1. Environmental Data Analysis with MatLab 2 nd Edition Lecture 22: Linear Approximations and Non Linear Least Squares

2. Lecture 01Using MatLab Lecture 02Looking At Data Lecture 03Probability and Measurement Error Lecture 04Multivariate Distributions Lecture 05Linear Models Lecture 06The Principle of Least Squares Lecture 07Prior Information Lecture 08Solving Generalized Least Squares Problems Lecture 09Fourier Series Lecture 10Complex Fourier Series Lecture 11Lessons Learned from the Fourier Transform Lecture 12Power Spectra Lecture 13Filter Theory Lecture 14Applications of Filters Lecture 15Factor Analysis Lecture 16Orthogonal functions Lecture 17Covariance and Autocorrelation Lecture 18Cross-correlation Lecture 19Smoothing, Correlation and Spectra Lecture 20Coherence; Tapering and Spectral Analysis Lecture 21Interpolation Lecture 22Linear Approximations and Non Linear Least Squares Lecture 23Adaptable Approximations with Neural Networks Lecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-Tests Lecture 24 Confidence Limits of Spectra, Bootstraps SYLLABUS

3. 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

4. Taylor Series and Linear Approximations

5. polynomial approximation to a function y(t) in the neighborhood of a point t 0

6. polynomial approximation to a function y(t) in the neighborhood of a point t 0 find coefficients by taking derivatives

7. polynomial approximation to a function y(t) in the neighborhood of a point t 0 find coefficients by taking deriatives evaluate at t 0 0 0 0

8. polynomial approximation to a function y(t) in the neighborhood of a point t 0

9. polynomial approximation to a function y(t) in the neighborhood of a point t 0 Taylor series

10. Taylor Series Linear approximation ≈

11. example

14. Linear approximation

16. r (λ 1,L 1 ) (λ 2,L 2 ) example: distances on a sphere measured in terms of central angle, r

17. exact formula: 6 trig functions approximate formula: 1 trig function and 1 square root

18. (λ 2,L 2 =0) (λ 1 =0,L 1 =0) linear quadratic

19. application to estimates of variance

20. spectral analysis scenario measure angular frequency, m want confidence bounds on corresponding period, T

21. exact (but difficult) method assume m is Normally-distributed, p(m) work out the distribution p(T) compute its mean and variance by integration

22. approximate (and easy) method assume m is Normally-distributed with mean m est work out a linear approximation of T in neighborhood of m est use formula for error propagation for a linear functions

23. consider small fluctuations about the estimated angular frequency so T est

25. application to least squares

26. Goal Solve non-linear problems of the form by generalized least squares

27. Taylor series of predicted data

28. Taylor expansion of predicted data with and

29. Taylor expansion of predicted data with and linearized equation

30. Taylor expansion of total error

32. gradient vector curvature matrix

33. linearized least squares

34. minimize error

35. linearized least squares minimize error

36. linearized least squares minimize error linear theory

37. linearized least squares minimize error linear theory

38. linearized least squares minimize error linear theory

39. linearized least squares minimize error linear theory

40. linearized least squares guess for the solution

41. linearized least squares trial solution deviation of data from prediction of trial solution

42. linearized least squares trial solution deviation of data from prediction of trial solution linearized data kernel

43. linearized least squares trial solution deviation of data from prediction of trial solution linearized data kernel correction to solution

44. linearized least squares trial solution deviation of data from prediction of trial solution linearized data kernel correction to solution updated solution

45. linearized least squares repeat

46. prior information = written in terms of the unknown

47. modification of generalized least squares

48. example of generalized least squares

49. sinusoid of unknown amplitude & frequency superimposed on a constant background level

50. example of generalized least squares normalized unknowns, so m i ≈1 sinusoid of unknown amplitude & frequency superimposed on a constant background level amplitude background level frequency

51. compute derivatives & evaluate in neighborhood of a guess m ω a