Modelling conditional probability distributions for periodic variables

Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce two novel techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite.

Modelling conditional probability distributions for periodic variables

Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we apply two novel techniques to the problem of extracting the distribution of wind vector directions from radar catterometer data gathered by a remote-sensing satellite.

Modelling conditional probability distributions for periodic variables

Most conventional techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce three related techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite.

Mixture density networks

Minimization of a sum-of-squares or cross-entropy error function leads to network outputs which approximate the conditional averages of the target data, conditioned on the input vector. For classifications problems, with a suitably chosen target coding scheme, these averages represent the posterior probabilities of class membership, and so can be regarded as optimal. For problems involving the prediction of continuous variables, however, the conditional averages provide only a very limited description of the properties of the target variables. This is particularly true for problems in which the mapping to be learned is multi-valued, as often arises in the solution of inverse problems, since the average of several correct target values is not necessarily itself a correct value. In order to obtain a complete description of the data, for the purposes of predicting the outputs corresponding to new input vectors, we must model the conditional probability distribution of the target data, again conditioned on the input vector. In this paper we introduce a new class of network models obtained by combining a conventional neural network with a mixture density model. The complete system is called a Mixture Density Network, and can in principle represent arbitrary conditional probability distributions in the same way that a conventional neural network can represent arbitrary functions. We demonstrate the effectiveness of Mixture Density Networks using both a toy problem and a problem involving robot inverse kinematics.

Estimating conditional probability densities for periodic variables

Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we introduce three novel techniques for tackling such problems, and investigate their performance using synthetic data. We then apply these techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite.

Estimating conditional volatility with neural networks

It is well known that one of the obstacles to effective forecasting of exchange rates is heteroscedasticity (non-stationary conditional variance). The autoregressive conditional heteroscedastic (ARCH) model and its variants have been used to estimate a time dependent variance for many financial time series. However, such models are essentially linear in form and we can ask whether a non-linear model for variance can improve results just as non-linear models (such as neural networks) for the mean have done. In this paper we consider two neural network models for variance estimation. Mixture Density Networks (Bishop 1994, Nix and Weigend 1994) combine a Multi-Layer Perceptron (MLP) and a mixture model to estimate the conditional data density. They are trained using a maximum likelihood approach. However, it is known that maximum likelihood estimates are biased and lead to a systematic under-estimate of variance. More recently, a Bayesian approach to parameter estimation has been developed (Bishop and Qazaz 1996) that shows promise in removing the maximum likelihood bias. However, up to now, this model has not been used for time series prediction. Here we compare these algorithms with two other models to provide benchmark results: a linear model (from the ARIMA family), and a conventional neural network trained with a sum-of-squares error function (which estimates the conditional mean of the time series with a constant variance noise model). This comparison is carried out on daily exchange rate data for five currencies.

On the annealed VC entropy for margin classifiers: A statistical mechanics study

Using techniques from Statistical Physics, the annealed VC entropy for hyperplanes in high dimensional spaces is calculated as a function of the margin for a spherical Gaussian distribution of inputs.

A new entropy measure based on the Renyi entropy rate using Gaussian kernels

The concept of entropy rate is well defined in dynamical systems theory but is impossible to apply it directly to finite real world data sets. With this in mind, Pincus developed Approximate Entropy (ApEn), which uses ideas from Eckmann and Ruelle to create a regularity measure based on entropy rate that can be used to determine the influence of chaotic behaviour in a real world signal. However, this measure was found not to be robust and so an improved formulation known as the Sample Entropy (SampEn) was created by Richman and Moorman to address these issues. We have developed a new, related, regularity measure which is not based on the theory provided by Eckmann and Ruelle and proves a more well-behaved measure of complexity than the previous measures whilst still retaining a low computational cost.

Robust control of nonlinear stochastic systems by modelling conditional distributions of control signals

We introduce a novel inversion-based neuro-controller for solving control problems involving uncertain nonlinear systems that could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. In this work a novel robust inverse control approach is obtained based on importance sampling from these distributions. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The performance of the new algorithm is illustrated through simulations with example systems.

Conditional distribution modeling for estimating and exploiting uncertainty in control systems

This paper presents a general methodology for estimating and incorporating uncertainty in the controller and forward models for noisy nonlinear control problems. Conditional distribution modeling in a neural network context is used to estimate uncertainty around the prediction of neural network outputs. The developed methodology circumvents the dynamic programming problem by using the predicted neural network uncertainty to localize the possible control solutions to consider. A nonlinear multivariable system with different delays between the input-output pairs is used to demonstrate the successful application of the developed control algorithm. The proposed method is suitable for redundant control systems and allows us to model strongly non Gaussian distributions of control signal as well as processes with hysteresis.

