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Presentation Transcript

Contents

- What is forecasting ?
- Linear Regression
- Avoiding Overfitting
- Differencing
- ARMA models
- Sparse ARMA models
- Case Studies

2

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

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

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

18

Background On Filters (Appendix B)

- We need to understand how to use discrete filters.
- Example: write the Matlab command for

24

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

Fitting an ARMA Process

- Called the Box-Jenkins method
- Difference the data until stationary
- Examine ACF to get a feeling of order (p,q)
- Fit an ARMA model using maximum likelihood

51

Fitting an ARIMA Model

Apply Scientific Method

1. make stationary and normal (how ?)

2. bound orders p,q

3. fit an ARMA model to Yt -

i.e. Yt -» ARMA

4. compute residuals and verify white noise and normal

Fitting an ARMA modelPb is :

given orders p,q

given (x1, …xn) (transformed data)

compute the parameters of an ARMA (p,q) model that maximizes the likelihood

Q:What are the parameters ?

A: the mean , the polynomial coefficients k and k , the noise variance 2

52

This is a non-linear optimization problem

- Maximizing the likelihood is a non-linear optimization problems
- Usually solved by iterative, heuristic algorithms, may converge to a local maximummay not converge
- Some simple, non MLE, heuristics exist for AR or MA models
- Ex: fit the AR model that has the same theoretical ACF as the sample ACF
- Common practice is to bootstrap the optimization procedure by starting with a “best guess”
- AR or MA fit, using heuristic above

53

Fitting ARMA Model is Same as Minimizing One-Step ahead prediction error

54

Example

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Forecasting with ARMA

- Assume Yt is fitted to an ARMA process
- The prediction problem is: given Y1=y1,…,Yt=yt find the conditional distribution of Yt+h
- We know it is normal, with a mean that depends on (y1,…,yt) and a variance that depends only on the fitted parameters of the ARMA process
- There are many ways to compute this; it is readily done by Matlab

59

Forecasting Formulae for ARIMA

- Y = original data
- X = differenced data, fitted to an ARMA model
- Obtain point prediction for X usingwhatwejustsaw
- Apply Proposition 6.4.1 to obtain point prediction for Y
- Apply formula for predictioninterval
- There are severalothermethods, but theymay have numericalproblems. Seecomments in the lecture notes afterprop 6.5.2

62

Improve Confidence Interval If Residuals are not Gaussian (but appear to be iid)

- Assume residuals are not gaussian but are iid
- How can we get confidence intervals ?
- Bootstrap by sampling from residuals

64

6. Sparse ARMA Models

- Problem: avoid many parameters when the degree of the A and C polynomials is high, as in the previous example
- Based on heuristics
- Multiplicative ARIMA, constrained ARIMA
- Holt Winters

67

Sparse models give less accurate predictions but have much fewer parameters and are simple to fit.

Constrained ARIMA

(corrected or not)

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Conclusion

- Forecasting is useful when savings matter; for example
- Save money on server space rental
- Save energy
- Capturing determinism is perhaps most important and easiest
- Prediction intervals are useful to avoid gross mistakes
- Re-scaling the data may help
- … à vous de jouer.

93

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