A joint renewal process used to model event based data

Mergenthaler, Wolfgang; Jaroszewski, Daniel; Feller, Sebastian; Laumann, Larissa Article A joint renewal process used to model event based data Decision Analytics Provided in Cooperation with: Springer Nature Suggested Citation: Mergenthaler, Wolfgang; Jaroszewski, Daniel; Feller, Sebastian; Laumann, Larissa (2016) : A joint renewal process used to model event based data, Decision Analytics, ISSN 2193-8636, Springer, Heidelberg, Vol. 3, Iss. 1, pp. 1-18, https://doi.org/10.1186/s40165-016-0019-9 This Version is available at: https://hdl.handle.net/10419/161806 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ A joint renewal process used tomodel event based data Wolfgang Mergenthaler* , Daniel Jaroszewski, Sebastian Feller and Larissa Laumann Model Renewal processes have been a frequent object of analysis in early studies of stochastic processes, see Cox (1962), for instance. Only recently the idea of parallel renewal processes receives more attention, see Borgelt and Picado-Muino (2012), Gaigalas (2003), Kai etal. (2014), CRC (1994), Kallen etal. (2010), Truccolo (2005), Modir etal. (2010). However, little emphasis has been given to the subject of stochastic dependence between processes so far, with few exceptions such as shown in Borgelt and Picado-Muino (2012) or Truccolo (2005), Modir etal. (2010). Spike train analysis is an active neurobiological research area calling for parallel renewal processes. The latter paper emphasizes stochastic dependence between point processes described by conditionally independent intensity functions. In the same spirit, stochastic dependence between events will be at the core of the present paper in combination with a linear damage model in a condition Abstract In many industrial situations, where systems must be monitored using data recorded throughout a historical period of observation, one cannot fully rely on sensor data, but often only has event data to work with. This, in particular, holds for legacy data, whose evaluation is of interest to systems analysts, reliability planners, maintenance engineers etc. Event data, herein defined as a collection of triples containing a time stamp, a failure code and eventually a descriptive text, can best be evaluated by using the paradigm of joint renewal processes. The present paper formulates a model of such a process, which proceeds by means of state dependent event rates. The system state is defined, at each point in time, as the vector of backward times, whereby the backward time of an event is the time passed since the last occurrence of this event. The present paper suggests a mathematical model relating event rates linearly to the backward times. The parameters can then be estimated by means of the method of moments. In a subsequent step, these event rates can be used in a Monte-Carlo simulation to forecast the numbers of occurrences of each failure in a future time interval, based on the current system state. The model is illustrated by means of an example. As forecasting system malfunctions receives increasingly more attention in light of modern conditionbased maintenance policies, this approach enables decision makers to use existing event data to implement state dependent maintenance measures. Keywords: Renewal processes, Linear damage accumulation, Renewal equation, Moment method Mathematics Subject Classification: 45A05, 60G99, 60K15 Open Access © 2016 Mergenthaler et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. RESEARCH Mergenthaler et al. Decis. Anal. (2016) 3:2 DOI 10.1186/s40165-016-0019-9 *Correspondence: mergenthaler@ frankfurt-consulting.de FCE Frankfurt Consulting Engineers GmbH, Altmuensterstrasse 2d, 65439 