Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies

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Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies

M. P. Mignolet, Christian Soize

To cite this version:

M. P. Mignolet, Christian Soize. Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies. Probabilistic Engineering Mechanics, Elsevier, 2008, 23 (2-3), pp.267-278. �10.1016/j.probengmech.2007.12.027�. �hal-00685147�

Nonparametric Stochastic Modeling Of Linear Systems With Pre- scribed Variance Of Several Natural Frequencies

M.P. Mignolet

Arizona State University, Tempe, AZ 85287-6106, USA

C. Soize

Université de Marne-la-Vallée, 77454 Marne-la-Vallée, France

ABSTRACT: A complete probabilistic model of random positive definite matrices is developed that incorpo- rates constraints on the standard deviations of a set of its eigenvalues. The model is in particular applicable to the representation of the mass and stiffness matrices of random dynamic systems of which certain natural fre- quencies are observed. The model development is based on the maximization of the entropy under a set of constraints representing the prescribed eigenvalue standard deviations, the mean matrix being given, and the existence of the mean Frobenius norm of the inverse of the random matrix. The efficient simulation of sam- ples of random matrices according to the proposed model is discussed in details. Finally, examples of applica- tion validate the above concepts and demonstrate the usefulness of the proposed model.

Keywords: structural dynamics, random systems, random matrices, maximum entropy, probabilistic model 1 INTRODUCTION

The stochastic modeling and simulation of random multi degree of freedom systems has often in the past been accomplished by postulating joint prob- ability density functions of the components of their mass, stiffness, and/or damping matrices or, equiva- lently, of their corresponding natural frequencies, damping ratios, and mode shapes (e.g. Ghanem and Spanos, 1991, Kleiber et al, 1992, Rivas-Guerra and Mignolet, 2004, Schueller, 1997). However, a dif- ferent approach has recently been proposed (Soize, 2000, 2001) and validated (see Soize, 2005, for a re- view) in which the probabilistic model of the mass, stiffness, and/or damping matrices is not assumed but rather determined to maximize the entropy under the constraints (i) that these matrices are positive definite, (ii) that their mean values are prescribed, and (iii) of an overall measure of variation (variance of the norms of the matrices prescribed). This ap- proach has been named nonparametric since no pa- rameter value is selected by the user. It will be fur- ther qualified of unconstrained to differentiate it from its constrained extension presented below.

The unconstrained approach is applicable to a broad range of situations in which little, i.e. only a single measure of variability, is known about the randomness of the system. In such cases, the uncon- strained approach is particularly advantageous as it

permits to derive, on a rational basis, a full stochas- tic model of the system. There are however various other situations in which more than a single measure of randomness is known, e.g. when tests have been conducted. Natural frequencies are the most standard information obtained from dynamic tests and thus estimates of the mean and variance of the first few natural frequencies of the system may realistically be available. Since the mean natural frequencies are likely to be close to the values obtained for the de- sign configurations, it is typically the variances which provide the most valuable information about the system randomness. They should thus be incor- porated in the stochastic model of the system.

In this light, the goal of the present investigation is to extend the formulation of the unconstrained nonparametric approach to allow for additional con- straints on the standard deviations of some of the ei- genvalues of the mass, damping, and/or stiffness ma- trices. This novel approach will be referred to as constrained in the remainder of the paper.

Note that the expected domain of application of these concepts is the low frequency range in which the natural frequencies are distinguishable and ob- servable.

2 ENTROPY MAXIMIZATION AND CON- STRAINTS

It is desired here to simulate realizations of symmet- ric positive definite random matrices, e.g. the mass and/or stiffness matrix of a linear dynamic system, the properties of which, i.e. eigenvalues, eigenvec- tors, components, etc., are all random. This require- ment necessitates the specification of the joint prob- ability density function of all elements of the matrix.

In most practical problems, however, this informa- tion is not available - only some moments and/or marginal probability density functions are likely to be available. In the absence of the exact distribution, it is then appropriate to ask what are the desirable features of this distribution. In this context, note that the design of structural systems is often robust, i.e.

that small perturbations in their geometrical and ma- terial properties do not alter significantly the prob- ability of failure/fatigue life of the system consid- ered (see Rivas-Guerra and Mignolet, 2004 for a notable counterexample in turbomachinery). It is thus desirable to dispose of a probabilistic model which places particular emphasis on “larger” devia- tions from the design conditions. Equivalently, this model should have a large value of the entropy as defined by

∫ ( ) ( )

Ω

−

= p x p x dx

S _X ln _X (1)

where X denotes the vector of random variables con- sidered of joint probability density function ^pX

( )

^x . Further, x denotes the realized values of X, and Ω the domain of support of ^pX

( )

^x .

