Observation impact in data assimilation: the effect of non-Gaussian observation error.

Data assimilation methods which avoid the assumption of Gaussian error statistics are being developed for geoscience applications. We investigate how the relaxation of the Gaussian assumption affects the impact observations have within the assimilation process. The effect of non-Gaussian observation error (described by the likelihood) is compared to previously published work studying the effect of a non-Gaussian prior. The observation impact is measured in three ways: the sensitivity of the analysis to the observations, the mutual information, and the relative entropy. These three measures have all been studied in the case of Gaussian data assimilation and, in this case, have a known analytical form. It is shown that the analysis sensitivity can also be derived analytically when at least one of the prior or likelihood is Gaussian. This derivation shows an interesting asymmetry in the relationship between analysis sensitivity and analysis error covariance when the two different sources of non-Gaussian structure are considered (likelihood vs. prior). This is illustrated for a simple scalar case and used to infer the effect of the non-Gaussian structure on mutual information and relative entropy, which are more natural choices of metric in non-Gaussian data assimilation. It is concluded that approximating non-Gaussian error distributions as Gaussian can give significantly erroneous estimates of observation impact. The degree of the error depends not only on the nature of the non-Gaussian structure, but also on the metric used to measure the observation impact and the source of the non-Gaussian structure.

The representation of conditional expectations of non-Gaussian noise

For Wiener spaces conditional expectations and $L^{2}$-martingales w.r.t. the natural filtration have a natural representation in terms of chaos expansion. In this note an extension to larger classes of processes is discussed. In particular, it is pointed out that orthogonality of the chaos expansion is not required.

Measures of observation impact in non-Gaussian data assimilation

ABSTRACT Non-Gaussian/non-linear data assimilation is becoming an increasingly important area of research in the Geosciences as the resolution and non-linearity of models are increased and more and more non-linear observation operators are being used. In this study, we look at the effect of relaxing the assumption of a Gaussian prior on the impact of observations within the data assimilation system. Three different measures of observation impact are studied: the sensitivity of the posterior mean to the observations, mutual information and relative entropy. The sensitivity of the posterior mean is derived analytically when the prior is modelled by a simplified Gaussian mixture and the observation errors are Gaussian. It is found that the sensitivity is a strong function of the value of the observation and proportional to the posterior variance. Similarly, relative entropy is found to be a strong function of the value of the observation. However, the errors in estimating these two measures using a Gaussian approximation to the prior can differ significantly. This hampers conclusions about the effect of the non-Gaussian prior on observation impact. Mutual information does not depend on the value of the observation and is seen to be close to its Gaussian approximation. These findings are illustrated with the particle filter applied to the Lorenz ’63 system. This article is concluded with a discussion of the appropriateness of these measures of observation impact for different situations.

Fast identification algorithms for Gaussian process model

A class identification algorithms is introduced for Gaussian process(GP)models.The fundamental approach is to propose a new kernel function which leads to a covariance matrix with low rank,a property that is consequently exploited for computational efficiency for both model parameter estimation and model predictions.The objective of either maximizing the marginal likelihood or the Kullback–Leibler (K–L) divergence between the estimated output probability density function(pdf)and the true pdf has been used as respective cost functions.For each cost function,an efficient coordinate descent algorithm is proposed to estimate the kernel parameters using a one dimensional derivative free search, and noise variance using a fast gradient descent algorithm. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

Production and purification of xylooligosaccharides from oil palm empty fruit bunch fibre by a non-isothermal process

Oil palm empty fruit bunches (OPEFB) fibre, a by-product generated from non-woody, tropical perennial oil palm crop was evaluated for xylooligosaccharides (XOS) production. Samples of OPEFB fibre were subjected to non-isothermal autohydrolysis treatment using a temperature range from 150 to 220 °C. The highest XOS concentration, 17.6 g/L which relayed from solubilisation of 63 g/100 g xylan was achieved at 210 °C and there was a minimum amount of xylose and furfural being produced. The chromatographic purification which was undertaken to purify the oligosaccharide-rich liquor resulted in a product with 74–78% purity, of which 83–85% was XOS with degree of polymerisation (DP) between 5 and 40.

