Environmental Data Analysis with MatLab
2nd Edition
Lecture 22:
Linear Approximations
and
Non Linear Least Squares
SYLLABUS
Lecture 01
Lecture 02
Lecture 03
Lecture 04
Lecture 05
Lecture 06
Lecture 07
Lecture 08
Lecture 09
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 22
Lecture 23
Lecture 24
Using MatLab
Looking At Data
Probability and Measurement Error
Multivariate Distributions
Linear Models
The Principle of Least Squares
Prior Information
Solving Generalized Least Squares Problems
Fourier Series
Complex Fourier Series
Lessons Learned from the Fourier Transform
Power Spectra
Filter Theory
Applications of Filters
Factor Analysis
Orthogonal functions
Covariance and Autocorrelation
Cross-correlation
Smoothing, Correlation and Spectra
Coherence; Tapering and Spectral Analysis
Interpolation
Linear Approximations and Non Linear Least Squares
Adaptable Approximations with Neural Networks
Hypothesis testing
Hypothesis Testing continued; F-Tests
Confidence Limits of Spectra, Bootstraps
Goals of the lecture
learn how to make linear approximations
of non-linear functions
apply liner approximations to error estimation
apply liner approximations to least squares
Taylor Series and Linear Approximations
polynomial approximation to a function y(t)
in the neighborhood of a point t0
polynomial approximation to a function y(t)
in the neighborhood of a point t0
find coefficients by taking derivatives
polynomial approximation to a function y(t)
in the neighborhood of a point t0
evaluate at t0
0
find coefficients by taking deriatives
0
0
polynomial approximation to a function y(t)
in the neighborhood of a point t0
polynomial approximation to a function y(t)
in the neighborhood of a point t0
Taylor series
Taylor Series
Linear approximation
≈
example
example
example
example
Linear approximation
example: distances on a sphere
(λ1,L1)
(λ2,L2)
r
measured in terms of central angle, r
exact formula: 6 trig functions
approximate formula: 1 trig function and 1 square root
(λ2,L2=0)
(λ1=0,L1=0)
application to estimates of variance
spectral analysis scenario
measure angular frequency, m
want confidence bounds on corresponding period, T
exact (but difficult) method
assume m is Normally-distributed, p(m)
work out the distribution p(T)
compute its mean and variance by integration
approximate (and easy) method
assume m is Normally-distributed with mean mest
work out a linear approximation of T in neighborhood of mest
use formula for error propagation for a linear functions
consider small fluctuations about the
estimated angular frequency
Test
so
application to least squares
Goal
Solve non-linear problems of the form
by generalized least squares
Taylor series of predicted data
Taylor expansion of predicted data
with
and
Taylor expansion of predicted data
linearized equation
with
and
Taylor expansion of total error
Taylor expansion of total error
Taylor expansion of total error
gradient
vector
curvature
matrix
linearized least squares
linearized least squares
minimize error
linearized least squares
minimize error
linearized least squares
minimize error
linear theory
linearized least squares
minimize error
linear theory
linearized least squares
minimize error
linear theory
linearized least squares
minimize error
linear theory
linearized least squares
guess for the solution
linearized least squares
trial solution
deviation of data from
prediction of trial solution
linearized least squares
trial solution
deviation of data from
prediction of trial solution
linearized data kernel
linearized least squares
trial solution
deviation of data from
prediction of trial solution
linearized data kernel
correction
to solution
linearized least squares
trial solution
deviation of data from
prediction of trial solution
linearized data kernel
correction
to solution
updated solution
linearized least squares
repeat
prior information
written in terms of the unknown
=
modification of generalized least squares
example of generalized least squares
example of generalized least squares
sinusoid of unknown amplitude & frequency
superimposed on a constant background level
example of generalized least squares
sinusoid of unknown amplitude & frequency
superimposed on a constant background level
normalized unknowns, so mi≈1
frequency
amplitude
background
level
compute derivatives & evaluate in neighborhood of a guess
mωa