Modelling and Analysis of Some Time Series

The thesis deals with some of the non-linear Gaussian and non-Gaussian time models and mainly concentrated in studying the properties and application of a first order autoregressive process with Cauchy marginal distribution. In this thesis some of the non-linear Gaussian and non-Gaussian time series models and mainly concentrated in studying the properties and application of a order autoregressive process with Cauchy marginal distribution. Time series relating to prices, consumptions, money in circulation, bank deposits and bank clearing, sales and profit in a departmental store, national income and foreign exchange reserves, prices and dividend of shares in a stock exchange etc. are examples of economic and business time series. The thesis discuses the application of a threshold autoregressive(TAR) model, try to fit this model to a time series data. Another important non-linear model is the ARCH model, and the third model is the TARCH model. The main objective here is to identify an appropriate model to a given set of data. The data considered are the daily coconut oil prices for a period of three years. Since it is a price data the consecutive prices may not be independent and hence a time series based model is more appropriate. In this study the properties like ergodicity, mixing property and time reversibility and also various estimation procedures used to estimate the unknown parameters of the 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.

A minimum approximate-BER beamforming approach for PSK modulated wireless systems

A beamforming algorithm is introduced based on the general objective function that approximates the bit error rate for the wireless systems with binary phase shift keying and quadrature phase shift keying modulation schemes. The proposed minimum approximate bit error rate (ABER) beamforming approach does not rely on the Gaussian assumption of the channel noise. Therefore, this approach is also applicable when the channel noise is non-Gaussian. The simulation results show that the proposed minimum ABER solution improves the standard minimum mean squares error beamforming solution, in terms of a smaller achievable system's bit error rate.

Verification of cloud-fraction forecasts

Cloud radar and lidar can be used to evaluate the skill of numerical weather prediction models in forecasting the timing and placement of clouds, but care must be taken in choosing the appropriate metric of skill to use due to the non- Gaussian nature of cloud-fraction distributions. We compare the properties of a number of different verification measures and conclude that of existing measures the Log of Odds Ratio is the most suitable for cloud fraction. We also propose a new measure, the Symmetric Extreme Dependency Score, which has very attractive properties, being equitable (for large samples), difficult to hedge and independent of the frequency of occurrence of the quantity being verified. We then use data from five European ground-based sites and seven forecast models, processed using the ‘Cloudnet’ analysis system, to investigate the dependence of forecast skill on cloud fraction threshold (for binary skill scores), height, horizontal scale and (for the Met Office and German Weather Service models) forecast lead time. The models are found to be least skillful at predicting the timing and placement of boundary-layer clouds and most skilful at predicting mid-level clouds, although in the latter case they tend to underestimate mean cloud fraction when cloud is present. It is found that skill decreases approximately inverse-exponentially with forecast lead time, enabling a forecast ‘half-life’ to be estimated. When considering the skill of instantaneous model snapshots, we find typical values ranging between 2.5 and 4.5 days. Copyright c 2009 Royal Meteorological Society

Geostatistical analysis of rainfall

Rainfall can be modeled as a spatially correlated random field superimposed on a background mean value; therefore, geostatistical methods are appropriate for the analysis of rain gauge data. Nevertheless, there are certain typical features of these data that must be taken into account to produce useful results, including the generally non-Gaussian mixed distribution, the inhomogeneity and low density of observations, and the temporal and spatial variability of spatial correlation patterns. Many studies show that rigorous geostatistical analysis performs better than other available interpolation techniques for rain gauge data. Important elements are the use of climatological variograms and the appropriate treatment of rainy and nonrainy areas. Benefits of geostatistical analysis for rainfall include ease of estimating areal averages, estimation of uncertainties, and the possibility of using secondary information (e.g., topography). Geostatistical analysis also facilitates the generation of ensembles of rainfall fields that are consistent with a given set of observations, allowing for a more realistic exploration of errors and their propagation in downstream models, such as those used for agricultural or hydrological forecasting. This article provides a review of geostatistical methods used for kriging, exemplified where appropriate by daily rain gauge data from Ethiopia.

