Transcript Document
Forecasting
JY Le Boudec 1
Contents
1. What is forecasting ?
2. Linear Regression 3. Avoiding Overfitting 4. Differencing 5. ARMA models 6. Sparse ARMA models 7. 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
4
2. Linear Regression
Simple, for simple cases Based on extrapolating the explanatory variables 5
6
7
8
Estimation and Forecasting
In practice we estimate from y, …, y t When computing the forecast, we pretend estimation error is known, and thus make an 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
10
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
3. The Overfitting Problem
The best model is not necessarily the one that fits best 12
Prediction for the better model
This is the overfitting problem 13
How to avoid overfitting
Method 1: use of test data Method 2: information criterion 14
15
16
Best Model for Internet Data, polynomial of degree up to 2
17
d = 1
18
Best Model for Internet Data, polynomial of degree up to 10
19
4. Differencing the Data
20
21
22
Point Predictions from Differenced Data
23
Background On Filters (Appendix B)
We need to understand how to use discrete filters.
Example: write the Matlab command for 24
25
A simple filter
Q: compute X back from Y 26
27
28
29
Impulse Response
30
31
32
A filter with stable inverse
33
How is this prediction done ?
This is all very intuitive 34
35
Prediction assuming differenced data is iid
36
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
38
39
Compare the Two
Linear Regression with 3 parameters + variance Assuming differenced data is iid 40
41
5. Using ARMA Models
When the differenced data appears stationary but not iid 42
Test of iid-ness
43
44
ARMA Process
45
46
ARMA Processes are Gaussian (non iid)
47
48
49
AR I MA Process
50
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 Y t i.e. Y t » ARMA 4. compute residuals and verify white noise and normal Fitting an ARMA model Pb is : given orders p,q given (x 1 , …x likelihood n ) (transformed data) compute the parameters of an ARMA (p,q) model that maximizes the
Q:What are the parameters ?
A: the mean
, the polynomial coefficients variance
2
k and
k , the noise
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 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 53
Fitting ARMA Model is Same as Minimizing One-Step ahead prediction error
54
Best Model Order
55
Check the Residuals
56
Example
57
58
Forecasting with ARMA
Assume Y t is fitted to an ARMA process The prediction problem is: given Y 1 =y 1 ,…,Y t =y t find the conditional distribution of Y t+h We know it is normal, with a mean that depends on (y 1 ,…,y t ) 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
60
61
Forecasting Formulae for ARIMA
Y = original data X = differenced data, fitted to an ARMA model 1.
2.
3.
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 62
63
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
65
With gaussian assumption With bootstrap from residuals 66
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
68
Holt Winters Model 1: EWMA
69
70
EWMA is OK when there is no trend and no periodicity
71
72
73
74
75
76
77
(corrected or not) Constrained ARIMA Sparse models give less accurate predictions but have much fewer parameters and are simple to fit.
78
7. Case Studies
79
80
81
82
83
84
85
86
87
h = 1 88
89
h = 2 90
91
log h = 1 92
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
94