MachineLearningOnCoursera

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

\[\begin{aligned}
J(X,\Theta) = \dfrac{1}{2} \sum_{(i,j):r(i,j)=1}((\theta^{(j)})^{T}x^{(i)}-y^{(i,j)})^2 + \dfrac{\lambda}{2}[\sum_{i=1}^{n_{m}}\sum_{k=1}^{n}(x_k^{(i)})^2 + \sum_{j=1}^{n_\mu} \sum_{k=1}^n(\theta_{k}^{(j)})^2]\\
J(X,\Theta) = \dfrac{1}{2}Sum\{(X\Theta'-Y).*R\} + \dfrac{\lambda}{2}(Sum\{\Theta.^2\} + Sum\{X.^2\}\\
\end{aligned}\]
\[\begin{aligned}
\dfrac{\partial J}{\partial X} = ((X\Theta'-Y).*R) \Theta + \lambda X\\
\dfrac{\partial J}{\partial \Theta} = ((X\Theta'-Y).*R)'X + \lambda \Theta
\end{aligned}\]