Testing of Sensor Condition Using Gaussian Mixture Model

Ladislav Jirsa and Lenka Pavelkov

Institute of Information Theory and Automation, Czech Academy of Sciences

Pod Vod

arenskou v

ı 4, Prague, Czech republic

Keywords:

Sensor Faults, Bayesian Statistics, Gaussian Mixture, Dynamic Weights.

Abstract:

The paper describes a method of sensor condition testing based on processing of data measured by the sensor

using a Gaussian mixture model with dynamic weights. The procedure is composed of two steps, off-line

and on-line. In off-line stage, fault-free learning data are processed and described by a probabilistic mixture

of regressive models (mixture components) including a transition table between active components. It is

assumed that each component characterises one property of data dynamics and just one component is active

in each time instant. In on-line stage, tested data are used for transition table estimation compared with the

fault-free transition table. The crossing of given level of difference announces a possible fault.

1 INTRODUCTION

Fault detection plays an important role in today’s in-

dustry. An industrial plant has many possible fault

sources, e.g., sensors, actuators, hardware compo-

nents, communication lines. There exist a large

amount of approaches and solutions, mainly tailored

for a particular system, see e.g. (Isermann, 2011).

Sensors belong to basic units of an industrial

plant. Their faults may be critical for correct control

of a system or decision of proper operational state.

The analytical redundancy is a frequently used

method in fault detection. It is based on indirect

measurement of the variable of interest and requires

a model of the concerned physical system. The infor-

mation on the variable is obtained by other available

quantities as inputs of the system model, giving the re-

quired variable as output. For example, in (Walambe

et al., 2010), air-breathing combustion system of an

aircraft engine is modelled, the model is fed by a set

of signals from various sensors and the residual sig-

nal is processed by a bank of extended Kalman ﬁl-

ters, each one corresponding to one type of a sensor

fault. State of aircraft engine is similarly diagnosed

in (Wei and Yingqing, 2009), whereas in (Lu et al.,

2012), hardware redundancy is added, when signals

from two physical sensors are compared against the

output of the engine model and tested for two types

of faults.

Concerning particular classes of system model,

analysis of existence and inference of explicit re-

lations for state estimation are elaborated e.g.

a discrete-time linear systems with state delays for

probabilistic sensor gain faults (He et al., 2008) or

continuous Lipschitz nonlinear systems of three or

more sensors using linear matrix inequalities (Raja-

man and Ganguli, 2004).

Absence of the explicit system and fault model

can be substituted by methods based on learning,

e.g. support vector machines used for classiﬁcation

of faults into multiple classes (Wang et al., 2014b)

in metal cutting industry or, in the same application

area, (auto)regressive model of multisensory informa-

tion processed by cointegration method, used for pre-

diction of tool wear (Wang et al., 2014a). Application

of Gaussian processes and R

enyi entropy is shown

in (Bo

skoski et al., 2013) for bearing fault prognos-

tics and estimation of faultless lifetime. To classify

the states, both faultless and faulty data must be pro-

cessed.

A question of extracting features contained in cor-

rect data has been considered. This paper proposes

an alternative approach to a sensor condition moni-

toring. A generic data-based probabilistic methodol-

ogy not requesting a system and fault model is de-

veloped, detecting situations when a single sensor in

question provides data that are not in accordance with

the historical experience. The applicability of the ap-

proach is intended, initially, just for a sensor monitor-

ing. Also, only one sensor (data source, quantity) is

explicitly considered without dependencies on other

quantities.

550

Jirsa L. and Pavelkova L..

Testing of Sensor Condition Using Gaussian Mixture Model.

DOI: 10.5220/0005063605500558

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 550-558

ISBN: 978-989-758-039-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

The work is included in application of probabilis-

tic methods in cold metal rolling industry, see (Dede-

cius and Jirsa, 2010), (Ettler et al., 2011), (Ettler

et al., 2013), (Jirsa et al., 2013), (Ettler and Dede-

cius, 2013). There, an industrial plant is described

by a hierarchical structure where a sensor is one of

low-level units. Data generated by a sensor can be

distorted in two ways: (a) faulty sensor under correct

process condition, (b) correct sensor under faulty pro-

cess condition. We do not distinguish between these

two cases and we focus on output data properties, de-

spite of cause or localization of eventual fault.

