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The parameter estimation methods for the nonlinear exponential autoregressive model are investigated in this paper. We develop a forgetting factor gradient parameter estimation algorithm for improving the estimation accuracy. For the purpose of improving the identification accuracy further, a forgetting factor multi-innovation stochastic gradient algorithm is derived by using the multi-innovation theory. The effectiveness of the proposed algorithms is proved by a simulation example.

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model, based on the Gaussian likelihood conditional on initial values. We give conditions on the parameters such that the process X t is fractional of order d and cofractional of order d — b; that is, there exist vectors β for which βʹX t is fractional of order d — b and no other fractionality order is possible. For b = 1, the model nests the I(d — 1) vector autoregressive model. We define the statistical model by 0 < b ≤ d, but conduct inference when the true values satisfy 0 ≤ d₀ — b₀ < 1/2 and b₀ ≠ 1/2, for which ${{\\mathrm{\\beta }}^{\\prime }}_{0}{\\mathrm{X}}_{\\mathrm{t}}$ is (asymptotically) a stationary process. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end, we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are independent and identically distributed with suitable moment conditions and initial values are bounded. Because the limit is deterministic, this implies uniform convergence in probability of the conditional likelihood function. If the true value b₀ > 1/2, we prove that the limit distribution of ${\\mathrm{T}}^{{\\mathrm{b}}_{0}}(\\hat{\\mathrm{\\beta }}-{\\mathrm{\\beta }}_{0})$ is mixed Gaussian, while for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II. If b₀ < 1/2, all limit distributions are Gaussian or chi-squared. We derive similar results for the model with d = b, allowing for a constant term.

Nowadays speech synthesis or text to speech (TTS), an ability of system to produce human like natural sounding voice from the written text, is gaining popularity in the field of speech processing. For any TTS, intelligibility and naturalness are the two important measures for defining the quality of a synthesized sound which is highly dependent on the prosody modeling using acoustic model of synthesizer. The purpose of this review survey is firstly to study and analyze the various approaches used traditionally (articulatory synthesis, formant synthesis, concatenative speech synthesis and statistical parametric techniques based on hidden Markov model) and recently (statistical parametric based on deep learning approaches) for acoustic modeling with their pros and cons. The approaches based on deep learning to build the acoustic model has significantly contributed to the advancement of TTS as models based on deep learning are capable of modelling the complex context dependencies in the input data. Apart from these, this article also reviews the TTS approaches for generating speech with different voices and emotions to makes the TTS more realistic to use. It also addresses the subjective and objective metrics used to measure the quality of the synthesized voice. Various well known speech synthesis systems based on autoregressive and non-autoregressive models such as Tacotron, Deep Voice, WaveNet, Parallel WaveNet, Parallel Tacotron, FastSpeech by global tech-giant Google, Facebook, Microsoft employed the architecture of deep learning for end-to-end speech waveform generation and attained a remarkable mean opinion score (MOS).

