You-xiang CUI, Lei SUN*, Li-hui SUI, Xiao-min XU and Jun KANG
Department of Quality Management Engineering, Businness School, Shanghai Dianji University
No.1350 Ganlan Road, Lingang New City, Pudong District, Shanghai 201306, China
a[email protected], b[email protected], c [email protected]
* Corresponding author
Keywords: Random shock model, Time series, ARIMA models
Abstract. There are an unlimited number of ways in which a process can be nonstationary. However, the types of economic and industrial series that we wish to analyze frequently exhibit a particular kind of homogeneous nonstationary behaviort. A random shock model for a system whose state deteriorates continuously is introduced and analyzed. This paper is also discuss the truncated form of the random shock model.
Introduction
In today’s technological world, nearly everyone depends upon the continued functioning of a wide array of complex machinery and equipment for our everyday safety,security, mobility and economic welfare. The term white noise is sometimes used in the context of phylogenetically based statistical methods to refer to a lack of phylogenetic pattern in comparative data.
In statistics and econometrics one often assumes that an observed series of data values is the sum of a series of values generated by a deterministic linear process, depending on certain independent (explanatory) variables, and on a series of random noise values. Then regression analysis is used to infer the parameters of the model process from the observed data, e.g. by ordinary least squares, and to test the null hypothesis that each of the parameters is zero against the alternative hypothesis that it is non-zero. Hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and the same Gaussian probability distribution — in other words, that the noise is white. If there is non-zero correlation between the noise values underlying different observations then the estimated model parameters are still unbiased, but estimates of their uncertainties (such as confidence intervals) will be biased (not accurate on average). This is also true if the noise is heteroskedastic — that is, if it has different variances for different data points.
Alternatively, in the subset of regression analysis known as time series analysis there are often no explanatory variables other than the past values of the variable being modeled (the dependent variable). In this case the noise process is often modeled as a moving average process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.
There are an unlimited number of ways in which a process can be nonstationary. However, the types of economic and industrial series that we wish to analyze frequently exhibit a particular kind of homogeneous nonstationary behavior that can be represented by a stochastic model.
An important class of stochastic models for describing time series, which has received a great deal of attention, comprises what are called stationary models. Stationary models assume that the process remains in statistical equilibrium with probabilistic properties that do not change over time, in particular varying about a fixed constant mean level and with constant variance. However, forecasting has been of particular importance in industry, business, and economics, where many time series are often better represented as nonstationary and, in particular, as having no natural constant mean level over time.
A general linear stochastic model is described that supposes a time series to be generated by a
2016 International Conference on Artificial Intelligence and Computer Science (AICS 2016) ISBN: 978-1-60595-411-0
that use parameters parsimoniously. Parsimony may often be achieved by representation of the linear process in terms of a small number of autoregressive and moving average terms. The properties of the resulting autoregressive–moving average (ARMA) models are discussed in preparation for their use in model building.
Linear Filter Model
The stochastic models we employ are based on the idea that an observable time series ztin which successive values are highly dependent can frequently be regarded as generated from a series of independent “shocks” at. These shocks are random drawings from a fixed distribution, usually assumed normal and having mean zero and variance a2. Such a sequence of independent random
variables at, at1,at2 …is called a white noise process,as shown in Figure 1. (B)
White noise
at zt
Figure 1 Representation of a time series as the output from a linear filter
The white noise process at is supposed transformed to the process zt by what is called a linear
filter . The linear filtering operation simply takes a weighted sum of previous random shocks at, so that
t 1 t1 2 t2
t a a a
z
t a B) (
(1)
In general, is a parameter that determines the “level” of the process, and is the linear operator that
transforms at into zt and is called the transfer function of the filter. The model can allow for a
flexible range of patterns of dependence among values of the process {zt} expressed in terms of the
independent (unobservable) random shocks at .
Random Shock Form of the Model
A linear model can be written as the output zt from the linear filter,whose input is white noise, or a
sequence of uncorrelated shocks at with mean 0 and common variance
2
a
.It is sometimes useful to express the ARIMA model . in particular, the weights will be needed in calculate the variance of the forecast errors. However, since the nonstationary ARIMA processes are not in statistical
equilibrium over time, they cannot be assumed to extend infinitely into the past, and hence an infinite representation will not be possible. But a related finite truncated form, which will be discussed subsequently, always exists. We now show that the weights for an ARIMA process may be obtained directly from the difference equation form of the model.
operate on both sides of (1)with the generalized autoregressive operator (B), we obtain
t
t B B a
z
B) ( ) ( )
(
(2)
.the weights may be obtained by equating coefficients of B in the expansion
qq d
p d
p B B B B B
B
1
2 2 1
1 1 1
1
(3)
the jweights of the ARIMA process can be determined recursively through the equations j d p j d p j j
j
1 1 2 2
j>0 (4)
The weights satisfy the homogeneous difference equation defined by the generalized autoregressive operator 0 ) 1 )( ( )
(Bj B B dj
(5)
Since |φ| < 1, the weights j tend to A0 for large j, so that shocks at−j, which entered in the
remote past, receive a constant weight A0. However, the representation is strictly not valid because
the infinite sum on the right does not converge in any sense; that is, the weights j are not absolutely
summable as in the case of a stationary process. A related truncated version of the random shock form of the model is always valid. The infinite random shock form an ARIMA process, even though this form is strictly not convergent device to represent the valid truncated form.