As a data description method, probabilistic mix-

tures of regressive models (components) with normal

noise are used. It is assumed that in each time in-

stant only one component (model) describes the cur-

rent data. If the data behaviour changes and an-

other component becomes suitable for its description,

a transition between the components occurs, in other

words, data description switches from one model to

another. Transition probability between components

in the mixture, i.e. transition table, represents dy-

namic weights of particular component in the mix-

ture.

The mixture components describe particular prop-

erties of processed data. A component can be inter-

preted in one of two ways:

• Each component represents a particular mode of

the system generating the data. These modes are

deﬁned in advance by expert. For example, fre-

quency of engine rotation increases, frequency is

constant, frequency decreases. Road is empty,

jammed or closed. Patient is male or female.

Number of components corresponds to the num-

ber of modes and each component has a clearly

deﬁned meaning.

• Each component represents a particular property

of the data in some sense. This property has not

been known in advance and it has been detected

during the mixture estimation, according to the

model structure, prior information and other con-

ditions. Meaning of such a component may not be

clearly interpreted as in the previous case.

However, these aproaches can be combined, i.e. a

part of components can be deﬁned in advance and a

part can be detected ad hoc. In this work, the sec-

ond approach is adopted: nothing is a priori known

about data and resulting components describe a par-

ticular property, e.g. mean and variance in case of

static model or a particular dynamic feature in case of

dynamic model of the given order.

The proposed procedure is composed of two steps,

off-line and on-line.

In off-line stage, historical fault-free learning data

are processed to estimate an off-line transition table

between the mixture components.

In on-line stage, the tested data are matched

against the mixture components and on-line transition

table is estimated. Compared to the off-line one, the

tested data are declared either correct or faulty.

The paper is organized as follows: Section 2 de-

scribes methodology of data description and estima-

tion of mixture parameters, Section 3 contains exper-

iments with data matching and faults simulation.

2 METHODOLOGY

Throughout the paper, this notation is used: vec-

tors are represented by columns, their elements are

in text enumerated in brackets as x = [x

,...,x

ℓ

]

′

where

′

denotes transposition and ℓ

represents length

of the vector x. Symbol x

∗

denotes set of x-

values, x

is the value of x at discrete-time instant

t ∈ t

∗

≡

{

1,2,...,T

}

,T < +∞. Set of time la-

belled quantities x

up to time t is denoted as x (t) ≡

{

t−1

,...,x

}

, where x

represents prior value

or expert knowledge, possibly empty, f (x|s) is a prob-

ability (density) function (pdf) of x conditioned by s,

random variable is not formally distinguished from its

value. The point estimate (mean value) of x is denoted

by ˆx.

2.1 Model of the Data

Data processing focuses on single scalar variable

(data channel) without any consideration of explicit

dependencies to other variables. We use autoregres-

sive (AR) model, generally of m-th order, with normal

noise, optionally with offset (absolute term).

The data sequence

{

,...,y

}

(or simply

data) is modelled by

= ϑ

′

+ e

(1)

with vector of constant but unknown regression co-

efﬁcients ϑ and regression vector ψ

= [y

t−1

t−2

..., y

t−m

,1]

′

. If the trailing 1 is present, then offset

is added to the model, if omitted, then the model is

purely autoregressive. In case of ψ

= [1]

′

, the model

is static.

The term e

represents a white normal noise. It

is described by Gaussian pdf with zero mean and un-

known but constant variance r, i.e. f (e

) = N

(0,r).

It is related both to a noise on y

and to imperfect

model, with the assumptions mentioned above.

TestingofSensorConditionUsingGaussianMixtureModel

551

2.2 Gaussian Mixture

Data distribution is approximated by a Gaussian mix-

ture (K

arn

y et al., 2005) with dynamic weights of

components (Nagy et al., 2011). For its construction,

it is assumed that each component represents one of

n operating modes of the data generating system. The

component, that is active in time instant t, is labelled

by c

∈ c

∗

≡

{

1,...,n

}

. In other words, the c

-th com-

ponent generates the data vector Ψ

c;t

≡

{

,ψ

′

c;t

}

′

time t.

Unlike in (1), the data vector ψ

c;t

is labelled by c

because each component can be generally described

by a model with different structure.

2.2.1 Component Model

Mixture model describes n different operating modes,

c-th component describes behaviour in c-th mode, c ∈

∗

= {1,...,n}. Its form is

f (y

|c,y(t − 1),Θ

) ≡ f (y

|c,ψ

c;t

,Θ

) =

= N

(

′

c;t

)

. (2)

The parameter Θ = (ϑ,r), where r is the noise vari-

ance.

The conjugate prior is the Gauss-inverse-Wishart

pdf

f (ϑ,r|y(t)) ≡ f (ϑ, r|V,ν) =

= I (V,ν)

−1

−

(ν+ℓ

+2)

×exp

{

−

(

[−1,ϑ

′

]V [−1, ϑ

′

]

′

)}

, (3)

where I (V,ν) is normalization integral, V and ν are

ﬁnite sufﬁcient statistics, tr is a matrix trace, V is an

extended information matrix—symmetric and posi-

tive deﬁnite matrix having the size of the extended

regression vector Ψ. The statistic ν is a data counter—

positive scalar. The term −1 appears as a unit coefﬁ-

cient by y

of opposite sign to ϑ when expressing e

using (1).

Indices c and t were omitted at ϑ, r , V , ν and ψ.

2.2.2 Pointer Model

The pointers c

are assumed to evolve according to the

model

f (c

t−1

,y(t − 1), α, Θ) ≡ f (c

t−1

,α) = α

t−1

(4)

where α

t−1

are transition probabilities that the sys-

tem will be in mode c

in time t, if it was in mode

t−1

in time t − 1. It holds for the stochastic matrix

i j

≡ α

j|i

≥ 0,

∑

j∈c

∗

j|i

= 1, ∀i, j ∈ c

∗

. (5)

The conjugate prior is the Dirichlet pdf

f (α|y(t)) = B(κ

)

−1

∏

∈c

∗

∏

t−1

∈c

∗

t−1

, (6)

where κ

is n × n matrix of sufﬁcient statistics and

B(κ

) is the normalization integral.

2.2.3 Mixture Model

Assuming known parameters Θ and α, the mixture

model is a marginal pdf of the joint pdf of y

, c

and

t−1

f (y

|y(t − 1), α,Θ) =

∑

∈c

∗

∑

t−1

∈c

∗

f (y

t−1

|y(t − 1), α,Θ)=

∑

∈c

∗

∑

t−1

∈c

∗

f (y

,y(t − 1), Θ

) ×

× f (c

t−1

,α) f (c

t−1

|y(t − 1)). (7)

The chain rule f (a,b|c) = f (a|b,c) f (b|c) and the

marginalization rule for a discrete variable f (a|c) =

∑

f (a,b|c) were used here.

2.3 Mixture Estimation

In case of known parameters Θ and α, the formula (7)

can be used directly.

In case of unknown parameters Θ and α, as a tech-

nical approximation used in this paper, the relation (7)

can be used with substituted point estimates of the

unknown parameters Θ and α. This is equivalent to

f (Θ|y(t − 1)) ≈ δ(Θ −

Θ), where δ is Dirac distribu-

tion and

Θ is the point estimate of Θ (the same holds

for α). This approach makes algorithms simpler and

faster but artiﬁcially increases precision of the model

by neglecting parameters’ uncertainty (Nagy, 2014).

The fully Bayesian approach requests unknown

parameters to be included into the joint pdf (7) as

random variables and treated consistently. The the-

ory and procedure is described in detail in (Nagy

et al., 2011). However, it exhibits higher computa-

tional complexity and posterior pdfs of unknown pa-

rameters must be approximated anyway to preserve

the prior forms (3) and (6).

2.3.1 Estimation of Regression Coeffcients

Let us assume availability of the statistics V

c;t−1

and κ

t−1

from the previous time step. The ex-

tended information matrix V can be decomposed as

V = L

′

DL with

L =

[

1 0

yψ

]

, D =

[

0 D

]

, (8)

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

552

where L is a positive deﬁnite lower triangular matrix

with unit diagonal and D is a diagonal matrix with

positive entries.

