A common model of vibratory diagnostics objects is the stochastic difference schemes, and theirs parametrical identification is carried out least squares and least absolute deviations techniques. It is well known that these techniques are unstable under stochastic heterogeneity of observable process, specifically, in the presence of outliers. One way to make the stable parametrical identification of vibratory diagnostics objects is implementation of generalized least absolute deviations method based on concave loss function. Obtained requirements to the loss function guaranteeing the steadiness evaluation, algorithms of identification and examples are presente

Alexander N. Tyrsin

Anatoly V. Panyukov

English

Panyukov, Anatoly V.

Tyrsin, Alexander N.

Journal of Mechatronics and Artificial Intelligence in Engineering

 350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV 1, A, A. N. TYRSIN2, B    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  142 350. Stable parametric identification of vibratory diagnostics objects  A. V. Panyukov1, a, A. N. Tyrsin2, b 1 South-Ural State University, 76, Lenina ave., Chelyabinsk, 454080, Russia  2 Chelyabinsk State University, 129, Bratyev Kashirinykh str., Chelyabinsk, 454021, Russia  E-mail: apav@susu.ac.ru, bat2001@yandex.ru   (Received 15 April 2008; accepted 13 June 2008)      Abstract. A common model of vibratory diagnostics objects is the stochastic difference schemes, and theirs paramet ical identification is carried out least squares and least absolute deviations techniques. It is well known that these techniques are unstable under stochastic heterogeneity of observabl  process, specifically, in the presence of outliers. One way to make the stable parametrical identification of vibrato y diagnostics objects is implementation of generalized least absolute deviations method based on concave loss function. Obtained requirements to the loss function guaranteei g the steadiness evaluation, algorithms of identification and examples are presented.   Keywords: autoregression; generalized least absolute deviations method; linear stochastic difference scheme; random vibration; stable evaluation of autoregression model factors; weighted least absolute deviations method.   Introduction  At present wide experience to create methods of vibratory diagnostic and rupture life forecast is accumulated. One of exploitable mathematical model for these problems is linear stochastic difference scheme   1, 1, 2, ...Lk j k j kjy a y k L Lξ−=− = = + +∑ ,               (1)   (1) in which yl = y(l∆), l = 0, 1, 2, 3, …, are data vibration at point of time l∆; ∆ is sampling interval; {ξk} are unobservable stochastic processes; aj, j = 1, 2,…, L are the model parameters appointed at the design stage; L  is the size of log. It is significant that unobservable random values ξk are in accepting independent values under diagnostic problems.  For example it is the established fact [1] that random vibration of single mass linear mechanical system satisfies Eq.(1) under 2L = , i.e. it is second-order autoregression process   1 1 2 2 , 3,4,5,k k k ky a y a y kξ− −= + + = K .          (2)   (2) Autoregression factors a1, a2 of this process are unambiguously interdependent with resonance frequency and damping decrement of the system i.e.  10 2021 1arccos ; ln( )2 22af afaδπ= = − −∆ ∆−.      (3)  Object design characteristics and interdependent model Eq. (1) factors aj, j = 1, 2,…, L are denatured under development of structure degradation processes. System malfunction may be diagnosed by variations with time of coefficients aj being estimated by data vibration signals yk. To decrease false alarm and aim omission risks it iissue of the day that the problem is taking into account overidentification and heterogeneousness of data vibration signals under estimating of coefficients aj. Sources of overidentification and heterogeneousness of data vibration are a) part of sampling may be mismatch to the accepted model because of embryonic defect; b) instability of error variance of measuring; c) availability of outliers in the middle of yk; d) multiplicative nature of noises kξ . It is well known that the least squares and least ab olute deviations techniques (LST and LADT) are unstable under stochastic heterogeneity of observable processes, specifically, in the presence of outliers [2] . One way of doing the stable parametrical identification of vibratory diagnostics objects is implementation of generalized least absolute deviations technique (GLADT) based on concave loss function [3] .  Generalized least absolute deviations method  Follow [3] we define GLADT estimation of parameters aj, j = 1, 2,…, L for model (1) and data {yk : k = 1, 2, 3, …, n} as  ( ) ( )1 2 11, , , arg minLnLL k l k llk La a a y a yρ −=∈ = += = −∑ ∑a Ra K , (4)  where function ρ(x) is monotone increasing and twice differentiable for all nonnegative x, ρ(0) = 0, and ρ″(x) ≤ 0.  It is proved [3] that all local minimums of the goal function of problem (4) form set:   350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV 1, A, A. N. TYRSIN2, B    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  143( ){{ }nkkkkkkkyayaaaULLLllklkL≤<<<≤=∈== ∑=−...1:,...,,;:,...,,10101)()()k(2)(1kkkk. (5)  Hence we may look for a  by means of solving CnL systems of L linear equations and choosing optimal vector from U. Under L = 1, 2, 3 such enumeration of possibilities is realizable.  Let us consider interrelation between GLADT and weight least absolute deviations technique (WLADT). Statement 1.  { }( ) 110 : 1, 2, , arg minLnLk k k l k llk Lp k L L n p y a y U−=∈ = + ∀ ≥ = + + − ∈  ∑ ∑a RK.(6)  Proof. As is easy to see problem (6) is optimization of the piecewise linear convex function. The introduction of slack variables leads us to linear programming problem   111, ,1, ,min : , 0, 1,2, ,LL nnLk k k k l k l k kla ak Lu up u u y a y u u k L n+−== + ⋅ − ≤ − ≤ ≥ = +  ∑ ∑KKK.(7)  Task (7) has the canonical form, n – 1 variables and 3(n – L – 1) constraints including the nonnegativity constraints for variables uk. Let m be equal to the number of zero values of variables uk at optimal solution of the task, i.e. exactly m conditions   1Lk l k lly a y −==∑   are held. Then active constraints are m nonnegativity constraints for zero variables uk, m common constraints with this variables uk, and n – m – L – 1 constraints for positive variables uk. General number of the active constraints equals to n + m – L – 1. On the other hand it is necessary number of active constraints for the optimal basic solution no less than n – 1. Therefore we have m ≥ L.  Statement 1 is proved. Statement 2.   ( ) { }( )1 21 2( ) ( ) ( )( ) ( ) ( )11( , , , ) 0 : 1, 2, , :( , , , ) arg min .LL LknLk k l k llk La a a U p k L L na a a p y a yk k kk k ka R−=∈ = +∀ ∈ ∃ ≥ = + += −∑ ∑K KK                         (8) Proof. Let ( )χ ∗k  be characteristic function for set k (5). The set ( ){ }: 1,2, ,ip i i L nχ= = +k K  is hold condition of the statement for given k .  Statement 2 is proved. Ascertained interrelation between GLADT and WLADT permits to ground GLADT estimation for stable evaluation of autoregression model factors under availability of outliers in the middle of yk. Groundlessness of LADT estimation under such conditions is proved in [2]  In the same place existence of stable WLADT estimation is ascertained, but general way to look of proper weights { }: 1,2, ,kp k L n= + K  is not presented.  Further we describe the implementation of GLADT for stable evaluation of autoregression model factors. Like in [2] we apply the influence functional presented in [4] .  