Flörsheim am Main, Germany Page 2 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 based maintenance context. A formal representation of a parallel renewal process is given in Modir etal. (2010), whereby the authors look at the process from the point of view of an abstract Poisson process with state-dependent event rates, conditionally independent given the history of the process. Kai etal. (2014) is another example of a biomedical application of a parallel renewal process, whereby the individual occurrence rates of neural spikes depend not only on the neuron in question, but also on a set of neighboring neurons. A Monte-Carlo algorithm is used to construct a parallel renewal process based on the event rates identified. The paper is structured as follows: The remaining sections of this chapter describe the process, the event rates and general modelling assumptions. Chapter 2 deals with the multidimensional renewal equation. After presenting the most general case with stochastic dependence, the special case without stochastic dependence between processes is considered and an asymptotic result for the expected number of cumulated events is derived. We proceed to show that there is a continuous transition from the case of stochastic dependence to the case of stochastic independence with one individual parameter tending to zero. Chapter 3 deals with technical details of the estimation problem used to find the model parameters. Chapter 4 illustrates the numerical findings by means of an example. Input data The input data in the case of an event oriented data model consists, from the practical point of view, of a list of records, say, with each record containing a failure code, a date time object and some explanatory text. From the mathematical point of view, however, it is sufficient to •classify the different failure codes, •index each class and •transform each date time object into a real number representing the occurrence time for the respective event. This process results in a list of occurrence times, grouped according to failure classes as shown: The occurrence times will then be used to construct a joint renewal process. The process Let E:= {1, ...,n} be the set of all failure codes, i.e. events. Then, at each point in time t∈IR+ let Xi(t) be the backward time of event i∈E and define the vector of backward times as (1) T = T 1,1 ,...,T 1,N 1 ............ Tn,1, ... ,Tn,Nn (2) X(t)={X1(t),...,Xn(t)}T Page 3 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 where The probability for an event i∈E to occur in the time interval (t,t+dt) conditionally upon trajectory X(t) is then given by In this equation it is assumed, that events are conditionally independent on each other, if the system state is given. i(X(t)) is an event rate or an intensity function such as defined in Press (2007). The following stochastic differential equation can now be proven for X(t): Theorem1 Let ei,i∈E be the unit vector in direction i. Then the following holds: whereby w.p. stands for “with probability”. Proof Each of the events i∈E occurs with probability i(X(t))dt +o(dt) , in which case all of the events except event i age by an incremental amount of time dt and the backward time of event i is reset to 0. No event, therefore, occurs with probability 1−i∈Ei(X(t))dt +o(dt) . More than one event occurs with probability o(dt) only.  