Nonlinear Time-Fractional Differential Equations in Combustion Science

MSC 2010: 34A08 (main), 34G20, 80A25

Improving the predictability of time series by chaotic parameter estimation

Limited literature regarding parameter estimation of dynamic systems has been identified as the central-most reason for not having parametric bounds in chaotic time series. However, literature suggests that a chaotic system displays a sensitive dependence on initial conditions, and our study reveals that the behavior of chaotic system: is also sensitive to changes in parameter values. Therefore, parameter estimation technique could make it possible to establish parametric bounds on a nonlinear dynamic system underlying a given time series, which in turn can improve predictability. By extracting the relationship between parametric bounds and predictability, we implemented chaos-based models for improving prediction in time series. ^ This study describes work done to establish bounds on a set of unknown parameters. Our research results reveal that by establishing parametric bounds, it is possible to improve the predictability of any time series, although the dynamics or the mathematical model of that series is not known apriori. In our attempt to improve the predictability of various time series, we have established the bounds for a set of unknown parameters. These are: (i) the embedding dimension to unfold a set of observation in the phase space, (ii) the time delay to use for a series, (iii) the number of neighborhood points to use for avoiding detection of false neighborhood and, (iv) the local polynomial to build numerical interpolation functions from one region to another. Using these bounds, we are able to get better predictability in chaotic time series than previously reported. In addition, the developments of this dissertation can establish a theoretical framework to investigate predictability in time series from the system-dynamics point of view. ^ In closing, our procedure significantly reduces the computer resource usage, as the search method is refined and efficient. Finally, the uniqueness of our method lies in its ability to extract chaotic dynamics inherent in non-linear time series by observing its values. ^

Concentrations and d13C compositions of n-alkanes, n-alcohols, n-alkanoic acids in a 34-month time series of suspended sediments from the outflow of the Congo River

The concentrations, distributions, and stable carbon isotopes (d13C) of plant waxes carried by fluvial suspended sediments contain valuable information about terrestrial ecosystem characteristics. To properly interpret past changes recorded in sedimentary archives it is crucial to understand the sources and variability of exported plant waxes in modern systems on seasonal to inter-annual timescales. To determine such variability, we present concentrations and d13C compositions of three compound classes (n-alkanes, n-alcohols, n-alkanoic acids) in a 34-month time series of suspended sediments from the outflow of the Congo River. We show that exported plant-dominated n-alkanes (C25-C35) represent a mixture of C3 and C4 end members, each with distinct molecular distributions, as evidenced by an 8.1 ± 0.7 per mil (±1Sigma standard deviation) spread in d13C values across chain-lengths, and weak correlations between individual homologue concentrations (r = 0.52-0.94). In contrast, plant-dominated n-alcohols (C26-C36) and n-alkanoic acids (C26-C36) exhibit stronger positive correlations (r = 0.70-0.99) between homologue concentrations and depleted d13C values (individual homologues average <= -31.3 per mil and -30.8 per mil, respectively), with lower d13C variability across chain-lengths (2.6 ± 0.6 per mil and 2.0 ± 1.1 per mil, respectively). All individual plant-wax lipids show little temporal d13C variability throughout the time-series (1 Sigma <= 0.9 per mil), indicating that their stable carbon isotopes are not a sensitive tracer for temporal changes in plant-wax source in the Congo basin on seasonal to inter-annual timescales. Carbon-normalized concentrations and relative abundances of n-alcohols (19-58% of total plant-wax lipids) and n-alkanoic acids (26-76%) respond rapidly to seasonal changes in runoff, indicating that they are mostly derived from a recently entrained local source. In contrast, a lack of correlation with discharge and low, stable relative abundances (5-16%) indicate that n-alkanes better represent a catchment-integrated signal with minimal response to discharge seasonality. Comparison to published data on other large watersheds indicates that this phenomenon is not limited to the Congo River, and that analysis of multiple plant-wax lipid classes and chain lengths can be used to better resolve local vs. distal ecosystem structure in river catchments.