Using monthly returns to model conditional heteroscedasticity

This empirical study examines the extent of non-linearity in a multivariate model of monthly financial series. To capture the conditional heteroscedasticity in the series, both the GARCH(1,1) and GARCH(1,1)-in-mean models are employed. The conditional errors are assumed to follow the normal and Student-t distributions. The non-linearity in the residuals of a standard OLS regression are also assessed. It is found that the OLS residuals as well as conditional errors of the GARCH models exhibit strong non-linearity. Under the Student density, the extent of non-linearity in the GARCH conditional errors was generally similar to those of the standard OLS. The GARCH-in-mean regression generated the worse out-of-sample forecasts.

Conditional volatility and firm size:an empirical analysis of UK equity portfolios

The techniques and insights from two distinct areas of financial economic modelling are combined to provide evidence of the influence of firm size on the volatility of stock portfolio returns. Portfolio returns are characterized by positive serial correlation induced by the varying levels of non-synchronous trading among the component stocks. This serial correlation is greatest for portfolios of small firms. The conditional volatility of stock returns has been shown to be well represented by the GARCH family of statistical processes. Using a GARCH model of the variance of capitalization-based portfolio returns, conditioned on the autocorrelation structure in the conditional mean, striking differences related to firm size are uncovered.

A novel entropy measure for analysis of the electrocardiogram

There has been much recent research into extracting useful diagnostic features from the electrocardiogram with numerous studies claiming impressive results. However, the robustness and consistency of the methods employed in these studies is rarely, if ever, mentioned. Hence, we propose two new methods; a biologically motivated time series derived from consecutive P-wave durations, and a mathematically motivated regularity measure. We investigate the robustness of these two methods when compared with current corresponding methods. We find that the new time series performs admirably as a compliment to the current method and the new regularity measure consistently outperforms the current measure in numerous tests on real and synthetic data.

Analysis of nocturnal oxygen saturation recordings using kernel entropy to assist in sleep apnea-hypopnea diagnosis

In this study, a new entropy measure known as kernel entropy (KerEnt), which quantifies the irregularity in a series, was applied to nocturnal oxygen saturation (SaO 2) recordings. A total of 96 subjects suspected of suffering from sleep apnea-hypopnea syndrome (SAHS) took part in the study: 32 SAHS-negative and 64 SAHS-positive subjects. Their SaO 2 signals were separately processed by means of KerEnt. Our results show that a higher degree of irregularity is associated to SAHS-positive subjects. Statistical analysis revealed significant differences between the KerEnt values of SAHS-negative and SAHS-positive groups. The diagnostic utility of this parameter was studied by means of receiver operating characteristic (ROC) analysis. A classification accuracy of 81.25% (81.25% sensitivity and 81.25% specificity) was achieved. Repeated apneas during sleep increase irregularity in SaO 2 data. This effect can be measured by KerEnt in order to detect SAHS. This non-linear measure can provide useful information for the development of alternative diagnostic techniques in order to reduce the demand for conventional polysomnography (PSG). © 2011 IEEE.

Conditional osmotic stress in yeast:a system to study transport through aquaglyceroporins and osmostress signaling

The accumulation and transport of solutes are hallmarks of osmoadaptation. In this study we have employed the inability of the Saccharomyces cerevisiae gpd1Δ gpd2Δ mutant both to produce glycerol and to adapt to high osmolarity to study solute transport through aquaglyceroporins and the control of osmostress-induced signaling. High levels of different polyols, including glycerol, inhibited growth of the gpd1Δ gpd2Δ mutant. This growth inhibition was suppressed by expression of the hyperactive allele Fps1-AΔ of the osmogated yeast aquaglyceroporin, Fps1. The degree of suppression correlated with the relative rate of transport of the different polyols tested. Transport studies in secretory vesicles confirmed that Fps1-Δ1 transports polyols at increased rates compared with wild type Fps1. Importantly, wild type Fps1 and Fps1-Δ1 showed similarly low permeability for water. The growth defect on polyols in the gpd1Δ gpd2Δ mutant was also suppressed by expression of a heterologous aquaglyceroporin, rat AQP9. We surmised that this suppression was due to polyol influx, causing the cells to passively adapt to the stress. Indeed, when aquaglyceroporin-expressing gpd1Δ gpd2Δ mutants were treated with glycerol, xylitol, or sorbitol, the osmosensing HOG pathway was activated, and the period of activation correlated with the apparent rate of polyol uptake. This observation supports the notion that deactivation of the HOG pathway is closely coupled to osmotic adaptation. Taken together, our "conditional" osmotic stress system facilitates studies on aquaglyceroporin function and reveals features of the osmosensing and signaling system. © 2005 by The American Society for Biochemistry and Molecular Biology, Inc.