前言

第一部分是关于线性回归
具体的概念其实在课程视频中都讲的比较清楚，但是在接触第一个练习之前，我一直不知道实际上如何coding,但是通过这种类似程序填空的形式能帮助我们理解这个实现。

单变量线性回归

1----读取数据并绘制图形

%% ======================= 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.m
plotData(X, y);

2-----计算COST FUNCTION----J(/theta)

%% =================== Part 3: Cost and Gradient descent ===================

X = [ones(m, 1), data(:,1)]; % Add a column of ones to x
theta = zeros(2, 1); % initialize fitting parameters

% Some gradient descent settings
iterations = 1500;
alpha = 0.01;

fprintf(\'\nTesting the cost function ...\n\')
% compute and display initial cost
J = 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 function
J = computeCost(X, y, [-1 ; 2]);
fprintf(\'\nWith theta = [-1 ; 2]\nCost computed = %f\n\', J);
fprintf(\'Expected cost value (approx) 54.24\n\');

具体的cost function的计算如下

function J = computeCost(X, y, theta)
%COMPUTECOST Compute cost for linear regression
%   J = COMPUTECOST(X, y, theta) computes the cost of using theta as the
%   parameter for linear regression to fit the data points in X and y

% Initialize some useful values
m = length(y); % number of training examples

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

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
%               You should set J to the cost.
H=X*theta;
J=sum(1/(2*m)*((H-y).^2));

% =========================================================================

end

需要注意上面的

H=X*theta;

这里我们采用向量化的方法，相比于for循环的迭代，能较大的提升程序的简洁性以及效率。这里在吴恩达老师在讲复习线性代数矩阵乘法中有提及。点这里进入这一节的传送门。
3------使cost function最小化，求出/theta
**这里使用的是梯度下降的方法，在线性回归模型中，还有另一种方法叫做 正规方程法 **
正规方程法
这里是在多变量里用到的正规方程法，单变量当然也适用。需要注意X扩展了一列值为1的向量。
梯度下降法（左为单特征/即n=1，右为多特征变量/即n>=1）

（可选）4—可视化
其实我认为在线性回归（包括后面的逻辑回归），目的就是求出那个参数，求出参数，问题就可以解决了，就可以用这个 假设方程H去预测值

%% ============= Part 4: Visualizing J(theta_0, theta_1) =============
fprintf(\'Visualizing J(theta_0, theta_1) ...\n\')

% Grid over which we will calculate J
theta0_vals = linspace(-10, 10, 100);
theta1_vals = linspace(-1, 4, 100);

% initialize J_vals to a matrix of 0\'s
J_vals = zeros(length(theta0_vals), length(theta1_vals));

% Fill out J_vals
for 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);
    end
end


% 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 flipped
J_vals = J_vals\';
% Surface plot
figure;
surf(theta0_vals, theta1_vals, J_vals)
xlabel(\'\theta_0\'); ylabel(\'\theta_1\');

% Contour plot
figure;
% Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100
contour(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);