Time Series Analysis Method)

 

The time series prediction method is an extended prediction of historical data, also known as the extended Prediction Method of history. It is a method of extending the process and regularity of social and economic phenomena reflected by the time series to predict their development trend.

Definition

Based on historical statistical data, this paper summarizes the method for predicting the demand power consumption of the relationship between the power load development level and time sequence. Simple average, weighted average, and moving average. Application discipline

Electricity (level 1 Discipline); Power System (level 2 discipline)

1 Overview

It includes general statistical analysis (such as auto-correlation analysis and spectral analysis), establishment and inference of statistical models, and the optimal prediction, control, and filtering of time series. Classic statistical analysis assumes that data sequences are independent, while time series analysis focuses on the interdependence of data sequences. For example, the first month, the second month ,......, The rainfall of the nth month can be predicted by the time series analysis method.

With the development of computer-related software, mathematical knowledge is no longer an empty talk theory. Time Series Analysis is mainly based on mathematical statistics and other knowledge, and relevant mathematical knowledge is applied in related applications.

2 references

Reference: Dictionary of Scientific and Technical Methods

A time series is a set of numbers in chronological order. Time Series Analysis uses this series to apply mathematical statistics to predict the future development of things. Time Series Analysis is one of the quantitative prediction methods. Its basic principle is to acknowledge the continuity of things. By applying past data, we can speculate on the Development Trend of things. Second, taking into account the randomness of things development. The development of any thing may be affected by accidental factors. Therefore, we must use the weighted average method in statistical analysis to process historical data. This method is simple and easy to master, but has poor accuracy. It is generally only applicable to short-term prediction. Time series prediction generally reflects three types of actual changes: trend changes, periodic changes, and random changes.

Time Series Analysis is a theory and method for establishing a mathematical model based on the time series data observed by the system through curve fitting and parameter estimation. It generally uses curve fitting and parameter estimation methods (such as non-linear least square method. Time Series Analysis is often used in macro control of national economy, comprehensive regional development planning, enterprise management, market potential forecast, weather forecast, hydrological forecast, earthquake precursor forecast, crop disease and insect disaster forecast, Environment pollution Control, ecological balance, astronomy and oceans.

3 elements

A time series is usually composed of four elements: Trend, seasonal change, cyclic fluctuation, and irregular fluctuation.

Trend: a continuous upward or downward change in a time series over a long period of time.

Seasonal changes: periodic fluctuations that occur repeatedly within one year. It is the result of various factors, such as climate conditions, production conditions, holidays, or people's customs and habits.

Cyclical fluctuations: periodic changes in the time series that show a non-fixed length. The cycle of cyclical fluctuations may last for a period of time, but unlike the trend, it is not a constant change in a single direction, but an alternate fluctuation with the same fluctuations.

Irregular fluctuations: random fluctuations in the time series after the trend, seasonal changes, and cycle fluctuations are removed. Irregular fluctuations are usually included in the time series, resulting in a wave-shaped or vibrating change in the time series. A sequence containing only random fluctuations is also called a stable sequence.

4 basic steps

The basic steps of time series modeling are as follows: ① use observation, investigation, statistics, sampling, and other methods to obtain the dynamic time series data of the observed system. ② Perform Correlation Analysis Based on Dynamic Data and find self-correlation functions. The related graph shows the changing trend and cycle, and shows the hops and inflection points. A hop is an observation that is inconsistent with other data. If the hop point is a correct observed value, it should be taken into account during modeling. If it is abnormal, it should be adjusted to the expected value. Inflection point refers to the point where the time sequence suddenly changes from the rising trend to the decreasing trend. If there is an inflection point, different models must be used to segment the time series during modeling, for example, the threshold regression model. ③ Identify an appropriate random model and perform curve fitting. That is, use a general random model to fit the observed data of the time series. For short or simple time series, the trend model and seasonal model can be fitted with errors. For a stable time series, fitting can be performed by using the general ARMA model (Autoregressive moving average model) and its special situations such as the autoregressive moving average model, the moving average model or the combined-ARMA model. When there are more than 50 observed values, the ARMA model is generally used. For a non-stable time series, we must first perform a Difference Operation on the observed time series to form a stable time series, and then use an appropriate model to fit the difference sequence.

5 main purpose System Description

Based on the time series data obtained from the system observation, the system is objectively described using the curve fitting method. System Analysis

When the observed values are taken from more than two variables, the changes in a time series can be used to describe the changes in another time series, so as to gain a deep understanding of the mechanism of generating a given time series. Predict the future

Generally, the ARMA model is used to fit a time series and predict the future value of the time series. Decision Making and Control

According to the time series model, the input variables can be adjusted to keep the system development process on the target value, that is, the necessary control can be performed when the process is predicted to deviate from the target.

