在逻辑回归中绘制决策边界

Question

我在一个看起来像这样的小数据集上运行逻辑回归：

在实施梯度下降和成本 function 之后，我在预测阶段获得了 89% 的准确率，但是我想确保一切正常，所以我试图 plot 分隔两个数据集的决策边界线。

下面我展示了显示成本 function 和 theta 参数的图表。 可以看出，目前我正在错误地打印决策边界线。

当我缩小决策边界图时，我可以看到以下内容：

我的决策边界绘制在数据集下方。 需要注意的一件事是我使用了特征缩放。

以下是我使用的代码：

主程序

%% 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 ====================
%  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);

% 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;


%% ============ 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);

%Normalize Feature
[X_norm mu sigma] = featureNormalize(X);

% Add intercept term to x and X_test
X = [ones(m, 1) X];
X_norm = [ones(m, 1) X_norm];

% Initialize fitting parameters
initial_theta = zeros(n + 1, 1);

% Compute and display initial cost and gradient
J = computeCostgrad(X_norm, y, initial_theta);

fprintf('Cost at initial theta (zeros): %f\n', J);
fprintf('Expected cost (approx): 0.693\n');


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

%% ============= Part 2a: Gradient Descent =====================
alpha=0.1;
iter=1000;
[theta, J_hist]=gradientDescent(initial_theta, X_norm, y, alpha, iter);
fprintf('Theta found by gradient descent:\n');
fprintf('%f\n', theta);

% Plot the convergence graph
figure;
plot(1:numel(J_hist), J_hist, '-b', 'LineWidth', 2);
xlabel('Nnumelumber of iterations');
ylabel('Cost J');



% 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;

%% ============== 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);
pred_admit=[45 85];
norm_pred_admit=[1,(pred_admit-mu)./sigma];
prob = norm_pred_admit*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_norm);

fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);
fprintf('Expected accuracy (approx): 89.0\n');
fprintf('\n');

计算成本梯度

function [J] = computeCostgrad(X, y, theta)
  % Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;


prediction=sigmoid(X*theta);
prob1=-y'*log(prediction);
prob0=(1-y')*log(1-prediction);
J=1/m*(prob1-prob0);
endfunction

梯度下降

function [theta, J_hist] = gradientDescent(theta, X, y, alpha, iter)

   m=length(y);
   J_hist=zeros(iter, 1);
  for (i=1:iter)
  prediction=sigmoid(X*theta);
  err=prediction-y;
  newDecrement = (alpha * (1/m) * err' * X); 
  theta=theta-newDecrement';
  J_hist(i)=computeCostgrad(X,y,theta);
  end

endfunction

绘图决策边界

function plotDecisionBoundary(theta, X, y)
plotData(X(:,2:3), y);
hold on

if size(X, 2) <= 3
    % Only need 2 points to define a line, so choose two endpoints
    plot_x = [min(X(:,2))-2,  max(X(:,2))+2];

    % Calculate the decision boundary line
    plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));

    % Plot, and adjust axes for better viewing
    plot(plot_x, plot_y)

    % Legend, specific for the exercise
    legend('Admitted', 'Not admitted', 'Decision Boundary')
    axis([30, 100, 30, 100])
else
    % Here is the grid range
    u = linspace(-1, 1.5, 50);
    v = linspace(-1, 1.5, 50);

    z = zeros(length(u), length(v));
    % Evaluate z = theta*x over the grid
    for i = 1:length(u)
        for j = 1:length(v)
            z(i,j) = mapFeature(u(i), v(j))*theta;
        end
    end
    z = 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 off

end

特征归一化

function [X_norm, mu, sigma] = featureNormalize(X)

X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));


mu=mean(X);
sigma=std(X);
X_norm1=(X(:,1)-mu(1))/sigma(1);
X_norm2=(X(:,2)-mu(2))/sigma(2);
X_norm=[X_norm1,X_norm2];

任何人都可以帮助我正确绘制决策边界。 我认为在绘制决策边界时计算 yintercept 存在一些错误。

Answer 1

因为您使用了特征缩放，所以您的权重与原始数据不匹配。

您需要将X_norm传递给您的plotDecisionBoundary function，而不是原始数据X ，如下所示：

plotDecisionBoundary(theta, X_norm, y);

同样，当您想要预测一个新示例时，您首先需要使用您计算的相同值对其进行缩放以标准化您的训练示例。

此外，在plotDecisionBoundary中， axis([30, 100, 30, 100])适合X而不是X_norm 。 因此，您需要更改它以适应X_norm的范围（这只是为了舒适，您始终可以通过更改缩放来更改它，直到找到线）。