Then it holds for parameter estimates

ϑ = L

−1

yψ

and ˆr =

ν − 2

. (9)

2.3.2 Estimation of Transition Probabilities

Point estimate of α (see (4)) equals

t−1

;t−1

∑

∈c

∗

t−1

;t−1

. (10)

The probability w

t−1

that the system was at time

t in the mode c

and at time t − 1 in the mode c

t−1

with known (measured) output y

, is

t−1

= f (y

,ψ

c;t

)

t−1

, (11)

where f (y

,ψ

c;t

) is likelihood function for the

estimation step (parametrised model of the data as

a function of parameters with ﬁxed data) and w

t−1

unconditional probability that the system was at time

t − 1 in the mode c

t−1

The probability w

of mode c

at time t is

∑

t−1

∈c

∗

t−1

≡ f (c

|y(t)). (12)

2.3.3 Update of Statistics

For models of components, the statistics of the c-th

component are updated according to these formulae:

time update

c;t|t−1

= λ

c;t−1

+ (1 − λ

(13)

c;t|t−1

= λ

c;t−1

+ (1 − λ

)ν

(14)

data update

c;t

= V

c;t|t−1

+ w

c;t

′

c;t

, (15)

c;t

= ν

c;t|t−1

+ w

(16)

where λ

is forgetting factor for components, 0 <

≤ 1, and V

, ν

are optional alternative statistics

deﬁned by the user to stabilize the update. The higher

is, the more information from the previous data is

kept in the statistics and the less shift of Θ in time is

allowed with new data.

Prior matrix V

c;0

is chosen as diagonal with small

positive values to guarantee regularity, prior ν

c;0

should be small positive, too.

For model of pointers, the statistic κ is updated in

the following way

time update

t−1

;t|t−1

= λ

t−1

;t−1

+ (1 − λ

)κ

(17)

data update

t−1

= κ

t−1

;t|t−1

+ w

t−1

(18)

where λ

is forgetting factor for transition probabili-

ties, 0 < λ

≤ 1, and κ

stabilizing alternative matrix

deﬁned by the user.

All elements of prior matrix κ

t−1

can be cho-

sen as a small positive constant.

2.3.4 Notes on Approximation

Except of the adopted approach of substituting point

estimates of unknown parameters into conditions of

pfds, there is one more issue to be pointed out.

The data update (18) is actually approximation in

the situation when we are uncertain about the pointers

t−1

and c

, which are substituted by their conditional

probabilities w

t−1

. This approach is called quasi-

Bayes (QB) and consists in approximation of the Kro-

necker δ(c

,c) by its mean value E

[δ(c

,c)] = w

the true c

is unknown (K

arn

y et al., 2005).

Another possible approximation is to substitute

the unknown pointers c

t−1

and c

by indices of com-

ponents with maximum value of likelihood (ML)

within the mixture, i.e.

= arg max

c∈c

∗

f (y

|c,ψ

c;t

). (19)

Then, only the corresponding element of κ

t−1

incremented by 1 in (18). We tried both these ap-

proaches.

2.3.5 Notes on Estimation Algorithm

The computation starts with prior statistics V

c;0

, ν

c;0

and κ

t−1

. Probabilities w

are chosen as uniform.

The time step begins with point estimation of pa-

rametes

and

t−1

using (9) and (10). Then, ma-

trix w

t−1

is obtained by (11) and it is used to get

unconditional probablilities w

by (12) as weights for

update of components’ statistics in (15) and (16). Fi-

nally, the transition probability statistic is updated us-

ing conditional probability w

t−1

according to (17)

and (18), t is incremented by 1 and new time step with

a new data vector is started.