Consistently with the above discussion, a prob- abilistic model of uncertain nxn matrices A has been formulated (Soize, 2000, 2001) to maximize the value of the entropy S

∫ ( ) ( )

Ω

−

= p a p a da

S _A ln _A (2)

given the following physical constraints:

∫ ( )

Ω

=1 a d a

p_A (3)

[ ] ∫ ( )

Ω

=

= a p a da A A

E _A (4)

and

∫ [ ( ) ] ( )

Ω

ν

= finite det

ln a p_A a da (5)

where denotes the operation of mathematical

expectation,

[]

^.

E

( )

^a

p_A is the joint probability density function of the elements of A, and det(A) is its de- terminant.

The first two of the above constraints correspond to the normalization of the total probability to 1 (Eq.

(3)) and the specification of the mean matrix (Eq.

(4)). The third one, Eq. (5), implies the existence of the mean squared Frobenius norm of the inverse ma- trix A⁻¹(see Soize (2000,2001) for discussion). To apply this approach to the simulation of random mass, stiffness, and damping matrices of dynamical systems, it is further required to ensure both the symmetry and positive definiteness of every realized matrix A. This is achieved by introducing the Cho- lesky decomposition of A, i.e.

A L~L~^T

= (6) where L~

is an lower triangular matrix with non- negative diagonal elements and ^T denotes the op- eration of matrix transposition. The domain of sup- port Ω of the obtained probability density functio then such that the elements

n is L~ij

belong to for

(

−∞,+∞

)

j

i ≠ and

[

0,+∞

)

^for i= j, i.e.

{

^{= ; ,}

[

^{, ∈=}

(

^{1−,...,∞,+∞:}

)

^{, >}

]

^∩

^[

^∈

^[

^0,+∞

^{) ]} }

^.

= Ω

ii ij

T ij

L j i L

n j

i L L L a

The maximization of the entropy, Eq. (2), under the constraints of Eqs (3)-(6) yields a closed form expression for the joint probability density function of the elements of the random matrix A(Soize, 2000, 2001). Further, this distribution only depends on the single parameter ν so that only a broad knowledge of the matrix uncertainty needs to be known or can be enforced. In some situations how- ever, e.g. when considering insertable turbomachin- ery blades, tests may have been performed that pro- vide more information on the system variability. In the context of structural dynamics, such tests will of- ten focus on the natural frequencies of the system and will likely result in estimates of the variance of the first few natural frequencies. In such circum- stances, it is highly desirable to dispose of a prob- abilistic model of the corresponding mass and stiff- ness matrices that accurately accounts for all the available information. If only one natural frequency is observed, its variance as estimated by the tests can serve for the determination of the parameter ν, Eq. (5), corresponding to the mass and/or stiffness matrix model. However, to account for two or more variances, it is necessary to extend the formulation of Eq. (2)-(5) by introducing additional constraints that can reflect the knowledge on the natural fre-

quencies.

To address this extension, consider the general- ized eigenvalue problems

i i

i B

Aϕ =λ ϕ (7) and

i i

i B

Aφ =λ~ φ

(8) where B is a deterministic symmetric, positive defi- nite matrix. In the ensuing discussions, it will be as- sumed that the eigenvectors

ϕi and

φi are normal- ized with respect to B so that

ϕ^T_i Bϕ_i =1 _i

i T

i Aϕ =λ

ϕ (9)

and

φ^T_i Bφ_i =1 _i

i T

i Aφ =λ

φ ~

. (10) Constraining the variance of the eigenvalues directly is unfortunately extremely challenging be- cause of the lack of an exact expression for the natu- ral frequencies of the random matrix

λi

A in terms of its elements. Accordingly, an indirect approach will be selected here which relies on simple constraints that are akin to the second order moments of the ei- genvalues. More specifically, it will be assumed that the value of _⎢⎣^⎡

(

^{φ φ}i

)

²_⎥⎦^⎤

T

i A

E is specified, i.e.

( ) ( ∫ ) ^{( )}

Ω

λ

= φ

φ

⎥⎦=

⎢⎣ ⎤

⎡φ φ ^{2 2 2}~²

i i i A

T i i

T

i A a p a da s

E

(11)

where s_i, ⁱ_∈^{I ⊆}

[ ]

^1,n , are m known positive con- stants and

φi are the eigenvectors of the mean ma- trix A corresponding to the m eigenvalues of which the variance is known. For exampl if the variances of the three lowest eigenvalues of

e,

A have been es- timated, the n

φi will in turn be the eigenvector of the mean matrix A corresponding to its three lowest eigenvalues.

