A Bayesian approach to estimating the regression coefficients of a multinomial logit model

A Bayesian approach to estimation of the regression coefficients of a multinominal logit model with ordinal scale response categories is presented. A Monte Carlo method is used to construct the posterior distribution of the link function. The link function is treated as an arbitrary scalar function. Then the Gauss-Markov theorem is used to determine a function of the link which produces a random vector of coefficients. The posterior distribution of the random vector of coefficients is used to estimate the regression coefficients. The method described is referred to as a Bayesian generalized least square (BGLS) analysis. Two cases involving multinominal logit models are described. Case I involves a cumulative logit model and Case II involves a proportional-odds model. All inferences about the coefficients for both cases are described in terms of the posterior distribution of the regression coefficients. The results from the BGLS method are compared to maximum likelihood estimates of the regression coefficients. The BGLS method avoids the nonlinear problems encountered when estimating the regression coefficients of a generalized linear model. The method is not complex or computationally intensive. The BGLS method offers several advantages over Bayesian approaches. ^

NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Analysis of life expectancy using linear and nonlinear models

Life expectancy has consistently increased over the last 150 years due to improvements in nutrition, medicine, and public health. Several studies found that in many developed countries, life expectancy continued to rise following a nearly linear trend, which was contrary to a common belief that the rate of improvement in life expectancy would decelerate and was fit with an S-shaped curve. Using samples of countries that exhibited a wide range of economic development levels, we explored the change in life expectancy over time by employing both nonlinear and linear models. We then observed if there were any significant differences in estimates between linear models, assuming an auto-correlated error structure. When data did not have a sigmoidal shape, nonlinear growth models sometimes failed to provide meaningful parameter estimates. The existence of an inflection point and asymptotes in the growth models made them inflexible with life expectancy data. In linear models, there was no significant difference in the life expectancy growth rate and future estimates between ordinary least squares (OLS) and generalized least squares (GLS). However, the generalized least squares model was more robust because the data involved time-series variables and residuals were positively correlated. ^

Planimetric ice-flow convergence and flow-orthonormal strain rate across the Antarctic Ice Sheet, with links to model result files

Fast-flowing ice streams discharge most of the ice from the interior of the Antarctic Ice Sheet coastward. Understanding how their tributary organisation is governed and evolves is essential for developing reliable models of the ice sheet's response to climate change. Despite much research on ice-stream mechanics, this problem is unsolved, because the complexity of flow within and across the tributary networks has hardly been interrogated. Here I present the first map of planimetric flow convergence across the ice sheet, calculated from satellite measurements of ice surface velocity, and use it to explore this complexity. The convergence map of Antarctica elucidates how ice-stream tributaries draw ice from the interior. It also reveals curvilinear zones of convergence along lateral shear margins of streaming, and abundant convergence ripples associated with nonlinear ice rheology and changes in bed topography and friction. Flow convergence on ice-stream tributaries and their feeding zones is markedly uneven, and interspersed with divergence at distances of the order of kilometres. For individual drainage basins as well as the ice sheet as a whole, the range of convergence and divergence decreases systematically with flow speed, implying that fast flow cannot converge or diverge as much as slow flow. I therefore deduce that flow in ice-stream networks is subject to mechanical regulation that limits flow-orthonormal strain rates. These properties and the gridded data of convergence and flow-orthonormal strain rate in this archive provide targets for ice- sheet simulations and motivate more research into the origin and dynamics of tributarization.

