# Week Six

## F Score

\begin{aligned} P &= &\dfrac{2}{\dfrac{1}{P}+\dfrac{1}{R}}\ &= &2 \dfrac{PR}{P+R} \end{aligned}

# Week Seven

## Support Vector Machine

### Cost Function

\begin{aligned} &\min_{\theta}\lbrack-\dfrac{1}{m}{\sum_{y_{i}\in Y, x_{i} \in X}{y_{i} \log h(\theta^{T}x_{i})}+(1-y_{i})\log (1-h(\theta^{T}x_{i}))+\dfrac{\lambda}{2m} \sum_{\theta_{i} \in \theta}{\theta_{i}^{2}}}\rbrack\ &\Rightarrow \min_{\theta}[-\sum_{y_{i} \in Y,x_{i} \in X}{y_{i} \log{h(\theta^{T}x_{i})}+(1-y_{i})\log(1-h(\theta^{T}x_{i}}))+\dfrac{\lambda}{2}\sum_{\theta_{i} \in \theta }{\theta^2_{i}}]\ &\Rightarrow\min_{\theta}[C\sum_{y_{i} \in Y,x_{i} \in X}{y_{i} \log{h(\theta^{T}x_{i})}+(1-y_{i})\log(1-h(\theta^{T}x_{i}}))+\sum_{\theta_{i} \in \theta }{\theta^2_{i}}]\\end{aligned}

C is somewhat $$\dfrac{1}{\lambda}$$.

• Large C:

• lower bias, high variance
• Small C:
• Higher bias, low variance
• Large $$\sigma^2$$: Features $$f_{i}$$ vary more smoothly.
• Higher bias, low variance
• Small $$\sigma^2$$: Features $$f_{i}$$ vary more sharply.
• Lower bias, high variance.

\begin{aligned} & \dfrac{1}{2} \sum_{\theta_{i} \in \theta}{\theta_{i}^2}\&s.t&\theta^{T}x_{i} \geq 1, if\ y_{i} = 1&\&&\theta^{T}x_{i} \leq -1, if\ y_{i} = 0& \end{aligned}

### PS

If features are too many related to m, use logistic regression or SVM without a kernel.

If n is small, m is intermediate, use SVM with Gaussian kernal.

If n is small, m is large, add more features and use logistic regression or SVM without a kernel.

# Week Eight

## K-means

### Cost Function

It try to minimize

$\min_{\mu}{\dfrac{1}{m} \sum_{i=1}^{m} ||x^{(i)} - \mu_{c^{(i)}}}||^2$

For the first loop, minimize the cost function by varing the centorid. For the second loop, it minimize the cost funcion with cetorid fixed and realign the centorid of every x in the training set.

### Initialize

Initialize the centorids randomly. Randomly select k samples from the training set and set the centorids to these random selected samples.

It is possible that K-meas fall into the local minimum, So repeat to initialize the centorids randomly until the cost(distortion) is suitable for your purposes.

K-means converge all the time and it will not increase the cost during the training processs. More centoirds will decease the cost, if not, the k-means must fall into the local minimum and reinitialize the centorid until the cost is less.

## PCA (Principal Component Analysis)

Restruct x from z meeting the below nonequation

$1-\dfrac{\dfrac{1}{m} \sum_{i=1}^{m}||x^{(i)}-x^{(i)}_{approximation}||^2}{\dfrac{1}{m} \sum_{i=1}^{m} ||x^{(i)}||^2} \geq 0.99$

PS:

the nonequation can be equal to the below

\begin{aligned} [U, S, D] &= svd(sigma)\ U_{reduce} &= U(:, 1:k)\ z &= U_{reduce}‘ * x\ x_{approximation} &= U_{reduce} * x\\\ S &= \left( \begin{array}{ccc} s_{11}&0&\cdots&0\ 0&s_{22}&\cdots&0\ \vdots&\vdots&\ddots&\vdots\ 0&0&\cdots&s_{nn} \end{array} \right)\\\ \dfrac{\sum_{i=1}^{k}s_{ii}^2}{\sum_{i=1}^{m} s_{ii}^2} &\geq 0.99 \end{aligned}

# Week Nine

## Anomaly Detection

### Gaussian Distribution

Multivariate Gaussian Distribution takes the connection of different variants into account

$p(x) = \dfrac{1}{(2\pi)^{\frac{n}{2}}|\Sigma|^{\frac{1}{2}}}e^{-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)}$

Single variant Gaussian Distribution is a special example of Multivariate Gaussian Distribution, where

$\Sigma = \left(\begin{array}{ccc} \sigma_{11}&&&&\ &\sigma_{22}&&&\ &&\ddots&&\ &&&\sigma_{nn}&\\end{array}\right)$

When training the Anomaly Detection, we can use Maximum Likelihood Estimation

\begin{aligned} \mu &= \dfrac{1}{m} \sum_{i=1}^{m}x^{(i)}\ \Sigma &= \dfrac{1}{m} \sum_{i=1}^{m} (x^{(i)}-\mu)(x^{(i)}-\mu)^{T} \end{aligned}

When we use single variant anomaly detection, the numerical cost is much cheaper than multivariant. But may need to add some new features to distinguish the normal and non-normal.

### Recommender System

#### Cost Function

\begin{aligned} J(X,\Theta) = \dfrac{1}{2} \sum_{(i,j):r(i,j)=1}((\theta^{(j)})^{T}x^{(i)}-y^{(i,j)})^2 + \dfrac{\lambda}{2}[\sum_{i=1}^{n_{m}}\sum_{k=1}^{n}(x_k^{(i)})^2 + \sum_{j=1}^{n_\mu} \sum_{k=1}^n(\theta_{k}^{(j)})^2]\ J(X,\Theta) = \dfrac{1}{2}Sum\{(X\Theta‘-Y).*R\} + \dfrac{\lambda}{2}(Sum\{\Theta.^2\} + Sum\{X.^2\}\\end{aligned}

\begin{aligned} \dfrac{\partial J}{\partial X} = ((X\Theta‘-Y).*R) \Theta + \lambda X\ \dfrac{\partial J}{\partial \Theta} = ((X\Theta‘-Y).*R)‘X + \lambda \Theta \end{aligned}