PIES time series in the North Atlantic at the Mid-Atlantic Ridge (2006-2010)

In the western North Atlantic, warm and saline water is brought by the North Atlantic Current (NAC) from the subtropics into the subpolar gyre. Four inverted echo sounders with high precision pressure sensors (PIES) were moored between 47°40' N and 52°30' N to study the main pathways of the NAC from the western into the eastern basin. The array configuration that forms three segments (northern, central, and southern) allows partitioning of the NAC and some assessment of NAC flow paths through the different Mid-Atlantic Ridge fracture zones. We exploit the correlation between the NAC transport measured between 2006 and 2010 and the geostrophic velocity from altimeter data to extend the time series of NAC transports to the period from 1992 to 2013. The mean NAC transport over the entire 21 years is 27 ± 5 Sv, consisting of 60% warm water of subtropical origin and 40% subpolar water. We did not find a significant trend in the total transport time series, but individual segments had opposing trends, leading to a more focused NAC in the central subsection and decreasing transports in the southern and northern segments. The spectral analysis exhibits several significant peaks. The two most prominent are around 120 days, identified as the time scale of meanders and eddies, and at 4-9 years, most likely related to the NAO. Transport composites for the years of highest and lowest NAO indices showed a significantly higher transport (+2.9 Sv) during strong NAO years, mainly in the southern segment.

Hierarchical clustering with multiple-height branch-cut applied to short time-series gene expression data.

Rigid adherence to pre-specified thresholds and static graphical representations can lead to incorrect decisions on merging of clusters. As an alternative to existing automated or semi-automated methods, we developed a visual analytics approach for performing hierarchical clustering analysis of short time-series gene expression data. Dynamic sliders control parameters such as the similarity threshold at which clusters are merged and the level of relative intra-cluster distinctiveness, which can be used to identify "weak-edges" within clusters. An expert user can drill down to further explore the dendrogram and detect nested clusters and outliers. This is done by using the sliders and by pointing and clicking on the representation to cut the branches of the tree in multiple-heights. A prototype of this tool has been developed in collaboration with a small group of biologists for analysing their own datasets. Initial feedback on the tool has been positive.

Change-point detection in environmental time series based on the informational approach

In this study, the Schwarz Information Criterion (SIC) is applied in order to detect change-points in the time series of surface water quality variables. The application of change-point analysis allowed detecting change-points in both the mean and the variance in series under study. Time variations in environmental data are complex and they can hinder the identification of the so-called change-points when traditional models are applied to this type of problems. The assumptions of normality and uncorrelation are not present in some time series, and so, a simulation study is carried out in order to evaluate the methodology’s performance when applied to non-normal data and/or with time correlation.

A space-time hybrid hourly rainfall model for derived flood frequency analysis

For derived flood frequency analysis based on hydrological modelling long continuous precipitation time series with high temporal resolution are needed. Often, the observation network with recording rainfall gauges is poor, especially regarding the limited length of the available rainfall time series. Stochastic precipitation synthesis is a good alternative either to extend or to regionalise rainfall series to provide adequate input for long-term rainfall-runoff modelling with subsequent estimation of design floods. Here, a new two step procedure for stochastic synthesis of continuous hourly space-time rainfall is proposed and tested for the extension of short observed precipitation time series. First, a single-site alternating renewal model is presented to simulate independent hourly precipitation time series for several locations. The alternating renewal model describes wet spell durations, dry spell durations and wet spell intensities using univariate frequency distributions separately for two seasons. The dependence between wet spell intensity and duration is accounted for by 2-copulas. For disaggregation of the wet spells into hourly intensities a predefined profile is used. In the second step a multi-site resampling procedure is applied on the synthetic point rainfall event series to reproduce the spatial dependence structure of rainfall. Resampling is carried out successively on all synthetic event series using simulated annealing with an objective function considering three bivariate spatial rainfall characteristics. In a case study synthetic precipitation is generated for some locations with short observation records in two mesoscale catchments of the Bode river basin located in northern Germany. The synthetic rainfall data are then applied for derived flood frequency analysis using the hydrological model HEC-HMS. The results show good performance in reproducing average and extreme rainfall characteristics as well as in reproducing observed flood frequencies. The presented model has the potential to be used for ungauged locations through regionalisation of the model parameters.

