吴恩达《机器学习》

4.1 Multiple features(variables)

notation
$n$n = number of feature
$x^{\left(i\right)}$x(i) = input (features) of $i$ith training example
$x_j^{\left(i\right)}$xj(i)​ = value of features $j$j in $i$ith training example

hypothesis:

previously: $h_{\theta }\left(x\right)={\theta }_0+{\theta }_{1}x$hθ​(x)=θ0​+θ1​x
Now: $h_{\theta }\left(x\right)={\theta }_0+{\theta }_1{x}_1+{\theta }_2{x}_2+\cdots +{\theta }_n{x}_n={\theta }^{T}x$hθ​(x)=θ0​+θ1​x1​+θ2​x2​+⋯+θn​xn​=θTx
for convenience of notation,define $x_0=1$x0​=1
$X=\left(\begin{array}{c}{x}_0\\ x_1\\ x_2\\ \dots \\ x_n\end{array}\right)\in R^{n+1}\theta =\left(\begin{array}{c}{\theta }_0\\ {\theta }_1\\ {\theta }_2\\ \dots \\ {\theta }_n\end{array}\right)\in R^{n+1}$X=⎝⎜⎜⎜⎜⎛​x0​x1​x2​…xn​​⎠⎟⎟⎟⎟⎞​∈Rn+1θ=⎝⎜⎜⎜⎜⎛​θ0​θ1​θ2​…θn​​⎠⎟⎟⎟⎟⎞​∈Rn+1
conclusion:
multivariate linear regression

4.2 Gradient Descent for multiple variables

hypothesis: