.5的 p(index)=1;

下面是簡化寫法，也是matlab推薦的寫法，速度更快

%或者寫成 index=sigmoid(X*theta)>=0.5; p(index)=1;

代碼

function p = predict(theta, X) %PREDICT Predict whether the label is 0 or 1 using learned logistic %regression parameters theta % p = PREDICT(theta, X) computes the predictions for X using a % threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)m = size(X, 1); % Number of training examples% You need to return the following variables correctly p = zeros(m, 1);% ====================== YOUR CODE HERE ====================== % Instructions: Complete the following code to make predictions using % your learned logistic regression parameters. % You should set p to a vector of 0's and 1's %index=sigmoid(X*theta)>=0.5; p(index)=1;% =========================================================================end

然后是計算準確率
這里準確率的計算是通過比較預測值p和真實值y差異的平均值。若p==y，則結果=1，若p≠y,則結果=0，這樣所有的結果累加，然后取平均值，計算出來的就是準確率。比如100個數據中，有90個是預測值等于真實值，然后平均值就是（901+100）/100=0.9。

fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);

結果

For a student with scores 45 and 85, we predict an admission probability of 0.776291 Expected value: 0.775 +/- 0.002Train Accuracy: 89.000000 Expected accuracy (approx): 89.0

數據集
ex2data1.txt

34.62365962451697,78.0246928153624,0 30.28671076822607,43.89499752400101,0 35.84740876993872,72.90219802708364,0 60.18259938620976,86.30855209546826,1 79.0327360507101,75.3443764369103,1 45.08327747668339,56.3163717815305,0 61.10666453684766,96.51142588489624,1 75.02474556738889,46.55401354116538,1 76.09878670226257,87.42056971926803,1 84.43281996120035,43.53339331072109,1 95.86155507093572,38.22527805795094,0 75.01365838958247,30.60326323428011,0 82.30705337399482,76.48196330235604,1 69.36458875970939,97.71869196188608,1 39.53833914367223,76.03681085115882,0 53.9710521485623,89.20735013750205,1 69.07014406283025,52.74046973016765,1 67.94685547711617,46.67857410673128,0 70.66150955499435,92.92713789364831,1 76.97878372747498,47.57596364975532,1 67.37202754570876,42.83843832029179,0 89.67677575072079,65.79936592745237,1 50.534788289883,48.85581152764205,0 34.21206097786789,44.20952859866288,0 77.9240914545704,68.9723599933059,1 62.27101367004632,69.95445795447587,1 80.1901807509566,44.82162893218353,1 93.114388797442,38.80067033713209,0 61.83020602312595,50.25610789244621,0 38.78580379679423,64.99568095539578,0 61.379289447425,72.80788731317097,1 85.40451939411645,57.05198397627122,1 52.10797973193984,63.12762376881715,0 52.04540476831827,69.43286012045222,1 40.23689373545111,71.16774802184875,0 54.63510555424817,52.21388588061123,0 33.91550010906887,98.86943574220611,0 64.17698887494485,80.90806058670817,1 74.78925295941542,41.57341522824434,0 34.1836400264419,75.2377203360134,0 83.90239366249155,56.30804621605327,1 51.54772026906181,46.85629026349976,0 94.44336776917852,65.56892160559052,1 82.36875375713919,40.61825515970618,0 51.04775177128865,45.82270145776001,0 62.22267576120188,52.06099194836679,0 77.19303492601364,70.45820000180959,1 97.77159928000232,86.7278223300282,1 62.07306379667647,96.76882412413983,1 91.56497449807442,88.69629254546599,1 79.94481794066932,74.16311935043758,1 99.2725269292572,60.99903099844988,1 90.54671411399852,43.39060180650027,1 34.52451385320009,60.39634245837173,0 50.2864961189907,49.80453881323059,0 49.58667721632031,59.80895099453265,0 97.64563396007767,68.86157272420604,1 32.57720016809309,95.59854761387875,0 74.24869136721598,69.82457122657193,1 71.79646205863379,78.45356224515052,1 75.3956114656803,85.75993667331619,1 35.28611281526193,47.02051394723416,0 56.25381749711624,39.26147251058019,0 30.05882244669796,49.59297386723685,0 44.66826172480893,66.45008614558913,0 66.56089447242954,41.09209807936973,0 40.45755098375164,97.53518548909936,1 49.07256321908844,51.88321182073966,0 80.27957401466998,92.11606081344084,1 66.74671856944039,60.99139402740988,1 32.72283304060323,43.30717306430063,0 64.0393204150601,78.03168802018232,1 72.34649422579923,96.22759296761404,1 60.45788573918959,73.09499809758037,1 58.84095621726802,75.85844831279042,1 99.82785779692128,72.36925193383885,1 47.26426910848174,88.47586499559782,1 50.45815980285988,75.80985952982456,1 60.45555629271532,42.50840943572217,0 82.22666157785568,42.71987853716458,0 88.9138964166533,69.80378889835472,1 94.83450672430196,45.69430680250754,1 67.31925746917527,66.58935317747915,1 57.23870631569862,59.51428198012956,1 80.36675600171273,90.96014789746954,1 68.46852178591112,85.59430710452014,1 42.0754545384731,78.84478600148043,0 75.47770200533905,90.42453899753964,1 78.63542434898018,96.64742716885644,1 52.34800398794107,60.76950525602592,0 94.09433112516793,77.15910509073893,1 90.44855097096364,87.50879176484702,1 55.48216114069585,35.57070347228866,0 74.49269241843041,84.84513684930135,1 89.84580670720979,45.35828361091658,1 83.48916274498238,48.38028579728175,1 42.2617008099817,87.10385094025457,1 99.31500880510394,68.77540947206617,1 55.34001756003703,64.9319380069486,1 74.77589300092767,89.52981289513276,1

