Mathematical Modeling--- Forecast Method: Time Series Analysis

Time series analysis — time series data

For data: time series data
For the same object data from different time continuous observation made in
three parts

• Describe the past
• Analysis rule
• Fortune-telling

Three major models

• Seasonal decomposition
• Exponential smoothing method
• ARIMA model

Components

1. Time element
2. Numerical elements

Preprocessing: when there are missing values

• The method when the missing value is in the middle
  
  

Model 1 — Time series decomposition model

Time value change decomposition four kinds of changes

1. Long-term trend $T$
   has been affected by the long-term trend for a long period of time, showing a continuous rise or fall
2. Seasonal Trend $The\; change\; of\; S$
   from season (it can be in quarter, month, week and other time units, but not in year) makes the indicator change periodically
3. Cycle change $C$
   takes several years as the cycle, showing wave-like cyclical changes, manifested as increasing and decreasing alternately
4. Irregular changes $I$
   can not predict or unpredictable

Two models from four changes

$TSCI$

• Product model
• Additive model
  

SPSS operation

Step 1: Define the time variable

Step 2: Make time series diagram and analyze

• The timeline label is the time variable defined in the previous step
• Timeline option: You can mark the corresponding timeline in the generated graph
• Format: the format of the drawing

After drawing the picture, you can modify the fill color of the picture, etc.

• Analysis based on time series graphs
  

Step 3: Seasonal decomposition

Cycle is less than 1 year

• Result analysis:
  Four new variables will be obtained, corresponding to
  

• Seasonal factors in results

1. Cumulative model— $T+S+C+I==Variableamount$
   Seasonal factor SS of cumulative modelThe sum of$S$ is 0
   In the cycle, each seasonal factor represents the relationship with the annual average value. The value higher or lower than the seasonal factor is
   set as the sales volume
   
2. Multiplication model— $T\ast S\ast C\ast I==Variableamount$
   Seasonal factor SS of the multiplicative modelThe product of$S$ is 1
   In the cycle, each seasonal factor represents the relationship with the annual average, the percentage value higher or lower

Step 4: Draw a time series diagram after seasonal decomposition

1. Modify the name of the new variable
2. Analysis -> Time Series Forecast -> Sequence Diagram
   
   Pay attention to modify the line color of the diagram and the graphic background

Step 5: Forecast

If it is difficult to predict directly, it is more difficult to directly predict sales

• However, the straight line (I) (T+C+I) (S) (T+C) in the figure can be predicted
  .$T+S+C+I==Variableamount$
  Multiplication model— $T\ast S\ast C\ast I==Variableamount$

Model 2-Exponential Smoothing Model-Multiple Model Types

Simple model

Disadvantages: because of the principle, only the value of the future period can be predicted

Linear trend model and Brown (Brown) linear trend model

Damping trend model

Proposed on the Holt model

Simple seasonal

• $[$ $]$ Is the rounding symbol

Winter additive model

Winter Multiplication Model

ARIMA 与 SARIMA

• $\xi$ is the white noise sequence, and the white noise residual test will generally be performed to obtain the value

General steps

Import data, time variable must be defined

Draw a time series graph

• Difference: First-order difference can be obtained.
  If it is ARIMA(p,1,q), the graph after first-order difference can be drawn
  

View the optimal model given by SPSS

• Only select the dependent variable because it is a univariate sequence analysis
• Will get an optimal model, and then the analysis can be based on the optimal model analysis
• All outlier options can be checked

Need to check option

Statistics -> Parameter Estimated Value
Graph -> Fitted Value, ACF PACF, Predicted Confidence Interval and Fitted Confidence Interval-plus the latter two graphs may be
stored vaguely -> Predicted value, upper limit of confidence interval, confidence interval Lower limit
forecast -> you can specify the date and confidence interval of the forecast (the significance level in the figure is $a=5$ )

Precautions

Evaluation index

Model fit in the output results

Write paper

1. Describe the data-whether there are missing values, data trends, whether there are seasonal changes in the data
   (you can write according to the optimal model)
2. Eliminate outliers
3. Draw a sequence diagram
   Analysis -> Time series forecasting -> Sequence diagram
4. Explain the working principle of SPSS's expert modeler and choose an optimal model
   
5. Write the obtained model expression and parameter estimation into the model
   
   

• Other delay values ​​are not displayed, then 0
• $X_t$Represents residual, predicted value $-$ the true value
  

6. Residual test for white noise
   

• There are no more than two straight lines in the PCF and PACF graphs, indicating that there is no significant difference from 0 (that is, white noise $X=0$)
• When the significance of the Q test is greater than 0.5, we say that the null hypothesis cannot be rejected

7. Through smooth $R^2$，$R^2,$or standardize BIC to detect the quality of the model