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Detection and Analysis of Impulse Point Sequences on Correlated

Disturbance Phone

G. Filaretov , A. AvshalumovMoscow Power Engineering Institute,

Moscow Research Institute of Cybernetic Medicine

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Setting of ProblemLet the observable discrete process z is the sum: z = x +

i

0 100 200 300 400 500 600 700 800 900 1000-3-2-101234

x t

t

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4i t

t

0 100 200 300 400 500 600 700 800 900 1000-3-2-101234

z t

t

x (disturbance) - the discrete correlated Gaussian process with standard autocorrelation function , r(k)= 0, 1, 2,….; k=0,1,2,…;

i - the Poisson impulse sequence with distribution function of intervals between impulses .)exp()( ии tt

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Additional conditions

• Distribution function of impulses amplitudes is known up to parameters.• Impulses number M is too small in comparison with common discrete observations N (not more, than 10 – 15 %).

The main task: with help of observable realization z- to detect impulses i and to estimate unknown intensity of impulse point sequence;- to estimate parameters distribution function of impulses amplitudes.

Proposed method for detection and parameters estimation of Poisson impulse sequence includes two stages: Stage 1 intended for detection and position localization of impulse sequence points. Stage 2 intended for estimation of all unknown parameters.

),...,,;()( 21 kии AfAf

k ,...,, 21

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Stage 1 - detection and position localization of impulses sequence points.

Algorithm is based on detection of statistically significant deviation observable value zt from point zt* , which is found by linear interpolation in two neighbour points zt-1 and zt+1

Presence of impuls in point zt , if: Here - Gaussian distribution quantile, appropriated to

confidence probability P. The equivalent formula: , where -

the second order difference for time moment t+1.

211*

tttttt

zzzzzy

yPt uy 1

Pu 1

)2(1)2(1 Pt u )2(

1t

5

In common form this algorithm of detection includes next sequence of operations:

•Computing of the first order , and the second order differences , for time series

•Estimating of variance for time series . •Choice of confidence probability P (usually P = 0,90 -

0,99).•Fixing of impulses with help of criteria. •Elimination of fixed impulses from time series : all these

points are replaced by new values, which are computed with help of linear interpolation in two neighbour points.

Practically it is necessary to organize iterative regime of this algorithm work. The iterative process is lasted until new anomalous points will not find out during the latest iteration. As a result position of all found impulses on discrete time scale

is fixed.

1 ttt zz Nt ,2

1)2(

ttt Nt ,3Nzzz ,...,, 21 2

)2( )2(

t

*1

*2

*1 ,...,, mttt

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Stage 2 - estimation of all unknown parameters

For every found point the impulse amplitude is computed as deviation of observation in this point from value, which is found by linear interpolation in two neighbour points:

; j = 1, 2,…, m.

After this the amplitudes distribution function histogram with ordinates ; i = 1, 2,… is built. Extracting of subset I, which contains substantively warped values of histogram ordinates on account of limited sensitivity of the detection anomalous observations algorithm. This extraction can be made or visual with help of histogram, or by formal exception of histogram intervals from zone . Value is computed for corrected zt after last iteration.

*)( *jtj ztA

2

11 **

jj ttzz

)( иAf

im

)2(1 Pu

)2(

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Estimation of distribution function parameters

, using values , which not belong subset I. Determination of ordinates for histogram intervals, which belong subset I, using function

building of corrected histogram.

Computing of loss coefficient KI in fixation of impulses,

connected with limited sensitivity of the detection algorithm

KI =

This coefficient defines the relativity decrease of impulses numbering comparison with its real value.

)( иAf

k ˆ,...,ˆ,ˆ 21 imim

)ˆ,...,ˆ,ˆ;( 21 kиAf

Ii Ii

ii mm

mˆˆ

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Determination of time intervals between neighbor detection points ; j = 1, 2,…, m. As impulses with amplitudes near zero cannot be detected, selected point process differs from real: it is more rare. However on account of random location of nondetected points, this process remains Poisson impulse process , but with another intensity

< . Estimating of intensity parameter with help of maximum likelihood method:

Determination of corrected estimation for intensity parameter

. Proposed algorithm works out the formulated above problem in full volume.

**1

*jjиj ttt

* *1

1

**̂

m

jиjtm

IK*̂ˆ

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MODEL EXAMPLE• Process is formed by quadruple passing of discrete white

noise through inertia element ( constant time – 10 discrete time units). The observed realization length - N= 20020 discrete volumes.

• Poisson impulse point process includes 917 points. Model (empirical) volume of intensity .

• Distribution function of impulse amplitudes – exponential –

046,0ˆ М

);( 1иAf )exp( 11 A

2501

10

Stage 1• The first iteration - were detected 565 points (critical value =1,28).• The second iteration - were detected 158 points (critical value = 3,09).• The third iteration was not fulfilled. • In total 723 points from 917 were detected.

Stage 2• Amplitudes are estimated and the appropriate

distribution histogram is built:

Pu 1

Pu 1

)( *jtA

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• As the subset I we choose points, belonging to the first

histogram interval (number of these point is equal 164). • Using the all other histogram intervals with help of nonlinear estimation method we find approximation for dependence of histogram ordinates from interval centers and also unknown parameter of exponential (under condition) amplitude distribution :

;

• With help of the last formula we define the estimation for points number in the first grouping interval or another words – for

subset I : = 359. Corrected histogram with using of

practically the same as the initial histogram for A.

• The loss coefficient KI in fixation of anomalous points was computed: KI = 0,788.

)ˆexp( 1Abm 87,461b 8,2501̂

1m̂

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• Time intervals between detected points were established. How it was waited, we have distribution of exponential type with the intensity parameter estimation for this distribution : =0,0361.

Corrected estimation of intensity parameter is computed:

Summing up, it is possible to say, that proposed method of detection and analysis is very effective. The relative error for amplitude mean estimation is less than 1%, аnd for intensity parameter of Poisson impulse point process – about 2%. It is substantively that impulse point process capacity is less 0,1% from capacity of correlated stochastic process, on phone which this point process is detected and analyzed.

*̂

0458,0788,00361,0ˆ

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CONCLUSION

• It is necessary to underline that quality of end result depends on peculiarities of concrete applied task (what kind are characteristics of process , impulse component, function

13and so on).

• Areas of possible application proposed method can be various. In particular, it was used for aims of medical diagnostic as means of heart rhythm infringements.

tx )( иAf