regularized logistic regression正則化邏輯回歸

問題背景：預測芯片是否可以通過質量檢測。
數據集：過去芯片的兩項檢測結果的歷史數據

數據可視化

%% Initialization clear ; close all; clc%% Load Data % The first two columns contains the X values and the third column % contains the label (y).data = load('ex2data2.txt'); X = data(:, [1, 2]); y = data(:, 3);plotData(X, y);% Put some labels hold on;% Labels and Legend xlabel('Microchip Test 1') ylabel('Microchip Test 2')% Specified in plot order legend('y = 1', 'y = 0') hold off;

結果：芯片經歷兩次檢測的結果，調用的仍然是上文的 plotData函數

我們可以看到，該題不是一個線性的決策邊界，不能用線性函數來做。需要考慮多項式來做。

Part 1: Regularized Logistic Regression

從每一個特征點中提取出來更多的特征，我們使用$x_1$和$x_2$的最高六次多項式來進行映射。

從上面可以看到，我們把兩個特征（$x_1$和$x_2$）轉換成了一個（$28\times 1$）的向量
經過這個高維的向量，訓練出來的邏輯回歸分類器，它的決策邊界更加復雜，而且非線性。
對應的多項式特征轉變使用如下函數
mapFeature.m函數

function out = mapFeature(X1, X2) % MAPFEATURE Feature mapping function to polynomial features % % MAPFEATURE(X1, X2) maps the two input features % to quadratic features used in the regularization exercise. % % Returns a new feature array with more features, comprising of % X1, X2, X1.^2, X2.^2, X1*X2, X1*X2.^2, etc.. % % Inputs X1, X2 must be the same size %degree = 6; out = ones(size(X1(:,1))); for i = 1:degreefor j = 0:iout(:, end+1) = (X1.^(i-j)).*(X2.^j);end endend

下面計算正則化邏輯回歸的代價函數和梯度
代價函數為
需要注意的是，不需要正則化${\theta }_0$，即不需要正則化theta(1)，因為matlab下標是從1開始的。
對應的梯度函數

函數為
costFunctionReg.m文件
核心部分就是上面兩個公式的matlab實現

function [J, grad] = costFunctionReg(theta, X, y, lambda) %COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization % J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using % theta as the parameter for regularized logistic regression and the % gradient of the cost w.r.t. to the parameters. % 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 of a particular choice of theta. % You should set J to the cost. % Compute the partial derivatives and set grad to the partial % derivatives of the cost w.r.t. each parameter in thetafirst=y.*log(sigmoid(X*theta)); second=(1-y).*log(1-sigmoid(X*theta)); theta_1=[0;theta(2:end)];%把theta（1）拿掉，不參與正則化J=-sum((first+second))/m+lambda/(2*m)*(theta_1'*theta_1);%注意這里不是點乘，點乘得到的是向量，這里需要的是一個數：向量轉置*向量=數 %calculate the gradient %不參與正則化 grad=X'*(sigmoid(X*theta)-y)/m+lambda/m*theta_1;% =============================================================end