Stability of evaluation of autoregression model factors  Let ),,( 1 npn xx K−=x  be stationary in the wide sense time series. We observe autoregression model   1AR( ): ,Lk i k i kiL x a x kε−== + ∈∑ Z ,  where L is known order, a = (a1, … , ap) is vector of nonrandom factors, {εk} are independent identically distributed variates with non-degenerate distribution function. Let values  ,k k k ky x z kγξ= + ∈Z    be observable variables, {zγk} be independent identically distributed variates, {zγk} ~Bi(1, γ), 0 ≤ γ≤ 1, γ be obstruction level, {ξk} be independent identically distributed variates with distribution µξ from class Mξ; successions {xk}, {zγk}, { ξk} be independence. In that way we observe simple obstruction scheme of data by independence outliers. Under appearance obstruction (5) traditional estimations are inconsistent. For measuring of quality of estimation na) for vector a under observable data yn we suppose existence of convergence in probability n γ→Pa a) and equality aa =0 . Simple infinitesimal characteristic of estimation na) stability under data obstruction {yk} is vector  00( , ) limIF γγ ξ γµγ→−=a aa    named as influence functional for estimate na) [4] This functional characterizes the value of main linear member of asymptotic expansion of displacement   0 ( , ) ( )IF oγ γ ξµ γ γ− = +a a a .                            (9)   Condition of estimation stable is finite sensitivity to great mistake   (M , ) sup ( , )GES IFξ ξξ γ γ ξµµ∈Μ= < ∞a a .  350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV 1, A, A. N. TYRSIN2, B    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  144 In that case the main linear member of asymptotic expansion (7) is uniformly small for all obstructions and small γ.  Let GLADT estimation of parameters GLMna) be defined by way of algorithm (4). Qualitative assessment of stability for GLADT estimation GLMna) under outliers gives the following theorem. Theorem. Let time series (5) be observed. If loss function ρ(x) of algorithm (4) is so that ∞<ρ′≥)(sup 20xxx then (M , )GLMGES ξ γ < ∞a . Proof. The necessary conditions of minimum for task (4) are bridging set   ( )11( ) sign 0, 1, ,lnLa k k l k i k iiky y a y l Lρ ε − −==′ − ÷ =∑ ∑ K  (10)  It is proved in [2] that WLADT estimation obtained by the way solving of bridging set   LlyayyypLi ikikkknk,...,1,0)sign(,...,(1111=÷−∑∑ = −−−=         (11) under condition sup ( )pTyp∈< ∞Ryy y  has finite sensitivity to great outliers, i.e.   ∞<γξ ),(MGLMGES a .  Comparison of Eq. (10) and Eq. (11) as well as taking into account proved statements 1 and 2 implies assertion of the theorem under  20sup ( )xx xρ≥′ < ∞ .   Theorem is proved.  As is easy to see the concave increasing functions ( ) ( )1 exp ; arctan ; / 1x x x x− − +  are holding the theorem conditions, and functions ( )2; ; ; ln 1x x x x+  are not.  Computational experiments  Process AR(1) with one-sided outliers   010,, 0,1, , .,k k kk k k kxx x k My x zγα εξ−== + = = +K    Here 0.7α = ; εk  ~ N(0,σzk2); σzk2 are variates distributed uniformly in segment [0;2]; {zγk} are independent identically distributed variates; {zγk} ~Bi(1, γ), 10 ≤γ≤ , γ are an obstruction level; {ξk} are independent variates distributed uniformly in segment [50; 100]; M =1500 is the number of trials.  Test computer simulation is identification of autoregression factor α  at sight of signals yk evaluated according to Eq. (8). Identification algorithms are GLADT estimation (4) with the difference loss functions.  Simulation data is shown in Fig. 1.    