The objective of this chapter is to discuss the error propagation through computation from the error contained in the original data. We are not focusing on the measurement errors themselves, although the propagation of computational error is related to measurement errors. We also discuss the variability propagation with given mathematical relationships, or sensitivity of a system to a given parameter, which are all closely related to error propagation.

The objective of this chapter is to demonstrate the linkage between convolution and filtering and to discuss preliminary filtering with examples. The simplest low-pass filtering that allows low frequency to pass to the output is a “moving average,” which is essentially through a computation involving a rectangular window function (in this case, it is a filter), with a length determined by the cutoff frequency. The filtering action is accomplished by a convolution between the filter and the time series. However, moving average has its drawbacks because of the rectangular window effect or the side lobe effect. It is considered as a “poor man’s filter” because of the lack of sophistication in getting rid of the leakage from side lobes. One improvement over the moving average is using a non-rectangular window function in the convolution to reduce the sharp change at the edges. The basic ideas and examples presented here are useful in demonstrating how to do filtering with several MATLAB functions. When a low-pass filter is designed, a high-pass filter can be defined. With two or more low-pass filters, one can also design band-pass and band-stop filters. There are also other filters that can have various controls on the results.

This chapter discusses the transition between Fourier series and Fourier Transform, which is the tool for spectrum analysis. Generally, the use of linearly independent base functions allows a wide range of linear regression models that work in a least square sense such that the total error squared is minimized in finding the coefficients of the base functions. A special case is sinusoidal functions based on a fundamental frequency and all its harmonics up to infinity. This leads to the Fourier series for periodic functions. In this chapter, we start from the original Fourier series expression and convert the sinusoidal base functions to exponential functions. We can then consider the limit when the length of the function and the period of the original function approach infinity (so that the fundamental frequency approaches 0, including aperiodic functions), leading to the Fourier integral and Fourier Transform. We can then define the inverse Fourier Transform and establish the relationship between the coefficients of Fourier series and the discrete form Fourier Transform. All these are preparations for the fast Fourier Transform (FFT), an efficient algorithm of computation of the discrete Fourier Transform that is widely used in data analysis for oceanography and other applications.

This chapter discusses the drawback of Fourier analysis and the methods that can overcome its limitations. In general, Fourier analysis does not include information about time, particularly events. A slight modification of Fourier analysis can allow the addition of a dimension in time: by dividing the time series into smaller segments and doing the Fourier Transform for each segment, a method called short-time Fourier Transform (STFT) is introduced. Wavelet analysis is then discussed as a much better alternative to or replacement for STFT. It involves scaled and translated convolution with a short base function (short in the sense that it is essentially non-zero only in a finite interval). Wavelet analysis uses different base functions than the Fourier Transform. They are limited in time (unlike the infinitely long sinusoidal functions) and can be stretched or compressed to represent different scales (equivalent to frequencies). This method will allow the resolution of events at different times and different scales.

In this introductory chapter, we briefly go over the definitions of terms and tools we need for data analysis. Among the tools, MATLAB is the software package to use. The other tool is mathematics. Although much of the mathematics are not absolutely required before using this book, a person with a background in the relevant mathematics will always be better positioned with insight to learn the data analysis skills for real applications.

This chapter discusses some basic spherical trigonometry applicable to distance computations between points on the surface of the Earth, including in the ocean, particularly for large-scale problems in oceanography. Because of the curvature of the Earth, the plane geometry is not applicable.

The objective of this chapter is to discuss the concept of base functions and the basics of using some simple base functions to represent other functions. These base functions are needed in many commonly used analyses. An example of using base functions to approximate an almost arbitrary target function is the Taylor series expansion we have discussed. Here we say that an “almost arbitrary function” is not really arbitrary because, in theory, the target function must be differentiable an arbitrary number of times for the Taylor series expansion to be valid. The concept of linear independence of functions is important in understanding the selection of base functions.