Linear regression with vectorization

multiple features

我們考慮一個情況：

今天檸檬想要預測房地產的價格

不要問我為甚麼

可能最近他想要買房(?

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檸檬學長，那我們要怎麼~~做出一張酷酷的圖~~

喔不4，是做出更好的預測？

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其實並不難ㄛ

你們還記得上上?(不知道幾個)禮拜的教的ㄇ：

J(w, b) = \frac{1}{2m}\sum_{i = 1}^{m} (\hat{y}^{(i)} - y)^2

J(w, b) = \frac{1}{2m}\sum_{i = 1}^{m} (\hat{y}^{(i)} - y)^2

btw $\hat{y} = h_\theta(x)$ ~~因為 $\hat{y}$ 比較好打~~

我們要做的，便是改變 $\hat{y}$ 的算法

使我們能從不同的feature學習

i.e.

$\hat{y} = w_1x_1 + w_2x_2 + b$

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又或者，我們有更聰明的方法

如果你有認真聽我講矩陣的那堂課

它就是用在今天ㄉ

改變 $\hat{y}$ 的算法

這是個很棒的方法

但如果今天我們有不只兩個features要用

算 $\hat{y}$ 的柿子會變得很冗長

難道只能用手刻ㄌㄇ(手好酸

J(w, b) = \frac{1}{2m}\sum_{i = 1}^{m} (\hat{y}^{(i)} - y)^2

J(w, b) = \frac{1}{2m}\sum_{i = 1}^{m} (\hat{y}^{(i)} - y)^2

apply MULTIPLE FEATURES

我們可以改一下 $\hat{y}$ 的寫法：

$\hat{y} = \sum_{j = 1}^{n} (w_{j}x_{j}) + b$

看到這裡，你可能會覺得完全沒有比較好啊

莫名其妙，檸檬又在唬爛

J(w, b) = \frac{1}{2m}\sum_{i = 1}^{m} (\hat{y}^{(i)} - y)^2

J(w, b) = \frac{1}{2m}\sum_{i = 1}^{m} (\hat{y}^{(i)} - y)^2

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~~沒錯我真ㄉ在唬爛~~

如果我們令

$\hat{y}^{(i)} = \sum_{j = 1}^{n} (w_{j}x^{(i)}_{j}) + b$

x^{(i)} = \begin{bmatrix} x^{(i)}_{1} \\ \vdots \\ x^{(i)}_{n}\end{bmatrix},\ w = \begin{bmatrix}w_{1} & \cdots & w_{n}\end{bmatrix},\ b \in R

x^{(i)} = \begin{bmatrix} x^{(i)}_{1} \\ \vdots \\ x^{(i)}_{n}\end{bmatrix},\ w = \begin{bmatrix}w_{1} & \cdots & w_{n}\end{bmatrix},\ b \in R

我們便可以得到 $\hat{y}^{(i)} = wx + b$

stack up examples

如果我們令

X = \begin{bmatrix}x^{(1)}_{1} & \cdots & x^{(m)}_{1} \\ \vdots & \ddots & \vdots \\ x^{(1)}_{n} & \cdots & x^{(m)}_{n}\end{bmatrix},\ y = \begin{bmatrix}y^{(1)} & \cdots & y^{(m)}\end{bmatrix}

X = \begin{bmatrix}x^{(1)}_{1} & \cdots & x^{(m)}_{1} \\ \vdots & \ddots & \vdots \\ x^{(1)}_{n} & \cdots & x^{(m)}_{n}\end{bmatrix},\ y = \begin{bmatrix}y^{(1)} & \cdots & y^{(m)}\end{bmatrix}

那我們就可以改寫整個Cost function(程式上)

 import numpy as np
 
#如果我們已經有w, X, y
J = 1/(2*m) * np.sum((np.dot(w, X) + b - y) ** 2)

gradient descent

\frac{\partial J}{\partial w} = \frac{(wX + b - y)X^{T}}{m}

\frac{\partial J}{\partial w} = \frac{(wX + b - y)X^{T}}{m}

所以我直接給你們微分算出來的結果 (茶

\frac{\partial J}{\partial b} = \frac{\sum^{m}_{i = 1} (wX^{(i)} + b - y^{(i)})}{m}

\frac{\partial J}{\partial b} = \frac{\sum^{m}_{i = 1} (wX^{(i)} + b - y^{(i)})}{m}

Machine learning

社課 - 3

lecturer : lemon

topic : Vectorize

multiple features

multiple features

multiple features

multiple features

multiple features

multiple features

multiple features

apply MULTIPLE FEATURES

apply MULTIPLE FEATURES

apply MULTIPLE FEATURES

apply MULTIPLE FEATURES

apply MULTIPLE FEATURES

apply MULTIPLE FEATURES

apply MULTIPLE FEATURES

stack up examples

stack up examples

stack up examples

stack up examples

gradient descent

gradient descent

why to vectorize

why to vectorize

why to vectorize

大概就這樣ㄌ

Linear regression with vectorization

Linear regression with vectorization

lemonilemon

	import numpy as np

	#如果我們已經有w, X, y
	J = 1/(2m) np.sum((np.dot(w, X) + b - y) ** 2)

Machine learning

社課 - 3

lecturer : lemon

topic : Vectorize

Linear regression with vectorization

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