Dimensional Parameter Space Research Articles

Cosmological parameter estimation using particle swarm optimization

Obtaining the set of cosmological parameters consistent with observational data is an important exercise in current cosmological research. It involves finding the global maximum of the likelihood function in the multi-dimensional parameter space. Currently sampling based methods, which are in general stochastic in nature, like Markov-Chain Monte Carlo(MCMC), are being commonly used for parameter estimation. The beauty of stochastic methods is that the computational cost grows, at the most, linearly in place of exponentially (as in grid based approaches) with the dimensionality of the search space. MCMC methods sample the full joint probability distribution (posterior) from which one and two dimensional probability distributions, best fit (average) values of parameters and then error bars can be computed. In the present work we demonstrate the application of another stochastic method, named Particle Swarm Optimization (PSO), that is widely used in the field of engineering and artificial intelligence, for cosmological parameter estimation from WMAP seven years data. We find that there is a good agreement between the values of the best fit parameters obtained from PSO and publicly available code COSMOMC. However, there is a slight disagreement between error bars mainly due to the fact that errors are computed differently in PSO. Apart from presenting the results of our exercise, we also discuss the merits of PSO and explain its usefulness in more extensive search in higher dimensional parameter space.

Training Multi-Layer Perceptrons with the Unscented Kalman Particle Filter

Many Bayesian learning approaches to multi-layer perceptrons (MLPs) parameters optimization have been proposed such as the extended Kalman filter (EKF). In this paper, a sequential approach is applied to train the MLPs. Based on the particle filter, the approach named unscented Kalman particle filter (UPF) uses the unscented Kalman filter as proposal distribution to generate the importance sampling density. The UPF are devised to deal with the high dimensional parameter space that is inherent to neural network models. Simulation results show that the new algorithm performs better than traditional optimization methods such as the extended Kalman filter.

Refinements in the data analysis of VIRTIS-M-IR Venus nightside spectra

The successful long-duration radiation measurements performed by the VIRTIS instrument aboard ESA’s Venus Express spacecraft have provided an excellent collection of atmospheric and surface data that stand out due to their high temporal and spatial coverage of the planet and due to a high diversity of measurement and environmental conditions.A detailed investigation of the VIRTIS-M-IR data archive is described and possible improvements in the analysis of Venus night and dayside spectra are discussed. These improvements are both based on the application of new retrieval algorithms and on some calibration and data preprocessing refinements including a detector sensitivity investigation, an advanced straylight removal technique, and the allocation of a revised surface topography model customized to the intended surface and deep atmosphere parameter retrieval. Different data binning schemes are investigated and recommendations for their use, based on scientific objectives, are given.While the precise impacts on the retrieved parameters are difficult to summarize in a few numbers due to the sophisticated relations between the measurement space and the high dimensional parameter space, some general or exemplary trends are presented. Compared to retrieval results from spectra based on the earlier calibration and preprocessing pipeline, for instance CO and SO2 can display a column density change in the order of 10%, and the surface emissivity at 1.02μm can exhibit changes in the order of 0.2 and up to 0.36 in an example. Such changes can lead to different interpretations concerning atmospheric and geologic properties of Venus.The proposed refinements and the corresponding mathematical tools exploit the limits of what can be achieved with the available data. They enhance the retrieval reliability of atmospheric and surface parameters of Venus.Some of the tools described in the paper are not only very useful for VIRTIS data analyses, but should also provide interesting suggestions for a handling of data from other spaceborne experiments.

The marginal likelihood of dynamic mixture models

Analytical results for reducing the parameter space dimension when computing the marginal likelihood are given for the broad class of dynamic mixture models. These results allow the integration of scale parameters out of the likelihood by Kalman filtering and Gaussian quadrature. The method is simple and improves the accuracy of four marginal likelihood estimators, namely, the Laplace method, the Chib estimator, reciprocal importance sampling, and bridge sampling. For some empirically relevant cases like the local level and the local linear models, the marginal likelihood can be obtained directly without any posterior sampling. Implementation details are given in some examples. Two empirical applications illustrate the gain in accuracy achieved.