a) b) Fig. 1. Process AR(1) simulation data:   a) LST, LADT and GLADT estimations of autoregression factor α; b) Sample variance sa2 of the estimations  350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV 1, A, A. N. TYRSIN2, B    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  145Fig. 1. a demonstrates that LST and LADT estimations are inconsistent in spite of the fact that sample variance of these estimations is small (see Fig. 1.b). GLADT estimations with the loss functions satisfying theorem condition (lines 5, 6, 7, and 8) are consistent andhave small sample variance of ones. Vibrations of single mass linear mechanical system with resonance frequency f0 = 91.7Hz and damping decrement δ = 1.22 under sampling interval ∆ = 0.001sec satisfies second-order autoregression process (2) withfactors a1 = 1.5, and a2 = 0.8. Simulating vibration of this system we evaluate by AR(2) process  0 11 20,1,5 0,8 , 0,1, , ,,k k k kk k k kx xx x x k My x zγεξ− −= == − + = = +K        (12)  Here εk  ~ N(0,σzk2); σzk2 are variates distributed uniformly in segment [0;2]; {zγk} are independent identically distributed variates; {zγk} ~Bi(1, γ), 10 ≤γ≤ , γ are an obstruction level; {ξk} are independent variates distributed uniformly in segment [50; 100]; M =100 is number oftrials. Test computer simulation is identification of autoregression factors a1, and a2. at sight of signals yk evaluated according formulas (12). Identification algorithms are GLADT estimation (4) with the difference loss functions.  Simulation data is shown in Fig. 2. Again we observe that GLADT estimations with loss function satisfying the theorem condition are consistent only.     a)  b) Fig. 2. Single mass linear mechanical system simulation data:  ) LST, LADT and GLADT estimations of autoregrssion factor a1; b) LST, LADT and GLADT estimations of autoregression factor a2    Table 1. Results of tests   f0   ρ(x)=x2 ρ(x)=|x| ρ(x)=|x|0.5 ρ(x)=atan|x| ρ(x)=x2 ρ(x)=|x| ρ(x)=|x|0.5 ρ(x)=atan|x| 0 0,002 0,005 0,01 0,015 0,02 0,03 0,04 0,05 91,7976 146,8953        92,8006 90,5264 88,7691 70,9415      93,0717 89,3832 90,2000 92,1783 93,1951 98,0430 105,0666 109,1621 30,3857 93,1643 90,1766 91,1608 92,1390 90,9585 90,0422 91,6342 94,3107 92,0200 1,1529 2,9343        1,1873 1,8355 3,7423 7,9464      1,2738 1,3090 1,3433 1,4036 1,3781 1,4894 1,7251 2,3368 21,9591 1,2401 1,3600 1,2535 1,3204 1,2432 1,2760 1,1270 1,1538 1,3124   350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV 1, A, A. N. TYRSIN2, B    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  146 As indicated above (see Eq. (3)) autoregression factors a1, a2 of this process are unambiguously interdependent with resonance frequency and damping decrement of system. Estimation of resonance frequency f0 and damping decrement δ evaluated through autoregression factors (see Eq. (3)) are presented in Table 1. Empty strings in this table signify impossibility of the target calculation under current obstruction level γ. As may be seen from Table 1 the best technique in the absence of obstructions (γ = 0) is LST. But application of LST staves off calculations under γ = 0.005. On the other hand the application of GLADT estimation with the loss functions satisfying to the theorem condition enables to make stable estimations.   Summary  As the statements indicates that adduced theoretical proofs and computer simulation demonstrate possibility of stable parametric identification of vibratory diagnostic objects by usage of GLADT estimation with loss the function satisfying to the condition of the proved theorem.   Thanks  Authors feel appreciation to Russian Foundation for Basic Research for the financial supporting of the project # 07-01-96035-p_ypaл_a.  References  [1]  Karmalita V. A. Digital processing of random vibration. – Moscow: Mashinostroenie, 1986 (in Russian). [2]  Boldin M. V., Simonova G. I., Tyurin Yu. N. Sign statistical analysis of linear models. – Moscow: Nauka, 1997 (in Russian). [3]  Tyrsin A. N. Robust construction of regression models based on the generalized least absolute deviations method // Journal of Mathematical Sciences – 2006. Vol. 139, N 3. P. 6634–6642.  [4]  Martin R. D., Yohai V. J. Influence functionals for time series // Annals of Statistics, 1986. Vol. 14, N. 4. P.781–818.    

