SYSTEMS Identification
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Transcript of SYSTEMS Identification
SYSTEMSSYSTEMSIdentificationIdentification
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
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Lecture 9
Asymptotic Distribution of Parameter Estimators
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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Overview
If convergence is guaranteed, then * N
But, how fast does the estimate approach the limit?
What is the probability distribution of ? * N
The variance analysis of this chapter will reveal:
a) The estimate converges to at a rate proportional to
b) Distribution converges to a Gaussian distribution: N(0,Q)
c) Covariance matrix Q, depends on - The number of samples/data set size: N,
- The parameter sensitivity of the predictor: - The noise variance
*N
1
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Central Limit Theorem
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic Theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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Central Limit Theorem
The mathematical tool needed for asymptotic variance analysis is “Central Limit” theorems.
Example: Consider two independent random variable, X and Y, with the same uniform distribution, shown in Figure below.
Define another random variable Z as the sum of X and Y: Z=X+Y. we can obtain the distribution of Z, as :
dxxzfxfzf YXZ )()()(
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In general, the PDF of a random variable approaches a Gaussian
distribution, regardless of the PDF of each , as N gets larger.
N
iiX
1
Central Limit Theorem
Further, consider W=X+Y+Z. The resultant PDF is getting close to a Gaussian distribution
The resultant PDF is getting close to a Gaussian distribution.
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Central Limit Theorem
Let be a d-dimensional random variable with : ,...1,0, tX t
)( tXEm
]))([( Ttt mXmXEQ
Mean
Cov
Consider the sum of given by:mX t
N
tt mX
N1
N )(1
Y
Then, as N tends to infinity, the distribution of converges to the Gaussian distribution given by PDF:
NY
),0(YN N QNondistributi
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The Prediction-Error approach: Basic Theorem
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic Theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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M
NNN
DZV
),(minarg
),(
2
11),( 2
1
tN
ZVN
t
NN
0),( NNN ZV
Then, with prime denoting differentiation with respect to ,
Expanding around gives:*
)(),(),(),(0 ** NN
NN
NN
NN ZVZVZV
),( NNN ZV
is a vector “between” *& N
Applying the Central Limit Theorem, we can obtain the distribution of estimate as N tends to infinity.
Let be an estimate based on the prediction error method N
M
NNN
DZV
),(minarg
),(
2
11),( 2
1
tN
ZVN
t
NN
The Prediction-Error Approach
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The Prediction-Error Approach
),( NN ZV
),()],([)( *1* NN
NNN ZVZV
),(),(1
),( **
1
* ttN
ZVN
t
NN
Assume that is nonsingular, then:
** )(),(),( *
ty
d
dt
d
dt
Where as usual:
To obtain the distribution of , and must be computed as N tends to infinity.
* N
),( * N
N ZV
),()],([)( *1* NN
NNN ZVZV
),( NN ZV
N
NN
NNN ZV
d
dZV
|),(),(
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For simplicity, we first assume that the predictor is given by a linear regression:
Ttty )()|(
The actual data is generated by ( is the parameter vector of the true system)
)()()( 0* tetty T
So:
Therefore:
*
)(|)|( * ttyd
d
d
d T
)|()(),( tytyt
)()()()(),( 0*
0** tettett TT
N
t
NN tet
NZV
10
* )()(1
),(
The Prediction-Error Approach
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Let us treat as a random variable. Its mean is zero, since: tXtet )()( 0
The covariance is
Consider:
Appling the central limit Theorem:
The Prediction-Error Approach
0)]([)]([)]()([ 00 teEtEtetEm
RttEteEXX
stforssetetEmXmXEXX
TTst
TTst
Tst
02
0
00
])()([)]([)cov(
,0])()()()([]))([()cov(
)()(1
)(1
10
1
tetN
mXN
YN
t
N
ttN
),0(),( 0* RNZVNYN N
NN
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Next, compute :
And:
The Prediction-Error Approach
),( NNN ZV
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We obtain:
So:
),()],([)( *1* NN
NNN ZVZV
NasQNN N ),0()( *
The Prediction-Error Approach
RttN
ZV TN
tN
NNN
))()((1
lim),(1
),0(),( 0* RNZVNYN N
NN
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The extended result of estimate distribution is summarized in the following theorem, i.e. Ljun’g Textbook Theorem 9-1.