Mapping of the ASDEX Upgrade Operational Space for Disruption Prediction

The mapping of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional plasma parameter space of ASDEX Upgrade (AUG) has been performed using a 2-D self-organizing map (SOM), which reveals the map potentiality in data visualization. The proposed approach allows us the definition of simple displays capable of presenting meaningful information on the actual state of the plasma, but it also suggests to use the SOM as a disruption predictor. In this paper, various criteria have been studied to associate the risk of disruption of each cluster in the map to a disruption alarm threshold. The data for this study come from AUG experiments executed between July 2002 and November 2009. The prediction performance of the proposed system has been evaluated on a set of discharges different from those used for the map training, obtaining a good prediction success rate. A visual analysis of the predictor input signals has been performed for wrong predictions in order to identify possible common causes, and some criteria to increase prediction performance have been derived.

FERMI /LAT OBSERVATIONS OF 1ES 0229+200: IMPLICATIONS FOR EXTRAGALACTIC MAGNETIC FIELDS AND BACKGROUND LIGHT

We report the observation in the GeV band of the blazar 1ES 0229+200, which over recent years has become one the primary sources used to put constraints on the Extragalactic Background Light (EBL) and Extragalactic Magnetic Field (EGMF). We derive constraints on both the EBL and EGMF from the combined Fermi-HESS data set taking into account the direct and cascade components of the source spectrum. We show that the limit on the EBL depends on the EGMF strength and vice versa. In particular, an EBL density twice as high as that derived by Franceschini et al. (2008) is allowed if the EGMF is strong enough. On the other hand, an EGMF strength as low as 6x10^-18 G is allowed if the EBL density is at the level of the lower bound from the direct source counts. We present the combined EBL and EGMF limits on as an exclusion plot in two dimensional parameter space: EGMF strength vs. EBL density.

Characterizing new physics models by effective dimensionality of parameter space

We show that the dimension of the geometric shape formed by the phenomenologically valid points inside a multi-dimensional parameter space can be used to characterise different new physics models and to define a quantitative measure for the distribution of the points. We explain a simple algorithm to determine the box-counting dimension from a given set of parameter points, and illustrate our method with examples from different models that have recently been studied with respect to precision flavour observables.

Dense suspensions - solid-liquid interactions at the particle scale

Flows of solid–liquid suspensions span a multi–dimensional parameter space, with coordinates such as the Stokes number, the solids volume fraction, the density ratio, and Reynolds numbers. We are interested in systems with appreciable inertia effects - i.e., non–zero Stokes and Reynolds numbers - having density ratios of the order of one (typical for solid–liquid systems) and solids volume fractions of at least 0.1. Additional effects include strongly inhomogeneous solids distributions, non–Newtonian liquids, and sticky particles that tend to aggregate. This leads to a rich spectrum of interactions at the scale of individual particles. To reveal these we perform direct simulations of collections of a few thousand of particles carried by a liquid flow with resolution of the solid–liquid interfaces. For this we use the lattice–Boltzmann method supplemented with an immersed boundary approach.

CHAOS SEARCH IN FOURIER AMPLITUDE SENSITIVITY TEST

Work in Artificial Intelligence (AI) often involves search algorithms. In many complicated problems, however, local search algorithms may fail to converge into global optimization and global search procedures are needed. In this paper, we investigate the Fourier Amplitude Sensitivity Test (FAST) as an example of a global sensitivity analysis tool for complex, non-linear dynamical systems. FAST was originally developed based on the Fourier series expansion of a model output and on the assumption that samples of model inputs are uniformly distributed in a high dimensional parameter space. In order to compute sensitivity indices, the parameter space needs to be searched utilizing an appropriate (space-filling) search curve. In FAST, search curves are defined through learning functions, selection of which will heavily affect the global searching capacity and computational efficiency. This paper explores the characterization of learning functions involved in FAST and derives the underlying dynamical relationships with chaos search, which can provide new learning algorithms. This contribution has proven the general link that exists between chaos search and FAST, which helps us exploit the ergodicity of chaos search in AI applications.

