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Forecasting

Forecasting. JY Le Boudec. Contents. What is forecasting ? Linear Regression Avoiding Overfitting Differencing ARMA models Sparse ARMA models Case Studies. 1. What is forecasting ?. Assume you have been able to define the nature of the load

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Forecasting

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  1. Forecasting JY Le Boudec 1

  2. Contents • What is forecasting ? • Linear Regression • Avoiding Overfitting • Differencing • ARMA models • Sparse ARMA models • Case Studies 2

  3. 1. What is forecasting ? • Assume you have been able to define the nature of the load • It remains to have an idea about its intensity • It is impossible to forecast without error • The good engineer should • Forecast what can be forecast • Give uncertainty intervals • The rest is outside our control 3

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  5. 2. Linear Regression • Simple, for simple cases • Based on extrapolating the explanatory variables 5

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  9. Estimation and Forecasting • In practice we estimate  from y, …, yt • When computing the forecast, we pretend  is known, and thus make an estimation error • It is hoped that the estimation error is much less than the confidence interval for forecast • In the case of linear regression, the theorem gives the global error exactly • In general, we won’t have this luxury 9

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  11. We saw this already • A case where estimation error versus prediction uncertainty can be quantified • Prediction interval if model is known • Prediction interval accounting for estimation (t = 100 observed points) 11

  12. 3. The Overfitting Problem • The best model is not necessarily the one that fits best 12

  13. Prediction for the better model • This is the overfitting problem 13

  14. How to avoid overfitting • Method 1: use of test data • Method 2: information criterion 14

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  17. Best Model for Internet Data, polynomial of degree up to 2 17

  18. d = 1 18

  19. Best Model for Internet Data, polynomial of degree up to 10 19

  20. 4. Differencing the Data 20

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  23. Point Predictions from Differenced Data 23

  24. Background On Filters (Appendix B) • We need to understand how to use discrete filters. • Example: write the Matlab command for 24

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  26. A simple filter • Q: compute X back from Y 26

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  30. Impulse Response 30

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  33. A filter with stable inverse 33

  34. How is this prediction done ? • This is all very intuitive 34

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  36. Prediction assuming differenced data is iid 36

  37. Prediction Intervals • A prediction without prediction intervals is only a small part of the story • The financial crisis might have been avoided if investors had been aware of prediction intervals 37

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  40. Compare the Two Linear Regression with 3 parameters + variance Assuming differenced data is iid 40

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  42. 5. Using ARMA Models • When the differenced data appears stationary but not iid 42

  43. Test of iid-ness 43

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  45. ARMA Process 45

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  47. ARMA Processes are Gaussian (non iid) 47

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  50. ARIMA Process 50

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