Several comments can made in regards to the constraints of Eq. (11). First, these conditions in- volve second order moments, not variances, but this

se the mean values of switch is appropriate becau

(

i i

)

are already prescribed by Eq. (4). Next, as discussed above, these conditions do not generally relate exactly to the natural frequencies of the ran- dom matrix

T Aφ φ

A, but they do so when its eigenvectors are the same as those of its mean A. Finally, it should be noted that the specification of the con- straints of Eq. (11) provides freedom in the prob-

abilistic model of the random matrix A which can be used to match the known variances of the natural frequencies. This last issue will be discussed in more details further.

3 PROBABILISTIC MODEL DERIVATION Following the discussions of the previous sections, the proposed probabilistic model ^pA

( )

^a maximizes the entropy, S of Eq. (2), under the constraints of Eqs (3)-(5) and (11) as well as the symmetry and positive definiteness requirements of Eq. (6). Using Lagrange multipliers μ₀, μ~ , λ−1, and ~ the con-τ_i strained maximization of Eq. (2) is reduced to the unconstrained maximization of

( ) ( )

∫ ∫

Ω Ω ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧μ

− μ

−

=S p a da a p a da

S^* _{0 A} tr ~^T _A

( ) ∫ [ ( ) ] ( )

Ω

− λ

+ 1 lndet a p_A a da

( ) ^{( )}

∑ ∫

∈ Ω

φ φ τ

− a p_A a da

i T i I i

i

~ 2 (12)

Proceeding next by calculus of variations, it is shown that

( ) [ ( ) ] ( ) ( )

_⎥^⎥

⎦

⎤

⎢⎢

⎣

⎡− μ − τ φ φ

=

∑

∈

− λ

I

i i

T i i T

A a C a a a

p ~det ¹exp tr ~ ~ ²

(13) where C~ is the appropriate constant to satisfy the normalization condition, Eq. (3). It should be noted from Eq. (13) that this probabilistic model is inde- pendent of a rotation/change of coordinate system as the vectors

φi are fixed in space.

Before addressing the evaluation of the Lagrange multipliers, it is desired to simplify Eq. (13) and to address the positive definiteness requirement. In re- gards to simplifications, introduce first the matrix L such that

LT

L

A= and

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡Λ

=

Φ 0

~1/2

LT (14a,b)

where Φ denotes the nxm matrix formed by the m eigenvectors

φi, i∈I, and _Λ^~ is the corresponding diagonal matrix of eigenvalues. Note that the condi- tion of Eq. (14b) is introduced to simplify the con- straints of Eq. (11) as will be shown below. From Eqs (7)-(10), it can be proved that L can be ex- pressed in the partitioned form

⎥⎦⎤

⎢⎣⎡ ΦΛ

= A ⁻ L ~ ^1/2

D (15)

-m) matrix

where the nx(n D is any decomposition, e.g. Cholesky, of

DD^T = A−AΦΛ~⁻¹Φ^T A

. (16) tmultiplying Eq. (16) by

Pre- and pos Φ^T and Φ,

respectively, it is found that D^TΦ=0 Eq. (14b).

Next, expre

as required in ss the random matrix A as

A=LGL^T. (17)

hange of ra a

Proceeding with this c ndom vari bles, it is found that the probability density function of the elements of G is

( ) [ ] ( ) ( )

_⎥^⎥

⎦

⎤

⎢⎢

⎣

⎡− μ − τ

=

∑

=

−

λ m

i

ii T i

G g C g g g

p

1 2 1exp tr

det (18)

where C is a new normalization constant, L

L^T

=

μ μ~ , and ~ ~²

i i

i =τ λ

τ .

try and positive definite- To guarantee the symme

ness of G, and thus of A, the model of Eq. (18) is reformulated in terms of the elements of the lower triangular matrix H such that

G= HH^T . (19) ize (2000) a i

(20)

so that

As demonstrated in So , the J cob an of the transformation is

∏

= +

= ⁿ − l

l lln

n h

J

1

2 1

( )

( )

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝ τ ⎛

− μ

−

×

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

= ⎧

∑ ∑

∏

= =

=

− λ +

−

m i

i l

il T i

T n l

l lln H

h h

h h C h p

1

2 1

2 1

1 2

tr exp

(21)

where C is the appropriate normalization constant over the domain

( )

[ ] ^{[ [ ) ]}

{

^{:hij∈−∞,+∞,i> j∩ hii∈ 0,+∞}

}

^.

