The behaviour of the likelihood function for ARMA models

1984 ◽  
Vol 16 (4) ◽  
pp. 843-866 ◽  
Author(s):  
M. Deistler ◽  
B. M. Pötscher
Keyword(s):  
Parameter Space ◽  
Likelihood Function ◽  
Arma Models ◽  
Parameter Spaces ◽  
Continuity Properties ◽  
Spectral Components ◽  
Usual Parameter

The paper deals with some properties of the (Gaussian) likelihood function for multivariable ARMA models. Its behaviour at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed. We have not been able to show in general the existence of the maximum over the usual parameter spaces. However, the maximum always exists over a suitably enlarged parameter space (given that the data are non-degenerate), which includes parameters corresponding to processes with discrete spectral components.

The behaviour of the likelihood function for ARMA models

1984 ◽  
Vol 16 (04) ◽  
pp. 843-866 ◽  
Author(s):  
M. Deistler ◽  
B. M. Pötscher
Keyword(s):  
Parameter Space ◽  
Likelihood Function ◽  
Arma Models ◽  
Parameter Spaces ◽  
Continuity Properties ◽  
Spectral Components ◽  
Usual Parameter

The paper deals with some properties of the (Gaussian) likelihood function for multivariable ARMA models. Its behaviour at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed. We have not been able to show in general the existence of the maximum over the usual parameter spaces. However, the maximum always exists over a suitably enlarged parameter space (given that the data are non-degenerate), which includes parameters corresponding to processes with discrete spectral components.

Noninvertibility and Pseudo-Maximum Likelihood Estimation of Misspecified ARMA Models

1991 ◽  
Vol 7 (4) ◽  
pp. 435-449 ◽  
Author(s):  
B.M. Pötscher
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Function ◽  
Moving Average ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Arma Models ◽  
Limiting Behavior ◽  
Pseudo Likelihood ◽  
Moving Average Model ◽  
Pseudo Maximum Likelihood

Recently Tanaka and Satchell [11] investigated the limiting properties of local maximizers of the Gaussian pseudo-likelihood function of a misspecified moving average model of order one in case the spectral density of the data process has a zero at frequency zero. We show that pseudo-maximum likelihood estimators in the narrower sense, that is, global maximizers of the Gaussian pseudo-likelihood function, may exhibit behavior drastically different from that of the local maximizers. Some general results on the limiting behavior of pseudo-maximum likelihood estimators in potentially misspecified ARMA models are also presented.

scholarly journals Relation Between The Roughness, Linear Entropy and Visibility of A Quantum State, The Jaynes-Cummings Model.

2021 ◽  
Author(s):  
Adélcio Carlos de Oliveira ◽  
Mauricio Reis
Keyword(s):  
Quantum State ◽  
Parameter Space ◽  
Linear Entropy ◽  
Parameter Spaces ◽  
Dispersive Term

Abstract In this work, an analysis of the Jaynes-Cummings Model is conducted in the parameter spaces, composed of Roughness, Concurrence/Linear Entropy and Visibility. The analysis was carried out without including the effects of the environment and with the inclusion of a dispersive environment. As Roughness measures the state’s degree of non-classicality, its inclusion in the analysis allows to identify points in the dynamics that are not usually perceived by traditional analysis. It is observed that the parameter space is almost completely occupied when the dispersive term is small, and is concentrated in the region of less roughness and less purity as the dispersive coefficient is increased.

