We propose a Bayesian nonparametric approach to modelling and predicting a class of functional time series with application to energy markets, based on fully observed, noise-free functional data. Traders in such contexts conceive profitable strategies if they can anticipate the impact of their bidding actions on the aggregate demand and supply curves, which in turn need to be predicted reliably. Here we propose a simple Bayesian nonparametric method for predicting such curves, which take the form of monotonic bounded step functions. We borrow ideas from population genetics by defining a class of interacting particle systems to model the functional trajectory, and develop an implementation strategy which uses ideas from Markov chain Monte Carlo and approximate Bayesian computation techniques and allows to circumvent the intractability of the likelihood. Our approach shows great adaptation to the degree of smoothness of the curves and the volatility of the functional series, proves to be robust to an increase of the forecast horizon and yields an uncertainty quantification for the functional forecasts. We illustrate the model and discuss its performance with simulated datasets and on real data relative to the Italian natural gas market.
Indiana University Bloomington（印第安纳大学伯明顿分校）；Carnegie Mellon University（卡内基梅隆大学）；Santa Fe Institute（圣塔菲研究所）
We derive generalization error bounds for traditional time-series forecasting models. Our results hold for many standard forecasting tools including autoregressive models, moving average models, and, more generally, linear state-space models. These non-asymptotic bounds need only weak assumptions on the data-generating process, yet allow forecasters to select among competing models and to guarantee, with high probability, that their chosen model will perform well. We motivate our techniques with and apply them to standard economic and financial forecasting tools---a GARCH model for predicting …
The study of dependence between random variables is the core of theoretical and applied statistics. Static and dynamic copula models are useful for describing the dependence structure, which is fully encrypted in the copula probability density function. However, these models are not always able to describe the temporal change of the dependence patterns, which is a key characteristic of financial data. We propose a novel nonparametric framework for modelling a time series of copula probability density functions, which allows to forecast the entire function without the need of post-processing pr…
The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. In this paper we consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise consistent is still open. For finite-alphabet processes one may rely …
Semiparametric forecasting and filtering are introduced as a method of addressing model errors arising from unresolved physical phenomena. While traditional parametric models are able to learn high-dimensional systems from small data sets, their rigid parametric structure makes them vulnerable to model error. On the other hand, nonparametric models have a very flexible structure, but they suffer from the curse-of-dimensionality and are not practical for high-dimensional systems. The semiparametric approach loosens the structure of a parametric model by fitting a data-driven nonparametric model…
Point forecasting of univariate time series is a challenging problem with extensive work having been conducted. However, nonparametric probabilistic forecasting of time series, such as in the form of quantiles or prediction intervals is an even more challenging problem. In an effort to expand the possible forecasting paradigms we devise and explore an extrapolation-based approach that has not been applied before for probabilistic forecasting. We present a novel quantile Fourier neural network is for nonparametric probabilistic forecasting of univariate time series. Multi-step predictions are p…
In this paper, we propose two nonparametric methods used in the forecasting of functional time-dependent data, namely functional singular spectrum analysis recurrent forecasting and vector forecasting. Both algorithms utilize the results of functional singular spectrum analysis and past observations in order to predict future data points where recurrent forecasting predicts one function at a time and the vector forecasting makes predictions using functional vectors. We compare our forecasting methods to a gold standard algorithm used in the prediction of functional, time-dependent data by way …
In machine learning, a nonparametric forecasting algorithm for time series data has been proposed, called the kernel spectral hidden Markov model (KSHMM). In this paper, we propose a technique for short-term wind-speed prediction based on KSHMM. We numerically compared the performance of our KSHMM-based forecasting technique to other techniques with machine learning, using wind-speed data offered by the National Renewable Energy Laboratory. Our results demonstrate that, compared to these methods, the proposed technique offers comparable or better performance.