Theorem 1 Consider the estimate determined by:
Assume that the model structure is linear and uniformly stable and that the data set satisfies the quasi stationary requirements. Assume also that converges with probability 1 to a unique parameter vector involved in :
M
NNN
DZV
),(minarg
),(
2
11),( 2
1
tN
ZVN
t
NN
N
MD
Also we have:
And that:
Converge to with probability 1
0)( NV
The Prediction-Error Approach
*|),()|(1
lim)(1
*
tty
d
d
NV
N
tN
tm
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Where is the ensemble mean given by:
Then, the distribution of converges to the Gaussian distribution given by
)( NN
})],()][,({[.lim TNN
NN
NZVZVENQ
11 )]([)]([ VQVP
),0()( PNN N
The Prediction-Error Approach
tm
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As stated formally in Theorem 1, the distribution of converges to a Gaussian distribution for the broad class of system identification problems. This implies that the covariance of asymptotically converges to:
)( NN
PN
Cov N1ˆ
N
This is called the asymptotic covariance matrix and it depends not only on
(a) the number of samples/data set size: N, but also on (b) the parameter sensitivity of the predictor:
(c) Noise variance
)(),(),( ty
d
dt
d
dt
1*1* )]([)]([ VQVP
The Prediction-Error Approach
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Expression for the Asymptotic Variance
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic Theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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Let us compute the covariance once again for the general case:
Unlike the linear regression, the sensitivity is a function of θ,
Quadratic Criterion
)(),(}{ 000 tetandDc Assume that is a white noise with zero mean and variances . We have:0
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),(),()(),(),(),()( **0
*** ttEtetEttEV TT
),(),(),(),(1
lim
),()()(),(lim
**0
**
10
*
1 10
*2
ttEttEN
stetetEN
NQ
TTN
tN
Ts
N
t
N
sN
Similarly
Hence:1**
0 )],(),([ ttEP T
The asymptotic variance is therefore a) inversely proportional to the number of samples, b) proportional to the noise variance, and c) inversely related to the parameter sensitivity.
The more a parameter affects the prediction, the smaller the variance becomes .
PN
Cov N1ˆ
Quadratic Criterion
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1
1
]),(),(1
[
N
tN
TNNN tt
NP
),(2
11 2
1N
N
tN t
N
A very important and useful aspect of expressions for the asymptotic covariance matrix is that it can be estimated from data. Having N data points and determined we mat use:N
Since is not known, the asymptotic variance cannot be determined. N
sufficient data samples needed for assuming the model accuracy may be obtained.
Quadratic Criterion
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Consider the system
)()1()1()(: 00 tetutyatyS
)(tu )(0 te 0
Suppose that the coefficient for is known and the system is identified in the model structure
)1( tu
)1()1()( 0 tutyaty
)1()(ˆ),( tytyd
dt
and are two independent white noise with variances and respectively
We have:
atetutyatyM ).()1()1()(: 00
Or
Example : Covariance of LS Estimates
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Hence
)1(20
tyEP
002
0 )1(2)0()0( yvy RaRaR
0)0()0()1( 0 yuyy RRaR
To compute the covariance, square the first equation and take the expectation:
Multiplying the first equation by and taking expectation gives:
The last equality follows, since does not affect (due to the time delay )
20
02
1)0()1(
aRtyE y
0
200 1
a
NaCov N
)1( ty
)(ty)(tu
Hence:
Example : Covariance of LS Estimates
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Cov (a) = 0.0471
Example : Covariance of LS Estimates
Assume a=0.1,
Estimated values for parameter a, for 100 independent experiment using LSE, is shown in the bellow Figure.
,1 10
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Cov (a) = 0.0014
Example : Covariance of LS Estimates
,10 10 Now, assume a=0.1, ,10 10
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Cov (a) = 0.1204
Example : Covariance of LS Estimates
Now, assume a=0.1, ,1 100
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Example : Covariance of an MA(1) Parameter
Consider the system
is white noise with variance . The MA(1) model structure is used:0)}({ 0 te
)1()()( 000 tectety
)1()1()( tycctyccty
)1(),1(),1(),( tyctyctcct
0cc )1(
1
1),( 01
00
te
qcct
Nc
N
c
ctENcCov N
20
02
0 1
),(
1
Given the predictor 4.18:
Differentiation w.r.t c gives
At we have :
If is the PEM estimate of c:
)1()()( tcetety
)|(ˆ)(1)()()(1)()()|(ˆ tytyqCtyqAtuqBty
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),),,((),),,((),([1
),(1
tttttN
ZVN
t
NN
N
t
NN tt
NZV
1
),),,((1
),(
),,()( t ),,()( t
),),,((),(),),,((),),,((),(
),),,((),(),(),),,((),()(
ttEtttEtttE
tttEttttEVT
For general Norm ),,( t
We have:
Similarly:
We can use the asymptotic normality result in this more general form whenever required. The expression for the asymptotic covariance matrix is rather complicated in general.
