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Linear Regression

  • Linear regression: involves a response variable y and a single predictor variable x
    \[ y=w_{0}+w_{1}x \]
    where w0 (y-intercept) and w1 (slope) are regression coefficients
  • Method of least squares: estimates the best-fitting straight line

\[ w_{1}=\frac{\sum_{i=1}^{|D|}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{|D|}(x_{i}-\bar{x})^2} \]

\[ w_{0}=\bar{y}-w_{1}\bar{x} \]

  • Multiple linear regression: involves more than one predictor variable
    • Training data is of the form (X1, y1), (X2, y2),…, (X|D|, y|D|)
    • Ex. For 2-D data, we may have:
      \[ y=w+w_{1}x_{1}+w_{2}x_{2} \] 
    • Solvable by extension of least square method or using SAS, S-Plus
    • Many nonlinear functions can be transformed into the above

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