To evaluate the Lagrange multipliers

=

=

Ω h_ij,i,j 1,...,n

μ and

is ts s

τi, it necessary to first express the constrain of Eq (4) and (11) in terms of the elements of the random ma- trix H. Combining Eqs (4), (17), and (19), it is found that

E

[ ]

G = In (22) where I_n denotes the nxn identity matrix. The sim- plicity of this condition implies an equally simple

form of the matrix μ. Specifically, it can be shown that this matrix is di gonal and thus Eq. (21) reduces to

a

( ) ∏ ∑ ∑

= = = ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝ τ ⎛

⎟−

⎟

⎠

⎞

⎜⎜

⎝ μ ⎛

−

= ^m

i

i l

il i

i l

il ii

i p i ii

H h C h h h

p

1

2 1

2 1

) 2 ( exp

[ ]

{ } ∏ ∏ { [ ] }

∏

= +

−

= +

=

μ

−

× μ

−

× ⁿ

m i

i l

il ii il

n m i

ii i ii

p

ihii h C h

C

1 1 1

2 1

2 )

( exp exp

(23)

(24) om Eq. (23) that:

all independent d where C_i, i=1,..., n, and C_il, i=m+1,..., n; l=1,...,i-1, are appropriate normalization constants and

p

( )

i =n−i+²λ−¹. It is concluded fr

(i) the elements h_il , i>l and i>m are

of each other and independent of the other ele- ments h_il. Further, they are normally distribute with mean 0 and standard deviation

ii

il = μ

σ 1/ 2 .

(ii) the elements ii i>m, are all independent of ts

h ,

each other and independent of the other elemen hil. Further, they are distributed according to

( )

^{h C hp(i)exp}

[

^iihii2

]

p_{H ii} = _{i ii} −μ , 0hii ≥ (

ii 25)

e

)

wher

[ ( ) ]

( )

(

^{( ) 1 /2}

2 ^1/2

+ Γ

= μ ⁺

i C p

i p

i ii (26) and Γ

( )

^. denotes the Gamma function.

(iii) the elements h_il , l=1, ..., i for a given i∈1, dent of

⎢⎢

⎢

⎣

⎡

⎟⎟

⎜ ⎠

⎜ τ ⎝

⎟−

⎟

⎜ ⎠

⎜ μ ⎝

−

=

∑ ∑

=

= 1

2 1

) 2 ( exp

l il i

l il ii

i p i ii il

H h C h h h

p _il

(27)

[ ]

^m

are dependent on each other but indepen

the other elements h_il. Their joint distribution is

⎞ ⎤

⎛

⎞

⎛ ^{i i 2}

( )

⎥⎥

⎥

⎦ over the domain

( )

[

{

⁼

= _il, 1,..., : _il

i h l i h ^{∈−∞+∞ >}

]

^∩

[

^∈

[

^+∞

) ]}

Ω , ,i l h_ii 0, .

From the first observation, (i), it is concluded tha (iii) and th

sat- isfy the di

[ ]

Gil ⁼⁰ for i>l and i>m. Further, the observation t e symmetry of the distribution of Eq. (27) with respect to the origin imply similarly that

[ ]

Gil ⁼⁰

E for i>l and i≤ m. It then remains to agonal terms of the mean condition, Eq.

(22). For i>m,

[ ]

E

[ ] ^{( )} [ ]

²

1 2

2 1 1

1 _ii

i

l il ii

ii EH i EH

G

E

∑

=

μ +

−

=

=

= (28)

where ^E

[ ]

^Hii2 is obtained by integration of Eq. (27) as

(

^p

( )

^{i +1}

)

^/2μii. Combining this result and Eqs

(26) and (28), it is found that

2 1 2λ−

= +

μ n

ii for i>m. (29) At this point, it only remains to determine the La- grange multipliers μ_ii and τ_i for ^i∈

[ ]

^{1,m . This}

step is achieved by enforcing the conditions

E

[ ]

Gii ⁼¹ (30) and Eq. (11) or

^E

[ ]

^{Gii2 =si2 (31)}

d, it is usef

(32)

for i≤ m. To this en ul to proceed with the following change of variables that highlights the random element G_ii

