Lecture 22: Residuals, Multicollinearity, Inference

Least Squares Regression

  • definition of orthogonal
    • difference between Y vector and X theta hat is 0?
    • invert the matrix if full rank for solution

A Regression Model

  • X is a design matrix, first column is 1
  • theta is params

Residuals

  • difference Y and Y hat (estimate)

Seperating Signal and Noise

  • true signal + noise and prediction + residual

Residuals Sum to Zero

  • ?

  • The average of the fitted values is equal to average of the observed responses

  • orthogonal to the residuals?

Multiple R^2 and Overved Response and Fitted Values

  • Multiple R^2

  • Coefficient of Determination
    • variance of the fitted values
    • variance of observed responses
    • "percent of variance explained by the model"

Colinearity and the Meaning of Slope

  • Change in y per unit change in x_1 given all other variables held constant

  • colinearity: when a covariate can be predicted by a linear function of others

Inference and Assumptions of Randomness

  • our model can be expressed as intercept, weight of features, and error

  • We ha ve to estimate the weights (slopes)

  • how do we test theta_1 is 0?

Confidence Intervals for True Slope

  • could bootstrap, could build a confidence interval (?)