Stat 312: Lecture 19Linear RegressionMoo K. Chung[email protected]Nov 30, 2004Concepts1. Letxbe the speed of a car andybe the distance the cartraveled in an hour hour. Then we have modely=β0+β1x.Suppose we havenpaired measurements(xi, yi), i=1,· · ·, n.Since all measurement are supposed to benoisy, we introduce a noise term²in the above equation.Our modified stochastic model isy=β0+β1x+²,where²∼N(0, σ2). Since²is a random variable, weuseYinstead ofyfor convenience:Y=β0+β1x+².Note thatEY=β0+β1xandVY=σ2.2. Equivalently we can write the above linear model foreach paired measurement(xi, yj):Yj=β0+β1xj+²j,whereyjis the observed value of random variableYjand²j∼². Note thatEYj=β0+β1xj. Letˆβ0,ˆβ1beestimators ofβ0, β1. Then thepredicted valuesor fittedvalues are given byˆyj=ˆβ0+ˆβ1xj.The differences between the observationsyjand the pre-dicted valuesˆyjare called theresiduals(errors), i.e.rj=yj-ˆyj=yj-ˆβ0-ˆβ1xj.3.Least squares estimation. The least squares estimationis a method of estimating parametersβ0andβ1by min-imizing thesum of the squared errors(SSE):
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