Ecological analyses typically involve many interacting variables. Ecologists often specify lagged interactions in community dynamics (i.e. vector‐autoregressive models) or simultaneous interactions (e.g. structural equation models), but there is less familiarity with dynamic structural equation models (DSEM) that can include any simultaneous or lagged effect in multivariate time‐series analysis. We propose a novel approach to parameter estimation for DSEM, which involves constructing a Gaussian Markov random field (GMRF) representing simultaneous and lagged path coefficients, and then fitting this as a generalized linear mixed model to missing and/or non‐normal data. We provide a new R‐package dsem , which extends the ‘arrow interface’ from path analysis to represent user‐specified lags when constructing the GMRF. We also outline how the resulting nonseparable precision matrix can generalize existing separable models, for example, for time‐series and species interactions in a vector‐autoregressive model. We first demonstrate dsem by simulating a two‐species vector‐autoregressive model based on wolf–moose interactions on Isle Royale. We show that DSEM has improved precision when data are missing relative to a conventional dynamic linear model. We then demonstrate DSEM via two contrasting case studies. The first identifies a trophic cascade where decreased sunflower starfish has increased urchin and decreased kelp densities, while sea otters have a simultaneous positive effect on kelp in the California Current from 1999 to 2018. The second estimates how declining sea ice has decreased cold‐water habitats, driving a decreased density for fall copepod predation and inhibiting early‐life survival for Alaska pollock from 1963 to 2023. We conclude that DSEM can be fitted efficiently as a GLMM involving missing data, while allowing users to specify both simultaneous and lagged effects in a time‐series structural model. DSEM then allows conceptual models (developed with stakeholder input or from ecological expertise) to be fitted to incomplete time series and provides a simple interface for granular control over the number of estimated time‐series parameters. Finally, computational methods are sufficiently simple that DSEM can be embedded as component within larger (e.g. integrated population) models. We therefore recommend greater exploration and performance testing for DSEM relative to familiar time‐series forecasting methods.

Because of the uncertainty and randomness of wind speed, wind power has characteristics such as nonlinearity and multiple frequencies. Accurate prediction of wind power is one effective means of improving wind power integration. Because the traditional single model cannot fully characterize the fluctuating characteristics of wind power, scholars have attempted to build other prediction models based on empirical mode decomposition (EMD) or ensemble empirical mode decomposition (EEMD) to tackle this problem. However, the prediction accuracy of these models is affected by modal aliasing and illusive components. Aimed at these defects, this paper proposes a multi-frequency combination prediction model based on variational mode decomposition (VMD). We use a back propagation neural network (BPNN), autoregressive moving average (ARMA) model, and least squares support vector machine (LS-SVM) to predict high, intermediate, and low frequency components, respectively. Based on the predicted values of each component, the BPNN is applied to combine them into a final wind power prediction value. Finally, the prediction performance of the single prediction models (ARMA, BPNN, LS-SVM) and the decomposition prediction models (EMD and EEMD) are used to compare with the proposed VMD model according to the evaluation indices such as average absolute error, mean square error, and root mean square error to validate its feasibility and accuracy. The results show that the prediction accuracy of the proposed VMD model is higher.

Vector autoregressive (VAR) models aim to capture linear temporal interdependencies among multiple time series. They have been widely used in macroeconomics and financial econometrics and more recently have found novel applications in functional genomics and neuroscience. These applications have also accentuated the need to investigate the behavior of the VAR model in a high-dimensional regime, which provides novel insights into the role of temporal dependence for regularized estimates of the model's parameters. However, hardly anything is known regarding properties of the posterior distribution for Bayesian VAR models in such regimes. In this work, we consider a VAR model with two prior choices for the autoregressive coefficient matrix: a nonhierarchical matrix-normal prior and a hierarchical prior, which corresponds to an arbitrary scale mixture of normals. We establish posterior consistency for both these priors under standard regularity assumptions, when the dimension p of the VAR model grows with the sample size n (but still remains smaller than n). A special case corresponds to a shrinkage prior that introduces (group) sparsity in the columns of the model coefficient matrices. The performance of the model estimates are illustrated on synthetic and real macroeconomic datasets. Supplementary materials for this article are available online.

In recent years, efficient modeling and forecasting of electricity prices became highly important for all the market participants for developing bidding strategies and making investment decisions. However, as electricity prices exhibit specific features, such as periods of high volatility, seasonal patterns, calendar effects, nonlinearity, etc., their accurate forecasting is challenging. This study proposes a functional forecasting method for the accurate forecasting of electricity prices. A functional autoregressive model of order P is suggested for short-term price forecasting in the electricity markets. The applicability of the model is improved with the help of functional final prediction error (FFPE), through which the model dimensionality and lag structure were selected automatically. An application of the suggested algorithm was evaluated on the Italian electricity market (IPEX). The out-of-sample forecasted results indicate that the proposed method performs relatively better than the nonfunctional forecasting techniques such as autoregressive (AR) and naïve models.

