Nonstationarity in statistical process control — issues,cases, ideas

Statistical process control (SPC) is a powerful framework that is used in many industries to decrease process variability and to pinpoint special cause variation. Although a broad range of techniques have been developed to do so, often the real‐life situation does not fully comply with the basic assumptions that are made in SPC resulting in poor results. One of the main violations against the assumptions is the fact that industrial processes rarely behave in a stationary manner — this is evidently the case for biological processes but is also an important issue when monitoring industrial processes. Besides, the ever increasing amount of data, with a clear shift towards multivariate and even multiway quality control, makes the classical univariate approach not feasible anymore. These two observations pose important challenges to statisticians to develop novel SPC algorithms that are broadly applicable in modern industries. In this contribution we discuss both issues and use two very different case studies to show the reader recent directions and developments in the SPC landscape. Copyright © 2011 John Wiley & Sons, Ltd.     

Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process ?? on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of ??. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener?CIt? chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris?CFKG-inequalities for monotone functions of ??.     

Equivalent conditions of local asymptotics for the solutions of defective renewal equations, with applications

The paper obtains some equivalent conditions of local asymptotics for the solutions of defective renewal equations in the heavy-tailed case. As applications, the paper gives a different proof of a classical result about the local distribution of the supremum of a random walk. These results are also applied in examples involving the renewal function for terminating renewal processes and the age-dependent branching processes.     

Interaction Processes for Unions of Facets,the Asymptotic Behaviour with Increasing Intensity

In the series of models with interacting particles in stochastic geometry, a further contribution presents the facet process which is defined in arbitrary Euclidean dimension. In 2D, 3D specially it is a process of interacting segments, flat surfaces, respectively. Its investigation is based on the theory of functionals of finite spatial point processes given by a density with respect to a Poisson process. The methodology based on L ₂ expansion of the covariance of functionals of Poisson process is developed for U-statistics of facet intersections which are building blocks of the model. The importance of the concept of correlation functions of arbitrary order is emphasized. Some basic properties of facet processes, such as local stability and repulsivness are shown and a standard simulation algorithm mentioned. Further the situation when the intensity of the process tends to infinity is studied. In the case of Poisson processes a central limit theorem follows from recent results of Wiener-Ito theory. In the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of U-statistics and submodels of the facet process.     

Open problems in renewal, coupling and Palm theory

This note presents four sets of problems. The first suggests the possibility of a limit theory for null-recurrent renewal processes similar to the theory in the positive recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread out. The third concerns the coupling characterization of setwise convergence of distributions of stochastic processes to a stationary limit. The fourth concerns characterizations of mass-stationarity, a concept formalizing the intuitive idea that the origin is a typical location in the mass of a random measure. Mass-stationarity is an intrinsic characterization of Palm versions with respect to stationary random measures.     

Strong convergence of distributions of functionals of a sequence of processes,linearly generated by independent random variables

In the present paper we get sufficient conditions for the strong convergence of distributions of functionals of a sequence of stochastic processes, linearly generated by independent random variables, in the case when the distributions of these processes converge weakly to a Gaussian measure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 122–126, 1987.     

Malliavin calculus, geometric mixing, and expansion of diffusion functionals

Under geometric mixing condition, we presented asymptotic expansion of the distribution of an additive functional of a Markov or an ε-Markov process with finite autoregression including Markov type semimartingales and time series models with discrete time parameter. The emphasis is put on the use of the Malliavin calculus in place of the conditional type Cramér condition, whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach. In the second part, by means of the perturbation method and the operational calculus, we proved the geometric mixing property for non-symmetric diffusion processes, and presented a sufficient condition which is easily checked in practice. Accordingly, we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions, e.g., without the strong contractivity condition. Received: 7 September 1997 / Revised version: 17 March 1999     

Binary integer modeling of the traffic flow optimization problem,in the case of an autonomous transportation system

Our research aimed to optimize the transportation processes through the binary integer modeling of cooperative vehicle control by linking the dynamic traffic assignment approach and controlling the autonomous transport system. Our paper’s main contribution is a model transforming the optimal vehicle control problem into binary integer formulation, optimizing transport processes at the system level, and representing safety and dynamics related constraints on the vehicle level. Two small numerical case studies have illustrated the applicability and effectiveness of the model.     

The Geometric Markov Renewal Processes with Application to Finance

We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.     