COMPLETING FUZZY IF-THEN RULE BASES BY MEANS OF SMOOTHING SPLINES

2006 ◽  
Vol 14 (02) ◽  
pp. 235-244 ◽  
Author(s):  
THOMAS VETTERLEIN ◽  
MARTIN ŠTEPNICKA
Keyword(s):  
Fuzzy Sets ◽  
Parameter Space ◽  
Fuzzy Inference ◽  
Smoothing Splines ◽  
Partial Function ◽  
Rule Base ◽  
Parameter Spaces ◽  
Finite Dimensional ◽  
Rule Bases ◽  
Do So

A fuzzy if-then rule base may be viewed as a partial function between universes of fuzzy sets. For the construction of a fuzzy inference module, this partial function needs to be extended to a total one. Here, we propose a new method how to do so, making use of the method of smoothing splines. To this end, we identify the fuzzy sets with elements of a finite-dimensional real parameter space in an approximate way, using Perfilieva's fuzzy transforms. We then determine a function between two such parameter spaces by requiring that it reproduces the rule base as precise as possible and that it minimizes a parameter depending on its smoothness.

A GALLERY OF CHUA ATTRACTORS: PART III

2007 ◽  
Vol 17 (03) ◽  
pp. 657-734 ◽  
Author(s):  
ELEONORA BILOTTA ◽  
FAUSTO STRANGES ◽  
PIETRO PANTANO
Keyword(s):  
Parameter Space ◽  
Information Seeking ◽  
Ad Hoc ◽  
Three Dimensional ◽  
High Dimensional ◽  
Special Focus ◽  
Processes Of Change ◽  
Parameter Spaces ◽  
Chua’S Oscillator ◽  
Components Analysis

The visualization of patterns related to chaos is a challenge for those who are part of today's dynamical systems community, especially when we consider the aim of providing users with the ability to visually analyze and explore large, complex datasets related to chaos. Thus visualization could be considered a useful element in the discovery of unexpected relationships and dependencies that may exist inside the domain of chaos, both in the phase and the parameter spaces. In the second part of "A Gallery of Chua attractors", we presented an overview of forms which can only be produced by the physical circuit. In Part III, we illustrate the variety and beauty of the strange attractors produced by the dimensionless version of the system. As in our earlier work, we have used ad hoc methods, such as bifurcation maps and software tools, allowing rapid exploration of parameter space. Applying these techniques, we show how it is possible, starting from attractors described in the literature, to find new families of patterns, with a special focus on the cognitive side of information seeking and on qualitative processes of change in chaos, thus demonstrating that traditional categories of chaos exploration need to be renewed. After a brief introduction to dimensionless equations for Chua's oscillator, we show 150 attractors, which we represent using three-dimensional images, time series and FFT diagrams. For the most important patterns, we also report Lyapunov exponents. To show the position of dimensionless attractors in parameter space, we use parallel coordinate techniques that facilitate the visualization of high dimensional spaces. We use Principal Components Analysis (PCA) and Mahalanobis Distance to provide additional tools for the exploration and visualization of the structure of the parameter space.

scholarly journals Understanding hormonal crosstalk in Arabidopsis root development via emulation and history matching

2020 ◽  
Vol 19 (2) ◽  
Author(s):  
Samuel E. Jackson ◽  
Ian Vernon ◽  
Junli Liu ◽  
Keith Lindsey
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Parameter Space ◽  
History Matching ◽  
Parameter Spaces ◽  
Arabidopsis Root ◽  
Biologically Relevant ◽  
Hormonal Crosstalk ◽  
The Mathematical Model ◽  
Insight Into

AbstractA major challenge in plant developmental biology is to understand how plant growth is coordinated by interacting hormones and genes. To meet this challenge, it is important to not only use experimental data, but also formulate a mathematical model. For the mathematical model to best describe the true biological system, it is necessary to understand the parameter space of the model, along with the links between the model, the parameter space and experimental observations. We develop sequential history matching methodology, using Bayesian emulation, to gain substantial insight into biological model parameter spaces. This is achieved by finding sets of acceptable parameters in accordance with successive sets of physical observations. These methods are then applied to a complex hormonal crosstalk model for Arabidopsis root growth. In this application, we demonstrate how an initial set of 22 observed trends reduce the volume of the set of acceptable inputs to a proportion of 6.1 × 10−7 of the original space. Additional sets of biologically relevant experimental data, each of size 5, reduce the size of this space by a further three and two orders of magnitude respectively. Hence, we provide insight into the constraints placed upon the model structure by, and the biological consequences of, measuring subsets of observations.