∑

=

= ⁱ

l il

ii H

G

1 2

cos _i1 ii

ii G

H = Θ (33) H_i(i−1) = G_ii sinΘ_i1cosΘ_i2 (34) H_i(i−2) = G_ii sinΘ_i1sinΘ_i2cosΘ_i3 (35)

on till and so

H_i1= G_ii sinΘ_i1sinΘ_i2...sinΘ_i(i−1) (36) l=2,..,i-2, a

2 ,

0 . The acobian of can be found (e.

where Θi1∈

[

^0,π^/2

]

, ^Θil^∈

[ ]

^0,π , nd

[

π

)

∈

Θi(i₋1) J this transformation g. see Soize, 2000) to be

( )

( 2)

3 2

1 2

/ ⁻ i− i−

ii

g i

2

1 sin ...sin

2 sin

'= θi θi θi i₋

J (37)

Then, the joint probability density function of G_ii and Θ_il, l=1,..., i-1, is

(

^{g ,}

)

₂^{1C g( 2 3)/2exp}

[

²

]

p θ = n ii ii i ii

i ii il

ii

G g g

il

ii_Θ ^{+ λ−} −μ −τ

[ ]

( 2)

3 2 )

( 1 2

1 cos sin ...sin

sinθ ⁻ θ × θ ⁻ θ ₋

× _i ⁱ _i ^{p i} _i ⁱ _{i i} .

(3

n s

8) The above expression demonstrates that the random variables G_ii and Θ_il, l=1,..., i-1 are all independent of each other and that the joint probability density function of the angles Θ_il, l=1,..., i-1 does not de- pend on the values of τ This important observatio will be used in the nex ection. Further, the mar- ginal distribution of G_ii is

( )

i. t

[

²

]

2 / ) 2 (n i ii ii

G g C g

p ii = ^{+ λ−3} exp−μ_iig_ii −τ_ig_ii (39) here . For each value of i, the two para

w 0gii ≥ me-

ters μ_ii

4 SIMULATION OF RANDOM MATRICES The simulation of random matrices A according to the model derived above is achieved by first generat- ing random matrices H according to the joint distri- butions of Eqs (23), (25), (27), (38) and (39). On sample of

ce a H has b n generated, the correspo matrix

ee nding

G is determine from Eq. (19) and, finally realization of

d , a

A is obtained from q. (17). Thus, the simulation effort reduces to the generation of appro- priate samples of

E

H. From the observ ions drawn in the previous section, it is concluded that there besides the simulation of the Gaussian variates H_il i>m and i>l, three part ar issues. These are:

generation of the diagonal elements H_ii, i>m ac- cording to the distri ion of Eq. (25), (ii) the lation of the ratios H_il /G_ii, i≤ m, described by th angular variables in Eqs (33)-(38), and (iii) the g eration of samples of G_ii, i≤ m, according to Eq.

(39). These three issues are addressed in order be- low.

at

are,

icul (i) the

but simu-

e en- ,

4.1 Simulation of H , i>m _ii

The generation of samples of is simplified by considering the variable . Proceeding with the change of variables, it is found that the probability density function of is

Hii 2 ii ii

ii H

Y =μ

Yii

( ) [ ( ) ] [

ii

]

i p ii ii

Y y

i p y y

p ii −

+

= Γ ⁻ exp

2 / 1 ) (

2 / ) 1 ) ( (

, yii ≥0. (40) That is, is a Gamma distributed random variable for which efficient simulation algorithms exist, e.g.

see Devroye (1986). Once a sample of has been simulated according to the Gamma distribution, the corresponding value of , i > m, is found as

Yii

Yii

Hii

ii ii ii

H Y

= μ (41)

where μ_ii is given by Eq. (29).

4.2 Simulation of H_il / G_ii , i≤ m, l=1,..., i A first approach for the simulation of the random variables H_il / G_ii , l=1,..., i-1, and i≤ m, is to pro- ceed from the generalized spherical coordinates transformation of Eqs (33)-(36) and to generate in- dependent angles , l=1,..., i-1, according to the distributions

Θil

and _i are then finally solved from the con- straints given by Eqs (30) and (31).

τ

( )

2

[

1

]

^{( )}

1 1

1 ~ sin cos

1

i i p i

i i

i C

p_Θi θ = θ ⁻ θ ,

⎥⎦⎤

⎢⎣⎡ π

∈ θ_i1 0,2 (42) and

p_Θ

( )

θil =C^~il ^sinθil ^i−l−1

il (43)

with for l=2,..., i-2, and and where the coefficients

[

^π

∈

θ_il 0,

]

^θi(i₋1)^∈

[

^0,2π

)

C~il

are appropriate nor- malization constants.

