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Ordinary Least-Squares

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Outline Linear regression Geometry of least-squares Discussion of the Gauss-Markov theorem

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Ordinary Least-Squares One-dimensional regression

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Ordinary Least-Squares One-dimensional regression Find a line that represent the ”best” linear relationship:

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Ordinary Least-Squares One-dimensional regression Problem: the data does not go through a line

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Ordinary Least-Squares One-dimensional regression Problem: the data does not go through a line Find the line that minimizes the sum:

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Ordinary Least-Squares One-dimensional regression Problem: the data does not go through a line Find the line that minimizes the sum: We are looking for that minimizes

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Ordinary Least-Squares Matrix notation Using the following notations and

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Ordinary Least-Squares Matrix notation Using the following notations and we can rewrite the error function using linear algebra as:

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Ordinary Least-Squares Matrix notation Using the following notations and we can rewrite the error function using linear algebra as:

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Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters

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Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters

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Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters

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Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters and n measurements

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Ordinary Least-Squares Multidimentional linear regression Using a model with m parameters and n measurements

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Ordinary Least-Squares

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parameter 1

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Ordinary Least-Squares measurement n parameter 1

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Ordinary Least-Squares Minimizing

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Ordinary Least-Squares Minimizing

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Ordinary Least-Squares Minimizing is flat at

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Ordinary Least-Squares Minimizing is flat at

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Ordinary Least-Squares Minimizing is flat at does not go down around

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Ordinary Least-Squares Minimizing is flat at does not go down around

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Ordinary Least-Squares Positive semi-definite In 1-DIn 2-D

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Ordinary Least-Squares Minimizing

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Ordinary Least-Squares Minimizing

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Ordinary Least-Squares Minimizing

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Ordinary Least-Squares Minimizing Always true

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Ordinary Least-Squares Minimizing Always true The normal equation

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Ordinary Least-Squares Geometric interpretation

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Ordinary Least-Squares Geometric interpretation b is a vector in R n

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Ordinary Least-Squares Geometric interpretation b is a vector in R n The columns of A define a vector space range(A)

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Ordinary Least-Squares Geometric interpretation b is a vector in R n The columns of A define a vector space range(A) Ax is an arbitrary vector in range(A)

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Ordinary Least-Squares Geometric interpretation b is a vector in R n The columns of A define a vector space range(A) Ax is an arbitrary vector in range(A)

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Ordinary Least-Squares Geometric interpretation is the orthogonal projection of b onto range(A)

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Ordinary Least-Squares The normal equation:

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Ordinary Least-Squares The normal equation: Existence: has always a solution

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Ordinary Least-Squares The normal equation: Existence: has always a solution Uniqueness: the solution is unique if the columns of A are linearly independent

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Ordinary Least-Squares The normal equation: Existence: has always a solution Uniqueness: the solution is unique if the columns of A are linearly independent

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Ordinary Least-Squares Under-constrained problem

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Ordinary Least-Squares Under-constrained problem

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Ordinary Least-Squares Under-constrained problem

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Ordinary Least-Squares Under-constrained problem Poorly selected data One or more of the parameters are redundant

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Ordinary Least-Squares Under-constrained problem Poorly selected data One or more of the parameters are redundant Add constraints

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Ordinary Least-Squares How good is the least-squares criteria? Optimality: the Gauss-Markov theorem

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Ordinary Least-Squares How good is the least-squares criteria? Optimality: the Gauss-Markov theorem Let and be two sets of random variables and define:

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Ordinary Least-Squares How good is the least-squares criteria? Optimality: the Gauss-Markov theorem Let and be two sets of random variables and define: If

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Ordinary Least-Squares How good is the least-squares criteria? Optimality: the Gauss-Markov theorem Let and be two sets of random variables and define: If Then is the best unbiased linear estimator

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Ordinary Least-Squares a b eiei no errors in a i

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Ordinary Least-Squares a b eiei a b eiei no errors in a i errors in a i

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Ordinary Least-Squares a b homogeneous errors

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Ordinary Least-Squares a b a b homogeneous errors non-homogeneous errors

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Ordinary Least-Squares a b no outliers

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Ordinary Least-Squares a b a b no outliers outliers

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