2.2.2.1 Autocorrelation and Partial Autocorrelation (ACF & PACF)ACF
Summary
This ACF(Autocorrelation function) & PACF( Partial Autocorrelation function) tool is supported in the Time Series Analysis App. It is used to compute and plot the autocorrelations and the partial autocorrelations of a series.
Tutorial
This tutorial uses App’s built-in sample project. To open this sample OPJU file:
- Right click the Time Series Analysis App icon in the Apps Gallery and choose Show Samples Folder.
- A folder will open. Drag-and-drop the project file TSA Sample.opju into Origin.
ACF & PACF
- Expand Project Explorer docked on the left. Select folder Statistics and Test . The Book3 contains data
about Australian total wine sales by wine makers in bottles.
- Highlight Column A and B, and then click the Time Series Analysis App icon in the Apps Gallery.
- In the dialog, select Statistics and Test and ACF & PACF tool.
- Use the default dialog setting, click the OK button.
- Then you will get the report with Series, ACF and PACF three graphs.
Algorithm
Autocorrelation
Autocorrelation calculates the correlation between a time series and the time series with lags. It can be used to determine which terms to be included in ARIMA model.
This app calls nag_tsa_auto_corr (g13abc) function [1] to calculate autocorrelation.
For a time series , i=1, 2, ... n, the coefficient of lag k is:
where .
- Default Maximum Number of Lags
H0: The autocorrelation function is identically zero.
If P-value<0.05, the autocorrelation function is significantly different from zero.
- Standard Error of Autocorrelation
1. Independent model
2. Bartlett model
- t-value and Confidence Limits
Lower confidence limit at lag k:
Upper confidence limit at lag k:
H0: First k autocorrelations are identically zero.
Use distribution to calculate the P-value.
If P-value<0.05, first k autocorrelations are significantly different from zero.
Partial Autocorrelation
Partial autocorrelation calculates the correlation between a time series and the time series with lags excluding the influence of intermediate lags. It can be used to determine terms to include in ARIMA model.
This app calls nag_tsa_auto_corr_part (g13acc) function [3] to calculate partial autocorrelation.
For a time series , i=1, 2, ... n, partial autocorrelation coefficients can be solved by a recursive method [4]:
where ,
- is autocorrelation,
- is the predictor error variance ratio,
- is partial autocorrelation values, and is the autoregressive parameters of maximum order.
It was initialized by setting and .
- Standard Error of Partial Autocorrelation
- t-value and Confidence Limits
Lower confidence limit at lag k:
Upper confidence limit at lag k:
Reference
- nag_tsa_auto_corr (g13abc)
- George E. P. Box and Gwilym M. Jenkins (1976). Time Series Analysis: Forecasting and Control. (Revised Edition) Holden–Day
- nag_tsa_auto_corr_part (g13acc)
- J. Durbin (1960). The fitting of time series models. Rev. Inst. Internat. Stat. Vol.28, pp.233
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