Low-Cost Surrogate Modeling of Miniaturized Microwave Components Using Nested Kriging

2021 ◽  
Vol 35 (11) ◽  
pp. 1346-1347
Author(s):  
Anna Pietrenko-Dabrowska ◽  
Slawomir Koziel
Keyword(s):  
Parameter Space ◽  
Low Cost ◽  
Surrogate Modeling ◽  
Space Region ◽  
High Quality ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Modeling Methods ◽  
Microwave Components ◽  
Power Split

In the paper, a recently reported nested kriging methodology is employed for modeling of miniaturized microwave components. The approach is based on identifying the parameter space region that contains high-quality designs, and, subsequently, rendering the surrogate in this subset. The results obtained for a miniaturized unequal-power-split rat-race coupler and a compact three-section impedance transformer demonstrate reliability of the method even for highly-dimensional parameter spaces, as well as its superiority over conventional modeling methods.

The bandcount increment scenario. II. Interior structures

2008 ◽  
Vol 464 (2097) ◽  
pp. 2247-2263 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Parameter Space ◽  
Complex Interaction ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In this second part, we investigate fine substructures nested into the basic structures reported and explained in part I. It is demonstrated that the overall structure of the chaotic domain is caused by a complex interaction of bandcount increment, bandcount adding and bandcount doubling structures, whereby some of them are nested into each other ad infinitum leading to self-similar structures in the parameter space.

scholarly journals Mean Shift Cluster Recognition Method Implementation in the Nested Sampling Algorithm

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 185
Author(s):  
Martino Trassinelli ◽  
Pierre Ciccodicola
Keyword(s):  
Parameter Space ◽  
Likelihood Function ◽  
Search Algorithm ◽  
Probability Distributions ◽  
Mean Shift ◽  
Computation Time ◽  
Recognition Method ◽  
Nested Sampling ◽  
Bayesian Evidence ◽  
Cluster Recognition

Nested sampling is an efficient algorithm for the calculation of the Bayesian evidence and posterior parameter probability distributions. It is based on the step-by-step exploration of the parameter space by Monte Carlo sampling with a series of values sets called live points that evolve towards the region of interest, i.e., where the likelihood function is maximal. In presence of several local likelihood maxima, the algorithm converges with difficulty. Some systematic errors can also be introduced by unexplored parameter volume regions. In order to avoid this, different methods are proposed in the literature for an efficient search of new live points, even in presence of local maxima. Here we present a new solution based on the mean shift cluster recognition method implemented in a random walk search algorithm. The clustering recognition is integrated within the Bayesian analysis program NestedFit. It is tested with the analysis of some difficult cases. Compared to the analysis results without cluster recognition, the computation time is considerably reduced. At the same time, the entire parameter space is efficiently explored, which translates into a smaller uncertainty of the extracted value of the Bayesian evidence.

Shape Preserving Global Parameterization for CAD/CAM/CAE

2004 ◽  
Author(s):  
Chen-Han Lee ◽  
Lingyun Lu ◽  
Jon Dym ◽  
Guangyan Yin
Keyword(s):  
Parameter Space ◽  
Physical Space ◽  
Shape Preserving ◽  
Parameter Spaces ◽  
Individual Parameter ◽  
Global Parameter ◽  
Cad Cam ◽  
The Individual ◽  
Closed Shells

CAD/CAM/CAE applications often deal with open or closed shells of faces (surfaces). Each face has it’s own 2-D parameter space that may not be rectangular. In many applications we need to merge the individual parameter spaces into a single global parameter space that resembles the model shape in the 3-D physical space. In this paper, we present a method of building such shape-preserving global parameterization (of a shell of faces) that is suitable for CAD/CAM/CAE applications.

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