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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
Inferences about the betas
  • 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
14.4.5 Inference about
  • Recall Section 14.4 for the statistical
    properties of the Least Squares Estimates with
    some additional assumptions about the errors
    being

17
14.5 Multiple Linear Regression
  • This section will generalize Section 14.2 (Simple
    Linear Regression) by doing the Multiple Linear
    Regression thru an example of polynomial
    regression.
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