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ML (Methodologies, Taxonomy, What is it?, Hierarchical clustering, Feature…

- - - - Parameter tuning
- - - - Linear regression is used to predict the value of a continuous variable Y based on one or more input predictor variables X
      - Evoluation
        
        Root Mean Square Error
        $ E_{\text{rms}} = \sqrt{\frac{1}{N}\sum_{i=1}^N (y_i - \hat{y}_i)^2}) $
        where $ y_i $ is the true value and $ \hat{y}_i$ is the predicted value.
        
        Mean absolute error
        $ E_{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i| $
        
        Mean Squared Error
        $E_{MSE} = \frac{1}{N} \sum_{i=1}^N e_i^2 = \frac{1}{N} E_{SSE}$
        
        R-Squared
        $ E_R = 1 - \frac{\sum e_i^2}{\sum(y_i - \bar{y})^2} $
        where $ \bar y = \frac 1 N \sum y_i $
        
        Sum of squared Errors
        $E_{SSE} = \frac{e_1^2 + e_2^2 + \cdots + e_N^2} {N} = \frac{1}{N} \sum_{i=1}^N e_i^2$
        $ e = y_i - \hat{y_i} $
      - Problem
        
        Linear Solution
        
        Quadratic solution
        
        Cubic solution
        
        5-power solution
        
        n-th power solution
      - $$\hat{y}_i = f(x_i) = w_0 + w_1 x_i$$
      - Multiple Regression
        
        There are two or more predictors
        
        Problem
        
        Solution
        
        .
        
        $f(\mathbf{x}_i) = \mathbf{w}^T \mathbf{x}_i = w_0 + w_1 x_{i1} + \cdots + w_K x_{iK}$
        
        $\mathbf{w} = [w_0, w_1, \cdots, w_K]^T$
      - Models
        
        Linear
        
        Polynomial
        
        Sinusoids
        
        Radial basis functions
        
        Splines
        
        And many more! ...
      - Degrees of freedom
        
        A linear model $ y = w_0 + w_1x $ has two parameters and is inflexible, as it can only generate straight lines.
        
        A cubic model $y = w_0 + w_1x + w_2x^2 + w_3x^3 $ has 4 parameters and is more flexible than a linear one.
      - Under-fitting
        
        Over-fitting
        
        Just Right
        
        .