Note however that the above distributions are non standard and thus a different, easier approach was selected here. Specifically, it was observed in the previous section that the probability density func- tions of Eqs (42) and (43) do not depend on and thus they would be the same for =0. In this case however, the simulation of the random variables

is completely similar to the case i>m, i.e. the random variables , , are Gaussian variates with mean zero and standard deviation

τi

τi

Hil

Hil l≠i

μii

2 / 1

while H_ii = Y_ii /μ_ii where is a Gamma ran- dom variable. Note further that the appropriate value of

Yii

μii to be used is the one given by Eq. (29) to in- sure the consistency with p(i) of Eq. (24).

In view of these comments, the simulation of the terms H_il / G_ii , l=1,..., i-1, and i≤ m, can effi- ciently be accomplished as follows:

(a) generate H~_ii as

ii ii ii

H Y

= μ

~ ~

(44)

where μ_ii is given by Eq. (29) and Y~_ii is a Gamma distributed random variable of probabil- ity density function given by Eq. (40).

(b) generate H~_il

, l=1,..., i-1, as zero mean Gaussian random variables with standard deviation

μii

2 /

1 .

(c) form

∑

and the desired ratios

=

= ⁱ

l il

ii H

G

1

~2

~

ii il ii

il

G H G

H

~

~

= l=1,..., i . (45)

4.3 Simulation of G , i_ii ≤ ^m

The last step in the simulation of the random matrix H is the generation of the diagonal elements

according to their probability density function of Eq.

(39). This distribution is non standard and thus a dedicated simulation algorithm by rejection from the

Student’s t distribution with 3 degrees-of-freedom was devised (see Devroye (1986) for background).

Such an algorithm requires that there exists a con- stant such that

Gii

cθ

pG

( )

gii c pZ

(

gii for all (46)

ii ≤ θ θ

)

) )

≥0 gii

where is defined by Eq. (39) and is the probability density function of

in which Z is a random variable hav- ing the Student’s t distribution with 3 degrees-of- freedom. That is, the probability density function of Z is

(

ii

G g

p ii

(

ii

Z g

p _θ

Z y Z_θ = 0 +θ

( )

2 / 2 3

1 2 2 2

1 ⁻

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +

= z

z

p_Z z∈

(

−∞,+∞

)

(47) and thus,

( ) ( )

^3/2

2 0 2

2 2 1

2

1 ⁻

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

θ + −

= θ

θ

y g g

p_{Z ii} ⁱⁱ (48)

for . If values of , θ, and can be found for which Eq. (46) holds, then random de- viates can be generated as

(

^{−∞ +∞}

∈ ,

gii

)

cθ y₀

Gii

=Zθ

G_ii provided that ^c_θ ^pZ_θ

( )

^Z_θ ^U≤pG_ii

( )

^Z_θ

(49) in which U denotes a random variable uniform in the interval [0,1] and independent of . When the inequality in Eq. (49) is not satisfied, no sample is generated. Note further that 1/ is the probability that this inequality be satisfied, so that represents the average number of pairs of samples ( ,

Zθ

Gii

cθ

cθ

Zθ U ) that must be generated per value of to be simulated.

Accordingly, it is desired to have a value of as close as possible to 1.

Gii

cθ

The Student’s t distribution with 3 degrees-of- freedom was selected for the random variable Z first because it can easily be simulated as

( )

^⎟⎠

⎜ ⎞

⎝

⎛ −

= −

2 1 1

2 U

U

Z U (50)

where U is a random number uniformly distributed in [0,1]. Another advantage of the Student’s t distri- bution is that it led to values of that remained reasonably small, i.e. to efficient simulation algo- rithms, over a broad range of values of the parame- ters p(i) and .

cθ

/ 2_ii

i μ

τ

The approach selected here for the determination of the parameters , θ, and is to force the two distributions and to have their

cθ y₀

(

ii

G g

p ii

)

pZ_θ

(

gii

)

respective modes at the same value and to have

* 0

y g_ii =

( )

^{ii* Z}

( )

^ii*

G g c p g

p ii = θ θ . These 2 conditions will provide the values of and in terms of θ. This last parameter will then be selected to minimize while maintaining the inequality of Eq. (46) over the entire domain .

cθ y₀

cθ

≥0 gii

Matching the modes of the two distributions and leads directly to the condi- tion

(

ii

G g

p ii

)

pZ_θ

(

gii

)

2τ_i y₀² +μ_ii y₀ −q=0 (51) or

8 0

4

1 ₂

0 ⎜⎝⎛ μ + τ −μ ⎟⎠⎞≥

= τ ii i ii

i

q

y (52)

where the notation

2 3 2λ−

= n+

q (53) has been adopted for simplicity.

