Forecasting

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# Forecasting - PowerPoint PPT Presentation

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

JY Le Boudec

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

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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

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

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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

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• 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)

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

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Prediction for the better model
• This is the overfitting problem

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

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

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A simple filter
• Q: compute X back from Y

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

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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

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Compare the Two

Linear Regression with 3 parameters + variance

Assuming differenced data is iid

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

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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

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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

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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

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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

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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

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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

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Withbootstrapfromresiduals

• With gaussian assumption

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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

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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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log

h = 1

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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.

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Joyeuses

Pâques !

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