Long-period oscillations in GPS Up time series. Study over the European/Mediterranean area

The surface of the Earth is subjected to vertical deformations caused by geophysical and geological processes which can be monitored by Global Positioning System (GPS) observations. The purpose of this work is to investigate GPS height time series to identify interannual signals affecting the Earth’s surface over the European and Mediterranean area, during the period 2001-2019. Thirty-six homogeneously distributed GPS stations were selected from the online dataset made available by the Nevada Geodetic Laboratory (NGL) on the basis of the length and quality of the data series. The Principal Component Analysis (PCA) is the technique applied to extract the main patterns of the space and time variability of the GPS Up coordinate. The time series were studied by means of a frequency analysis using a periodogram and the real-valued Morlet wavelet. The periodogram is used to identify the dominant frequencies and the spectral density of the investigated signals; the second one is applied to identify the signals in the time domain and the relevant periodicities. This study has identified, over European and Mediterranean area, the presence of interannual non-linear signals with a period of 2-to-4 years, possibly related to atmospheric and hydrological loading displacements and to climate phenomena, such as El Niño Southern Oscillation (ENSO). A clear signal with a period of about six years is present in the vertical component of the GPS time series, likely explainable by the gravitational coupling between the Earth’s mantle and the inner core. Moreover, signals with a period in the order of 8-9 years, might be explained by mantle-inner core gravity coupling and the cycle of the lunar perigee, and a signal of 18.6 years, likely associated to lunar nodal cycle, were identified through the wavelet spectrum. However, these last two signals need further confirmation because the present length of the GPS time series is still too short when compared to the periods involved.

Variational and learning models for image and time series inverse problems

Inverse problems are at the core of many challenging applications. Variational and learning models provide estimated solutions of inverse problems as the outcome of specific reconstruction maps. In the variational approach, the result of the reconstruction map is the solution of a regularized minimization problem encoding information on the acquisition process and prior knowledge on the solution. In the learning approach, the reconstruction map is a parametric function whose parameters are identified by solving a minimization problem depending on a large set of data. In this thesis, we go beyond this apparent dichotomy between variational and learning models and we show they can be harmoniously merged in unified hybrid frameworks preserving their main advantages. We develop several highly efficient methods based on both these model-driven and data-driven strategies, for which we provide a detailed convergence analysis. The arising algorithms are applied to solve inverse problems involving images and time series. For each task, we show the proposed schemes improve the performances of many other existing methods in terms of both computational burden and quality of the solution. In the first part, we focus on gradient-based regularized variational models which are shown to be effective for segmentation purposes and thermal and medical image enhancement. We consider gradient sparsity-promoting regularized models for which we develop different strategies to estimate the regularization strength. Furthermore, we introduce a novel gradient-based Plug-and-Play convergent scheme considering a deep learning based denoiser trained on the gradient domain. In the second part, we address the tasks of natural image deblurring, image and video super resolution microscopy and positioning time series prediction, through deep learning based methods. We boost the performances of supervised, such as trained convolutional and recurrent networks, and unsupervised deep learning strategies, such as Deep Image Prior, by penalizing the losses with handcrafted regularization terms.

Interpretable Convolutional Neural Networks for Decoding and Analyzing Neural Time Series Data

Machine learning is widely adopted to decode multi-variate neural time series, including electroencephalographic (EEG) and single-cell recordings. Recent solutions based on deep learning (DL) outperformed traditional decoders by automatically extracting relevant discriminative features from raw or minimally pre-processed signals. Convolutional Neural Networks (CNNs) have been successfully applied to EEG and are the most common DL-based EEG decoders in the state-of-the-art (SOA). However, the current research is affected by some limitations. SOA CNNs for EEG decoding usually exploit deep and heavy structures with the risk of overfitting small datasets, and architectures are often defined empirically. Furthermore, CNNs are mainly validated by designing within-subject decoders. Crucially, the automatically learned features mainly remain unexplored; conversely, interpreting these features may be of great value to use decoders also as analysis tools, highlighting neural signatures underlying the different decoded brain or behavioral states in a data-driven way. Lastly, SOA DL-based algorithms used to decode single-cell recordings rely on more complex, slower to train and less interpretable networks than CNNs, and the use of CNNs with these signals has not been investigated. This PhD research addresses the previous limitations, with reference to P300 and motor decoding from EEG, and motor decoding from single-neuron activity. CNNs were designed light, compact, and interpretable. Moreover, multiple training strategies were adopted, including transfer learning, which could reduce training times promoting the application of CNNs in practice. Furthermore, CNN-based EEG analyses were proposed to study neural features in the spatial, temporal and frequency domains, and proved to better highlight and enhance relevant neural features related to P300 and motor states than canonical EEG analyses. Remarkably, these analyses could be used, in perspective, to design novel EEG biomarkers for neurological or neurodevelopmental disorders. Lastly, CNNs were developed to decode single-neuron activity, providing a better compromise between performance and model complexity.