Prediction of stable processes: Spectral and moving average representations

Summary For stable processes which are Fourier transforms of processes with independent increments, we obtain a Wold decomposition, we characterize their regularity and singularity, and, in the discrete-parameter case, we derive their linear predictors. In sharp contrast with the Gaussian case, regular stable processes which are Fourier transforms of processes with independent increments are not moving averages of stable motion.This research was supported by the Air Force Office of Scientific Research Grant No. F 49620 82C0009     

Rheology of a viscoelastic body under constant load,and prestressed with viscoelastic reinforcing

The present paper is a continuation of [1], which was concerned with a study of creep in a viscoelastic body and loss of stress in the viscoelastic reinforcing before application of an external load for the case of posttensioning of the reinforcing. Here we investigate the same rheological processes, but for the case where a constant external load has already been applied at a certain moment t₁. The equations obtained form a theoretical basis for designing, fabricating, and testing parts made of materials such as prestressed plastics.Mekhanika polimerov, Vol. 1, No. 1, pp. 50–60, 1965     

Diffusion approximation for storage processes

For a sequence of storage processes with general release rate functions which contain, as a special case, queuing processes, we show that under appropriate conditions, suitably normalized processes for storage processes converge to diffusions in the sense of law.     

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.     

On the uniqueness problem of two dimensional diffusion processes occurring in population genetics

Summary We shall investigate two dimensional diffusion processes occurring in population genetics. In order to construct such processes, we shall consider a martingale problem and discuss its uniqueness problem. Though the standard methods are not applicable in our case, we obtain the uniqueness in a fairly general case.     

Convexity,monotonicity, and gradient processes in Hilbert space

It is shown that in Hilbert spaces the gradient maps of convex functionals with uniformly bounded continuous second Frechét derivatives satisfy monotonicity conditions that insure that some convex combination of the identity, I, and I ? ▽f is either strictly contractive or at worst nonexpansive. This result leads to a complete resolution of the convergence question for a large class of associated gradient processes. In particular, weak convergence of the successive approximation sequence is established even in the singular case where f″ is not strictly positive at critical points of f.     

Race,SES, and mathematics achievement conflicts: how policy and political climate affect academic achievement

Socioeconomic status (SES) has been widely used as a determining factor to explain educational processes and outcomes such as mathematics academic achievement. Research has documented the links between SES and mathematics academic achievement. However, further understanding the complex relationship between contextual factors, such as policy, and its implications for these processes within an ideologically patriarchal society is paramount. After decades of United States school policy and reform—with the most recent focus on Common Core Standards—there continue to be inconsistent notions of what the “real” issues are and how to address those issues. This paper sets out to explore one specific case—House Bill 2281 (HB 2281) and, in effect, the banning of the Mexican–American Studies program in one school district in the US—in understanding the implications of policy in the shaping of the public education system. Implications for mathematics education research are explored.     

Some Problems of Local Hitting, Scaling, and Conditioning

The aim of this paper is to convey the basic ideas regarding some local hitting and conditioning properties of random measures. Some very general results are obtainable in the special case of simple point processes. Though much of this theory has no counterpart for general random measures, similar results do exist in two cases of special interest—for local time random measures and for Dawson–Watanabe superprocesses. This is an informal account of some of the basic hitting and conditioning properties common to all three cases. Precise statements and proofs will be provided elsewhere.     

On a multivariate central limit theorem for stationary bilinear processes

In 1957, Parzen proved a central limit theorem for a class of scalar processes which he called multilinear processes. In the present paper only stationary bilinear processes are considered, but the theory is generalized to the multivariate case.     

Stochastic processes with independent increments,taking values in a Hilbert space

Let $\text{H}$ be a real separable Hilbert space; let X(t), t?[0, 1], be a separable, stochastically continuous, $\text{H}$-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived.     

Coupled Continuous Time Random Maxima

Continuous Time Random Maxima (CTRM) are a generalization of classical extreme value theory: Instead of observing random events at regular intervals in time, the waiting times between the events are also random variables which have arbitrary distributions. In case that the waiting times between the events have infinite mean, the limit process that appears differs from the limit process that appears in the classical case. With a continuous mapping approach, we derive a limit theorem for the case that the waiting times and the subsequent events are dependent as well as for the case that the waiting times depend on the preceding events (in this case we speak of an Overshooting Continuous Time Random Maxima, abbr. OCTRM). We get the distribution functions of the limit processes and a formula for the Laplace transform in time of the CTRM and the OCTRM limit. With this formula we have another way to calculate the distribution functions of the limit processes, namely by inversion of the Laplace transform. Moreover, we present governing equations which in our case are time fractional differential equations whose solutions are the distribution functions of our limit processes.