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A joint renewal process used to model event based data
 Wolfgang Mergenthaler^{1}Email authorView ORCID ID profile,
 Daniel Jaroszewski^{1},
 Sebastian Feller^{1} and
 Larissa Laumann^{1}
 Received: 24 November 2015
 Accepted: 13 January 2016
 Published: 3 February 2016
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 MonteCarlo 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
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 PicadoMuino (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 PicadoMuino (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 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 statedependent 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 MonteCarlo 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

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.
The process
Theorem 1
Proof
Each of the events \(i \in \textit{E}\) occurs with probability \(\lambda _{i}(X(t))dt+o(dt)\), 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 \(1\sum _{i \in E} \lambda _{i}(X(t))dt+o(dt)\). More than one event occurs with probability o(dt) only. \(\square\)
The event rates
Corollary 1
Proof
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 \(\mathcal {F}_{t}\) be the sigma algebra generated by \(X(v), v \le t\) .
Preliminaries
Theorem 2
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(tX)), then the expected numbers are \((0,\ldots ,0)^T\). If it occurs during the time interval \([u,u+du)\) somewhere in the interval [0, t) and is of type \(i \in E\) (probability \(f_i(uX)du)\), then state X transforms into state \(X^{(i)}(X,u)\) and—therefore—the expected number, seen as a vector, is equal to \(e_i +\hat{C}(tuX^{(i)}(X,u))\). \(\square\)
Iterative approximation
Lemma 1
Proof
Please observe that (17) expresses the condition that the process will not “explode” at any time in a finite time interval.
A special case: stochastic independence
Assume (7) holds. Let \(\tilde{C}_i(t)\) be the solution of (14) under (7). The expected cumulated numbers of events then become independent. For each \(i \in E\) the following result can be proven, whereby \(\lambda := \lambda _i\) and \(\alpha := \alpha _{ii}\) has been set.
Corollary 2
Proof
The following conclusions are now easy to draw:
Corollary 3
Proof
(35) can immediately be derived from (22) by letting \(\lambda\) tend to zero. (36) must be concluded from equation (23), as (22) has been derived under the implicit assumption that \(\alpha \ne 0\) used in dividing the exponent by \(\alpha\), but the conclusion is straightforward. \(\square\)
Equation (35) has an interesting application. Assume (6) holds and, in addition \(\lambda =0\) can be safely assumed. In that case (35) allows to estimate \(\alpha\) by equating the slope of \(\hat{C}(t\alpha =0)\) with the coefficient of t.
A first order correction
Proposition 1
The estimation problem
Estimating \(\lambda\) and \(\alpha\) with the least squares principle
Functional equations

Differentiating (54) with respect to \(\lambda \, \text{and} \, \alpha _{i,j},i\in E,j \in E\)

Setting the results equal to zero and

Solving the resulting system of equations
Preparing the Numerical Solution
(55), when used in a minimization routine such as the FRPR method, may result in negative values for \(\lambda _{i}\) and \(\alpha _{i,j},i\in E,j\in E\). Both are undesirable effects. Negative values for \(\lambda _{i}\) would imply negative random components of the event rates, negative values for \(\alpha _{i,j},i \in E,j\in E\) would imply an unlikely healing effect, whereby the occurrence of events becomes less likely, the longer the backward time is. While this assumption is not entirely unlikely, it is not going to be considered any further in this paper.
Example
The problem at hand has its origins in monitoring trains in the German railroad industry. There is an abundance of historical data, however, some or most of it is discrete, event based material. Only slowly sensor data becomes available, less so because of technical reasons, but mostly due to the architectural database design complexity. This is why, parallel to stochastic time series analysis as used in evaluating sensor data, a mathematical model is needed to deal with event based data.
Figures 2 and 3 in Appendix 9 give two examples of approximating a cumulated event curve with an event rate model as given by (6). Both figures show the typical behaviour of this model in as far as the event rate grows quadratically with the backward time. This behaviour becomes very marked, when long backward times are observed. In a simulation context, this is not a critical phenomenon, because quadratically increasing failure rates make sure that untypically long interevent times become unlikely. Figure 3 also shows some typical behaviour of the model used: If there is enough structure in the time series and enough events, i.e. short backward times, then the model approximates the staircase structure represented by the cumulated event numbers to a sufficient degree of precision. As soon as backward times become long, event rates become prohibitively large. In Fig. 2 the following parameters have been used:\(n= 10,\lambda _i = 0.0001 + 0.0001*Z\) \(\alpha [i1,i] = 0.00001,\alpha [i,i] = 0.00001,\alpha [i,i+1] = 0.00001,\alpha [i,i+2] = 0.00001\). Figure 3 contains the approximation result of yet another example, whereby \(n = 10,\lambda _i = 0.0005 + 0.0005*Z\) \(\alpha [i1,i] = 0.0001 ,\alpha [i,i] = 0.0002, \alpha[i,i+1] = 0.0001,\alpha [i,i+2] = 0.000001\).
In both figures Z is a uniformly distributed random variable between 0 and 1. Please note the “quadratically explosive” nature of the expected cumulated event function with increasing backward times in Fig. 3.
Appendix
Appendix 1
Appendix 2
Appendix 3
Use the substitution \(v =\sqrt{\alpha }t+\frac{\lambda + s}{\sqrt{\alpha }}\), \(dv = \sqrt{\alpha } dt\), \(t=0\Rightarrow v = \frac{\lambda +s}{\sqrt{\alpha }}\), \(t=\infty \Rightarrow v =\infty\).
Appendix 4
Appendix 5
Appendix 6
Appendix 7
Appendix 8
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 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.
Declarations
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.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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