Next, matching the peak values of and requires that

(

ii

G g

p ii

)

θ

)

c pZ

(

gii

θ

^{cθ =2 2 θCi y0qexp}

[

^{−μii y0 −τi y02}

]

⁽⁵⁴⁾

which indicates that is proportional to θ while is independent of this parameter, see Eq. (52).

An acceptable simulation algorithm is obtained for all values of θ for which the inequality of Eq. (46) is satisfied. However, the most efficient of these algo- rithms is the one that minimizes the corresponding

, see Eq. (54). In view of the linearity of with respect to θ, it is concluded that the best algorithm is the one for which θ has the smallest possible value that guarantees the satisfaction of the inequality of Eq. (46) for all values of . To determine this value of θ, introduce first the function

cθ

y0

cθ c_θ

≥0 gii

( ) ( )

(

^{y w}

)

p

w y p c w

r

Gii

Z

+

= ^θ +

0

0 0

ln (55) and it is desired that r(w) be nonpositive for all

. From the above conditions, it is found that r(0)=0 and . Thus, a sufficient condition for r(w) to be nonpositive is that

y0

w≥−

( )

^{0 =0}

′ r

r′

( )

w ≥^0for w<⁰ and r′

( )

w ≤^0for w>⁰. (56) After differentiation of Eq. (55) and some algebraic manipulations, it is found that Eq. (56) is equivalent to

( )

(

0

)

²

2 0

2 4

2 3 w

w y

w y

i i

ii

τ − + τ + μ

≥ +

θ (57)

for all . Accordingly, the smallest value of θ corresponds to the maximum of the right-hand-

side of Eq. (57). In this regards, note from Eqs (51) and (52) that

y0

w≥−

w

y _i

i

ii + τ + τ

μ 4 ₀ 2 is always positive for all . Then, the right-hand-side of Eq.

(57) is monotonically increasing in and thus the maximum must occur for w > 0. Further,

y0

w≥−

[

y0^,0

]

w∈ −

( )

( )

( )

( ) ( )

(

0

)

²

0 2 0

0 0 0

2 0

0 0

4 4

4 12

9 4

max 3

) 58 2 (

4 max 3

y y w y

y w y

w w y

w y

i ii

i ii i

w ii

i i

w ii

τ + μ

τ + μ

= +

⎭⎬

⎫

⎩⎨

⎧ −

τ + μ

≤ +

⎭⎬

⎫

⎩⎨

⎧ −

τ + τ + μ

+

>

>

which suggests the value of θ as

( )

(

0

)

²

0 0

4 8

4 12

9

y y y

i ii

i ii

τ + μ

τ + μ

= +

θ . (59)

The above expression is simple but overestimates the maximum of the function on the right-hand-side of Eq. (57). This maximum can be obtained by dif- ferentiating and solving a cubic equation in w. Spe- cifically, it is found that

( )

(

^{4 2}

)

²

2

3 _*2

* 0

0 *

w w

y w y

i i

ii

− τ

+ τ + μ

+

=

θ (60)

where denotes the location of the maximum as given by

w*

( )

i

a w D

τ

= − 2

* (61) where

3 3 2 6

2 a

D a +

+ γ

= γ (62)

γ³ =108b+8a³+12 81b² +12ba³ (63) and

a=μ_ii +4τ_iy₀ b=3τ_i

(

μ_ii +2τ_iy0

)

. (64) A comparison of the simulation algorithms based on Eqs (59) and (60)-(64) will be presented in section 6.1.

The simulation of random values according to the probability density function of Eq. (39) then proceeds as follows. For given values of

Gii

μii and , the mode is first determined according to Eq.

(52), and θ and

τi

y0

Ci

c_θ/ are obtained in order from Eqs (59), or (60)-(64), and (54). After this prepara- tion phase, pairs of independent random numbers

(

^U,U

)

uniform in [0,1] are simulated. From U, a variate Z of the Student’s t distribution is obtained from Eq. (50) and the corresponding variable is determined as . It is next necessary to assess if the inequality of Eq. (49) is satisfied. To

Zθ

Z y Z_θ = 0+θ

this end, the term

[

^cθ/^Ci

]

^pZ_θ

^{( )}

^Zθ ^U is evaluated from the above values of c_θ/C_i and U and with

denoting the value of

)

, Eq. (48), for . The ratio

(

_θ

θ Z

p_Z

)

pZ_θ

(

gii

=Zθ

g_ii pG

( )

Z Cⁱ

ii _θ / is similarly obtained from Eq. (39). If

[

^c_θ/^Ci

]

^pZ_θ

^{( )}

^Z_θ ^U≤^pGii

^{( )}

^Z_θ /^Ci, (65) a value of is obtained as . Otherwise, the value of is rejected and no corresponding sample of is generated. Either way, the process is repeated starting with the simulation of a new pair of uniform random variables

Gii G_ii =Z_θ Zθ

Gii

(

^U,U

)

until the appro- priate number of samples of has been obtained.