The event rates The question now is, whether a plausible functional relationship of  on the system state X(t) can be found such that the parameters of this function can be efficiently estimated from the data available and such that this relationship can be used to generate realistic simulations of the joint renewal process serving as a reliable short-term forecast in the domain of up to one week, for instance. The following assumption will be used throughout this paper: (6) admits the interpretation, that each event rate consists of a random component i and a condition or state dependent component controlled by the parameters αi,j,i∈E,j∈E . The state dependent component is modelled such that the event rates are linearly dependent on the backward times (=ages) of the events with proportionality factors given by αi,j,i∈E,j∈E . Therefore we sometimes refer to the process as a process with linear damage accumulation. Figure1 shows schematically the dependence of the event rates on the vector of backward time. (3) X i (t)=t−max 0≤Ti,j≤t,1≤j≤Ni {T i,j },i∈E (4) P{Event ioccurs in (t,t+dt)|X(t)}=i(X(t))dt +o(dt) (5) X (t+dt)=X(t)+  j�=i ∗dt ∗ej−Xi(t)∗ei,w.p.(i(X(t))dt +o(dt )) X(t+dt)=X(t)+  j∈E ∗dt ∗ej,w.p.1−  i∈E i(X(t))dt +o(dt)  (6)  i(X(t)) =i+  j∈E αi,j∗Xj(t ) Page 4 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 Corollary 1 If the following holds then the individual renewal processes become independent. Proof Using (5) and (6) one obtains for all i∈E In (8) no cross-dependencies between different components of the vector X(t) can be observed.  The renewal equation Again, please note that the input data sample or, equivalently, the trajectory (2) has been observed and serves as input. Also, let Ft be the sigma algebra generated by X(v),v≤t . Preliminaries Conditionally upon this trajectory the inter-event time distribution holds. Assuming stochastic independence conditionally upon X ( v ) v≤t (9) immediately yields (7) αi,j=0, i∈E,j∈E,j�= i (8) X i(t + dt) = 0 w.p. i+αiiXi(t)dt +o(dt), Xi(t) + dt w.p. 1 − (i + αiiXi(t)dt) + o(dt) . (9) P {Ti≥u|Ft}=E   exp   −   iu+ � t+u t � j ∈ E αi,jXj(v)dv    |Ft   (10) R (t|X):= P{min i∈ETi≥u|Ft}=E   exp   − � i ∈ E (iu+ � t+u t � j ∈ E αi,jXj(v)dv)   |Ft   Fig. 1 Vector of backward times at time t Page 5 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 Define further Accordingly, fi(t|X)∗dt +o(dt) is the probability for an event of type i∈E to occur in the time interval (t,t+dt) conditionally upon the trajectory X(v)v≤t . Also let ˆ C(t|X) be the vector of expected numbers of renewals at time t, if the process starts in state X. The following then holds: Theorem2 Let Then Proof Equation (13), representing the expected number of renewals at time t under the condition that the process started in state X, can be conditioned upon the first occurrence of the event. If this occurs after time t (probability R(t|X)), then the expected numbers are (0, ...,0)T . If it occurs during the time interval [u,u+du) somewhere in the interval [0,t) and is of type i∈E (probability fi(u|X)du) , then state X transforms into state X(i)(X,u) and—therefore—the expected number, seen as a vector, is equal to ei+ˆ C(t−u|X ( i ) (X,u)) .  Iterative approximation In this section the individual components of ˆ C(t|X) will be considered one by one and the arguments  and α will be suppressed. Also, ˆ Ci(0|X)=0 will be assumed. Then the individual components of equation (13) can be written as Assume (14) has an iterative solution such that, for k=0, 1, 2, ... (11) F i(t|X):= P{min i∈ETi<u|Ft}=1−P{min i∈ETi≥u|Ft}= �t 0 fi(u|X) du fi(t|X):= R(t|X)×  i+ � j ∈ E αi,jXj(t)  (12) X ( i )(X,u):= X+  j�=i ej∗u−Xi∗ei,i∈E (13) ˆ C (t|X)=  i∈E Fi(t|X)ei+  i∈Et 0 fi(u|X)ˆ C(t−u|X(i)(X,u)) du (14) ˆ C i(t|X)=Fi(t|X)+  j ∈ E t 0 fj(u|X)ˆ Ci(t−u|X(j)(X,u)) du (15) ˆ C (0) i(t|X)=Fi(t|X) ˆ C (k+1) i(t|X)=Fi(t|X)+  j∈E t 0 fj(u|X)ˆ C(k) i(t−u|X(j)(X,u)) du Page 6 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 Then stages (1) and (2) of the iterative approximation can be written as (16) can be used to approximate the cumulated number of events for rare failure codes, and only in the near term environment. The advantage is, however, that those numbers take into consideration the initial condition X and therefore are in agreement with the requirements of “Condition Based Maintenance”. It will be shown now, that the iteration given in (15) converges. Lemma 1 For any T∈IR+ such that the following holds: Proof Let Then from (15) one obtains and therefore as shown in Appendix 1, proving the lemma in the limit for k→∞.  