Model and Cost Function

1 模型概述 - Model Representation

To establish notation for future use, we’ll use

• x(i)
  denote the “input” variables (living area in this example), also called input features, and
• y(i)
  denote the “output” or target variable that we are trying to predict (price).

A pair (x(i),y(i)) is called a training example
the dataset that we’ll be using to learn—a list of m training examples (x(i),y(i));i=1,...,m—is called a training set.
the superscript “(i)” in the notation is simply an index into the training set, and has nothing to do with exponentiation

• X
  denote the space of input values
• Y
  denote the space of output values

In this example

X = Y = R

To describe the supervised learning problem slightly more formally, our goal is,
given a training set, to learn afunction h : X → Yso that h(x) is a “good” predictor for the corresponding value of y.
For historical reasons, this function h is called a hypothesis. Seen pictorially, the process is therefore like this

• regression problem
  When the target variable that we’re trying to predict iscontinuous, such as in our housing example
• classification problem
  When y can take on only a small number of discrete values (such as if, given the living area, we wanted to predict if a dwelling is a house or an apartment, say)
  简单的介绍了一下数据集的表示方法，并且提出来h（hypothesis），即通过训练得出来的一个假设函数，通过输入x，得出来预测的结果y。并在最后介绍了线性回归方程

2 代价函数 - Cost Function

代价函数是用来测量实际值和预测值精确度的一个函数模型.
We can measure the accuracy of our hypothesis function by using acost function.
This takes an average difference (actually a fancier version of an average) of all the results of the hypothesis with inputs from x's and the actual output y's.

首先需要搞清楚假设函数和代价函数的区别
当假设函数为线性时，即线性回归方程，其由两个参数组成：theta0和theta1

我们要做的就是选取两个参数的值，使其代价函数的值达到最小化

J(θ0,θ1)=12m∑i=1m(y^i−yi)2=12m∑i=1m(hθ(xi)−yi)2

To break it apart, it is 1/2 x ̄ where x ̄ is the mean of the squares of hθ(xi)−yi , or the difference
between the predicted value and the actual value.
This function is otherwise called theSquared error function, or Mean squared error.
The mean is halved (1/2)as a convenience for the computation of the gradient descent, as the derivative term of the square function will cancel out the 1/2 term.
The following image summarizes what the cost function does:

3 代价函数(一)