Adaptive Hessian-Based Nonstationary Gaussian Process Response Surface Method for Probability Density Approximation with Application to Bayesian Solution of Large-Scale Inverse Problems

We develop an adaptive Hessian-based non-stationary Gaussian process (GP) response surface method for approximating a probability density function (pdf) that exploits its structure, particularly the Hessian of its negative logarithm. Of particular interest to us are pdfs that arise from the Bayesian solution of large-scale inverse problems, which imply very expensive-to-evaluate pdfs. The method can be considered as a piecewise adaptive Gaussian approximation in which a Gaussian tailored to the local Hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space. The task of efficiently partitioning the parameter space into subregions is done implicitly through Hessian-informed membership probability functions. The GP machinery is then employed to glue all local Gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density. The resulting response surface is also equipp...

Bifurcations in the Hodgkin–Huxley model exposed to DC electric fields

Diverse behaviors of the original Hodgkin–Huxley (HH) model, depending on the parameter values, have been studied extensively. This paper proposes modified HH equations exposed to externally applied extremely low frequency (ELF) electric fields. We investigate the effect of the DC electric fields on the dynamics of the modified HH model using bifurcation analysis. The obtained bifurcation sets partition the two dimensional parameter space, representing intensity of externally applied DC current and trans-membrane voltage induced by external DC electric fields, in terms of the qualitatively different behaviors of the HH model. Thus the neuronal information encodes the stimulus information, and vice versa. We also illustrate that the multi-stability phenomena in the HH model are associated with Hopf and double cycle bifurcations.

Zeroing in on supersymmetric radiation amplitude zeros

Radiation amplitude zeros have long been used to test the standard model. Here, we consider the supersymmetric radiation amplitude zero in chargino-neutralino associated production, which can be observed at the luminosity upgraded LHC. Such an amplitude zero only occurs if the neutralino has a large wino fraction and hence this observable can be used to determine the neutralino eigenstate content. We find that this observable can be measured by comparing the ${p}_{T}$ spectrum of the softest lepton in the trilepton ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ifmmode\pm\else\textpm\fi{}}{\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$ decay channel to that of a control process such as ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{+}{\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ensuremath{-}}$ or ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}{\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$. We test this technique on a previously generated model sample of the 19 dimensional parameter space of the phenomenological minimal supersymmetric standard model, and find that it is effective in determining the wino content of the neutralino.

Fast and Robust Circular Object Detection With Probabilistic Pairwise Voting

Accurate and efficient detection of circular objects in images is a challenging computer vision problem. Existing circular object detection methods can be broadly classified into two categories: voting based and maximum likelihood estimation (MLE) based. The former is robust to noise, however its computational complexity and memory requirement are high. On the other hand, MLE based methods (e.g., robust least squares fitting) are more computationally efficient but sensitive to noise, and can not detect multiple circles. This letter proposes Probabilistic Pairwise Voting (PPV), a fast and robust algorithm for circular object detection based on an extension of Hough Transform. The main contributions are threefold. 1) We formulate the problem of circular object detection as finding the intersection of lines in the three dimensional parameter space (i.e., center and radius of the circle). 2) We propose a probabilistic pairwise voting scheme to robustly discover circular objects under occlusion, image noise and moderate shape deformations. 3) We use a mode-finding algorithm to efficiently find multiple circular objects. We demonstrate the benefits of our approach on two real-world problems: 1) detecting circular objects in natural images, and 2) localizing iris in face images.

