Infinite-order Model Research Articles

Method of Dynamic VaR and CVaR Risk Measures Forecasting for Long Range Dependent Time Series on the Base of the Heteroscedastic Model

The paper proposes a new method of dynamic VaR and CVaR risk measures forecasting. The method is designed for obtaining the forecast estimates of risk measures for volatile time series with long range dependence. The method is based on the heteroskedastic time series model. The FIGARCH model is used for volatility modeling and forecasting. The model is reduced to the AR model of infinite order. The reduced system of Yule-Walker equations is solved to find the autoregression coefficients. The regression equation for the autocorrelation function based on the definition of a long-range dependence is used to get the autocorrelation estimates. An optimization procedure is proposed to specify the estimates of autocorrelation coefficients. The procedure for obtaining of the forecast values of dynamic risk measures VaR and CVaR is formalized as a multi-step algorithm. The algorithm includes the following steps: autoregression forecasting, innovation highlighting, obtaining of the assessments for static risk measures for residuals of the model, forming of the final forecast using the proposed formulas, quality analysis of the results. The proposed method is applied to the time series of the index of the Tokyo stock exchange. The quality analysis using various tests is conducted and confirmed the high quality of the obtained estimates.

Hawkes and INAR([formula omitted]) processes

In this paper, we discuss integer-valued autoregressive time series (INAR), Hawkes point processes, and their interrelationship. Besides presenting structural analogies, we derive a convergence theorem. More specifically, we generalize the well-known INAR(p), p∈N, time series model to a corresponding model of infinite order: the INAR(∞) model. We establish existence, uniqueness, finiteness of moments, and give formulas for the autocovariance function as well as for the joint moment-generating function. Furthermore, we derive a branching-process–as well as an AR(∞)–and an MA(∞) representation for the model. We compare Hawkes process properties with their INAR(∞) counterparts. Given a Hawkes process N, in the main theorem of the paper we construct an INAR(∞)-based family of point processes and prove its convergence to N. This connection between INAR and Hawkes models will be relevant in applications.

Truncation errors of assumed shape modeling for flexible-link manipulators

The assumed mode shape method has been widely used to derive finite degree-of-freedom dynamic models for flexible-link manipulators, which theoretically have infinite degree-of-freedom dynamics. For a single flexible manipulator, this approximation changes locations of the zeros of transfer functions between base torque and end-effector displacement. The change in locations of zeros considerably affects accuracy of the model and therefore the performance of model-based controllers. This article presents a comprehensive study on the change in locations of zeros due to the truncation associated with assumed mode shape method. It is shown that the locations of approximate zeros depend on four non-dimensional parameters, whereas the locations of analytical zeros depend on only two non-dimensional parameters. Approximate zeros are obtained from assumed mode shape method models, whereas analytical zeros are derived from infinite order models. A thorough study of the differences between approximate zeros and analytical zeros versus the number of mode shapes as well as all the physical parameters is performed. Moreover, guidelines are provided to select the numbers of mode shapes such that the approximate zeros become close to the analytical zeros. These guidelines can easily be used by control and modeling engineers, making them valuable for modeling and control of flexible robot manipulators.

Structural analysis of masonry elements in architecture

The present article establishes the vantage required, in the author’s view, to analyze masonry structure performance. Until the nineteenth century, the entire history of architecture revolved around the various styles of such structures. The introduction of high toughness materials (cast iron and steel) with the industrial revolution and the adoption of rigidity as a parameter for compatible analysis laid the grounds, both in engineering training and professional practice, for analytical methods based on elastic rigidity (stress-strain curve). Masonry structures cannot be analyzed on such a basis: stone or brick elasticity is not readily measured. As a result, they are viewed as mere aggregates of solids working under compression only, pursuant to a statically indeterminate model of infinite order as a result of their likewise infinite number of articulated bonds. Moreover, like all structures they change with time, rendering any analytical model trivial unless account is taken of the actual conditions (fissures or cracking) prevailing in the structure at any given time of its service life. This paper modestly attempts to clarify the assumptions on which analysis is built, evaluating the basic principles of equilibrium, the only valid grounds for such analysis, and the conditions in which such equilibrium can be reached. Yesterday’s builders drew exclusively from experience to ensure equilibrium: for them structural safety meant the absence of collapse. Today vector algebra, observation with high precision instruments and a priori determination of the conditions that may lead to collapse underlie accurate and reliable conjecture about the performance of masonry structures that tangibly establishes safety levels at least as precisely as sophisticated methods of analysis.

AN EXAMINATION OF ESTIMATED RESIDUALS IN A REGRESSION WITH AN INFINITE ORDER PARAMETRIC MODEL

We consider a linear regression with the error term that obeys an autoregressive model of infinite order and estimate parameters of the models. The parameters of the autoregressive model should be estimated based on estimated residuals obtained by means of the method of ordinary least squares, because the errors are unobservable. The consistency of the coefficients, variance and spectral density of the model obeyed by the error term is shown. Further, we estimate the coefficients of the linear regression by means of the method of estimated generalized least squares. We also show the consistency of the estimator. †Professor Toyooka sadly died at 8 April 1998. I, Usami, pray his soul may rest in peace. Most of the results in Section 3 had already been obtained before April 1998. Usami derived the results in Section 4.

Nonlinear AR modeling

A major reason for the success of linear autoregressive (AR) modeling is that Kolmogrorov proved that every linear system could be represented by a linear AR model of infinite order. The computation of a finite order AR approximation is, of course, the practical goal. In this paper, we prove that every nonlinear system with a Volterra series expansion can be represented as a nonlinear AR model of infinite order. Our method shows how an approximation to any desired order and degree can be achieved.