下面是調用函數，并且進行正則化邏輯回歸

%% =========== Part 1: Regularized Logistic Regression ============ % In this part, you are given a dataset with data points that are not % linearly separable. However, you would still like to use logistic % regression to classify the data points. % % To do so, you introduce more features to use -- in particular, you add % polynomial features to our data matrix (similar to polynomial % regression). %% Add Polynomial Features% Note that mapFeature also adds a column of ones for us, so the intercept % term is handled X = mapFeature(X(:,1), X(:,2));% Initialize fitting parameters initial_theta = zeros(size(X, 2), 1);%size(X, 2)%返回X第二維度的長度，也就是每一行的元素個數% Set regularization parameter lambda to 1 lambda = 1;% Compute and display initial cost and gradient for regularized logistic % regression [cost, grad] = costFunctionReg(initial_theta, X, y, lambda);fprintf('Cost at initial theta (zeros): %f\n', cost); fprintf('Expected cost (approx): 0.693\n'); fprintf('Gradient at initial theta (zeros) - first five values only:\n'); fprintf(' %f \n', grad(1:5)); fprintf('Expected gradients (approx) - first five values only:\n'); fprintf(' 0.0085\n 0.0188\n 0.0001\n 0.0503\n 0.0115\n');fprintf('\nProgram paused. Press enter to continue.\n'); pause;% Compute and display cost and gradient % with all-ones theta and lambda = 10 test_theta = ones(size(X,2),1); [cost, grad] = costFunctionReg(test_theta, X, y, 10);fprintf('\nCost at test theta (with lambda = 10): %f\n', cost); fprintf('Expected cost (approx): 3.16\n'); fprintf('Gradient at test theta - first five values only:\n'); fprintf(' %f \n', grad(1:5)); fprintf('Expected gradients (approx) - first five values only:\n'); fprintf(' 0.3460\n 0.1614\n 0.1948\n 0.2269\n 0.0922\n');fprintf('\nProgram paused. Press enter to continue.\n'); pause;

測試結果

Cost at initial theta (zeros): 0.693147 Expected cost (approx): 0.693 Gradient at initial theta (zeros) - first five values only:0.008475 0.018788 0.000078 0.050345 0.011501 Expected gradients (approx) - first five values only:0.00850.01880.00010.05030.0115Program paused. Press enter to continue.Cost at test theta (with lambda = 10): 3.164509 Expected cost (approx): 3.16 Gradient at test theta - first five values only:0.346045 0.161352 0.194796 0.226863 0.092186 Expected gradients (approx) - first five values only:0.34600.16140.19480.22690.0922Program paused. Press enter to continue.

Part 2: Regularization and Accuracies

正則化和準確性：嘗試不同的$\lambda$取值，看決策邊界如何變化和訓練集的準確性如何變化。

%% ============= Part 2: Regularization and Accuracies ============= % Optional Exercise: % In this part, you will get to try different values of lambda and % see how regularization affects the decision coundart % % Try the following values of lambda (0, 1, 10, 100). % % How does the decision boundary change when you vary lambda? How does % the training set accuracy vary? %% Initialize fitting parameters initial_theta = zeros(size(X, 2), 1);% Set regularization parameter lambda to 1 (you should vary this) lambda = 1;% Set Options options = optimset('GradObj', 'on', 'MaxIter', 400);% Optimize [theta, J, exit_flag] = ...fminunc(@(t)(costFunctionReg(t, X, y, lambda)), initial_theta, options);% Plot Boundary plotDecisionBoundary(theta, X, y); hold on; title(sprintf('lambda = %g', lambda))% Labels and Legend xlabel('Microchip Test 1') ylabel('Microchip Test 2')legend('y = 1', 'y = 0', 'Decision boundary') hold off;% Compute accuracy on our training set p = predict(theta, X);fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100); fprintf('Expected accuracy (with lambda = 1): 83.1 (approx)\n');

結果

對應的準確率

Train Accuracy: 88.983051 Expected accuracy (with lambda = 1): 83.1 (approx)

對應的準確率

Train Accuracy: 61.016949 Expected accuracy (with lambda = 1): 83.1 (approx)