Even though all data are processed, the resulting

mixture is usually not estimated satisfactorily after

one pass (iteration) of the data sequence. It is rec-

ommended to perform multiple iterations in this way:

• store prior statistics ν

c;0

and κ

t−1

• estimate the mixture using all the data, obtain

statistics V

c;T

, ν

c;T

and κ

t−1

• calculate λ

∑

c∈c

∗

c;0

−nν

)/(

∑

c∈c

∗

c;T

−nν

), where n is

number of components,

TestingofSensorConditionUsingGaussianMixtureModel

553

• calculate λ

∑

i, j∈c

∗

(

j|i;0

−κ

A,i j

)

∑

i, j∈c

∗

(

j|i;T

−κ

A,i j

)

• ﬂatten the mixture using time updates (13), (14)

and (17) using λ

and λ

• stabilizing terms (those with the alternative statis-

tics) in the time updates are necessary for numer-

ical reasons, e.g. 0 < ν

< ν

, V

low positive

diagonal, κ

a matrix ﬁlled with a positive con-

stant, according to the estimation performance,

• replace prior statistics by the ﬂattened posterior

statistics V

c;0|T

, ν

c;0|T

and κ

t−1

;0|T

and start

a new iteration.

This procedure improves prior information for the

new iteration which results in a better posterior es-

timate of the mixture. Iterations can be repeated un-

til the model converges according to a chosen crite-

rion (e.g. (20), for

αs of two subsequent iterations, or

other), usually 7×–10×, according to the order of the

model and number of estimated relevant components.

The number of components n is most often un-

known in advance. For mixture estimation, we

adopted this approach:

• choose initial number of components which is

much higher than the expected ﬁnal number (e.g.

n = 35),

• place the components uniformly into the domain

of f (y

|c,y(t − 1),Θ

), choose small variance r

(e.g. (max(y) − min(y))/(100n)), see (2),

• estimate the mixture using the description given

above,

• optionally cancel (remove) insigniﬁcant compo-

nents with lower weights, according to a cho-

sen criterion, to achieve compromise between low

number of components and satisfactory descrip-

tiveness of the mixture model.

Numerically stable and fast square root algo-

rithms, performing updates (13) and (15) directly on

matrices L and D, where V = L

′

DL (8), are available

e.g. in (K

arn

y et al., 2005). They guarantee positive

deﬁniteness of the extended information matrix V and

numerical manipulation with L and D is simpler and

faster. Direct decomposition of V to L and D may be

numerically unstable in some cases.

2.4 Comparison of Transition Tables

Having two matrices of statistics with the same di-

mensions, κ and ˜κ, we perform a simple comparison.

First, we get the point estimates of transition tables

and

α using (10), then calculate a value of the crite-

rion

ρ =

∑

i, j∈c

∗



j|i

−

j|i



(20)

and compare it against a chosen value

ρ > 0. If ρ <

ρ,

the matrices match. Criterion ρ can reach values from

0 (exact match) to 2 (total mismatch).

3 EXPERIMENTS

Experiments were performed on industrial data from

several cold rolling mills. As no faulty data were

available at the moment, several types of faults were

simulated by distortion of the data recorded in normal

operating conditions.

The initial purpose was to demontrate inﬂuence

of various situations on the criterion value ρ in (20).

Therefore, any critical value

ρ has not been proposed.

3.1 Estimation Procedure

Probabilistic mixture was estimated using data chosen

as the reference data describing faultless operation.

Then the components were ﬁxed, i.e. parameters

were kept constant. Using the same reference data,

the algorithm was run once more (with several itera-

tions), except that updates of the components’ statis-

tics V

and ν

(13), (14), (15) and (16) were disabled

and the statistics were left intact. Only α was esti-

mated. This parameter was denoted as the reference

transition table α

. Using the same procedure with

the ﬁxed components’ parameters for a different data

sequence, another transition table α was obtained, i.e.

each data sequence was characterized by its own tran-

sition table generated by a particular mixture model.

As mentioned in part 2.3.4, quasi-Bayes (QB)

or maximum likelihood (ML) approximation can be

used in (18). With QB, convergence of

α was rather

slow, whereas ML converged very rapidly within less

than 10 iterations. Therefore, the ML approximation

was used in computations.