Please observe that (17) expresses the condition that the process will not “explode” at any time in a finite time interval. (16) ˆ C (1) i(t|X)=Fi(t|X)+  j∈E t 0 fj(u|X)Fi(t−u|X(j)(X,u)) du ˆ C (2) i(t|X)=Fi(t|X)+ j∈Et 0 fj(u|X)Fi(t−u|X(j)(X,u)) du + j∈Et 0 fj(u|X) k∈Et−u 0 fk(v|X(j)(X,u)) ×Fi(t−u−v|X (k) (X (j) (X,u),v))dvdu (17) ρ :=  j∈ET 0 fj(u|X)du <1, X∈IRn + (18) lim k→∞ || ˆ C (k+1) (t | X) −ˆ C (k) (t | X) || = 0, 0 ≤ t ≤ T,X ∈ IR n + (19) D(k+1) (t|X):= ˆ C (k+1) (t|X)− ˆ C (k) (t|X),k=0, 1, 2, 3 ... γk:= sup t ∈[ 0,T ] sup X ∈ IRn + ×max l ∈ E D(k) l(t|X ) (20) D (k+1)(t|X)= t 0 du  i∈E fi(u|X)D(k)(t|X ) (21) ||D(k+1)(t|X)|| = ρ∗ ||D(k)(t|X)|| Page 7 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 A special case: stochastic independence Assume (7) holds. Let ˜ Ci(t) be the solution of (14) under (7). The expected cumulated numbers of events then become independent. For each i∈E the following result can be proven, whereby := i and α:= αii has been set. Corollary 2 whereby �(x) denotes the cumulative distribution function of the standard normal distribution. Proof Note that Let whereby the second equation above is proven in Appendix 2. Furthermore, defining and making use of the Laplace transformations introduced above yields Next, we compute Lφ(s) and Lf(s) . It is easy to see that (22) lim t →∞ ˜ Ci(t)= t exp  2 2α ∗ 2π α ∗ 1 − �   √ α  −1+o(1 ) (23) ˜ C i(t)= t 0 (+αu)exp −u+α u 2 2du +  t 0 (+αu)exp  −  u+α u2 2 ׈ Ci(t−u) du (24) φ( t):= t 0 (+αu)exp  −  u+α u 2 2 du = 1 − exp − t + α t2 2 (25) f(u):= (+αu)exp  −  u+αu 2 2  L C(s):= ∞ 0 exp(−st)˜ Ci(t)dt Lφ(s):= ∞ 0 exp(−st)φ(t)dt Lf(s) :=  ∞ 0 exp( − st)f(t)dt (26) L C(s)= L φ (s) 1 − L f (s ) (27) L φ(s) = 1 s− exp  (+s) 2 2α 2π α 1 − �  +s √α Page 8 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 see Appendix 3. Also, Lf(s) is shown to be expressed as as shown in Appendix 4. Now, upon using (26), (27) and (28) the following is obtained: which yields whereby—see Cox (1962)—O(1) is a function of s bounded as s→0 . According to Cox (1962), section 1.3, ˜ Ci(t) then satisfies  An equivalent proof can be obtained from one of the central results in renewal theory which states that whereby ¯ T is the expected renewal time, see chapter4 in Cox (1962). By definition Using substitutions in the style as shown above and properties of the incomplete Gamma function, as defined, for instance in Abramovitz and Stegun (1972),p. 262, one proves that Using (32) along with (34) proves the statement. The following conclusions are now easy to draw: Corollary 3 If =0 then (28) L f(s) = 1 − exp  (+s) 2 2α s  2π α 1 − �  +s √α (29) L C(s)= 1−sexp  (+s) 2 2α  2π α  1−�  +s √α  s2exp  (+s)2 2α  2π α  1 − �  +s √α  (30) lim s →0 LC(s)=    1 s2exp � 2 2α �� 2π α � 1 − � �  √α ��    −1 s+O(1 ) (31) lim t →∞ ˜ Ci(t)= t exp  2 2α ∗ 2π α ∗ 1 − �   √α −1+o(1 ) (32) lim t→∞ ˜ Ci(t) =t ¯ T (33) ¯ T=∞ 0 t( + α ∗ t)exp −  ∗ t + α ∗ t 2 2dt (34) ¯ T= exp   2 2α∗ 2π α 1 − �   √α (35) lim t→∞ ˆ C(t |  = 0) = 2  α 2π∗ t − 1 + O(1 ) Page 15 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 Appendix 5 Appendix 6 Defining one proves which is equivalent to (43). Appendix 7 (65) fi(t|X)=R(t|X)  i+� j∈E αi,j(u+Xi)   R(t|X)=exp  −� k∈E  kt+� j∈E αk,j (t+Xj)2 2    F i(t|X)=�t 0 