Stable parametric identification of vibratory diagnostics objects

Journal of Vibroengineering

 350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV1, A, A. N. TYRSIN2, B 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
142 
350. Stable parametric identification of vibratory diagnostics objects 
 
A. V. Panyukov1, a, A. N. Tyrsin2, b 
1
 South-Ural State University, 76, Lenina ave., Chelyabinsk, 454080, Russia  
2
 Chelyabinsk State University, 129, Bratyev Kashirinykh str., Chelyabinsk, 454021, Russia  
E-mail: apav@susu.ac.ru, bat2001@yandex.ru  
 
(Received 15 April 2008; accepted 13 June 2008) 
 
 
 
 
 
Abstract. A common model of vibratory diagnostics objects is the stochastic difference schemes, and theirs parametrical 
identification is carried out least squares and least absolute deviations techniques. It is well known that these techniques 
are unstable under stochastic heterogeneity of observable process, specifically, in the presence of outliers. One way to 
make the stable parametrical identification of vibratory diagnostics objects is implementation of generalized least absolute 
deviations method based on concave loss function. Obtained requirements to the loss function guaranteeing the steadiness 
evaluation, algorithms of identification and examples are presented.  
 
Keywords: autoregression; generalized least absolute deviations method; linear stochastic difference scheme; random 
vibration; stable evaluation of autoregression model factors; weighted least absolute deviations method.  
 
Introduction 
 
At present wide experience to create methods of vibratory 
diagnostic and rupture life forecast is accumulated. One of 
exploitable mathematical model for these problems is 
linear stochastic difference scheme  
 
1
, 1, 2, ...
L
k j k j k
j
y a y k L Lξ−
=
− = = + +∑ ,               (1) 
  (1) 
in which yl = y(l∆), l = 0, 1, 2, 3, …, are data vibration at 
point of time l∆; ∆ is sampling interval; {ξk} are 
unobservable stochastic processes; aj, j = 1, 2,…, L are the 
model parameters appointed at the design stage; L  is the 
size of log. It is significant that unobservable random 
values ξk are in accepting independent values under 
diagnostic problems.  
For example it is the established fact [1] that random 
vibration of single mass linear mechanical system satisfies 
Eq.(1) under 2L = , i.e. it is second-order autoregression 
process  
 
1 1 2 2 , 3, 4,5,k k k ky a y a y kξ− −= + + = K .          (2) 
  (2) 
Autoregression factors a1, a2 of this process are 
unambiguously interdependent with resonance frequency 
and damping decrement of the system i.e. 
 
1
0 2
02
1 1
arccos ; ln( )
2 22
af afa δπ= = − −∆ ∆−
.      (3) 
 
Object design characteristics and interdependent model 
Eq. (1) factors aj, j = 1, 2,…, L are denatured under 
development of structure degradation processes. System 
malfunction may be diagnosed by variations with time of 
coefficients aj being estimated by data vibration signals yk. 
To decrease false alarm and aim omission risks it is 
issue of the day that the problem is taking into account 
overidentification and heterogeneousness of data vibration 
signals under estimating of coefficients aj. Sources of 
overidentification and heterogeneousness of data vibration 
are a) part of sampling may be mismatch to the accepted 
model because of embryonic defect; b) instability of error 
variance of measuring; c) availability of outliers in the 
middle of yk; d) multiplicative nature of noises kξ . 
It is well known that the least squares and least absolute 
deviations techniques (LST and LADT) are unstable under 
stochastic heterogeneity of observable processes, 
specifically, in the presence of outliers [2] . One way of 
doing the stable parametrical identification of vibratory 
diagnostics objects is implementation of generalized least 
absolute deviations technique (GLADT) based on concave 
loss function [3] . 
 
Generalized least absolute deviations method 
 
Follow [3] we define GLADT estimation of parameters aj, 
j = 1, 2,…, L for model (1) and data {yk : k = 1, 2, 3, …, n} 
as  
( ) ( )1 2 1
1
, , , arg min
L
n
L
L k l k ll
k L
a a a y a yρ −=∈ = +
= = −∑ ∑
a R
a K , (4) 
 
where function ρ(x) is monotone increasing and twice 
differentiable for all nonnegative x, ρ(0) = 0, and ρ″(x) ≤ 0.  
It is proved [3] that all local minimums of the goal 
function of problem (4) form set: 
 
 350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV1, A, A. N. TYRSIN2, B 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
143
( ){
{ }nkkkkkkk
yayaaaU
LL
L
l
lklkL
≤<<<≤=∈
== ∑
=
−
...1:,...,,
;:,...,,
1010
1
)()()k(
2
)(
1
k
kkk
. (5) 
 
Hence we may look for a  by means of solving CnL 
systems of L linear equations and choosing optimal vector 
from U. Under L = 1, 2, 3 such enumeration of possibilities 
is realizable.  
Let us consider interrelation between GLADT and 
weight least absolute deviations technique (WLADT). 
Statement 1. 
 
