Book contents
- Time Series Data Analysis in Oceanography
- Time Series Data Analysis in Oceanography
- Copyright page
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Introduction to MATLAB
- 3 Time and MATLAB Functions for Time
- 4 Deterministic and Random Functions
- 5 Error and Variability Propagation
- 6 Taylor Series Expansion and Application in Error Estimate
- 7 Spherical Trigonometry and Distance Computation
- 8 A System of Linear Equations and Least Squares Method
- 9 Base Functions and Linear Independence
- 10 Generic Least Squares Method and Orthogonal Functions
- 11 Harmonic Analysis of Tide
- 12 Fourier Series
- 13 Fourier Transform
- 14 Discrete Fourier Transform and Fast Fourier Transform
- 15 Properties of Fourier Transform
- 16 More Discussion on the Harmonic Analysis and Fourier Analysis
- 17 Effect of Finite Sampling
- 18 Power Spectrum, Cospectrum, and Coherence
- 19 Window Functions for Reducing Side Lobes
- 20 Convolution, Filtering with the Window Method
- 21 Digital Filters
- 22 Rotary Spectrum Analysis
- 23 Short-Time Fourier Transform and Introduction to Wavelet Analysis
- 24 Empirical Orthogonal Function Analysis
- References
- Index
10 - Generic Least Squares Method and Orthogonal Functions
Published online by Cambridge University Press: 21 April 2022
- Time Series Data Analysis in Oceanography
- Time Series Data Analysis in Oceanography
- Copyright page
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Introduction to MATLAB
- 3 Time and MATLAB Functions for Time
- 4 Deterministic and Random Functions
- 5 Error and Variability Propagation
- 6 Taylor Series Expansion and Application in Error Estimate
- 7 Spherical Trigonometry and Distance Computation
- 8 A System of Linear Equations and Least Squares Method
- 9 Base Functions and Linear Independence
- 10 Generic Least Squares Method and Orthogonal Functions
- 11 Harmonic Analysis of Tide
- 12 Fourier Series
- 13 Fourier Transform
- 14 Discrete Fourier Transform and Fast Fourier Transform
- 15 Properties of Fourier Transform
- 16 More Discussion on the Harmonic Analysis and Fourier Analysis
- 17 Effect of Finite Sampling
- 18 Power Spectrum, Cospectrum, and Coherence
- 19 Window Functions for Reducing Side Lobes
- 20 Convolution, Filtering with the Window Method
- 21 Digital Filters
- 22 Rotary Spectrum Analysis
- 23 Short-Time Fourier Transform and Introduction to Wavelet Analysis
- 24 Empirical Orthogonal Function Analysis
- References
- Index
Summary
This chapter discusses a generic least squares method and a special situation when the base functions are orthogonal to each other, which makes the solution explicit; in addition, we learn that the essence of the least squares method can be viewed as a way to project the target function in a higher dimension onto a lower dimension formed by the base functions. The least squares method ensures that the error vector is “perpendicular” to the projected (or approximate) vector in the base function dimension (a lower dimension) and thus has the shortest “length” or minimized error. Although this chapter does not have much computation involved, it is very important for a good understanding of the meaning of many techniques and methods in the subsequent chapters.
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- Time Series Data Analysis in OceanographyApplications using MATLAB, pp. 178 - 189Publisher: Cambridge University PressPrint publication year: 2022