Modelling and forecasting homogeneous age-specific mortality rates of multiple countries could lead to improvements in long-term forecasting. Data fed into joint models are often grouped according to nominal attributes, such as geographic regions, ethnic groups, and socioeconomic status, which may still contain heterogeneity and deteriorate the forecast results. Our paper proposes a novel clustering technique to pursue homogeneity among multiple functional time series based on functional panel data modelling to address this issue. Using a functional panel data model with fixed effects, we can extract common functional time series features. These common features could be decomposed into two components: the functional time trend and the mode of variations of functions (functional pattern). The functional time trend reflects the dynamics across time, while the functional pattern captures the fluctuations within curves. The proposed clustering method searches for homogeneous age-specific mortality rates of multiple countries by accounting for both the modes of variations and the temporal dynamics among curves. We demonstrate that the proposed clustering technique outperforms other existing methods through a Monte Carlo simulation and could handle complicated cases with slow decaying eigenvalues. In empirical data analysis, we find that the clustering results of age-specific mortality rates can be explained by the combination of geographic region, ethnic groups, and socioeconomic status. We further show that our model produces more accurate forecasts than several benchmark methods in forecasting age-specific mortality rates.
Time series forecasting has gained lots of attention recently; this is because many real-world phenomena can be modeled as time series. The massive volume of data and recent advancements in the processing power of the computers enable researchers to develop more sophisticated machine learning algorithms such as neural networks to forecast the time series data. In this paper, we propose various neural network architectures to forecast the time series data using the dynamic measurements; moreover, we introduce various architectures on how to combine static and dynamic measurements for forecastin…
We consider forecasting functional time series of extreme values within a generalised extreme value distribution (GEV). The GEV distribution can be characterised using the three parameters (location, scale and shape). As a result, the forecasts of the GEV density can be accomplished by forecasting these three latent parameters. Depending on the underlying data structure, some of the three parameters can either be modelled as scalars or functions. We provide two forecasting algorithms to model and forecast these parameters. To assess the forecast uncertainty, we apply a sieve bootstrap method t…
When generating social policies and pricing annuity at national and subnational levels, it is essential both to forecast mortality accurately and ensure that forecasts at the subnational level add up to the forecasts at the national level. This has motivated recent developments in forecasting functional time series in a group structure, where static principal component analysis is used. In the presence of moderate to strong temporal dependence, static principal component analysis designed for independent and identically distributed functional data may be inadequate. Thus, through using the dyn…
The University of Texas at Dallas（得克萨斯大学达拉斯分校）；National Renewable Energy Laboratory (NREL)（国家可再生能源实验室）；IBM TJ Watson Research Center（IBM托马斯·J·沃森研究中心）
Solar forecasting accuracy is affected by weather conditions, and weather awareness forecasting models are expected to improve the performance. However, it may not be available and reliable to classify different forecasting tasks by using only meteorological weather categorization. In this paper, an unsupervised clustering-based (UC-based) solar forecasting methodology is developed for short-term (1-hour-ahead) global horizontal irradiance (GHI) forecasting. This methodology consists of three parts: GHI time series unsupervised clustering, pattern recognition, and UC-based forecasting. The dai…
Time series clustering based on prediction accuracy of global forecasting models
Research Center for Information and Communication Technologies (CITIC)（信息与通信技术研究中心（CITIC））；University of A Coruña（拉科鲁尼亚大学）；The University of Sydney Business School（悉尼大学商学院）
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…