Aggregated moving functional median in robust prediction of hierarchical functional time series - an application to forecasting web portal users behaviors
Cracow University of Economics（克拉科夫经济大学）；AGH University of Science and Technology（AGH科学技术大学）
Prediction for high dimensional time series is a challenging task due to the curse of dimensionality problem. Classical parametric models like ARIMA or VAR require strong modeling assumptions and time stationarity and are often overparametrized. This paper offers a new flexible approach using recent ideas of manifold learning. The considered model includes linear models such as the central subspace model and ARIMA as particular cases. The proposed procedure combines manifold denoising techniques with a simple nonparametric prediction by local averaging. The resulting procedure demonstrates a v…
Toxicologists are often concerned with determining the dosage to which an individual can be exposed with an acceptable risk of adverse effect. These types of studies have been conducted widely in the past, and many novel approaches have been developed. Parametric techniques utilizing ANOVA and nonlinear regression models are well represented in the literature. The biggest drawback of parametric approaches is the need to specify the correct model. Recently, there has been an interest in nonparametric approaches to tolerable dosage estimation. In this work, we focus on the monotonically decreasi…
We introduce twenty four two-parameter families of advanced time series forecasting functions using a new and nonparametric approach. We also introduce the concept of powering and derive nonseasonal and seasonal models with examples in education, sales, finance and economy. We compare the performance of our twenty four models to both Holt--Winters and ARIMA models for both nonseasonal and seasonal times series. We show in particular that our models not only do not require a decomposition of a seasonal time series into trend, seasonal and random components, but leads also to substantially lower…
Two nonparametric methods are presented for forecasting functional time series (FTS). The FTS we observe is a curve at a discrete-time point. We address both one-step-ahead forecasting and dynamic updating. Dynamic updating is a forward prediction of the unobserved segment of the most recent curve. Among the two proposed methods, the first one is a straightforward adaptation to FTS of the $k$-nearest neighbors methods for univariate time series forecasting. The second one is based on a selection of curves, termed \emph{the curve envelope}, that aims to be representative in shape and magnitude …
Probabilistic Computing Project Massachusetts Institute of Technology（麻省理工学院概率计算项目）
This article proposes a Bayesian nonparametric method for forecasting, imputation, and clustering in sparsely observed, multivariate time series data. The method is appropriate for jointly modeling hundreds of time series with widely varying, non-stationary dynamics. Given a collection of $N$ time series, the Bayesian model first partitions them into independent clusters using a Chinese restaurant process prior. Within a cluster, all time series are modeled jointly using a novel "temporally-reweighted" extension of the Chinese restaurant process mixture. Markov chain Monte Carlo techniques are…
Kernel analog forecasting (KAF), alternatively known as kernel principal component regression, is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data. This paper synthesizes descriptions of kernel methods and Koopman operator theory in order to provide a single consistent account of KAF. The framework presented here illuminates the property of the KAF method that, under measure-preserving and ergodic dynamics, it consistently approximates the conditional expectation of observables that are acted upon by the Koopman operator of the dynamical …
We propose a framework for general probabilistic multi-step time series regression. Specifically, we exploit the expressiveness and temporal nature of Sequence-to-Sequence Neural Networks (e.g. recurrent and convolutional structures), the nonparametric nature of Quantile Regression and the efficiency of Direct Multi-Horizon Forecasting. A new training scheme, *forking-sequences*, is designed for sequential nets to boost stability and performance. We show that the approach accommodates both temporal and static covariates, learning across multiple related series, shifting seasonality, future pla…
Forecasting the evolution of complex systems is one of the grand challenges of modern data science. The fundamental difficulty lies in understanding the structure of the observed stochastic process. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent non-stationary and nonlinear time series can be well approximated globally by an auto-regressive process of slowly diverging order. When linear prediction with ${\cal L}^2$ loss is concerned, the latter result facilitates a unified globally-optimal short-term forecasting theory for a w…
Autocovariance of the error term in a time series model plays a key role in the estimation and inference for the model that it belongs to. Typically, some arbitrary parametric structure is assumed upon the error to simplify the estimation, which inevitably introduces potential model-misspecification. We thus conduct nonparametric estimation of it. To avoid the difficult bandwidth selection issue under the traditional nonparametric truncation approach, this paper conducts the Bayesian estimation of its spectral density in a frequency domain. To this end, we consider two cases: fixed error varia…
In this paper, we consider the problem of modelling historical data on retail credit portfolio performance, with a view to forecasting future performance, and facilitating strategic decision making. We consider a situation, common in practice, where accounts with common origination date (typically month) are aggregated into a single vintage for analysis, and the data for analysis consists of a time series of a univariate portfolio performance variable (for example, the proportion of defaulting accounts) for each vintage over successive time periods since origination. An invaluable management t…
