Week Six

F Score

P = 2 1 P + 1 R = 2 P R P + R \begin{aligned} P &= &\dfrac{2}{\dfrac{1}{P}+\dfrac{1}{R}}\\ &= &2 \dfrac{PR}{P+R} \end{aligned} P==P1+R122P+RPR

Week Seven

Support Vector Machine

Cost Function

min ⁡ θ [ − 1 m ∑ y i ∈ Y , x i ∈ X y i log ⁡ h ( θ T x i ) + ( 1 − y i ) log ⁡ ( 1 − h ( θ T x i ) ) + λ 2 m ∑ θ i ∈ θ θ i 2 ] ⇒ min ⁡ θ [ − ∑ y i ∈ Y , x i ∈ X y i log ⁡ h ( θ T x i ) + ( 1 − y i ) log ⁡ ( 1 − h ( θ T x i ) ) + λ 2 ∑ θ i ∈ θ θ i 2 ] ⇒ min ⁡ θ [ C ∑ y i ∈ Y , x i ∈ X y i log ⁡ h ( θ T x i ) + ( 1 − y i ) log ⁡ ( 1 − h ( θ T x i ) ) + ∑ θ i ∈ θ θ i 2 ] \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} θmin[−m1yi∈Y,xi∈X∑yilogh(θTxi)+(1−yi)log(1−h(θTxi))+2mλθi∈θ∑θi2]⇒θmin[−yi∈Y,xi∈X∑yilogh(θTxi)+(1−yi)log(1−h(θTxi))+2λθi∈θ∑θi2]⇒θmin[Cyi∈Y,xi∈X∑yilogh(θTxi)+(1−yi)log(1−h(θTxi))+θi∈θ∑θi2]
C is somewhat 1 λ \dfrac{1}{\lambda} λ1.

Large C:
- lower bias, high variance
Small C:
- Higher bias, low variance
Large σ 2 \sigma^2 σ2: Features f i f_{i} fi vary more smoothly.
- Higher bias, low variance
Small σ 2 \sigma^2 σ2: Features f i f_{i} fi vary more sharply.
- Lower bias, high variance.
  1 2 ∑ θ i ∈ θ θ i 2 s . t θ T x i ≥ 1 , i f y i = 1 θ T x i ≤ − 1 , i f y i = 0 \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} 21θi∈θ∑θi2s.tθTxi≥1,if yi=1θTxi≤−1,if yi=0

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 ⁡ μ 1 m ∑ i = 1 m ∣ ∣ x ( i ) − μ c ( i ) ∣ ∣ 2 \min_{\mu}{\dfrac{1}{m} \sum_{i=1}^{m} ||x^{(i)} - \mu_{c^{(i)}}}||^2 μminm1i=1∑m∣∣x(i)−μc(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 − 1 m ∑ i = 1 m ∣ ∣ x ( i ) − x a p p r o x i m a t i o n ( i ) ∣ ∣ 2 1 m ∑ i = 1 m ∣ ∣ x ( i ) ∣ ∣ 2 ≥ 0.99 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 1−m1∑i=1m∣∣x(i)∣∣2m1∑i=1m∣∣x(i)−xapproximation(i)∣∣2≥0.99
PS:
the nonequation can be equal to the below
[ U , S , D ] = s v d ( s i g m a ) U r e d u c e = U ( : , 1 : k ) z = U r e d u c e ′ ∗ x x a p p r o x i m a t i o n = U r e d u c e ∗ x S = ( s 11 0 ⋯ 0 0 s 22 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ s n n ) ∑ i = 1 k s i i 2 ∑ i = 1 m s i i 2 ≥ 0.99 \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} [U,S,D]UreducezxapproximationS∑i=1msii2∑i=1ksii2=svd(sigma)=U(:,1:k)=Ureduce′∗x=Ureduce∗x=⎝⎜⎜⎜⎛s110⋮00s22⋮0⋯⋯⋱⋯00⋮snn⎠⎟⎟⎟⎞≥0.99

Week Nine

Anomaly Detection

Gaussian Distribution

Multivariate Gaussian Distribution takes the connection of different variants into account
p ( x ) = 1 ( 2 π ) n 2 ∣ Σ ∣ 1 2 e − 1 2 ( x − μ ) T Σ − 1 ( x − μ ) p(x) = \dfrac{1}{(2\pi)^{\frac{n}{2}}|\Sigma|^{\frac{1}{2}}}e^{-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)} p(x)=(2π)2n∣Σ∣211e−21(x−μ)TΣ−1(x−μ)
Single variant Gaussian Distribution is a special example of Multivariate Gaussian Distribution, where
Σ = ( σ 11 σ 22 ⋱ σ n n ) \Sigma = \left(\begin{array}{ccc} \sigma_{11}&&&&\\ &\sigma_{22}&&&\\ &&\ddots&&\\ &&&\sigma_{nn}&\\ \end{array}\right) Σ=⎝⎜⎜⎛σ11σ22⋱σnn⎠⎟⎟⎞
When training the Anomaly Detection, we can use Maximum Likelihood Estimation
μ = 1 m ∑ i = 1 m x ( i ) Σ = 1 m ∑ i = 1 m ( x ( i ) − μ ) ( x ( i ) − μ ) T \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} μΣ=m1i=1∑mx(i)=m1i=1∑m(x(i)−μ)(x(i)−μ)T
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

J ( X , Θ ) = 1 2 ∑ ( i , j ) : r ( i , j ) = 1 ( ( θ ( j ) ) T x ( i ) − y ( i , j ) ) 2 + λ 2 [ ∑ i = 1 n m ∑ k = 1 n ( x k ( i ) ) 2 + ∑ j = 1 n μ ∑ k = 1 n ( θ k ( j ) ) 2 ] J ( X , Θ ) = 1 2 S u m { ( X Θ ′ − Y ) . ∗ R } + λ 2 ( S u m { Θ . 2 } + S u m { X . 2 } \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} J(X,Θ)=21(i,j):r(i,j)=1∑((θ(j))Tx(i)−y(i,j))2+2λ[i=1∑nmk=1∑n(xk(i))2+j=1∑nμk=1∑n(θk(j))2]J(X,Θ)=21Sum{(XΘ′−Y).∗R}+2λ(Sum{Θ.2}+Sum{X.2}
∂ J ∂ X = ( ( X Θ ′ − Y ) . ∗ R ) Θ + λ X ∂ J ∂ Θ = ( ( X Θ ′ − Y ) . ∗ R ) ′ X + λ Θ \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} ∂X∂J=((XΘ′−Y).∗R)Θ+λX∂Θ∂J=((XΘ′−Y).∗R)′X+λΘ

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Machine Learning on Coursera

Week Six

F Score

Week Seven

Support Vector Machine

Cost Function

PS

Week Eight

K-means

Cost Function

Initialize

PCA (Principal Component Analysis)

Week Nine

Anomaly Detection

Gaussian Distribution

Recommender System

Cost Function

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