Effect of parameter change upon the extra-tropical atmospheric variability

Global climate models contain numerous parameters with uncertain values. In the context of climate simulation and prediction, it is relevant to obtain an estimate of the range of climate outcomes given the parameter uncertainty. Instead of randomly perturbing parameters, we determine parameter perturbations from short-term integrations that potentially have a high impact on the climate of the model. For this purpose we consider a dry, spectral quasi-geostrophic, three-level model on the sphere and its tangent linear and adjoint equations. With an empirical forcing, the model produces a fairly realistic simulation of the extra-tropical winter circulation. We allowed perturbations in a 1,449 dimensional parameter space. As a measure of impact on the climate we compute the change in the probability density function of the dominant patterns of variability. We find that the largest climate response in a set of 1,000 simulations with potentially high impact perturbations is much larger than the largest response in a similar set of simulations with randomly picked perturbations. We conclude that parameter sensitivity calculations based on short term integrations contain valuable information about the sensitivity of the model climate to parameter perturbations. The approach is feasible for state-of-the-art climate models provided that the tangent linear and adjoint equations are implemented.

SPECTRAL ENERGY DISTRIBUTION FITTING WITH MARKOV CHAIN MONTE CARLO: METHODOLOGY AND APPLICATION TOz= 3.1 Lyα-EMITTING GALAXIES

We present GalMC, a MCMC algorithm designed to fit the spectral energy distributions (SED) of galaxies to infer physical properties such as age, stellar mass, dust reddening, metallicity, redshift, and star formation rate. We describe the features of the code and the extensive tests conducted to ensure that our procedure leads to unbiased parameter estimation and accurate evaluation of uncertainties. We compare its performance to grid-based algorithms, showing that the efficiency in CPU time is ~ 100 times better for MCMC for a three dimensional parameter space and increasing with the number of dimensions. We use GalMC to fit the stacked SEDs of two samples of Lyman Alpha Emitters (LAEs) at redshift z=3.1. Our fit reveals that the typical LAE detected in the IRAC 3.6 micron band has age = 0.67 [0.37 - 1.81] Gyr and stellar mass = 3.2 [2.5 - 4.2] x 10^9 M_Sun, while the typical LAE not detected at 3.6 micron has age = 0.06 [0.01-0.2] Gyr and stellar mass = 2 [1.1 - 3.4] x 10^8 M_Sun. The SEDs of both stacks are consistent with the absence of dust. The data do not significantly prefer exponential with respect to constant star formation history. The stellar populations of these two samples are consistent with the previous study by Lai et al, with some differences due to the improved modeling of the stellar populations. A constraint on the metallicity of z=3.1 LAEs from broad-band photometry, requiring Z < Z_Sun at 95% confidence, is found here for the first time.

The influence of structure on the response properties of biologically plausible neural network models

Random sparse network architectures are intriguingly easy to construct, and, due to the lack of relevant structural studies, they have been difficult to disprove. Over the last decade, they have become the workhorse of computational studies of neuronal network dynamics. With the arrival of optical methods like channelrhodopsin and glutamate uncaging we can finally compare the predictions stemming from these architectures with responses measured in real cortical networks. Our model network is simulating L2/3 of a mouse barrel cortex column with three distinct neuronal populations: Excitatory pyramidal cells, fast-spiking inhibitory cells and non-fast-spiking inhibitory cells, modeled with biological plausible, cell-type specific adaptive exponential integrate and fire (AdEx) models [1]. For the connections between the three populations we used the probability of each connection and the distribution of synaptic weights, observed in biological measurements [2]. In parallel experiments and simulations, in which we stimulate groups of neurons and record the responses of the surrounding network, we can show that uniform random network models constructed from the statistics of pairwise recordings cannot capture the experimentally measured network responses. To improve the similarity between model and experiments, we investigate different ways to change the network architecture, for example by introducing correlations in the distribution of synaptic weights [3] or by modifying the in- and out-degree distributions of the network neurons. These changes are applied in a way that will still preserve the pairwise statistics in regards to the strengths and probabilities measured in biological experiments. In order to avoid an explosion of the number of usable parameters we are focusing on changes of the three most relevant types of connections, excitatory to excitatory, fast-spiking and non-fast-spiking, leading to a 12 dimensional parameter space. In order to visualize the high dimensional parameter space we rely on clutter based dimension reordering (CBDR) [4], a graphical technique used to build an intuitive representation of high dimensional data. This allows us to find not only the best network architecture in our parameter space but also to give an intuitive understanding of the impact of each parameter and of the interactions between parameters. The parameters with the highest impact on the response similarity between model and experiment are the in-degree and the input weight correlations in the connection from excitatory to fast-spiking neurons. Our work should contribute to a new, standard model architecture that will replace random networks as a tool to analyze biological network behavior.

