Model quality in L1 prediction error system identification

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Model quality in L1 prediction error system
identification

• J-C Carmona
• LSIS UMR-CNRS 6168Marseille France
• http//www.lsis.org/

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Agenda

• Introduction
• The prediction error framework
• Main statistical properties
• L1 Final Prediction Error criterion
• L1 Akaikes Informative Criterion
• Distance bound between 2 models
• Application experimental results
• Conclusion

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1. Introduction

• Model Quality
• model estimation step balance ( model
  error / noise influence)
• model validation
• - consistency with data not
  invalidation
• - non consistency
  invalidation
• Validation on fresh data (validation data)
• - not always possible
• - validation data are unused in
  estimation a poor estimation
• Idea predict the model behaviour from
  estimation data only
• FPE criterion
  AIC , MDL
• (Akaike)
  (Akaike) (Rissanen)

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• Introduction (2)
• ? Almost ever in L2 prediction error
  framework !!
• main advantage main
  disadvantages
• - L2 optimization simplicity - lack of
  robustness tail-problem (Rice et al 64)
• 
  - great statistical sensitivity / data
• ? L1 approach some interesting
  recent results
• - modeling uncertain systems
  via LP LSAD criterion
• (Gustafsson
  Mäkilä , Automatica, 96)
• - FPE criterion in L1
  identification
• (Carmona et
  al. , CDC02, 02)
• motivation

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2. The Prediction Error Framework
discrete-time SISO system S parameterized
model structure M one-step ahead predictor
prediction errors true system S0
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• The Prediction Error Framework (2)

estimation data set estimation criterion
estimation cost function - L2
approach LS criterion - L1
approach LSAD criterion
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3. Main Statistical Properties

• Convergence properties ( see L. Ljung
  1999)

let
where
then
i.e.
The estimate converges to  the best
approximation  in M
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• asymptotic properties of the estimate

q0
hyp 1 M is globally identifiable at
then ,
the true noise
normal distribution of the estimate
around its limit
and
with covariance matrix

where
(parameter vector gradient)
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• asymptotic properties of the estimate (2)

Some comments in L2 case (see L. Ljung 99)
, where l0 is the variance of the  true
noise . in L1 case, Q does not depend directly
on l0 winteresting in case of noisy
measurements zdiserves more investigations !


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4. L1 Final Prediction Error criterion
Validation criterion principle
prediction of the
(estimation criterion) on estimation data only
, i.e.
L1 FPE criterion
L2 FPE criterion
noise variance (to be evaluated!)
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5. L1 Akaike Information Criterion
Idea minimization of the information distance
model / true system distance Kullbach-Leibler
measure between their PDFs AIC
rule where (L2 case)
complexity term
( MDL criterion )
Rissanen rule
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• L1 Akaike Information Criterion (2)

L1 case
as for L1 FPE !
L2 case 2d/N , versus L1 case
d/N !!
first conclusions - FPE and AIC
criteria exist in L1 identification
- surprise thy are simpler than in L2
context weak influence
of the noise variance
complexity term less penalizing
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6. Distance between 2 estimated models
New problem and
are two (estimated) models.
What about their distance ?
assumptions - independent of the
estimation method (L2, L1, ) - based on
time observations and measurements only u(t),
y(t), e(t),
Assumption and comes
from the same data ZN Based on
an original result of L. Ljung and L. Guo (1997)
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• distance between 2 estimated models (2)

past inputs on horizon M
 information  matrix RN
supposed invertible (input PE)
hyp bounded input ? Cu Max u(t) , ?t ?
1,N Identification problem extra
mode investigation order increase 1 or 2
i.e. order G2 order G1 1 (or
2) rather  high frequency mode  ? tails of
impulse response for k ? M, ?1k ? ?2k
? ?k ? ?1k - ?2k ? 0 , let
, and
(input periodogram)
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• distance between 2 estimated models (3)

theorem
where impulse response tails
(assumption 1)
L(e j?) linear stable filter correlation
term residuals / past inputs
,
with ?(t) L(q) ?1(t) ?2(t)

( filtered residuals)
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• distance between 2 estimated models (3)

some comments
1. Only 1 term correlation
term ? simplicity !
2. Discussion on M / N in any case N the
largest possible !! assumption 1 ? M
large definition sum of
positive terms ? M small ? trade
off !!
we have
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7. Application and experimental results
the process

• Noise propagating through a semi-infinite duct
• Control oriented identification (Active Noise
  Control)
• Not really rational noise spectrum not really
  rational
• OE model structure M
• more precisely moles analysis ? d nA

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• application and experimental results (2)

• Comparison FPE criterion L1 / L2 estimation
• (use of standard algorithms)

.
17 seems to be the best order for both
criteria (notice the more marked  knee  in
L1 case)
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application and experimental results (3)
2. Bound analysis versus model order increase
(oder increase of 1 and 2 are presented)
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application and experimental results (4)
oder increase 2
L2 case odd values
L1 case odd values
L1 case even values
L2 case even values
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application and experimental results (5)
some comments
L1 estimation  important  variations for
" low " model orders ? process
dynamics captured step 1 ex 3 ? 4 ?
5 ? 6 , 9 ?10 and 11 ? 12 ?
physically confirmed (5 ? 6) ? tHP
step 2 ex 5 ? 7 and 12 ? 14
? physically confirmed (5 ? 7) ? 1st
acoustical mode
(12 ? 14) ? 2d
acoustical mode L2 estimation ? same
conclusions important  variations for "
high" model orders
Important difference ! Could we conclude "L1
gives better low order models / L2 gives better
high order models "   ??
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Conclusion

• we have shown that
• FPE, AIC rules are " simple in L1 case ?
  easy to use available (easy to use )
• 2. we have proposed
• a measure of the distance between 2 estimated
  models
• (the cost of the estimation
  ?!)
• 3. Some research " directions "  
• - optimization of the choice of M (past
  inputs horizon)
• - analyze the correlation between the
  terms (L1 case) and (L2 case)
• 
  versus
• low order model estimates in
  L1 case / high order in L2 case