Forecasting

JY Le Boudec

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Contents

1. What is forecasting ?2. Linear Regression

3. Avoiding Overfitting4. Differencing

5. ARMA models6. Sparse ARMA models

7. Case Studies

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1. What is forecasting ?

Assume you have been able to define the nature of the loadIt remains to have an idea about its intensity

It is impossible to forecast without error

The good engineer shouldForecast what can be forecast

Give uncertainty intervals

The rest is outside our control

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2. Linear Regression

Simple, for simple casesBased 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 errorIt 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 exactlyIn general, we won’t have this luxury

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We saw this alreadyA case where estimation error versus prediction uncertainty can be quantified

Prediction interval if model is knownPrediction 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

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This is the overfitting problem

How to avoid overfittingMethod 1: use of test dataMethod 2: information criterion

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

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d = 1

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

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4. Differencing the Data

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Point Predictions from Differenced Data

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Background On Filters (Appendix B)

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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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Impulse Response

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A filter with stable inverse

How is this prediction done ?

This is all very intuitive

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Prediction assuming differenced data is iid

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Prediction Intervals

A prediction without prediction intervals is only a small part of the storyThe financial crisis might have been avoided if investors had been aware of prediction intervals

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Compare the Two

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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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Test of iid-ness

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ARMA Process

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ARMA Processes are Gaussian (non iid)

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ARIMA Process

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Fitting an ARMA Process

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

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Fitting an ARIMA ModelApply Scientific Method

1. make stationary and normal (how ?)2. bound orders p,q3. 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

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

Q:What are the parameters ?

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

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Best Model Order

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Check the Residuals

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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

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Forecasting Formulae for ARIMA

Y = original dataX = differenced data, fitted to an ARMA model1. Obtain point prediction for X using what we just saw2. Apply Proposition 6.4.1 to obtain point prediction for Y3. Apply formula for prediction interval

There are several other methods, but they may have numerical problems. See comments in the lecture notes after prop 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 iidHow can we get confidence intervals ?

Bootstrap by sampling from residuals

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With bootstrap from residuals

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 heuristicsMultiplicative ARIMA, constrained ARIMAHolt Winters

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Holt Winters Model 1: EWMA

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EWMA is OK when there is no trend and no periodicity

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

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Constrained ARIMA

(corrected or not)

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7. Case Studies

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h = 1h = 1

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h = 2h = 2

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logh = 1log

h = 1

Conclusion

Forecasting is useful when savings matter; for exampleSave money on server space rentalSave 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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