ARIMA model predicts futture values based on past values

ARIMA

1. Box Jenkins Method of Forecasting Or ARIMA MODEL

2. Time Series Non Stationary No Trend Trend Stationary Seasonal Only Trend Overview

3. Stationary Series It has a constant mean It has a constant variance No Seasonality

4. Testing for Stationary Data Visual Tests Global vs Local Augmented Dicky Fuller or ADF test

5. White Noise It is a Time Series with: Mean = 0 Standard deviation is constant with time Lags are not auto correlated.

6. Testing for White Noise Visual Tests Global vs Local Check ACF

7. Auto Correlation The measurement of some value at a time period is dependent on the measurement of that value at the previous time period and the period before that and so on.

8. Partial Auto Correlation It is the direct component of the Auto Correlation. It is the direct effect of the variable at some previous time period on the value of the variable today. It helps us find the strength of the direct correlation between Y (t-1) and Y (t+1) after removing the indirect effects

9. ACF/PACF ACF (Autocorrelation Function) is a plot that summarizes the correlation of an observation with lag values. The X axis shows the lag and the Y axis shows the correlation coefficient between -1 and 1 for negative and positive correlation. PACF (Partial Autocorrelation Function) is the plot used to summarize the correlation of an observation with lag values that is not accounted for by other lagged observations at shorter time periods. It is the plot of the direct effect.

10. Auto Regressive Model Future demand is a function of the past demand. Future demand can be predicted based on the demand for the previous time period, and the time period before that, and before that, and so on. The values of the variable are auto-correlated, i.e. the values of variable Y at time period t are correlated with the values of Y at time period (t-1) and so on. Regression on it’s self. ‘p’ denotes the order of the Auto Regressive Model or the lag order. Auto-regression is regression of a variable on itself measured at different time points. Auto-regressive model with lag 1, AR(1), is given by Y t +1 =  Y t +  t +1

11. Moving Averages Model Future demand is a function of the past error. It is like Exponential Smoothing. Future demand could be a function of the error in the previous time period, or the time period before that also, or before that, and so on. ‘q’ denotes the size of the moving average window, also called the order of the Moving Average Model

12. Moving Averages Model Moving average (MA) processes are regression models in which the past residuals are used for forecasting future values of the time-series data. Moving average process of lag 1, MA(1), is given by Alternatively, a moving average process of lag 1 can be written as

13. ARMA/ARIMA Model Box and Jenkins proposed how to convert a non-stationary series to a stationary series using differencing. Taking a difference of two consecutive time periods removes the trend in the data. We could further take a difference of the difference, and so on. ‘d’ denotes the number of times that the raw observations are differenced, called the degree of differencing. It is combination of the AR and the MA models. ARMA Model ARIMA Model

14. Model Building Identification Estimation Diagnostic Checking Forecasting

15. ARIMA( p, d, q ) Model Building

16. Identification Check whether the time series is stationary. If not, check how many differences are required to make it stationary, i.e. what is the value of ‘d’. Identify the parameters of the ARMA model for the data, i.e. what are the values of ‘p’ and ‘q’.

17. Some Guidelines The model is AR if the ACF trails off after a lag and has a hard cut-off in the PACF after a lag. This is taken as the value of ‘p’. The model is MA if the PACF trails off after a lag and has a hard cut-off in the ACF after a lag. This lag value is taken as the ‘q’ value. The model is a mix of both AR and MA if both ACF and PACF trail off. In the ACF plot, if there is a positive correlation at lag 1, use the AR model. If there is a negative correlation at lag 1, use the MA model.

18. Diagnostic Checking Check the model for robustness and optimality. Check for: Overfitting - Make sure the model is not more complex than necessary Residual Errors – A. Should resemble White Noise B. Create ACF and PACF plots of the residual error time series to make sure there is no auto-correlation

19. Requirements for a Good Fit Normalized BIC should be minimum Ljung Box test should not be significant H : Model does not show lack of fit. H 1 : Model shows lack of fit. All coefficients should be significant ACF and PACF plots of the residual should be within limits.

20. White Noise Stationary - Visual Test Trend ADF ACF Idly No Mean and Variance not constant, Appears Not Stationary -7.15 Gradual decrease and then increase and sharp cutoff Dosa No Not Stationary - Variance not constant -5.86 Dies down extremely slowly So Not Stationary Chutney No Appears Stationary -7.3 Cuts off sharply So Stationary Sambhar No Appears Stationary -7.49 Cuts off sharply So Stationary Continental B/F No Appears Stationary -6.16 Cuts off sharply So Stationary North Indian B/F No May not be Stationary -7.3 Cuts off sharply, so possibly Stationary Omellette No Not Stationary - Mean and Variance not constant -3.85 Dies down extremely slowly So Not Stationary

21. Idly Model R Square RMSE MAPE Normalised BIC AIC BIC LB Test Model Parameters Significance Residual ACF and PACF Exponential Smoothing 0.261 0.24 6.12 7.81 3.67 739 742 Y Y Y (1,0,0) 0.21 6.29 8.53 3.76 749 755 Y Y Y (1,1,0) 0.06 6.66 8.05 3.88 N N Y (0,0,1) 0.15 6.53 9.28 3.84 759 764 N N Y (0,1,1) 0.14 6.35 7.75 3.77 741 747 Y Y Y

22. Chutney Model R Square RMSE MAPE Normalised BIC AIC BIC LB Test Model Parameters Significance Residual ACF and PACF (1,0,0) 0.2 10.0 6.0 4.7 856 861 Y Y Y (1,1,0) 0.1 10.9 6.6 4.9 870 875 Y N Y (0,0,1) 0.1 10.2 5.9 4.7 861 867 Y N Y Exponential Smoothing 0.39 0.1 10.3 6.2 4.7 858 860 Y Y Y

23. Omelette Model R Square RMSE MAPE Normalised BIC AIC BIC LB Test Model Parameters Significance Residual ACF and PACF Exponential Smoothing 0.618 0.572 3.44 20.59 2.51 608 610 Y Y Y (1,0,0) 0.584 3.41 21.83 2.53 611 616 Y Y Y (0,0,1) 0.388 30.09 655 660 N N Y (1,1,0) 0.525 3.66 21.62 2.68 613 618 Y Y Y (1,1,1) 0.621 3.33 21.48 2.55 608 616 Y Y Y

24. Dosa Model R Square RMSE MAPE Normalised BIC AIC BIC LB Test Model Parameters Significance Residual ACF and PACF Exponential Smoothing 0.419 0.26 9.81 23.70 4.61 846 849 Y Y Y (0,0,1) 0.20 10.21 27.24 4.73 863 868 N N Y (1,0,0) 0.29 9.61 23.79 4.60 849 855 Y N Y (1,1,0) 0.20 10.29 23.87 4.74 857 863 Y N Y (1,1,1) 0.28 9.80 23.72 4.69 847 855 Y Y Y (0,1,1) 0.26 9.89 23.48 4.67 848 854 Y Y Y

25. Continental B/F Model R Square RMSE MAPE Normalised BIC AIC BIC LB Test Model Parameters Significance Residual ACF and PACF (0,0,1) 0.28 4.93 9.15 3.27 693 698 Y Y Y (1,0,0) 0.28 4.95 8.93 3.28 693 699 Y Y Y (1,1,0) 0.06 5.44 9.08 3.47 712 718 Y N N (0,1,1) 0.07 5.43 9.13 3.47 711 717 N N Y

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