Note that Eq. (65) is equivalent to the inequality of Eq. (49) but is preferable to it because it does not re- quire the numerical evaluation of the normalization constant

Gii

Ci.

5 IDENTIFICATION OF NONPARAMETRIC MODEL PARAMETERS

5.1 Dispersion Parameter δ

It is desired next to characterize the overall variabil- ity of the random matrices A around their mean A. To this end, introduce the dispersion parameter δ as

⎥⎦⎤

⎢⎣⎡ −

=

δ² 1 ²

n F

I G

nE (66) where

V ²F =tr

[

VV^T

]

(67) is the square of the Frobenius norm of an arbitrary matrix V . Expanding, the right-hand-side of Eq.

(67) and taking Eq. (22) into account, it is found that ^{δ2 =} _n¹

{

^E

[

^tr

( )

^GGT

]

⁻ⁿ

}

(68) with

[ ( ) ] ^{∑ [}

=

= ⁿ

l k j i

jl il jk ik

T E H H H H

G G E

1 , , ,

tr

]

⁽⁶⁹⁾

where, by its lower triangular nature, for l >

i. The evaluation of the right-hand-side of Eq. (69) is accomplished by first separating the cases in which i=j and those in which i≠j. Further, it is recalled from Eq. (23) that and are independent if i≠j. Noting finally from Eq. (23) that

for k≠l, it is found that

=0 Hil

Hik H_jl

[

HikHil

]

=⁰ E

[ ( ) ] [ ] ( ) [ ]

2

1

2 1

,

2 2 1

tr 2

∑ ∑

∑

∑

= =

=

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣ + ⎡

−

=

n k

n k i

ik n

k i

ik n

i T ii

H E

H E G

E G

G E

. (70)

To evaluate the above expression, it should further be noted that ^E

[ ] [

^{Hik2 = EHi2(i−1)}

]

^{, for k≠}i, in view of the symmetry of the variables , l > i, in Eq.

(23) and further that these expectations are the same if i>m of i≤m in view of Eq. (30). Moreover, from Eqs (30), (33), and (34), it is found that

Hil

^E

[ ]

^{Hii2 =} _nⁿ₊⁺₂²_λ^λ₋⁻₁ⁱ (71) and

^E

[ ]

^{Hii2 =} _n+2¹_λ−1 ^{. (72)}

Introducing Eq. (71) and (72) into (70) and combin- ing the above results yields

^∑ [ ]

= + λ−

− λ

=

δ ⁿ

i

ii n

G n E

1 2 2

1 2 2

1 . (73)

Noting finally that ^E

[ ]

^{Gii2 =si2 for i≤} m (Eq. (31)) and ^E

[ ]

^{Gii2 =(n+2λ+1)/(n+2λ−1)} for i> m (from Eq. (25)), it is found that

( )( )

∑

= + λ− + λ +

− + +

=

δ ^m

i

i n

n n m s n

n 1 2 2

1 2

1 2 /

1

1 . (74)

5.2 Identification of the parameters λ, μ_ii, and τ_i The above derivations have been carried out in terms of the parameters λ, μ_ii, and but these coeffi- cients are not part of the original problem statement and thus they should, in principle, be evaluated in terms of the stated constraints, Eqs (5), (30), and (31). Note in this regard that the constraint of Eq.

(5) has two aspects: the finiteness of ν and its spe- cific value. The finiteness of ν guarantees the exis- tence of the mean squared Frobenius norm of the in- verse matrix

τi

−1

A while the specific value of this coefficient providing an overall measure of the ran- domness of the matrices A. In this light, ν and δ play a similar role and it is thus appropriate to re- place Eq. (5) by a fixed value of δ, Eq. (74), see So- ize (2001) for discussion. Proceeding in this manner would provide a direct expression for λ as

( )

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡δ − +

−

= + +

=

− λ +

∑

= m i

i n

s m n

n m q n

n

1 2

2 1

/ 2 2 1

2 1

2 . (75)