6. Specific Algorithms

The statistical rules of random data sequences are studied by using the random process theory and mathematical statistics method to solve practical problems. In most cases, random data is sequenced by time, so it is called a time series. It includes general statistical analysis (such as self-correlation analysis and spectral analysis), establishment and inference of statistical models, and optimal prediction, control, and filtering of random sequences. Classic statistical analysis assumes that data sequences are independent, while time series analysis focuses on the interdependence of data sequences. The latter is actually a statistical analysis of Random Processes of discrete indicators, so it can be seen as an integral part of Random Process statistics. For exampleX(T) Indicates the regionTMonths of rainfall ,{X(T),T= 1, 2 ,...} Is a time series. PairT= 1, 2 ,...,T, Recording the monthly rainfall dataX(1 ),X(2 ),...,X(T).TSample sequence. The time series analysis method can be used to analyze the rainfall of each month in the future.X(T+L)(L= 1, 2 ,...) . Time series analysis was applied to economic forecasting before the Second World War. During and after the Second World War, it was widely used in military science, space science, industrial automation, and other sectors.

In terms of mathematical methods, the statistical analysis of a stable random sequence (see the steady process) is developed in theory to form the basis of time series analysis. Frequency Domain Analysis

A time series can be seen as the superposition of various periodic disturbances. frequency domain analysis is used to determine the distribution of vibration energy in each cycle. Such distribution is called "spectrum" or "Power Spectrum ". Therefore, frequency domain analysis is also called spectral analysis. An important statistic in spectral analysis is a sequence cycle chart. When the sequence contains deterministic periodic componentsI(ω) Is one of the important contents of spectral analysis. In the rainfall sequence recorded by month, the sequenceX(T) Can be considered to contain a specific component of the cycle of 12, so the sequenceX(TCan be expressed as its cycle chart.I(ω.

When the spectral distribution function of the stable SequenceF(λ) Spectral density?(λ) (That is, power spectrum), available (2 π)-1i (λ) To estimate?(λ), It is?(λ. For example?(λ), AvailableI(ω).?(λ), The commonly used method is spectral window estimation.?(λ) Estimated values (λ) Is, formula is inWT (ω) Is called a spectral window function. Spectral window estimation is one of the important methods in practical application. Spectrum DistributionF(λ).I(ω. It is important to study the statistical properties of the above various estimator and improve the estimation method. Time Domain Analysis

It aims to determine the dependency between values of a sequence at different times, or determine the sequence's related structure. This structure uses the sequence auto-correlation function 0, 1 ,...) Is the auto-covariance function value of the sequence, M = EX(T) Is the mean of a stable sequence. M is often given using the following formula,Gamma(K),P(K) Estimation:, pass (K) Understand the sequence structure, called self-correlation analysis. It is a basic problem in correlation analysis to study their strong, weak conformances and their gradual distribution. Model Analysis

Since 1970s, the most widely used time series model has been the stable Auto-regression-Moving Average Model (ARMA model ). Its shape is: Medium ε (T) The mean is zero, and the variance isσRandom Sequences of two independent and same distribution; and σ 2 are the parameters of the model. They satisfy: For everything |Z| The plural of ≤1ZYes.PAndQIs the order of the model, not a negative integer. SpecialQWhen the value is 0, the preceding model is called an auto-regression model.PWhen the value is 0, it is called the moving average model. AccordingX(T) Is the statistical analysis of this model. For the stable sequence that meets the requirements of the ARMA model, linear optimal prediction and control have simple solutions, especially the self-regression model, which is more convenient to use. G. U. Ur in 1925 ~ In 1930, the concept of stable autoregressive was proposed. On September 18, 1943, Alibaba. Morgan and Alibaba. Wald published some theoretical results on the statistical methods of this model and their evolutionary properties. Generally, the statistical analysis of the ARMA model was developed after 1960s. EspeciallyP,QThe estimation of the value and its approximation theory appear later. In addition to the ARMA model, there are other models for analysis. The linear models are mature and closely related to the analysis of the ARMA model. Regression Analysis

If the time seriesX(T) Can be expressed as a deterministic component.Phi(T) And random componentsω(T), According to the sample valueX(1 ),X(2 ),...,X(T) To estimatePhi(T) And Analysisω(T) Is a regression analysis problem in time series analysis. Unlike classic regression analysis,ω(T) Is generally not independent of the same distribution, so a large amount of random process knowledge must be involved here. WhenPhi(T) Is an unknown linear combination of a finite number of known functions.ω(T) Is a stable sequence with zero mean, α 1, α 2 ,..., α S is an unknown parameter,Phi1 (T),Phi2 (T),...,PhiS (TIs a known function called linear regression model. Its Statistical analysis has been studied in depth. The rainfall model can be described as an example. Regression analysis includes: Whenω(T) When the statistical rule is known ,..., α S for estimation and PredictionX(T+L); Whenω(TWhen the statistical rule is unknown, both the above parameters must be estimated andω(T) For statistical analysis, such as spectral analysis and model analysis. Among these contents, an important topic is to prove α 1, α 2 ,..., Like the non-biased linear least variance estimation, the least square estimation of α S has the characteristics of conformances and the progressively normal distribution. Least Squares estimation J (1 ≤J≤ S) not involvedω(T).X(1 ),X(2 ),...,X(T) Directly calculate, and thus obtain (T) In time series analysis, insteadω(T. Theoretically, it has been proved that, under appropriate conditions, such substitution has a satisfactory approximation. Becauseω(T. Research in this area is still evolving.

For details about the optimal prediction, control, and filtering in time series analysis, see the stable process. The Research on multidimensional time series analysis has made some progress and has been applied to industrial production automation and economic analysis. In addition, statistics and analysis of non-linear models and non-parameter statistical analysis have gradually attracted attention.

Time Series Analysis Method)