Actually, after processing enough data from the

sequence into sufﬁcient statistics V , ν and κ , both QB

and ML approximations performed practically equiv-

alently, because the likelihood function was narrow

enough to single out one component, as in case of

ML. However, the initial phase of the iteration, when

likelihood assigns similar values to different compo-

nents, slows the convergence down, even, in some

cases, leads to inﬁnite loops.

The forgetting factors were set as λ

= λ

= 1,

i.e. estimation without forgetting, assuming constant

parameters.

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

554

For number of components n, prior statistics were

set to V

c,0

= 0.3n/data range, ν

c,0

= 25/n, κ

j|i;0

1/(2n

). Alternative statistics were set to V

diag(10

−7

), ν

= 1/(4n), κ

A, j|i

= 1/(20n

For experiments, we used data from metal cold

rolling mill. Total rolling force was chosen for a par-

ticular material and pass as learning data of sample

size 5 458. Data values were in the range ⟨0,5⟩. We

identiﬁed static component model (m = 0 + offset)

and AR component model of 2

order (m = 2 + off-

set). Static model converged with 14 and AR model

with 7 signiﬁcant components.

Histogram of the data is shown in Figure 1, the

mixture identiﬁed with static component model is

shown in Figure 2. Note that dynamic pointers are

used even in case of static component model.

0 1 2 3 4 5

100

150

200

250

Figure 1: Histogram of learning data.

0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.2

1.4

1.6

probaility

Figure 2: Static mixture describing the learning data.

3.2 Data Matching

First, learning data were, through their transition ta-

bles, matched against data describing the same quan-

tity, total rolling force, on the same rolling mill and

the same material during different metal strip passes

(data 1). Then, similar quantity, upper rolling force,

which is approximately half of the total force, was

tested in the same conditions as above (data 2). Last,

total rolling force measured on a different rolling mill

with a different material was processed (data 3).

The data were compared using mixtures with both

static and dynamic component models. The results

for static mixture model are shown in Table 1, for dy-

namic mixture model in Table 2.

Table 1: Data matching with static model (see text for ex-

planation).

min

mean

max

data 1 0.064 0.198 0.331

data 2 0.709 0.993 1.110

data 3 0.814 1.086 1.223

Table 2: Data matching with dynamic model (see text for

explanation).

min

mean

max

data 1 0.122 0.165 0.243

data 2 0.312 0.341 0.362

data 3 1.025 1.118 1.254

3.3 Simulation of Faults

As no faulty data were available at the moment, the

sensor faults had to be simulated. In all cases be-

low, learning data y

were modiﬁed to y

fault

, t ∈ t

∗

in a speciﬁed way. Such a pair of the data arrays,

and y

fault

, was used to generate a pair of transi-

tion tables. This was done both for static model and

for dynamic model. The pair of transition tables for

each model was compared using (20). The criteria ρ

(static model) and ρ

(dynamic model) are shown in

the tables.

3.3.1 Additive Noise

Data y

were modiﬁed by additive Gaussian noise e

fault

= y

+ e

∼ N (0,r

where r

is noise variance. Table 3 shows inﬂuence of

on data matching.

3.3.2 Additive Bias

Data y

were modiﬁed by additive bias b > 0,

fault

= y

+ b.

Table 4 shows inﬂuence of b on data matching.

TestingofSensorConditionUsingGaussianMixtureModel

555

Table 3: Additive noise.

−5

0.005 0.006

−4

0.034 0.032

−3

0.254 0.130

−2

0.738 0.724

Table 4: Additive bias.

b ρ

0.01 0.012 0.009

0.05 0.223 0.022

0.1 0.358 0.318

1 0.557 0.641

3.3.3 Additive Drift

Data y

were modiﬁed by additive drift d

fault

= y

+ d

max

Table 5 shows inﬂuence of d

max

on data matching.

Table 5: Additive drift.

max

0.001 0.001 0.004

0.01 0.003 0.005

0.1 0.040 0.017

1 0.201 0.101

2 0.302 0.146

3.3.4 Block Dropout

Data y

were modiﬁed by block dropout. The dropout

occurs in the interval ⟨t

+ h⟩, where h = T

100

Value p represents percentage of dropout related to

the sample size. Value t

was chosen randomly but

kept constant. The data values within the dropout in-

terval were set to zero.