With the advent of Big Data, nowadays in many applications databases containing large quantities of similar time series are available. Forecasting time series in these domains with traditional univariate forecasting procedures leaves great potentials for producing accurate forecasts untapped. Recurrent neural networks (RNNs), and in particular Long Short-Term Memory (LSTM) networks, have proven recently that they are able to outperform state-of-the-art univariate time series forecasting methods in this context when trained across all available time series. However, if the time series database …
Univariate time series often take the form of a collection of curves observed sequentially over time. Examples of these include hourly ground-level ozone concentration curves. These curves can be viewed as a time series of functions observed at equally spaced intervals over a dense grid. Since functional time series may contain various types of outliers, we introduce a robust functional time series forecasting method to down-weigh the influence of outliers in forecasting. Through a robust principal component analysis based on projection pursuit, a time series of functions can be decomposed int…
Functional time series whose sample elements are recorded sequentially over time are frequently encountered with increasing technology. Recent studies have shown that analyzing and forecasting of functional time series can be performed easily using functional principal component analysis and existing univariate/multivariate time series models. However, the forecasting performance of such functional time series models may be affected by the presence of outlying observations which are very common in many scientific fields. Outliers may distort the functional time series model structure, and thus…
When modeling sub-national mortality rates, it is important to incorporate any possible correlation among sub-populations to improve forecast accuracy. Moreover, forecasts at the sub-national level should aggregate consistently across the forecasts at the national level. In this study, we apply a grouped multivariate functional time series to forecast Australian regional and remote age-specific mortality rates and reconcile forecasts in a group structure using various methods. Our proposed method compares favorably to a grouped univariate functional time series forecasting method by comparing …
We address the problem of forecasting high-dimensional functional time series through a two-fold dimension reduction procedure. The difficulty of forecasting high-dimensional functional time series lies in the curse of dimensionality. In this paper, we propose a novel method to solve this problem. Dynamic functional principal component analysis is first applied to reduce each functional time series to a vector. We then use the factor model as a further dimension reduction technique so that only a small number of latent factors are preserved. Classic time series models can be used to forecast t…
We propose three methods for forecasting a time series modeled using a functional coefficient autoregressive model (FCAR) fit via spline-backfitted local linear (SBLL) smoothing. The three methods are a "naive" plug-in method, a bootstrap method, and a multistage method. We present asymptotic results of the SBLL estimation method for FCAR models and show the estimators are oracally efficient. The three forecasting methods are compared through simulation. We find that the naive method performs just as well as the multistage method and even outperforms it in some situations. We apply the naive a…
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 …
Accurate forecasting of zero coupon bond yields for a continuum of maturities is paramount to bond portfolio management and derivative security pricing. Yet a universal model for yield curve forecasting has been elusive, and prior attempts often resulted in a trade-off between goodness of fit and consistency with economic theory. To address this, herein we propose a novel formulation which connects the dynamic factor model (DFM) framework with concepts from functional data analysis: a DFM with functional factor loading curves. This results in a model capable of forecasting functional time seri…
An approach is presented for making predictions about functional time series. The method is applied to data coming from periodically correlated processes and electricity demand, obtaining accurate point forecasts and narrow prediction bands that cover high proportions of the forecasted functional datum, for a given confidence level. The method is computationally efficient and substantially different to other functional time series methods, offering a new insight for the analysis of these data structures.