We expect that some observers in perceptual signal detection experiments, such as radiologists, will make rational decisions, and therefore ratings from those observers are expected to form a convex ROC curve. However, measured and published curves are often not convex. This article examines the convexity-constrained nonparametric maximum likelihood estimator of the ROC curve given by Lloyd (2002). Like Lloyd we use the Pool Adjacent Violator Algorithm (PAVA) to construct the estimate of the convex curve. We present a direct proof that this estimate is a convex hull of the empirical ROC curve.…
North Carolina State University（北卡罗来纳州立大学）；University of Melbourne（墨尔本大学）
Short-term forecasting is an important tool in understanding environmental processes. In this paper, we incorporate machine learning algorithms into a conditional distribution estimator for the purposes of forecasting tropical cyclone intensity. Many machine learning techniques give a single-point prediction of the conditional distribution of the target variable, which does not give a full accounting of the prediction variability. Conditional distribution estimation can provide extra insight on predicted response behavior, which could influence decision-making and policy. We propose a techniqu…
For classifying time series, a nearest-neighbor approach is widely used in practice with performance often competitive with or better than more elaborate methods such as neural networks, decision trees, and support vector machines. We develop theoretical justification for the effectiveness of nearest-neighbor-like classification of time series. Our guiding hypothesis is that in many applications, such as forecasting which topics will become trends on Twitter, there aren't actually that many prototypical time series to begin with, relative to the number of time series we have access to, e.g., t…
Motivated by a broad range of potential applications, we address the quantile prediction problem of real-valued time series. We present a sequential quantile forecasting model based on the combination of a set of elementary nearest neighbor-type predictors called "experts" and show its consistency under a minimum of conditions. Our approach builds on the methodology developed in recent years for prediction of individual sequences and exploits the quantile structure as a minimizer of the so-called pinball loss function. We perform an in-depth analysis of real-world data sets and show that this …
This paper describes a method to estimate a production frontier that satisfies the axioms of monotonicity and concavity in a non-parametric Bayesian setting. An inefficiency term that allows for significant departure from prior distributional assumptions is jointly estimated in a single stage with parametric prior assumptions. We introduce heteroscedasticity into the inefficiency terms by local hyperplane-specific shrinkage hyperparameters and impose monotonicity using bound-constrained local nonlinear regression. Our minimum-of-hyperplanes estimator imposes concavity. Our Monte Carlo simulati…
Is Flow Matching Just Trajectory Replay for Sequential Data?
Department of Mathematics（数学系）；Nordita KTH Royal Institute of Technology and Stockholm University（KTH皇家理工学院与斯德哥尔摩大学联合研究所）；Department of Statistics（统计系）；International Computer Science Institute（国际计算机科学研究所）；Lawrence Berkeley National Laboratory（伯克利国家实验室）
Self-exciting Hawkes processes are used to model events which cluster in time and space, and have been widely studied in seismology under the name of the Epidemic Type Aftershock Sequence (ETAS) model. In the ETAS framework, the occurrence of the mainshock earthquakes in a geographical region is assumed to follow an inhomogeneous spatial point process, and aftershock events are then modelled via a separate triggering kernel. Most previous studies of the ETAS model have relied on point estimates of the model parameters due to the complexity of the likelihood function, and the difficulty in esti…
McCombs School of Business（麦克库姆斯商学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We propose Lomax delegate racing (LDR) to explicitly model the mechanism of survival under competing risks and to interpret how the covariates accelerate or decelerate the time to event. LDR explains non-monotonic covariate effects by racing a potentially infinite number of sub-risks, and consequently relaxes the ubiquitous proportional-hazards assumption which may be too restrictive. Moreover, LDR is naturally able to model not only censoring, but also missing event times or event types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a …
In many applications there is interest in estimating the relation between a predictor and an outcome when the relation is known to be monotone or otherwise constrained due to the physical processes involved. We consider one such application--inferring time-resolved aerosol concentration from a low-cost differential pressure sensor. The objective is to estimate a monotone function and make inference on the scaled first derivative of the function. We proposed Bayesian nonparametric monotone regression which uses a Bernstein polynomial basis to construct the regression function and puts a Dirichl…
The recent advent of smart meters has led to large micro-level datasets. For the first time, the electricity consumption at individual sites is available on a near real-time basis. Efficient management of energy resources, electric utilities, and transmission grids, can be greatly facilitated by harnessing the potential of this data. The aim of this study is to generate probability density estimates for consumption recorded by individual smart meters. Such estimates can assist decision making by helping consumers identify and minimize their excess electricity usage, especially during peak time…
Drug dilution (MIC) and disk diffusion (DIA) are the two most common antimicrobial susceptibility assays used by hospitals and clinics to determine an unknown pathogen's susceptibility to various antibiotics. Since only one assay is commonly used, it is important that the two assays give similar results. Calibration of the DIA assay to the MIC assay is typically done using the error-rate bounded method, which selects DIA breakpoints that minimize the observed discrepancies between the two assays. In 2000, Craig proposed a model-based approach that specifically models the measurement error and …