Truncated Form of the Random Shock Model
In order to define the notion of "white noise" in the theory of continuous-time signals, one must replace the concept of a "random vector" by a continuous-time random signal; that is, a random process that generates a function of a real-valued parameter.
The model can be express the current value zt of the process in terms of the t − k shocks at
1 1, , , t k
t a a
a
, which have entered the system since some time origin k <t. This time origin k might be the time at which the process was first observed.
The general model (B)zt (B)atis a difference equation with the solution
) ( )
(t k I t k C
zt k k
(6)
The solution of such equations closely parallels the solution of linear differential equations. The complimentary function Ck(tk)is the general solution of the homogeneous difference equation
0 ) ( )
(B Ck tk
(7)
In general, this solution will consist of a linear combination of certain functions of time. These functions are powers tj , real geometric (exponential) terms Gt and complex geometric (exponential) terms Dtsin(2f0tF) , where the constants G,f0 , and F are functions of the parameters (,) of the model. The coefficients that form the linear combinations of these terms can be determined so as to satisfy a set of initial conditions defined by the values of the process before time k + 1.
The particular integral Ik(tk) is any function that satisfies
t
k t k B a
I
B) ( ) ( )
(
(8)
In this expression B operates on t and not on k.Equation (8)is satisfied for t −k >q by
1 0 1 2 2 1 1 ) ( k t j k k t t t t j t jk t k a a a a a
I
t>k (9)
the time k + 1. Hence the truncated form of the random shock model for the ARIMA process is given by
1
0
) (
k t
j
k j t j
t a C t k
z
(10)
Any observation zt can be considered in relation to any previous time k and can be divided up into two additive parts. The first part Ck(t − k) is the component of zt , already determined at time k, and
indicates what the observations prior to time k + 1 had to tell us about the value of the series at time t . It represents the course that the process would take if at time k, the source of shocks at had been “switched off.” The second part, Ik(t − k), represents an additional component, unpredictable at time
k, which embodies the entire effect of shocks entering the system at time k. Hence, to specify an ARIMA process, one must specify the initializing component Ck(t − k) for some time origin k in the finite (but possibly remote) past,with the remaining course of the process being determined through the truncated random shock terms.
Acknowledgment
This research was financially supported by key course construction (Standardizing Engineering: 043206P1) and Management Science and Engineering (16YSXK02) in Shanghai Dianji University.
References
[1] Moskowitz, Ian H., et al. "Improved predictions of alarm and safety system performance through process and operator response‐time modeling." Aiche Journal (2016).
[2] Liu, Yu, H. Z. Huang, and H. Pham. "Reliability evaluation of systems with degradation and random shocks." IEEE (2008):328-333.
[3] Lim, Kyung Eun, J. S. Baek, and E. Y. Lee. "A random shock model for a continuously deteriorating system." Journal of Quality in Maintenance Engineering 11.3(2005):206-215.
[4] Box G E P, Jenkins G M, Reinsel G C. Time series analysis: forecasting and control[M]. Wiley. com. (2013)
[5] Tan, H., et al. "Short-Term Traffic Prediction Based on Dynamic Tensor Completion." IEEE Transactions on Intelligent Transportation Systems (2016):1-11.
[6] Yun, Fong Lim, Z. Yan, and W. Chen. "A quay crane system that self-recovers from random shocks." Flexible Services and Manufacturing Journal 27.4(2015):1-24.
[7] Chang, T. C., and F. F. Gan. "Charting techniques for monitoring a random shock process." Quality & Reliability Engineering 15.4(1999):295-301.
[8] Lim, Kyung Eun, J. S. Baek, and E. Y. Lee. "A random shock model for a continuously deteriorating system." Journal of Quality in Maintenance Engineering 11.3(2005):206-215.
[9] Mercier, Sophie, and H. P. Hai. "A Random Shock Model with Mixed Effect, Including Competing Soft and Sudden Failures, and Dependence." Methodology and Computing in Applied Probability 18.2(2016):377-400.
[10] Chen, P. "Economic Complexity: Fundamental Issues And Policy Implications† The Nature of Persistent Business Cycles: Random Shocks, Microfoundations, or Color." Time.dufe.edu.cn.
[12] Eliete Nascimento Pereira, and Cassius Tadeu Scarpin. "TIME SERIES FORECASTING BY USING A NEURAL ARIMA, MODEL BASED ON WAVELET DECOMPOSITION." Independent Journal of Management & Production 7.1(2016):págs. 252-270.
[13] Stone, Richard. "Random shocks in a simple growth model." Economic Modelling 1.3(1984):277-280.
[14] Aseev, S., K. Besov, and S. E. Ollus. "Optimal growth in a two-sector economy facing an expected random shock." Proceedings of the Steklov Institute of Mathematics 276.1(2012):4-34.
[15] Yu, Miaomiao, and Y. Tang. "Optimal replacement policy based on maximum repair time for a random shock and wear model." TOP (2016):1-15.
[16] Cui, Xue, and T. Shibata. "Investment strategies, random shock and asymmetric information (Financial Modeling and Analysis)." Rims Kokyuroku 1983(2016).