Estimation of Gaussian process regression model using probability distance measures

A new class of parameter estimation algorithms is introduced for Gaussian process regression (GPR) models. It is shown that the integration of the GPR model with probability distance measures of (i) the integrated square error and (ii) Kullback–Leibler (K–L) divergence are analytically tractable. An efficient coordinate descent algorithm is proposed to iteratively estimate the kernel width using golden section search which includes a fast gradient descent algorithm as an inner loop to estimate the noise variance. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

High-precision measurements of the co-polar correlation coefficient: non-Gaussian errors and retrieval of the dispersion parameter µ in rainfall

The co-polar correlation coefficient (ρhv) has many applications, including hydrometeor classification, ground clutter and melting layer identification, interpretation of ice microphysics and the retrieval of rain drop size distributions (DSDs). However, we currently lack the quantitative error estimates that are necessary if these applications are to be fully exploited. Previous error estimates of ρhv rely on knowledge of the unknown "true" ρhv and implicitly assume a Gaussian probability distribution function of ρhv samples. We show that frequency distributions of ρhv estimates are in fact highly negatively skewed. A new variable: L = -log10(1 - ρhv) is defined, which does have Gaussian error statistics, and a standard deviation depending only on the number of independent radar pulses. This is verified using observations of spherical drizzle drops, allowing, for the first time, the construction of rigorous confidence intervals in estimates of ρhv. In addition, we demonstrate how the imperfect co-location of the horizontal and vertical polarisation sample volumes may be accounted for. The possibility of using L to estimate the dispersion parameter (µ) in the gamma drop size distribution is investigated. We find that including drop oscillations is essential for this application, otherwise there could be biases in retrieved µ of up to ~8. Preliminary results in rainfall are presented. In a convective rain case study, our estimates show µ to be substantially larger than 0 (an exponential DSD). In this particular rain event, rain rate would be overestimated by up to 50% if a simple exponential DSD is assumed.

The case of negative day-ahead electricity prices

In recent years, Germany has significantly increased its share of electricity produced from renewable sources, which is mainly due to the Renewable Energy Act (EEG). The EEG substantially impacts the dynamics of intra-day electricity prices by increasing the likelihood of negative prices. In this paper, we present a non-Gaussian process to model German intra-day electricity prices and propose an estimation procedure for this model. Most importantly, our model is able to generate extreme positive and negative spikes. A simulation study demonstrates the ability of our model to capture the characteristics of the data.

Online bayesian inference in some time-frequency representations of non-stationary processes

The use of Bayesian inference in the inference of time-frequency representations has, thus far, been limited to offline analysis of signals, using a smoothing spline based model of the time-frequency plane. In this paper we introduce a new framework that allows the routine use of Bayesian inference for online estimation of the time-varying spectral density of a locally stationary Gaussian process. The core of our approach is the use of a likelihood inspired by a local Whittle approximation. This choice, along with the use of a recursive algorithm for non-parametric estimation of the local spectral density, permits the use of a particle filter for estimating the time-varying spectral density online. We provide demonstrations of the algorithm through tracking chirps and the analysis of musical data.

State estimation using the particle filter with mode tracking

A particle filter is a data assimilation scheme that employs a fully nonlinear, non-Gaussian analysis step. Unfortunately as the size of the state grows the number of ensemble members required for the particle filter to converge to the true solution increases exponentially. To overcome this Vaswani [Vaswani N. 2008. IEEE Trans Signal Process 56:4583–97] proposed a new method known as mode tracking to improve the efficiency of the particle filter. When mode tracking, the state is split into two subspaces. One subspace is forecast using the particle filter, the other is treated so that its values are set equal to the mode of the marginal pdf. There are many ways to split the state. One hypothesis is that the best results should be obtained from the particle filter with mode tracking when we mode track the maximum number of unimodal dimensions. The aim of this paper is to test this hypothesis using the three dimensional stochastic Lorenz equations with direct observations. It is found that mode tracking the maximum number of unimodal dimensions does not always provide the best result. The best choice of states to mode track depends on the number of particles used and the accuracy and frequency of the observations.

A Gaussian-mixture ensemble transform filter

We generalize the popular ensemble Kalman filter to an ensemble transform filter, in which the prior distribution can take the form of a Gaussian mixture or a Gaussian kernel density estimator. The design of the filter is based on a continuous formulation of the Bayesian filter analysis step. We call the new filter algorithm the ensemble Gaussian-mixture filter (EGMF). The EGMF is implemented for three simple test problems (Brownian dynamics in one dimension, Langevin dynamics in two dimensions and the three-dimensional Lorenz-63 model). It is demonstrated that the EGMF is capable of tracking systems with non-Gaussian uni- and multimodal ensemble distributions. Copyright © 2011 Royal Meteorological Society

Gaussian processes autoencoder for dimensionality reduction

Learning low dimensional manifold from highly nonlinear data of high dimensionality has become increasingly important for discovering intrinsic representation that can be utilized for data visualization and preprocessing. The autoencoder is a powerful dimensionality reduction technique based on minimizing reconstruction error, and it has regained popularity because it has been efficiently used for greedy pretraining of deep neural networks. Compared to Neural Network (NN), the superiority of Gaussian Process (GP) has been shown in model inference, optimization and performance. GP has been successfully applied in nonlinear Dimensionality Reduction (DR) algorithms, such as Gaussian Process Latent Variable Model (GPLVM). In this paper we propose the Gaussian Processes Autoencoder Model (GPAM) for dimensionality reduction by extending the classic NN based autoencoder to GP based autoencoder. More interestingly, the novel model can also be viewed as back constrained GPLVM (BC-GPLVM) where the back constraint smooth function is represented by a GP. Experiments verify the performance of the newly proposed model.

Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

Stochastic modelling of rainfall from satellite data

Satellite-based rainfall monitoring is widely used for climatological studies because of its full global coverage but it is also of great importance for operational purposes especially in areas such as Africa where there is a lack of ground-based rainfall data. Satellite rainfall estimates have enormous potential benefits as input to hydrological and agricultural models because of their real time availability, low cost and full spatial coverage. One issue that needs to be addressed is the uncertainty on these estimates. This is particularly important in assessing the likely errors on the output from non-linear models (rainfall-runoff or crop yield) which make use of the rainfall estimates, aggregated over an area, as input. Correct assessment of the uncertainty on the rainfall is non-trivial as it must take account of • the difference in spatial support of the satellite information and independent data used for calibration • uncertainties on the independent calibration data • the non-Gaussian distribution of rainfall amount • the spatial intermittency of rainfall • the spatial correlation of the rainfall field This paper describes a method for estimating the uncertainty on satellite-based rainfall values taking account of these factors. The method involves firstly a stochastic calibration which completely describes the probability of rainfall occurrence and the pdf of rainfall amount for a given satellite value, and secondly the generation of ensemble of rainfall fields based on the stochastic calibration but with the correct spatial correlation structure within each ensemble member. This is achieved by the use of geostatistical sequential simulation. The ensemble generated in this way may be used to estimate uncertainty at larger spatial scales. A case study of daily rainfall monitoring in the Gambia, west Africa for the purpose of crop yield forecasting is presented to illustrate the method.

Higher oxides of chlorine: absorption cross-sections of Cl2O6 and Cl2O4, the decomposition of Cl2O6, and the reactions of OClO with O and O-3

The absorption cross-sections of Cl2O6 and Cl2O4 have been obtained using a fast flow reactor with a diode array spectrometer (DAS) detection system. The absorption cross-sections at the wavelengths of maximum absorption (lambda(max)) determined in this study are those of Cl2O6: (1.47 +/- 0.15) x 10(-17) cm(2) molecule(-1), at lambda(max) = 276 nm and T = 298 K; and Cl2O4: (9.0 +/- 2.0) x 10(-19) cm(2) molecule(-1), at lambda(max) = 234 nm and T = 298 K. Errors quoted are two standard deviations together with estimates of the systematic error. The shapes of the absorption spectra were obtained over the wavelength range 200-450 nm for Cl2O6 and 200-350 nm for Cl2O4, and were normalized to the absolute cross-sections obtained at lambda(max) for each oxide, and are presented at 1 nm intervals. These data are discussed in relation to previous measurements. The reaction of O with OCIO has been investigated with the objective of observing transient spectroscopic absorptions. A transient absorption was seen, and the possibility is explored of identifying the species with the elusive sym-ClO3 or ClO4, both of which have been characterized in matrices, but not in the gas-phase. The photolysis of OCIO was also re-examined, with emphasis being placed on the products of reaction. UV absorptions attributable to one of the isomers of the ClO dimer, chloryl chloride (ClClO2) were observed; some Cl2O4 was also found at long photolysis times, when much of the ClClO2 had itself been photolysed. We suggest that reports of Cl2O6 formation in previous studies could be a consequence of a mistaken identification. At low temperatures, the photolysis of OCIO leads to the formation of Cl2O3 as a result of the addition of the ClO primary product to OCIO. ClClO2 also appears to be one product of the reaction between O-3 and OCIO, especially when the reaction occurs under explosive conditions. We studied the kinetics of the non-explosive process using a stopped-flow technique, and suggest a value for the room-temperature rate coefficient of (4.6 +/- 0.9) x 10(-19) cm(3) molecule(-1) s(-1) (limit quoted is 2sigma random errors). The photochemical and thermal decomposition of Cl2O6 is described in this paper. For photolysis at k = 254 nm, the removal of Cl2O6 is not accompanied by the build up of any other strong absorber. The implications of the results are either that the photolysis of Cl2O6 produces Cl-2 directly, or that the initial photofragments are converted rapidly to Cl-2. In the thermal decomposition of Cl2O6, Cl2O4 was shown to be a product of reaction, although not necessarily the major one. The kinetics of decomposition were investigated using the stopped-flow technique. At relatively high [OCIO] present in the system, the decay kinetics obeyed a first-order law, with a limiting first-order rate coefficient of 0.002 s(-1). (C) 2004 Elsevier B.V. All rights reserved.