{ }( ) 1
1
0 : 1, 2, , arg min
L
n
L
k k k l k ll
k L
p k L L n p y a y U−=∈ = +
 
∀ ≥ = + + − ∈ 
 
∑ ∑
a R
K
.(6) 
 
Proof. As is easy to see problem (6) is optimization of the 
piecewise linear convex function. The introduction of slack 
variables leads us to linear programming problem  
 
1
1
1
, , 1
, ,
min : , 0, 1,2, ,
L
L n
n L
k k k k l k l k kla a k Lu u
p u u y a y u u k L n
+
−=
= +
 
⋅ − ≤ − ≤ ≥ = + 
 
∑ ∑
K
K
K
.(7) 
 
Task (7) has the canonical form, n – 1 variables and 3(n –
 L – 1) constraints including the nonnegativity constraints 
for variables uk. Let m be equal to the number of zero 
values of variables uk at optimal solution of the task, i.e. 
exactly m conditions  
 
1
L
k l k lly a y −==∑  
 
are held. Then active constraints are m nonnegativity 
constraints for zero variables uk, m common constraints 
with this variables uk, and n – m – L – 1 constraints for 
positive variables uk. General number of the active 
constraints equals to n + m – L – 1. On the other hand it is 
necessary number of active constraints for the optimal 
basic solution no less than n – 1. Therefore we have m ≥ L.  
Statement 1 is proved. 
Statement 2.  
 
( ) { }( )1 2
1 2
( ) ( ) ( )
( ) ( ) ( )
1
1
( , , , ) 0 : 1, 2, , :
( , , , ) arg min .
L
L L
k
n
L
k k l k ll
k L
a a a U p k L L n
a a a p y a y
k k k
k k k
a R
−=∈ = +
∀ ∈ ∃ ≥ = + +
= −∑ ∑
K K
K
 
                        (8) 
Proof. Let ( )χ ∗k  be characteristic function for set k (5). 
The set ( ){ }: 1,2, ,ip i i L nχ= = +k K  is hold condition 
of the statement for given k .  
Statement 2 is proved. 
Ascertained interrelation between GLADT and 
WLADT permits to ground GLADT estimation for stable 
evaluation of autoregression model factors under 
availability of outliers in the middle of yk. Groundlessness 
of LADT estimation under such conditions is proved in [2]  
In the same place existence of stable WLADT estimation is 
ascertained, but general way to look of proper weights 
{ }: 1,2, ,kp k L n= + K  is not presented.  
Further we describe the implementation of GLADT for 
stable evaluation of autoregression model factors. Like in 
[2] we apply the influence functional presented in [4] . 
 
Stability of evaluation of autoregression model factors 
 
Let ),,( 1 npn xx K−=x  be stationary in the wide sense 
time series. We observe autoregression model  
 
1
AR( ): ,Lk i k i kiL x a x kε−== + ∈∑ Z , 
 
where L is known order, a = (a1, … , ap) is vector of 
nonrandom factors, {εk} are independent identically 
distributed variates with non-degenerate distribution 
function. Let values 
 
,k k k ky x z k
γξ= + ∈Z  
  
be observable variables, {zγk} be independent identically 
distributed variates, {zγk} ~Bi(1, γ), 0 ≤ γ≤ 1, γ be 
obstruction level, {ξk} be independent identically 
distributed variates with distribution µξ from class Mξ; 
successions {xk}, {zγk}, {ξk} be independence. In that way 
we observe simple obstruction scheme of data by 
independence outliers. 
Under appearance obstruction (5) traditional 
estimations are inconsistent. For measuring of quality of 
estimation na
)
 for vector a under observable data yn we 
suppose existence of convergence in probability 
n γ→
Pa a
)
 and equality aa =0 . 
Simple infinitesimal characteristic of estimation na
)
 
stability under data obstruction {yk} is vector 
 
0
0
( , ) limIF γγ ξ γµ γ→
−
=
a a
a  
  
named as influence functional for estimate na
)
 [4] This 
functional characterizes the value of main linear member 
of asymptotic expansion of displacement  
 
0 ( , ) ( )IF oγ γ ξµ γ γ− = +a a a .                            (9) 
  