Asymptotic Variance for general Norms.
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)(),,(),),,((1
),(1
twithttN
ZVConsiderN
t
NN
0))(( teE Assume that . Then under assumption after straightforward calculations:
MS
100 )],(),()[( ttEkP T
20
20
))](([
))](([)(
teE
teEk
02
02 )()(,1)()(,
2
1)( tEeksoeandee Clearly for for quadratic
The choice of in the criterion only acts as scaling of the covariance matrix
Asymptotic Variance for general Norms.
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Frequency-Domain Expressions for the Asymptotic Variance
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic Theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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Frequency-Domain Expressions for the Asymptotic Variance.
The asymptotic variance has different expression in the frequency domain, which we will find useful for variance analysis and experiment design.
Let transfer function and noise model be consolidated into a matrix
The gradient of T, that is, the sensitivity of T to θ, is
For a predictor, we have already defined W(q,θ,) and z(t), s.t.
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Therefore the predictor sensitivity is given by
Where
Substituting in the first equation:
Frequency-Domain Expressions for the Asymptotic Variance.
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At (the true system), note and0 )(),( 00 tet
where
Ttetutx )]()([)( 00
Let be the spectrum matrix of :)(0
x )(0 tx
Using the familiar formula:
Frequency-Domain Expressions for the Asymptotic Variance.
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For the noise spectrum,
Using this in equation below:
We have:
The asymptotic variance in the frequency domain.
Frequency-Domain Expressions for the Asymptotic Variance.
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Distribution of Estimation for the correlation Approach
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic Theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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The Correlation Approach
0),(ˆ
NN
DN Zfsol
M
N
tF
NN tt
NZf
1
),(),(1
),(
),()(),()(0 ,N
NNNN
NN
NN ZfZfZf
),()(),( tqLtF )(),()(),(),( tuqKtyqKt uy
By Taylor expansion we have:
),( * t ),( * NN ZV ),( * t
),( NN Zf
This is entirely analogous with the previous one obtained for PE approach, with he difference that in is replaced with in .
We shall confine ourselves to the case study in Theorem 8.6, that is, and linearly generated instruments. We thus have:
)(
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The Correlation Approach
Theorem : consider by 0),(ˆ
NN
DN Zfsol
M
),( t
});,(),,(),,(),,({ Muyuy DqKd
dqK
d
dqKqK
Assume that is computed for a linear, uniformly stable model structureAnd that:
NaspwN 1..*
)( *f
NasZEfN NN ,0),( *
),0()( * PNAsN N
1*1* )]([)]([ fQfP
),(),(.lim ** NTN
NN
NZfZfENQ
TF tqLtEttEf )],()()[,(),(),()(
is a uniformly stable family of filters.
Assume also that that is nonsingular and that
Then
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The Correlation Approach
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Example : Covariance of an MA(1) Parameter
Consider the system
is white noise with variance . Using PLR method:0)}({ 0 te
)1()()( 000 tectety
0cc )1(1
1),( 01
00
te
qcctAt we have :
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Distribution of Estimation for the Instrumental Variable Methods
Topics to be covered include:
Central Limit Theorem
The Prediction-Error approach: Basic Theorem
Expression for the Asymptotic Variance
Frequency-Domain Expressions for the Asymptotic Variance
Distribution of Estimation for the correlation Approach
Distribution of Estimation for the Instrumental Variable Methods
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Instrumental Variable Methods
Suppose the true system is given as
Where e(t) is white noise with variance independent of {u(t)}. then0
is independent of {u(t)} and hence of if the system operates in open loop. Thus is a solution to:
),( * t
0
We have:
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Instrumental Variable Methods
To get an asymptotic distribution, we shall assume it is the only solution to . MD
Introduce also the monic filter
Intersecting into these Eqs.,
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Consider the system
)()1()1()(: 00 tetutyatyS
)(tu )(0 te 0
Suppose that the coefficient for is known and the system is identified in the model structure.
Let a be estimated by the IV method using:
)1( tu
and are two independent white noise with variances and respectively
Example : Covariance of LS Estimates
By comparing the above system with
We have
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Instrumental Variable Methods