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## Ch14: Linear Least Squares

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### Ch14: Linear Least Squares 14.1: INTRO: Fitting a pth-order polynomial will require finding (p+1) coefficients from the data. Thus, a straight line (p=1) is obtained ... – PowerPoint PPT presentation

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Title: Ch14: Linear Least Squares

1
Ch14 Linear Least Squares
• 14.1 INTRO
• Fitting a pth-order polynomial will require
finding (p1) coefficients from the data. Thus,
a straight line (p1) is obtained thru its slope
and intercept.
• LS (Least Squares) method finds parameters by
minimizing the sum of the squared deviations of
the fitted values from the actual observations.

2
Predicting y (responsedependent) from x
(predictorindependent)
• Formula

3
14.2 Simple Linear Regression(linear in the
parameters)
• Regression is NOT fitting line but E(YXx)
• 14.2.1 Properties of the estimated slope
Intercept

4
Variance-Covariance of the betas
• Under the assumptions of Theorem A

5
• In the previous result,

6
14.2.2 Assessing the Fit
• Recall, that the residuals are the differences
between the observed and the fitted values
• Residuals are to be plotted versus the x-values.
• Ideal plot should look like a horizontal blur
that is to say that one can reasonably model it
as linear.
• Caution the errors have zero mean and are said
to be homoscedastic constant variance
independently of the predicator x. That is to
say

7
Steps in Linear Regression
• Fit the Regression Model (Mathematics)
• Pick a method Least Squares or else
• Plot the data Y versus g(x)
• Compute regression estimates residuals
• Check for linearity outliers (plot residuals)
• More diagnostics (beyond the scoop of this class)
• Statistical Inference (Statistics)
• Check for error assumptions
• Check for normality (if not transform data)
• If nonlinear form, then (beyond the scoop of this
class)
• Least Squares Java applet
• http//www.math.tamu.edu/FiniteMath/Classes/LeastS
quares/LeastSquares.html

8
14.2.3 Correlation Regression
• A close relation exists between Correlation
Analysis Fitting straight lines by the Least
Squares method.

9
14.3 Matrix approach to Linear Least Squares
• Weve already fitted straight lines (p1).
• What if p gt 1 ? ? Investigate some Linear
Algebra tools

10
Formulation of the Least Squares problem
11
14.4 Statistical Properties of Least Squares
Estimates
• 14.4.1 Vector-valued Random Variables

12
Cross-covariance matrix
13
14.4.2 Mean and Covariance of Least Squares
Estimates
14
14.4.3 Estimation of the common variance for the
random errors
• In order to make inference about , one must
get an estimate of the parameter (if
unknown).

15
14.4.4 Residuals Standardized Residuals
16