Condition of estimation stable is finite sensitivity to great 
mistake  
 
(M , ) sup ( , )GES IF
ξ ξ
ξ γ γ ξ
µ
µ
∈Μ
= < ∞a a . 
 350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV1, A, A. N. TYRSIN2, B 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
144 
In that case the main linear member of asymptotic 
expansion (7) is uniformly small for all obstructions and 
small γ.  
Let GLADT estimation of parameters GLM
n
a
)
 be defined 
by way of algorithm (4). Qualitative assessment of stability 
for GLADT estimation GLMna
)
 under outliers gives the 
following theorem. 
Theorem. Let time series (5) be observed. If loss 
function ρ(x) of algorithm (4) is so that 
∞<ρ′
≥
)(sup 2
0
xx
x
 then (M , )GLMGES ξ γ < ∞a . 
Proof. The necessary conditions of minimum for task (4) 
are bridging set  
 
( )1
1
( ) sign 0, 1, ,
l
n
L
a k k l k i k ii
k
y y a y l Lρ ε − −=
=
′ − ÷ =∑ ∑ K  (10) 
 
It is proved in [2] that WLADT estimation obtained by the 
way solving of bridging set  
 
Ll
yayyyp L
i ikikkk
n
k
,...,1
,0)sign(,...,(
111
1
=
÷−∑∑ = −−−
=         (11) 
under condition sup ( )p Ty p∈ < ∞R yy y  has finite 
sensitivity to great outliers, i.e.  
 
∞<γξ ),(M GLMGES a . 
 
Comparison of Eq. (10) and Eq. (11) as well as taking into 
account proved statements 1 and 2 implies assertion of the 
theorem under 
 
2
0
sup ( )
x
x xρ
≥
′ < ∞ .  
 
Theorem is proved.  
As is easy to see the concave increasing functions 
( ) ( )1 exp ; arctan ; / 1x x x x− − +  are holding the 
theorem conditions, and functions ( )2; ; ; ln 1x x x x+  
are not. 
 
Computational experiments 
 
Process AR(1) with one-sided outliers  
 
0
1
0,
, 0,1, , .
,
k k k
k k k k
x
x x k M
y x zγ
α ε
ξ
−
=

= + =
 = +
K  
  
Here 0.7α = ; εk  ~ N(0,σzk2); σzk2 are variates distributed 
uniformly in segment [0;2]; {zγk} are independent 
identically distributed variates; {zγk} ~Bi(1, γ), 10 ≤γ≤ , 
γ are an obstruction level; {ξk} are independent variates 
distributed uniformly in segment [50; 100]; M =1500 is the 
number of trials.  
Test computer simulation is identification of 
autoregression factor α  at sight of signals yk evaluated 
according to Eq. (8). Identification algorithms are GLADT 
estimation (4) with the difference loss functions.  
Simulation data is shown in Fig. 1. 
 
  
a) b) 
Fig. 1. Process AR(1) simulation data:   a) LST, LADT and GLADT estimations of autoregression factor α; b) Sample variance sa2 of 
the estimations 
 350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV1, A, A. N. TYRSIN2, B 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
145
Fig. 1. a demonstrates that LST and LADT estimations are 
inconsistent in spite of the fact that sample variance of 
these estimations is small (see Fig. 1.b). GLADT 
estimations with the loss functions satisfying theorem 
condition (lines 5, 6, 7, and 8) are consistent and have 
small sample variance of ones. 
Vibrations of single mass linear mechanical system 
with resonance frequency f0 = 91.7Hz and damping 
decrement δ = 1.22 under sampling interval ∆ = 0.001sec 
satisfies second-order autoregression process (2) with 
factors a1 = 1.5, and a2 = 0.8. Simulating vibration of this 
system we evaluate by AR(2) process 
 
0 1
1 2
0,
1,5 0,8 , 0,1, , ,
,
k k k k
k k k k
x x
x x x k M
y x zγ
ε
ξ
− −
= =

= − + =
 = +
K        (12) 
 
Here εk  ~ N(0,σzk2); σzk2 are variates distributed uniformly 
in segment [0;2]; {zγk} are independent identically 
distributed variates; {zγk} ~Bi(1, γ), 10 ≤γ≤ , γ are an 
obstruction level; {ξk} are independent variates distributed 
uniformly in segment [50; 100]; M =100 is number of 
trials. 
Test computer simulation is identification of 
autoregression factors a1, and a2. at sight of signals yk 
evaluated according formulas (12). Identification 
algorithms are GLADT estimation (4) with the difference 
loss functions.  
Simulation data is shown in Fig. 2. 
Again we observe that GLADT estimations with loss 
function satisfying the theorem condition are consistent 
only.  
 
