機器學習（二）

一、模型
線性迴歸

in supervised learning,have a data set(called a training set)-> to predict the price of the house

symbols of supervised learning:
m ->訓練樣本的數量
x ->輸入變量（feature）
y->目標變量


二、代價函數

what we want to do is come up with values for parameter
(minimization problem)->minimize the square difference
(h(x)-y) ->sum over my training set,sum from i equals 1 to M of the square difference between the prediction

in actually,cost function:


depend on your training set,get a cost function looks like that:

使用等高線：

三、梯度下降

• start with some parameters
• keep changing to reduce cost function until end up at a minimum
  可能由於初始點的不同，導致最後結果的差異：
  
  
  learning rate：how big a step we take downhill with gradient descent
• if too small,gradient descent can be slow
• if too large,gradient descent can overshot the minimum,it may fail to converge,or even diverge
• as we approach a local minimum,gradient descent will automatically take small steps,so,no need to decrease learning rate over time

需要先更新temp後，再賦值，否則temp1是按照更新後的值進行計算所得到的。

simplified situation：


“Batch” Gradient Descent:each step of gradient descent uses all the training examples.

some other of gradient descent that are not batch versions,look at small subsets of the training sets at a time.

四、多元特徵變量

多個特徵量表達式可以寫爲;



五、多元梯度下降（特徵縮放）
idea：make sure features are on a similar scale

it turns out gradient descent will just have a much harder time(反覆振盪）

特徵值除以最大值：

get every feature into approximately (-1,1)

歸一化：

即（x-average value）/（range value）

六、多元梯度下降法（學習率）

• debugging：how to make sure gradient descent is working correctly(cost function decrease after every iteration)
  
• how to choose learning rate(for sufficiently small learning rate,cost function decrease on every iteration)

除了線性迴歸，還可以使用多項式迴歸等其他擬合方法