Age model for ODP Leg 175 sites

Upwelling along the western coast of Africa south of the equator may be partitioned into three major areas, each having its own dynamics and history: (1) the eastern equatorial region, comprising the Congo Fan and the area of Mid-Angola; (2) the Namibia upwelling system, extending from the Walvis Ridge to Lüderitz; and (3) the Cape Province region, where upwelling is subdued. The highest nutrient contents in thermocline waters are in the northern region, the lowest in the southern one. Wind effects are at a maximum near the southern end of the Namibia upwelling system, and maximum productivity occurs near Walvis Bay, where the product between upwelling rate and nutrient content of upwelled waters is at a maximum. In the Congo/Angola region, opal tends to follow organic carbon quite closely in the Quaternary record. However, organic carbon has a strong precessional component, while opal does not. Despite relatively low opal content, sediments off Angola show the same patterns as those off the Congo; thus, they are part of the same regime. The spectrum shows nonlinear interference patterns between high- and low-latitude forcing, presumably tied to thermocline fertility and wind. On Walvis Ridge, as in the Congo-Angola region, the organic matter record behaves normally; that is, supply is high during glacial periods. In contrast, interglacial periods are favorable for opal deposition. The pattern suggests reduction in silicate content of the thermocline during glacial periods. The reversed phase (opal abundant during interglacials) persists during the entire Pleistocene and can be demonstrated deep into the Pliocene, not just on Walvis Ridge but all the way to the Oranje River and off the Cape Province. From comparison with other regions, it appears that silicate is diminished in the global thermocline, on average, whenever winds become strong enough to substantially shorten the residence time of silicate in upper waters (Walvis Hypothesis, solving the Walvis Paradox of reversed phase in opal deposition). The central discovery during Leg 175 was the documentation of a late Pliocene opal maximum for the entire Namibia upwelling system (early Matuyama Diatom Maximum [MDM]). The maximum is centered on the period between the end of the Gauss Chron and the beginning of the Olduvai Chron. A rather sharp increase in both organic matter deposition and opal deposition occurs near 3 Ma in the middle of the Gauss Chron, in association with a series of major cooling steps. As concerns organic matter, high production persists at least to 1 Ma, when there are large changes in variability, heralding subsequent pulsed production periods. From 3 to 2 Ma, organic matter and opal deposition run more or less parallel, but after 2 Ma opal goes out of phase with organic matter. Apparently, this is the point when silicate becomes limiting to opal production. Thus, the MDM conundrum is solved by linking planetary cooling to increased mixing and upwelling (ramping up to the MDM) and a general removal of silicate from the upper ocean through excess precipitation over global supply (ramping down from the MDM). The hypothesis concerning the origin of the Namibia opal acme or MDM is fundamentally the same as the Walvis Hypothesis, stating that glacial conditions result in removal of silicate from the thermocline (and quite likely from the ocean as a whole, given enough time). The Namibia opal acme, and other opal maxima in the latest Neogene in other regions of the ocean, marks the interval when a cooling ocean selectively removes the abundant silicate inherited from a warm ocean. When the excess silicate is removed, the process ceases. According to the data gathered during Leg 175, major upwelling started in the late part of the late Miocene. Presumably, this process contributed to the drawing down of carbon dioxide from the atmosphere, helping to prepare the way for Northern Hemisphere glaciation.

Trade Coefficients and the Role of Elasticity in a Spatial CGE Model Based on the Armington Assumption

The Armington Assumption in the context of multi-regional CGE models is commonly interpreted as follows: Same commodities with different origins are imperfect substitutes for each other. In this paper, a static spatial CGE model that is compatible with this assumption and explicitly considers the transport sector and regional price differentials is formulated. Trade coefficients, which are derived endogenously from the optimization behaviors of firms and households, are shown to take the form of a potential function. To investigate how the elasticity of substitutions affects equilibrium solutions, a simpler version of the model that incorporates three regions and two sectors (besides the transport sector) is introduced. Results indicate: (1) if commodities produced in different regions are perfect substitutes, regional economies will be either autarkic or completely symmetric and (2) if they are imperfect substitutes, the impact of elasticity on the price equilibrium system as well as trade coefficients will be nonlinear and sometimes very sensitive.

Column Generation Algorithms for Nonlinear Optimization II: Numerical Investigations

García et al. present a class of column generation (CG) algorithms for nonlinear programs. Its main motivation from a theoretical viewpoint is that under some circumstances, finite convergence can be achieved, in much the same way as for the classic simplicial decomposition method; the main practical motivation is that within the class there are certain nonlinear column generation problems that can accelerate the convergence of a solution approach which generates a sequence of feasible points. This algorithm can, for example, accelerate simplicial decomposition schemes by making the subproblems nonlinear. This paper complements the theoretical study on the asymptotic and finite convergence of these methods given in [1] with an experimental study focused on their computational efficiency. Three types of numerical experiments are conducted. The first group of test problems has been designed to study the parameters involved in these methods. The second group has been designed to investigate the role and the computation of the prolongation of the generated columns to the relative boundary. The last one has been designed to carry out a more complete investigation of the difference in computational efficiency between linear and nonlinear column generation approaches. In order to carry out this investigation, we consider two types of test problems: the first one is the nonlinear, capacitated single-commodity network flow problem of which several large-scale instances with varied degrees of nonlinearity and total capacity are constructed and investigated, and the second one is a combined traffic assignment model

A constitutive model for the in-plane mechanical behavior of nonwoven fabrics

A constitutive model is presented for the in-plane mechanical behavior of nonwoven fabrics. The model is developed within the context of the finite element method and provides the constitutive response for a mesodomain of the fabric corresponding to the area associated to a finite element. The model is built upon the ensemble of three blocks, namely fabric, fibers and damage. The continuum tensorial formulation of the fabric response rigorously takes into account the effect of fiber rotation for large strains and includes the nonlinear fiber behavior. In addition, the various damage mechanisms experimentally observed (bond and fiber fracture, interfiber friction and fiber pull-out) are included in a phenomenological way and the random nature of these materials is also taken into account by means of a Monte Carlo lottery to determine the damage thresholds. The model results are validated with recent experimental results on the tensile response of smooth and notched specimens of a polypropylene nonwoven fabric.