Ensemble prediction for nowcasting with a convection-permitting model—I: description of the system and the impact of radar-derived surface precipitation rates

A key strategy to improve the skill of quantitative predictions of precipitation, as well as hazardous weather such as severe thunderstorms and flash floods is to exploit the use of observations of convective activity (e.g. from radar). In this paper, a convection-permitting ensemble prediction system (EPS) aimed at addressing the problems of forecasting localized weather events with relatively short predictability time scale and based on a 1.5 km grid-length version of the Met Office Unified Model is presented. Particular attention is given to the impact of using predicted observations of radar-derived precipitation intensity in the ensemble transform Kalman filter (ETKF) used within the EPS. Our initial results based on the use of a 24-member ensemble of forecasts for two summer case studies show that the convective-scale EPS produces fairly reliable forecasts of temperature, horizontal winds and relative humidity at 1 h lead time, as evident from the inspection of rank histograms. On the other hand, the rank histograms seem also to show that the EPS generates too much spread for forecasts of (i) surface pressure and (ii) surface precipitation intensity. These may indicate that for (i) the value of surface pressure observation error standard deviation used to generate surface pressure rank histograms is too large and for (ii) may be the result of non-Gaussian precipitation observation errors. However, further investigations are needed to better understand these findings. Finally, the inclusion of predicted observations of precipitation from radar in the 24-member EPS considered in this paper does not seem to improve the 1-h lead time forecast skill.

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.

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.

A new floating model level scheme for the assimilation of boundary layer top inversions: the univariate assimilation of temperature.

The assimilation of observations with a forecast is often heavily influenced by the description of the error covariances associated with the forecast. When a temperature inversion is present at the top of the boundary layer (BL), a significant part of the forecast error may be described as a vertical positional error (as opposed to amplitude error normally dealt with in data assimilation). In these cases, failing to account for positional error explicitly is shown t o r esult in an analysis for which the inversion structure is erroneously weakened and degraded. In this article, a new assimilation scheme is proposed to explicitly include the positional error associated with an inversion. This is done through the introduction of an extra control variable to allow position errors in the a priori to be treated simultaneously with the usual amplitude errors. This new scheme, referred to as the ‘floating BL scheme’, is applied to the one-dimensional (vertical) variational assimilation of temperature. The floating BL scheme is tested with a series of idealised experiments a nd with real data from radiosondes. For each idealised experiment, the floating BL scheme gives an analysis which has the inversion structure and position in agreement with the truth, and outperforms the a ssimilation which accounts only for forecast a mplitude error. When the floating BL scheme is used to assimilate a l arge sample of radiosonde data, its ability to give an analysis with an inversion height in better agreement with that observed is confirmed. However, it is found that the use of Gaussian statistics is an inappropriate description o f t he error statistics o f t he extra c ontrol variable. This problem is alleviated by incorporating a non-Gaussian description of the new control variable in the new scheme. Anticipated challenges in implementing the scheme operationally are discussed towards the end of the article.

Flow structure and near-field dispersion in arrays of building-like obstacles

Dispersion in the near-field region of localised releases in urban areas is difficult to predict because of the strong influence of individual buildings. Effects include upstream dispersion, trapping of material into building wakes and enhanced concentration fluctuations. As a result, concentration patterns are highly variable in time and mean profiles in the near field are strongly non-Gaussian. These aspects of near-field dispersion are documented by analysing data from direct numerical simulations in arrays of building-like obstacles and are related to the underlying flow structure. The mean flow structure around the buildings is found to exert a strong influence over the dispersion of material in the near field. Diverging streamlines around buildings enhance lateral dispersion. Entrainment of material into building wakes in the very near field gives rise to secondary sources, which then affect the subsequent dispersion pattern. High levels of concentration fluctuations are also found in this very near field; the fluctuation intensity is of order 2 to 5.

