Ch 3.1: Linear Regression

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• 2.2 Assessing Model Accuracy

• Least squares coefficient estimates for linear regression

• Residual sum of squares (RSS)

Predict \(\beamer@setmathcolor Y\) on a single predictor variable \(\beamer@setmathcolor X\) \begin {equation*} Y \approx \beta _0 + \beta _1 X \end {equation*} ”\(\beamer@setmathcolor \approx \)” .... “is approximately modeled as”

Predict \(\beamer@setmathcolor Y\) on a single predictor variable \(\beamer@setmathcolor X\) \begin {equation*} Y \approx \beta _0 + \beta _1 X \end {equation*} ”\(\beamer@setmathcolor \approx \)” .... “is approximately modeled as”

\begin {equation*} \min _{\beta _0,\beta _1} \sum _i (y_i - \hat \beta _0 - \hat \beta _1 x_i)^2 \end {equation*}

\begin {align*} \frac {\partial RSS}{\partial \hat \beta _0} & = -2 \sum _i (y_i - \hat {\beta }_0 - \hat {\beta }_1x_i) = 0 \\ \frac {\partial RSS}{\partial \hat \beta _1} & = -2 \sum _i x_i(y_i - \hat \beta _0 - \hat \beta _1x_i) = 0 \end {align*}

\begin {align*} \hat \beta _1 &= \frac {\sum _{i=1}^n(x_i-\overline x) (y_i - \overline y)} {\sum _{i=1}^n (x_i - \overline x)^2}\\ \hat \beta _0 & = \overline y - \hat \beta _1 \overline x \end {align*}

• More on simple linear regression!

• Evaluation of model etc.

• Homework 1

  • Due Sun, Jan 25