A Computational Model for the Stimulation of Rat Sciatic Nerve Using a Transverse Intrafascicular Multichannel Electrode

Neuroprostheses based on electrical stimulation could potentially help disabled persons. They are based on neural interface that aim at creating an intimate contact with neural cells. The efficacy of neuroprostheses can be improved by increasing the selectivity of the neural interfaces used to stimulate specific subsets of cells. Selectivity is strongly influenced by interface design. Computer models can be useful for exploring the high dimensional space of design parameters with the aim to provide guidelines for the development of more efficient electrodes, with minimal animal use and optimization of manufacturing processes. The purpose of this study was to implement a realistic model of the performance of a transverse intrafascicular multichannel electrode (TIME) implanted into the rat sciatic nerve. A realistic finite element method (FEM) model was developed taking into account the anatomical and physiological features of the rat sciatic nerve. Electric potentials were calculated and interpolated voltages were applied to the model of a rat sciatic nerve axon, based on experimental biophysical data. Results indicate that high intra-fascicular and inter-fascicular selectivity values with low current levels can be achieved with TIMEs. The selectivity of TIMEs was also compared to an extraneural electrode, showing that higher selectivity with less current can be obtained. Using this model, the robustness of electrode performances for translational and rotational displacements were evaluated.

Performance and Robustness of the Monte Carlo Importance Sampling Algorithm Using Parallelized S-ADAPT for Basic and Complex Mechanistic Models

The Monte Carlo Parametric Expectation Maximization (MC-PEM) algorithm can approximate the true log-likelihood as precisely as needed and is efficiently parallelizable. Our objectives were to evaluate an importance sampling version of the MC-PEM algorithm for mechanistic models and to qualify the default estimation settings in SADAPT-TRAN. We assessed bias, imprecision and robustness of this algorithm in S-ADAPT for mechanistic models with up to 45 simultaneously estimated structural parameters, 14 differential equations, and 10 dependent variables (one drug concentration and nine pharmacodynamic effects). Simpler models comprising 15 parameters were estimated using three of the ten dependent variables. We set initial estimates to 0.1 or 10 times the true value and evaluated 30 bootstrap replicates with frequent or sparse sampling. Datasets comprised three dose levels with 16 subjects each. For simultaneous estimation of the full model, the ratio of estimated to true values for structural model parameters (median [5-95% percentile] over 45 parameters) was 1.01 [0.94-1.13] for means and 0.99 [0.68-1.39] for between-subject variances for frequent sampling and 1.02 [0.81-1.47] for means and 1.02 [0.47-2.56] for variances for sparse sampling. Imprecision was ≤25% for 43 of 45 means for frequent sampling. Bias and imprecision was well comparable for the full and simpler models. Parallelized estimation was 23-fold (6.9-fold) faster using 48 threads (eight threads) relative to one thread. The MC-PEM algorithm was robust and provided unbiased and adequately precise means and variances during simultaneous estimation of complex, mechanistic models in a 45 dimensional parameter space with rich or sparse data using poor initial estimates.

Variance-Based Global Sensitivity Analysis via Sparse-Grid Interpolation and Cubature

AbstractThe stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional (random) parameter space. In addition to providing highly accurate stochastic solutions, the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters. In this paper, we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol’. This method allows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids. We discuss convergence of this method, apply it to several test cases and compare to existing methods. As a result which may be of independent interest, we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema. This allows one to manipulate the sparse grid collocation results in a highly efficient manner.

Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications

We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasi-Monte Carlo methods, which use deterministically chosen sample points in an appropriate (usually high-dimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated Karhunen–Loève expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O ( N - 1 / 2 ) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasi-Monte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 10 6 are reported.