# A Survey on Change Detection and Time Series Analysis with Applications
## 1 Introduction  
A periodic oscillation can be described by a smooth sinusoidal function. It may be a function of time or distance. Consider the sinusoidal wave \(f(t)=A\sin(2\pi\omega t+\theta)\), where \(f\) is a function of time \(t\). The parameters \(A\), \(\omega\) and \(\theta\) are called the amplitude, cyclic frequency or number of cycles per unit time and phase of the sinusoid, respectively. This function is periodic with period \(T=1/\omega\)[1, 2]. A time series is an ordered sequence of data points measured at discrete time intervals that may not be equally spaced or may contain data gaps, and so the time series is unequally spaced or unevenly sampled. In certain experiments, measurements may also have uncertainties introduced by random noise and thus may also be unequally weighted [3, 4]. In many areas of research, such as geodesy, geophysics, geodynamics, astronomy and speech communications, researchers deal with series of data points measured over time or distance to study the periodicity and/or power of certain constituents. For example, in a speech record, one may be interested in certain signals in the record, such as the voice of a person obscured by other simultaneously recorded constituents, noise from musical instruments and the environment. It is often difficult to study these signals by direct analysis of the time series; however, decomposing the time series into another domain may result in detecting the constituents of interest much easier. Furthermore, the physical series of data are usually a combination of various sinusoidal waves, such as light emitted from a star, sound waves, ocean waves, and thus certain phenomena may be studied more accurately by decomposing the data into a new domain, e.g., frequency or time-frequencydomains. The decomposition of a series of data points into sinusoidal waves of various frequencies may be imagined as dispersing sunlight to waves of different frequencies representing different colors when it passes through a triangular prism, e.g., a rainbow. In other words, a triangular prism separates the sunlight to different wavelengths; the longer wavelength is red, and the shorter wavelength is blue. The word spectrum was first referred to the range of colors observed when white light was refracted through a prism. If all statistical properties of a time series, i.e., the mean, variance, all higher-order statistical moments and auto-correlation function, do not change in time, then the time series is called stationary. A time series is called non-stationary if it violates at least one of the assumptions of stationarity [2]. A non-stationary time series may contain systematic noise, such as trends, datum shifts, and jumps, indicating that its mean value is not constant in time. The second moments, central and mixed moments, of the time series values form a symmetric matrix called the covariance matrix [5; 6]. In certain fields, such as geodesy, geophysics, and astronomy, a time series is usually associated with a covariance matrix, which means that the time series is stationary or non-stationary in its second statistical moments. In geodynamics applications, seismic noise may contaminate the time series of interest or certain components of the time series may exhibit variable frequency, such as linear, quadratic, exponential or hyperbolic chirps. To solve problems regarding heat transfer and vibrations, [PERSON] [7] introduced the Fourier series and showed that certain functions can be written as an infinite sum of harmonics or sinusoids. The Fourier transform and Fourier's law are also named after him. The Fourier transform basically decomposes a time series into the frequency domain using sinusoidal basis functions. The amplitudes of the sinusoids for a set of frequencies generate a spectrum like the light spectrum. The Fourier transform is one of the equations that changed our world. Since then, numerous Fourier-based methods have been proposed by researchers for various purposes, such as time series analysis.