其中
plotDecisionBoundary.m函數如下，繪制決策邊界的函數代碼

function plotDecisionBoundary(theta, X, y) %PLOTDECISIONBOUNDARY Plots the data points X and y into a new figure with %the decision boundary defined by theta % PLOTDECISIONBOUNDARY(theta, X,y) plots the data points with + for the % positive examples and o for the negative examples. X is assumed to be % a either % 1) Mx3 matrix, where the first column is an all-ones column for the % intercept. % 2) MxN, N>3 matrix, where the first column is all-ones% Plot Data plotData(X(:,2:3), y); hold onif size(X, 2) <= 3% Only need 2 points to define a line, so choose two endpointsplot_x = [min(X(:,2))-2, max(X(:,2))+2];% Calculate the decision boundary lineplot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));% Plot, and adjust axes for better viewingplot(plot_x, plot_y)% Legend, specific for the exerciselegend('Admitted', 'Not admitted', 'Decision Boundary')axis([30, 100, 30, 100]) else% Here is the grid rangeu = linspace(-1, 1.5, 50);v = linspace(-1, 1.5, 50);z = zeros(length(u), length(v));% Evaluate z = theta*x over the gridfor i = 1:length(u)for j = 1:length(v)z(i,j) = mapFeature(u(i), v(j))*theta;endendz = z'; % important to transpose z before calling contour% Plot z = 0% Notice you need to specify the range [0, 0]contour(u, v, z, [0, 0], 'LineWidth', 2) end hold offend

數據集
ex2data2.txt文件中內容如下

0.051267,0.69956,1 -0.092742,0.68494,1 -0.21371,0.69225,1 -0.375,0.50219,1 -0.51325,0.46564,1 -0.52477,0.2098,1 -0.39804,0.034357,1 -0.30588,-0.19225,1 0.016705,-0.40424,1 0.13191,-0.51389,1 0.38537,-0.56506,1 0.52938,-0.5212,1 0.63882,-0.24342,1 0.73675,-0.18494,1 0.54666,0.48757,1 0.322,0.5826,1 0.16647,0.53874,1 -0.046659,0.81652,1 -0.17339,0.69956,1 -0.47869,0.63377,1 -0.60541,0.59722,1 -0.62846,0.33406,1 -0.59389,0.005117,1 -0.42108,-0.27266,1 -0.11578,-0.39693,1 0.20104,-0.60161,1 0.46601,-0.53582,1 0.67339,-0.53582,1 -0.13882,0.54605,1 -0.29435,0.77997,1 -0.26555,0.96272,1 -0.16187,0.8019,1 -0.17339,0.64839,1 -0.28283,0.47295,1 -0.36348,0.31213,1 -0.30012,0.027047,1 -0.23675,-0.21418,1 -0.06394,-0.18494,1 0.062788,-0.16301,1 0.22984,-0.41155,1 0.2932,-0.2288,1 0.48329,-0.18494,1 0.64459,-0.14108,1 0.46025,0.012427,1 0.6273,0.15863,1 0.57546,0.26827,1 0.72523,0.44371,1 0.22408,0.52412,1 0.44297,0.67032,1 0.322,0.69225,1 0.13767,0.57529,1 -0.0063364,0.39985,1 -0.092742,0.55336,1 -0.20795,0.35599,1 -0.20795,0.17325,1 -0.43836,0.21711,1 -0.21947,-0.016813,1 -0.13882,-0.27266,1 0.18376,0.93348,0 0.22408,0.77997,0 0.29896,0.61915,0 0.50634,0.75804,0 0.61578,0.7288,0 0.60426,0.59722,0 0.76555,0.50219,0 0.92684,0.3633,0 0.82316,0.27558,0 0.96141,0.085526,0 0.93836,0.012427,0 0.86348,-0.082602,0 0.89804,-0.20687,0 0.85196,-0.36769,0 0.82892,-0.5212,0 0.79435,-0.55775,0 0.59274,-0.7405,0 0.51786,-0.5943,0 0.46601,-0.41886,0 0.35081,-0.57968,0 0.28744,-0.76974,0 0.085829,-0.75512,0 0.14919,-0.57968,0 -0.13306,-0.4481,0 -0.40956,-0.41155,0 -0.39228,-0.25804,0 -0.74366,-0.25804,0 -0.69758,0.041667,0 -0.75518,0.2902,0 -0.69758,0.68494,0 -0.4038,0.70687,0 -0.38076,0.91886,0 -0.50749,0.90424,0 -0.54781,0.70687,0 0.10311,0.77997,0 0.057028,0.91886,0 -0.10426,0.99196,0 -0.081221,1.1089,0 0.28744,1.087,0 0.39689,0.82383,0 0.63882,0.88962,0 0.82316,0.66301,0 0.67339,0.64108,0 1.0709,0.10015,0 -0.046659,-0.57968,0 -0.23675,-0.63816,0 -0.15035,-0.36769,0 -0.49021,-0.3019,0 -0.46717,-0.13377,0 -0.28859,-0.060673,0 -0.61118,-0.067982,0 -0.66302,-0.21418,0 -0.59965,-0.41886,0 -0.72638,-0.082602,0 -0.83007,0.31213,0 -0.72062,0.53874,0 -0.59389,0.49488,0 -0.48445,0.99927,0 -0.0063364,0.99927,0 0.63265,-0.030612,0

總結

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