fi(u|X)du ˜ fi(t|X)=˜ R(t|X)(i+αi,i(t+Xi)) ˜ R(t|X)=exp �−�it+αi,i t2 2�� ˜ Fi(t) =� t 0 fi(u)du (66) r i:= � t 0   R(u|X)   i+ � j ∈ E αi,j(u+Xj)   −˜ R(u|X) � i+αi,i(u+Xi) �  (67) t i:= � k ∈ E � t 0 duR(u|X)   k+ � j ∈ E αk,j(u+Xj)   (ˆ Ci(t−u|X(k)(u,X )) (68) w i:= � k∈E�t 0 du  R(u,X)  k+� j∈E αk,j(u+Xj)   −˜ R(u,X)(k+αk,k(u+Xk)) ט Ci(t−u|X(k)(u,X))   (69) ˆ C (t|X)− ˜ C(t)=  i∈E ei[ri+ti+wi ] (70) Ai=�t 0 du exp � −� k∈E�ku+αk,k (u+Xk)2 2� × exp  −� k∈E� j∈E,j�=k αk,j (u+Xj)2 2  i+� j∈E αi,j(u+Xj)   −(i+αi,i(u+Xi)) � Page 16 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 Now, for the sake of simplicity, let Xj=0, j∈E . This yields Therefore Appendix 8 Again, by letting Xl=0, l∈E one obtains , after some regrouping Therefore (71) Ai=�t 0 exp �−� k∈E�ku+αk,k u2 2��  � j∈E,j�=i αi,j u − � k∈E � j∈E,j�=k αk,j � i u2 2+˜αi u3 2 �   +o(¯α) (72) | Ai|≤ ¯α t 0 udu +ni 2 t 0 u2du +n¯α 2˜αi t 0 u3du +o(¯ α) =¯ α  t2 2+ ni t3 6+ n ˜ αi t4 8+ o( ¯ α) (73) fk(u|X)−˜ fk(u,X)=exp �−� l∈E�lu+αl,l u2 2��  � j∈E,j�=k αk,ju −k � l∈E � j∈E,j�=l αl,j u2 2− � j∈E αk,j � j∈E,j�=l αl,j u3 2  +o(¯ α) (74) | Bi|≤  k∈E t 0 du  ¯αu+nk¯α u 2 2+n˜αk∗¯α∗u 3 2  ˜ Ci(t−u|X(k)(u,X)) +o(¯ α) ¯α k∈Et 0 duu+nk u2 2+n˜αk u3 3t−u ˜ Ti +o(¯α) =¯α ˜ Ti  k∈E t3 6+nk t4 24 +n˜αk t5 40  +o(¯α) Page 17 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 Appendix 9: Figures Conclusions This paper deals with a joint renewal process, whose component processes are coupled via failure rates depending linearly on the vector of backward times. It is shown such such a process can be described by a multidimensional renewal equation. In the case of stochastic independence an asymptotic approximation for the limiting cumulative number of events is derived. It is also shown, how the component processes become independent with one single quantity tending to zero. The model parameters can be estimated using the least squares principle. In order to prevent parameters such as rates and 0 2 4 6 8 10 12 14 16 18 20 1 91 181 271 361 451 541 631 721 811 901 991 1081 1171 1261 1351 1441 1531 1621 1711 1801 1891 1981 2071 2161 2251 2341 2431 2521 2611 2701 2791 2881 2971 3061 3151 3241 3331 3421 3511 3601 3691 3781 3871 3961 4051 4141 4231 4321 4411 4501 4591 4681 4771 4861 4951 Fig. 2 Approximating cumulated event numbers through a linear damage model 0 2 4 6 8 10 12 14 16 1 89 177 265 353 441 529 617 705 793 881 969 1057 1145 1233 1321 1409 1497 1585 1673 1761 1849 1937 2025 2113 2201 2289 2377 2465 2553 2641 2729 2817 2905 2993 3081 3169 3257 3345 3433 3521 3609 3697 3785 3873 3961 4049 4137 4225 4313 4401 4489 4577 4665 4753 4841 4929 Fig. 3 Approximating cumulated event numbers through a linear damage model Page 18 of 18 Mergenthaler et al. Decis. Anal. (2016) 3:2 damage parameters from becoming negative, one can temporarily use the squares of the parameters as the decision variables in the least squares functional equations. A numerical example shows how the cumulative number of events is approximated by a continuous function. The vector of backward times is by no means the only possible state variable to be used in a linear model. Rather, any statistic can be used, such as, for instance, the sliding average of the cumulated event numbers over a given embedding window. Authors’ contribution WM suggested the use of the paradigm of a Joint Renewal Process as a Model for Event Based Data. SF helped in elaborating the multidimensional analogon of the renewal equation. DJ was inbstrumental in carrying out most of the Laplace transformations. LL programmed the Least Square MInimization in order to determine the model parameters. All authors read and approved the final manuscript. 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