Table 6 shows inﬂuence of p on data matching.

4 CONCLUSION

The paper presents a method for comparison of scalar

data arrays. The data are described by a probabilis-

tic Gaussian mixture with dynamic component point-

ers and represented as a corresponding transition ta-

ble between the components, according to the given

Table 6: Block dropout.

p[%] ρ

0.1 0.001 0.006

1 0.002 0.006

5 0.006 0.010

10 0.018 0.017

20 0.021 0.020

mixture model. It is a case of unsupervised trans-

formation of data into a multidimensional feature ap-

plied for classiﬁcation. Two component models were

used: static component model (offset only) and dy-

namic component model (AR model of 2

order with

offset).

Matching of data sequences was quantiﬁed to

demonstrate the sensitivity of the method to spe-

ciﬁc faults and situations. No classiﬁcation of cor-

rect/faulty data was performed yet due to unconvinc-

ing results.

To represent faults, additive noise, bias and drift

were simulated. As they demonstrated behaviour of

the method, in our point of view, sufﬁciently, mul-

tiplicative faults were not included in the study al-

though they occur in practice as well.

The data values were in the interval ⟨0, 5⟩, which

is obvious from Figures 1 and 2.

4.1 Performance of the Models

Static model is naturally more sensitive to data dif-

ference because it ignores data dynamics, which re-

sults in higher values of the criterion ρ and higher

dispersion when processing multiple data sequences.

This property can be seen in Tables 1 and 2, data 1

and 2. On the other hand, if the data dynamics is com-

pletely different from the one of the learning data, the

dynamic model is more sensitive (with lower disper-

sion), see the same tables, data 3.

Sensitivity of static model to simulated faults is

generally higher as well. The exception is the lowest

values of noise and drift. In case of block dropouts,

sensitivity depends in a more complex way on the

droput size, which, however, both static and dynamic

models reﬂect very mildly.

The sensitivity of the method is limited by lack

of external information on data added to its construc-

tion. For speciﬁc data, which may be suitable for the

method application using a speciﬁc model, this au-

tonomous property can be of advantage. Dynamic

component model of corresponding order would fo-

cus on dynamic properties rather than absolute scale.

On the other hand, estimation success of dynamic

component model is increased if transitions are rare.

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

556

It was observed that for a particular data sequence,

usually one or two structures of dynamic models

yielded a sufﬁcient number of signiﬁcant compo-

nents. This property indicates that the method might

“choose” the suitable dynamic features for the given

data. However, it requires further testing.

4.2 Open Problems

Generic aiming and ﬂexibility of the described

method is balanced by its limited performance. Prac-

tical applicability of the method may be enhanced by

considering several topics, among others

• assuming bounded noise rather than Gaussian,

• reconsidering advantages and disadvantages of

dynamic component models,

• validating of the mixture by other means than con-

vergence of its transition table,

• trying a better criterion for comparison (e.g.

Kullback-Leibler divergence of Dirichlet pdfs,

dynamic clustering in parameter space etc.)

• employing components of different dynamic or-

der,

The QB approximation is more consistent with the

Bayesian methodology used in this work than the ML

one. Therefore, ﬁxing the convergence issue in case

of QB approximation is desirable.

As the method is unsupervised in the described

phase, no critical value of

ρ was set up. The question

is how to make this set up methodically and gener-

ally. The method is based on occurrence of speciﬁc

features with a speciﬁc set of data sequences and ad-

equate model. Stability of these features within the

set indicates data similarity. The question is whether

these features can be extracted by the method or

added externally by an expert.

The supervision is a crucial topic. It might im-

prove the performance and set the critical value of

but, on the other hand, lead to tailored solutions with

loss of intended generality.

Other models than AR or other technology could

be taken into account as well.

Last but not least, the method is based on extract-

ing information from data. Hence, a wider set of data,

including those with real faults, should be studied.

ACKNOWLEDGEMENTS

This research has been supported by EUROSTARS

Project 7D12004 E!7662.

REFERENCES

skoski, P., Ga

sperin, M., and Petelin, D. (2013). Bearing

fault prognostics using R

enyi entropy based features

and Gaussian process models. Mechanical Systems

and Signal Processing. Submitted.

