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Environmental Data Analysis with MatLab Lecture 6: The Principle of Least Squares

# Environmental Data Analysis with MatLab Lecture 6: The Principle of Least Squares

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Environmental Data Analysis with MatLab

Lecture 6:The Principle of Least Squares

Lecture 01 Using MatLabLecture 02 Looking At DataLecture 03 Probability and Measurement Error Lecture 04 Multivariate DistributionsLecture 05 Linear ModelsLecture 06 The Principle of Least SquaresLecture 07 Prior InformationLecture 08 Solving Generalized Least Squares ProblemsLecture 09 Fourier SeriesLecture 10 Complex Fourier SeriesLecture 11 Lessons Learned from the Fourier TransformLecture 12 Power SpectraLecture 13 Filter Theory Lecture 14 Applications of Filters Lecture 15 Factor Analysis Lecture 16 Orthogonal functions Lecture 17 Covariance and AutocorrelationLecture 18 Cross-correlationLecture 19 Smoothing, Correlation and SpectraLecture 20 Coherence; Tapering and Spectral Analysis Lecture 21 InterpolationLecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-TestsLecture 24 Confidence Limits of Spectra, Bootstraps

SYLLABUS

purpose of the lecture

estimate model parameters using the

principle of least-squares

part 1

the least squares estimation of model parameters and their covariance

the prediction error

motivates us to define an error vector, e

prediction error in straight line case

-6 -4 -2 0 2 4 6-15

-10

-5

0

5

10

15

x

dplot of linedata01.txt

auxiliary variable, x

data

, d

dipre

diobs ei

total errorsingle number summarizing the error

sum of squares of individual errors

principle of least-squares

that minimizes

least-squares and probability

suppose that each observation has a Normal p.d.f.

2

for uncorrelated datathe joint p.d.f. is just the product of

the individual p.d.f.’s

least-squares formula for E suggests a link

between probability and least-squares

now assume that Gm predicts the mean of d

minimizing E(m) is equivalent to maximizing p(d)

Gm substituted for d

the principle of least-squaresdetermines the m

that makes the observations “most probable”

in the sense of maximizingp(dobs)

the principle of least-squaresdetermines the model parameters

that makes the observations “most probable”

(provided that the data are Normal)

this isthe principle of maximum likelihood

a formula for mestat the point of minimum error, E

∂E / ∂mi = 0so solve this equation for mest

Result

where the result comes fromE =

so

unity when k=jzero when k≠jsince m’s are independent

use the chain rule

so just delete sum over j and replace j with k

which gives

covariance of mestmest is a linear function of d of the form mest = M dso Cm = M Cd MT, with M=[GTG]-1GTassume Cd uncorrelated with uniform variance, σd

2

then

two methods of estimating the variance of the data

posterior estimate: use prediction error

prior estimate: use knowledge of measurement technique

the ruler has 1mm tic marks, so σd≈½mm

posterior estimates are overestimates when the model is poor

reduce N by M since an M-parameter model can exactly

fit N data

confidence intervals for the estimated model parameters

(assuming uncorrelated data of equal variance)

soσmi = √[Cm]ii

and

m=mest±2σmi (95% confidence)

MatLab script for least squares solution

mest = (G’*G)\(G’*d);Cm = sd2 * inv(G’*G);sm = sqrt(diag(Cm));

part 2

exemplary least squares problems

Example 1: the mean of data

the constant

will turn out to be the mean

usual formula for the mean

variance decreases with number of data

m1est = d = 2σd± √N (95% confidence)

formula for mean formula for covariance

combining the two into confidence limits

Example 2: fitting a straight line

intercept

slope

[GTG]-1=(uses the rule)

intercept and slope are uncorrelated

when the mean of x is zero

keep in mind that none of this algrbraic manipulation is needed if we just compute

using MatLab

Generic MatLab scriptfor least-squares problems

mest = (G’*G)\(G’*dobs);dpre = G*mest;e = dobs-dpre;E = e’*e;sigmad2 = E / (N-M);covm = sigmad2 * inv(G’*G);sigmam = sqrt(diag(covm));mlow95 = mest – 2*sigmam;mhigh95 = mest + 2*sigmam;

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-40

-20

0

2040

time, days

obs t

em

p,

C

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-40

-20

0

20

40

time, days

pre

tem

p,

C

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-40

-20

0

2040

time, days

err

or,

C

d(t)obs

d(t)pre

error, e(t)time t, days

time t, days

time t, days

Example 3:modeling long-term trend and annual cycle in

Black Rock Forest temperature data

the model:

long-term trend annual cycle

Ty=365.25; G=zeros(N,4); G(:,1)=1; G(:,2)=t; G(:,3)=cos(2*pi*t/Ty); G(:,4)=sin(2*pi*t/Ty);

MatLab script to create the data kernel

prior variance of databased on accuracy of thermometerσd = 0.01 deg C

posterior variance of databased on error of fitσd = 5.60 deg C

huge difference, since the model does not include diurnal cycle of weather patterns

long-term slope

95% confidence limits based on prior variancem2 = -0.03 ± 0.00002 deg C / yr95% confidence limits based on posterior variancem2 = -0.03 ± 0.00460 deg C / yrin both cases, the cooling trend is significant, in the sense that the confidence intervals do not include zero or positive slopes.

However

The fit to the data is poor, so the results should be used with caution. More effort needs to be put into developing a better model.

part 3

covariance and the shape of the error surface

m1est

0 4m20

4

mest

m1

m2est

solutions within the region of low error are almost as good as mest

small range of m2

large range of m1

E(m)mi

miest

near the minimum the error is shaped like a parabola. The curvature of the parabola

controls the with of the region of low error

near the minimum, the Taylor series for the error is:

curvature of the error surface

starting with the formula for error

we compute its 2nd derivative

but

so

curvature of the error surface

covariance of the model parameters

the covariance of the least squares solution

is expressed

in the shape of the error surface

E(m)mi

miest

E(m)mi

miest

large variance

small variance