 
 
a)  b) 
Fig. 2. Single mass linear mechanical system simulation data:  a) LST, LADT and GLADT estimations of autoregression 
factor a1; b) LST, LADT and GLADT estimations of autoregression factor a2 
 
 
 
Table 1. Results of tests  
 
f0  
 
ρ(x)=x2 ρ(x)=|x| ρ(x)=|x|0.5 ρ(x)=atan|x| ρ(x)=x2 ρ(x)=|x| ρ(x)=|x|0.5 ρ(x)=atan|x| 
0 
0,002 
0,005 
0,01 
0,015 
0,02 
0,03 
0,04 
0,05 
91,7976 
146,8953 
 
 
 
 
 
 
 
92,8006 
90,5264 
88,7691 
70,9415 
 
 
 
 
 
93,0717 
89,3832 
90,2000 
92,1783 
93,1951 
98,0430 
105,0666 
109,1621 
30,3857 
93,1643 
90,1766 
91,1608 
92,1390 
90,9585 
90,0422 
91,6342 
94,3107 
92,0200 
1,1529 
2,9343 
 
 
 
 
 
 
 
1,1873 
1,8355 
3,7423 
7,9464 
 
 
 
 
 
1,2738 
1,3090 
1,3433 
1,4036 
1,3781 
1,4894 
1,7251 
2,3368 
21,9591 
1,2401 
1,3600 
1,2535 
1,3204 
1,2432 
1,2760 
1,1270 
1,1538 
1,3124 
 
 350. STABLE PARAMETRIC IDENTIFICATION OF VIBRATORY DIAGNOSTICS OBJECTS.  A. V. PANYUKOV1, A, A. N. TYRSIN2, B 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
146 
As indicated above (see Eq. (3)) autoregression factors 
a1, a2 of this process are unambiguously interdependent 
with resonance frequency and damping decrement of 
system. Estimation of resonance frequency f0 and damping 
decrement δ evaluated through autoregression factors (see 
Eq. (3)) are presented in Table 1. Empty strings in this 
table signify impossibility of the target calculation under 
current obstruction level γ. 
As may be seen from Table 1 the best technique in the 
absence of obstructions (γ = 0) is LST. But application of 
LST staves off calculations under γ = 0.005. On the other 
hand the application of GLADT estimation with the loss 
functions satisfying to the theorem condition enables to 
make stable estimations.  
 
Summary 
 
As the statements indicates that adduced theoretical proofs 
and computer simulation demonstrate possibility of stable 
parametric identification of vibratory diagnostic objects by 
usage of GLADT estimation with loss the function 
satisfying to the condition of the proved theorem.  
 
Thanks 
 
Authors feel appreciation to Russian Foundation for Basic 
Research for the financial supporting of the project # 07-
01-96035-p_ypaл_a. 
 
References 
 
[1]  Karmalita V. A. Digital processing of random vibration. – 
Moscow: Mashinostroenie, 1986 (in Russian). 
[2]  Boldin M. V., Simonova G. I., Tyurin Yu. N. Sign 
statistical analysis of linear models. – Moscow: Nauka, 
1997 (in Russian). 
[3]  Tyrsin A. N. Robust construction of regression models 
based on the generalized least absolute deviations method // 
Journal of Mathematical Sciences – 2006. Vol. 139, N 3. 
P. 6634–6642.  
[4]  Martin R. D., Yohai V. J. Influence functionals for time 
series // Annals of Statistics, 1986. Vol. 14, N. 4. P.781–
818. 
 
 
 


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Stable parametric identification of vibratory diagnostics objects

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