Dedecius, K. and Jirsa, L. (2010). Bayesian Soft Sensing in

Cold Sheet Rolling. In Proc. of the 6th Int. Workshop

on Data-Algorthms-Decision Making. UTIA AV

CR.

Ettler, P. and Dedecius, K. (2013). Overview of bounded

support distributions and methods for Bayesian treat-

ment of industrial data. In Proceedings of the 10th

international conference on informatics in control,

automation and robotics (ICINCO), pages 380–387,

Reykjavik, Iceland.

Ettler, P., Juricic, D., Dedecius, K., and Preglej, A. (2011).

Advanced soft sensor technology to be used for cold

rolling mills. In Emerging Technologies & Factory

Automation (ETFA), 2011 IEEE 16th Conference on,

pages 1–4. IEEE.

Ettler, P., Puchr, I., and Dedecius, K. (2013). Bayesian

model mixing for cold rolling mills: Test results. In

Fikar, M. and Kvasnica, M., editors, Proceedings of

the 2013 International Conference on Process Control

(PC), pages 359–364,

Strbsk

e Pleso, Slovakia.

He, X., Wang, Z., and Zhou, D. H. (2008). State estima-

tion for time-delay systems with probabilistic sensor

gain reductions. Asia-Paciﬁc Journal of Chemical En-

gineering, 3(6):712–716.

Isermann, R. (2011). Fault Diagnosis Applications: Model

Based Condition Monitoring, Actuators, Drives, Ma-

chinery, Plants, Sensors, and Fault-tolerant Systems.

Springer Verlag.

Jirsa, L., Pavelkov

a, L., and Dedecius, K. (2013). Pre-

liminaries of probabilistic hierarchical fault detection.

In Preprints of the 3rd International Workshop on

Scalable Decision Making held in conjunction with

ECML/PKDD 2013, pages 1–14, Prague, Czech re-

public.

arn

y, M., B

ohm, J., Guy, T. V., Jirsa, L., Nagy, I., Ne-

doma, P., and Tesa

r, L. (2005). Optimized Bayesian

Dynamic Advising: Theory and Algorithms. Springer,

London.

Lu, F., Huang, J., and Xing, Y. (2012). Fault diagnostics

for turbo-shaft engine sensors based on a simpliﬁed

on-board model. Sensors, 12(8):11061–11076.

Nagy, I. (2014). Advanced statistical methods

and their applications.

CVUT FD, Prague.

http://www.fd.cvut.cz/personal/nagyivan/PhDLoad/

LecturesPhD.pdf, In Czech.

Nagy, I., Suzdaleva, E., K

arn

y, M., and Mlyn

rov

a, T.

(2011). Bayesian estimation of dynamic ﬁnite mix-

tures. Int. Journal of Adaptive Control and Signal

Processing, 25(9):765–787.

Rajaman, R. and Ganguli, A. (2004). Sensor fault diag-

nostics for a class of non-linear systems using linear

matrix inequalities. International Journal of Control,

77(10):920–930.

Walambe, R., Gupta, N., Bhange, N., Ananthkrishnan, N.,

Choi, J., Park, I., and Yoon, H. (2010). Novel redun-

TestingofSensorConditionUsingGaussianMixtureModel

557

dant sensor fault detection and accommodation algo-

rithm for an air-breathing combustion system and its

real-time implementation. Defence Science Journal,

60(1).

Wang, G., Liu, C., Cui, Y., and Feng, X. (2014a). Tool wear

monitoring based on cointegration modelling of mul-

tisensory information. International Journal of Com-

puter Integrated Manufacturing, 27(5):479–487.

Wang, G., Yang, Y., Zhang, Y., and Xie, Q. (2014b). Vi-

bration sensor based tool condition monitoring using

ν support vector machine and locality preserving pro-

jection. Sensors and Actuators A: Physical, 209(0):24

– 32.

Wei, X. and Yingqing, G. (2009). Aircraft engine sen-

sor fault diagnostics based on estimation of engine’s

health degradation. Chinese Journal of Aeronautics,

22(1):18 – 21.

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

558