Forecasting
JY Le Boudec
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Contents
1. What is forecasting ?
2. Linear Regression
3. Avoiding Overfitting
4. Differencing
5. ARMA models
6. 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 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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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)
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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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Best Model for Internet Data, polynomial of
degree up to 2
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d=1
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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)
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
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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 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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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 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 model
Pb 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 maximum
may 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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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 data
X = differenced data, fitted to an ARMA model
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Obtain point prediction for X using what we just saw
Apply Proposition 6.4.1 to obtain point prediction for Y
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 iid
How can we get
confidence intervals ?
Bootstrap by sampling
from residuals
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With gaussian
assumption
With bootstrap from
residuals
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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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Holt Winters Model 1: EWMA
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EWMA is OK when there is no trend and no
periodicity
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Constrained ARIMA
Sparse models give less
accurate predictions but
have much fewer
parameters and are simple
to fit.
(corrected or not)
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7. Case Studies
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h=1
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h=2
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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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