Kraichnan–Leith–Batchelor similarity theory and two-dimensional inverse cascades

We study the scaling properties and Kraichnan–Leith–Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids (α-turbulence models) simulated at resolution 8192x8192. We consider α=1 (surface quasigeostrophic flow), α=2 (2D Euler flow) and α=3. The forcing scale is well resolved, a direct cascade is present and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both α=1 and α=2. The active scalar field for α=3 contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction −(7−α)/3 in all three systems. Since we stop the simulations well before the cascades have reached the domain scale, vortex formation and spectral steepening are not due to condensation effects; nor are they caused by large-scale dissipation, which is absent. One- and two-point p.d.f.s, hyperflatness factors and structure functions indicate that the inverse cascades are intermittent and non-Gaussian over much of the inertial range for α=1 and α=2, while the α=3 inverse cascade is much closer to Gaussian and non-intermittent. For α=3 the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling ℰ(k)∝k−2 (α=1) and ℰ(k)∝k−5/3 (α=2) in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation (α=1 and α=2) and non-realizability (α=3) cause 2D inverse cascades to deviate from the KLB predictions, but that the flow between the vortices exhibits KLB scaling and non-intermittent statistics for α=1 and α=2.

An Ensemble Kalman Smoother for Nonlinear Dynamics

It is for mally proved that the general smoother for nonlinear dynamics can be for mulated as a sequential method, that is, obser vations can be assimilated sequentially during a for ward integration. The general filter can be derived from the smoother and it is shown that the general smoother and filter solutions at the final time become identical, as is expected from linear theor y. Then, a new smoother algorithm based on ensemble statistics is presented and examined in an example with the Lorenz equations. The new smoother can be computed as a sequential algorithm using only for ward-in-time model integrations. It bears a strong resemblance with the ensemble Kalman filter . The difference is that ever y time a new dataset is available during the for ward integration, an analysis is computed for all previous times up to this time. Thus, the first guess for the smoother is the ensemble Kalman filter solution, and the smoother estimate provides an improvement of this, as one would expect a smoother to do. The method is demonstrated in this paper in an intercomparison with the ensemble Kalman filter and the ensemble smoother introduced by van Leeuwen and Evensen, and it is shown to be superior in an application with the Lorenz equations. Finally , a discussion is given regarding the properties of the analysis schemes when strongly non-Gaussian distributions are used. It is shown that in these cases more sophisticated analysis schemes based on Bayesian statistics must be used.

CONSTRAINING THE NONEXTENSIVE MASS FUNCTION OF HALOS FROM BAO, CMB AND X-RAY DATA

Clusters of galaxies are the most impressive gravitationally-bound systems in the universe, and their abundance (the cluster mass function) is an important statistic to probe the matter density parameter (Omega(m)) and the amplitude of density fluctuations (sigma(8)). The cluster mass function is usually described in terms of the Press-Schecther (PS) formalism where the primordial density fluctuations are assumed to be a Gaussian random field. In previous works we have proposed a non-Gaussian analytical extension of the PS approach with basis on the q-power law distribution (PL) of the nonextensive kinetic theory. In this paper, by applying the PL distribution to fit the observational mass function data from X-ray highest flux-limited sample (HIFLUGCS), we find a strong degeneracy among the cosmic parameters, sigma(8), Omega(m) and the q parameter from the PL distribution. A joint analysis involving recent observations from baryon acoustic oscillation (BAO) peak and Cosmic Microwave Background (CMB) shift parameter is carried out in order to break these degeneracy and better constrain the physically relevant parameters. The present results suggest that the next generation of cluster surveys will be able to probe the quantities of cosmological interest (sigma(8), Omega(m)) and the underlying cluster physics quantified by the q-parameter.

An information-theoretic approach to statistical dependence: Copula information

We discuss the connection between information and copula theories by showing that a copula can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. We define the information excess as a measure of deviation from a maximum-entropy distribution. The idea of marginal invariant dependence measures is also discussed and used to show that empirical linear correlation underestimates the amplitude of the actual correlation in the case of non-Gaussian marginals. The mutual information is shown to provide an upper bound for the asymptotic empirical log-likelihood of a copula. An analytical expression for the information excess of T-copulas is provided, allowing for simple model identification within this family. We illustrate the framework in a financial data set. Copyright (C) EPLA, 2009

HFD groups in the Solovay model

Hajnal and Juhasz proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelof. The example constructed is a topological subgroup H subset of 2(omega 1) that is an HFD with the following property (P) the projection of H onto every partial product 2(I) for I is an element of vertical bar omega(1)vertical bar(omega) is onto. Any such group has the necessary properties. We prove that if kappa is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on 2(kappa), there is an HFD topological group in 2(omega 1) which has property (P). Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.