### (C) the basic methods of time series analysis

1. model selection and modeling basic steps

(1) modeling basic steps

1) with observation, survey, sampling, obtaining time series dynamic data.

2) Make correlation plots to study the trend and period of change and to be able to find jump points and inflection points. The inflection point is the point at which the time series suddenly changes from an upward trend to a downward trend, and if there is an inflection point, a different model must be used to fit the time series in segments when modeling.

3) Identify a suitable stochastic model for curve fitting.

(2) Model selection

For short or simple time series, trend and seasonal models with errors can be used for fitting. For smooth time series, they can be fitted with autoregressive (AR) models, moving average (MA) models, or combinations thereof, such as autoregressive moving average (ARMA) models.

A pure AR model implies that an observation of a variable consists of a linear combination of its previous p observations plus a random error term, as if it were regressing on itself, hence the name autoregressive model.

MA models imply that an observation of a variable consists of a linear combination of the current and previous n random errors.

ARMA models are generally used when there are more than 50 observations.

For non-stationary time series, the series is first differentiated (Difference, i.e., each observation is subtracted from its predecessor or periodic value) into a stationary time series and then fitted with an appropriate model. This ARMA model integrated by the Difference method is called AutoregressiveIntegratedMovingAverage (ARIMA), or ARIMA model for short (Zhang Wentong, 2002; Xue Wei, 2005; G.E.P. Boxetal., 1994).

The ARIMA model requires that the time series satisfy the conditions of smoothness and reversibility, i.e., the mean of the series does not increase or decrease with time, and the variance of the series does not change with time. However, since the stratigraphic elemental content we are concerned with changes as a time series with trend and periodic components, none of which are smooth, it is necessary to differencing them to eliminate these components that make the series not smooth. So we choose the more powerful ARIMA model.

2. Smoothness and periodicity study

Some mathematical models have to test whether the periodic change is a smooth process, i.e., its statistical properties do not change over time, we can identify the smoothness and periodicity of the series based on the series plot, autocorrelation function plot, partial autocorrelation function plot, and spectral density plot. When the sequence diagram shows obvious segmentation characteristics can be used to calculate the segmentation method, if the segmentation of each segment of the spectral map is basically the same or similar, it is considered that the process is smooth, otherwise it is non-smooth.

ACF (Autocorrelationsfunction) is a simple and conventional correlation coefficient describing the correlation coefficients between the current observations of the sequence and the observations in front of the sequence; and PACF (Partialautocorrelationsfunction) is a function that after controlling for other influences in the sequence, measures the correlation between the current values of the sequence and a certain value of the sequence. measures the degree of correlation between the current value of the series and some previous value.

Smooth processes have autocorrelation and partial autocorrelation coefficients that are a function of the time interval, independent of the time point of origin, and both decay in some way towards 0.

Sequences are non-smooth when the ACF maintains a positive correlation for many periods and when the value of the ACF usually decreases very slowly to zero.

The autocorrelation-partial autocorrelation function of the series is symmetric, i.e., it reflects periodic variation characteristics.

3. Spectral analysis

The deterministic periodic function X(t) (let the period be T) can be expressed as the sum of sine and cosine functions of some different frequencies by Fourier series expansion under certain conditions (Chen Lei et al., 2001), which is assumed to be a finite term, namely:

Quaternary environmental geochemistry of Dongting Lake Area

where the frequency fk=k/T, k=1, 2, …, N.

The above equation shows that if the difference in phase is put aside, the period variation of such functions depends entirely on the frequency and amplitude of each cosine function component. In other words, we can characterize the fluctuation of X(t) by the following function:

Quaternary Environmental Geochemistry of Dongting Lake Area

The function p(f) and the function X(t) express the same periodic fluctuation, and they are actually equivalent, except that they are described from two different perspectives of the frequency domain and time domain. Call p(f) the power spectral density function of X(t), or spectral density for short. It not only reflects the periodicity of each intrinsic component in X(t), but also shows the importance of each of these periodic components in the overall X(t). Specifically, the spectral density function plot should show a more pronounced bump at the corresponding frequency of each periodic component in X(t), and the larger the amplitude of the component, the higher the peak, and the greater the overall impact on X(t).

In fact, regardless of whether the problem itself is periodic or uncertain (such as a continuous stochastic process or a time series) can be described in the frequency domain using a similar approach, except that the form and meaning of the representation is much more complex than above. The spectral analysis of time series is to estimate the spectral density function of the time series, to find the main periodic components of the series, through the analysis of the components to achieve the main periodic fluctuations of the time series to grasp the characteristics.

According to the theory of spectral analysis, for a smooth time series ｛Xt｝, if its self-covariance function R(k) meets |R(k)| <+∞, then its spectral density function h(f) must exist and have a Fuchs' transform relationship with R(k), i.e., the standardized spectral density of the smooth time series ｛Xt｝, p(f) is a Fuchs' transform of the autocorrelation function r(k). Since p(f) is a dimensionless relative value, it is easier to analyze and compare in many cases.

How to estimate the spectral density or the standard spectral density function from a time series ｛Xt, t=1, 2, …, n｝ given by a real problem is the main problem to be solved by spectral analysis. This book uses Tukey-Hanning (Tukey-Hanning) window spectral estimation method.

### What are the disadvantages of time series analysis?

Advantages: from the time series can be found in the variable changes in the characteristics of the trend and the development of the law, so that the variables of the future changes in the effective prediction.

Disadvantages: in the application of time series analysis method of market forecasting should pay attention to the market phenomenon of the future development of the law of change and the level of development, may not necessarily be completely consistent with its history and the current development of the law of change. Between the sequence forecasting method for highlighting the time series temporarily does not take into account the impact of external factors, and thus there is a forecast error defects, when encountered in the outside world, there are often large changes, there will be a large deviation.

The basic characteristics:

1, trend: a variable with the progress of time or independent variable changes, showing a relatively slow and long-term continued rise, fall, stay the same nature of the change tendency, but the magnitude of change may not be equal.

2, periodicity: a factor due to external influences with the natural seasons of the alternation of peaks and troughs of the law.

3, stochastic: individual random changes, the overall statistical pattern.

4, comprehensive: the actual change is the superposition or combination of several changes. Forecasts try to filter out irregular changes, highlighting the trend and cyclical changes.