Threshold Autoregressive (TAR) models have been widely used by statisticians for non-linear time series forecasting during the past few decades, due to their simplicity and mathematical properties. On the other hand, in the forecasting community, general-purpose tree-based regression algorithms (forests, gradient-boosting) have become popular recently due to their ease of use and accuracy. In this paper, we explore the close connections between TAR models and regression trees. These enable us to use the rich methodology from the literature on TAR models to define a hierarchical TAR model as a regression tree that trains globally across series, which we call SETAR-Tree. In contrast to the general-purpose tree-based models that do not primarily focus on forecasting, and calculate averages at the leaf nodes, we introduce a new forecasting-specific tree algorithm that trains global Pooled Regression (PR) models in the leaves allowing the models to learn cross-series information and also uses some time-series-specific splitting and stopping procedures. The depth of the tree is controlled by conducting a statistical linearity test commonly employed in TAR models, as well as measuring the error reduction percentage at each node split. Thus, the proposed tree model requires minimal external hyperparameter tuning and provides competitive results under its default configuration. We also use this tree algorithm to develop a forest where the forecasts provided by a collection of diverse SETAR-Trees are combined during the forecasting process. In our evaluation on eight publicly available datasets, the proposed tree and forest models are able to achieve significantly higher accuracy than a set of state-of-the-art tree-based algorithms and forecasting benchmarks across four evaluation metrics.

Variable selection has played a fundamental role in regression analysis. Spatial autoregressive model is a useful tool in econometrics and statistics in which context variable selection is necessary but not adequately investigated. In this paper, we consider conducting variable selection in spatial autoregressive models with a diverging number of parameters. Smoothly clipped absolute deviation penalty is considered to obtain the estimators. Moreover the dimension of the covariates are allowed to vary with sample size. In order to attenuate the bias caused by endogeneity, instrumental variable is adopted in the estimation procedure. The proposed method can do parametric estimation and variable selection simultaneously. Under mild conditions, we establish the asymptotic and oracle property of the proposed estimators. Finally, the performance of the proposed estimation procedure is examined via Monte Carlo simulation studies and a data set from a Boston housing price is analyzed as an illustrative example.

Sediment transport in rivers and channels can be understood as a diffusive phenomenon, where sediment particles separate as they move downstream. This diffusive behavior can be Fickian or anomalous (superdiffusive, subdiffusive, ballistic, etc.), depending on the time evolution of the sediment displacement variance. Many authors have observed transitions from ballistic to Fickian or subdiffusive regimes as the observation timescale increases, aligning with the conceptual model of Nikora et al. (2002), . Despite progress, the mechanisms driving these transitions remain unclear. To investigate them, we simulate a flat‐bed channel entraining sediment using Direct Numerical Simulations (DNS) to solve the flow and a point‐particle Discrete Element Method (DEM) to resolve particle dynamics, including collisions. The DNS–DEM algorithm is two‐way coupled, with the flow responding to particles through a cell‐based projection of drag forces. In parallel, we implement stochastic models calibrated from DNS‐DEM results, including linear and non‐linear advection‐diffusion equations and autoregressive Markov models (correlated/non‐correlated with Gaussian/non‐Gaussian distributions). We study eight cases spanning Shields numbers from 0.03 to 0.85, focusing on the evolution of the mean, variance, skewness, and kurtosis of particle displacement. We observe a ballistic regime at short timescales and near‐Fickian at longer ones, though subdiffusion is also present at low Shields numbers. Variance, skewness, and kurtosis approach Fickian behavior as time increases, with convergence rates depending on the Shields number. We find that particle motion correlation—an indirect measure of particle inertia—drives the transition from ballistic to Fickian regimes. In contrast, transitions to subdiffusive states are governed by resting times, not particle inertia.