Monash University（莫纳什大学）；University of Melbourne（墨尔本大学）；Uber Technologies Inc（优步技术公司）
With large quantities of data typically available nowadays, forecasting models that are trained across sets of time series, known as Global Forecasting Models (GFM), are regularly outperforming traditional univariate forecasting models that work on isolated series. As GFMs usually share the same set of parameters across all time series, they often have the problem of not being localised enough to a particular series, especially in situations where datasets are heterogeneous. We study how ensembling techniques can be used with generic GFMs and univariate models to solve this issue. Our work sys…
There are many forecasting related packages in R with varied popularity, the most famous of all being \texttt{forecast}, which implements several important forecasting approaches, such as ARIMA, ETS, TBATS and others. However, the main issue with the existing functionality is the lack of flexibility for research purposes, when it comes to modifying the implemented models. The R package \texttt{smooth} introduces a new approach to univariate forecasting, implementing ETS and ARIMA models in Single Source of Error (SSOE) state space form and implementing an advanced functionality for experiments…
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 …
Age-specific mortality rates are often disaggregated by different attributes, such as sex, state and ethnicity. Forecasting age-specific mortality rates at the national and sub-national levels plays an important role in developing social policy. However, independent forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider reconciling forecasts of age-specific mortality rates, extending the methods of Hyndman et al. (2011) to functional time series, where age is considered as a continuum. The grouped functional time series me…
Classical time series models have serious difficulties in modeling and forecasting the enormous fluctuations of electricity spot prices. Markov regime switch models belong to the most often used models in the electricity literature. These models try to capture the fluctuations of electricity spot prices by using different regimes, each with its own mean and covariance structure. Usually one regime is dedicated to moderate prices and another is dedicated to high prices. However, these models show poor performance and there is no theoretical justification for this kind of classification. The mer…
Age-specific mortality rates are often disaggregated by different attributes, such as sex, state, ethnic group and socioeconomic status. In making social policies and pricing annuity at national and subnational levels, it is important not only to forecast mortality accurately, but also to ensure that forecasts at the subnational level add up to the forecasts at the national level. This motivates recent developments in grouped functional time series methods (Shang and Hyndman, 2017) to reconcile age-specific mortality forecasts. We extend these grouped functional time series forecasting methods…
This paper presents a novel time series clustering method, the self-organising eigenspace map (SOEM), based on a generalisation of the well-known self-organising feature map (SOFM). The SOEM operates on the eigenspaces of the embedded covariance structures of time series which are related directly to modes in those time series. Approximate joint diagonalisation acts as a pseudo-metric across these spaces allowing us to generalise the SOFM to a neural network with matrix input. The technique is empirically validated against three sets of experiments; univariate and multivariate time series clus…
Functional data clustering is to identify heterogeneous morphological patterns in the continuous functions underlying the discrete measurements/observations. Application of functional data clustering has appeared in many publications across various fields of sciences, including but not limited to biology, (bio)chemistry, engineering, environmental science, medical science, psychology, social science, etc. The phenomenal growth of the application of functional data clustering indicates the urgent need for a systematic approach to develop efficient clustering methods and scalable algorithmic imp…
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）
Environmental data often take the form of a collection of curves observed sequentially over time. An example of this includes daily pollution measurement curves describing the concentration of a particulate matter in ambient air. These curves can be viewed as a time series of functions observed at equally spaced intervals over a dense grid. The nature of high-dimensional data poses challenges from a statistical aspect, due to the so-called `curse of dimensionality', but it also poses opportunities to analyze a rich source of information to better understand dynamic changes at short time interv…
We introduce a novel functional time series methodology for short-term load forecasting. The prediction is performed by means of a weighted average of past daily load segments, the shape of which is similar to the expected shape of the load segment to be predicted. The past load segments are identified from the available history of the observed load segments by means of their closeness to a so-called reference load segment, the later being selected in a manner that captures the expected qualitative and quantitative characteristics of the load segment to be predicted. Weak consistency of the su…
When modeling sub-national mortality rates, we should consider three features: (1) how to incorporate any possible correlation among sub-populations to potentially improve forecast accuracy through multi-population joint modeling; (2) how to reconcile sub-national mortality forecasts so that they aggregate adequately across various levels of a group structure; (3) among the forecast reconciliation methods, how to combine their forecasts to achieve improved forecast accuracy. To address these issues, we introduce an extension of grouped univariate functional time series method. We first conside…
Measles presents a unique and imminent challenge for epidemiologists and public health officials: the disease is highly contagious, yet vaccination rates are declining precipitously in many localities. Consequently, the risk of a measles outbreak continues to rise. To improve preparedness, we study historical measles data both pre- and post-vaccine, and design new methodology to forecast measles counts with uncertainty quantification. We propose to model the disease counts as an integer-valued functional time series: measles counts are a function of time-of-year and time-ordered by year. The c…
Generating interpretable visualizations from complex data is a common problem in many applications. Two key ingredients for tackling this issue are clustering and representation learning. However, current methods do not yet successfully combine the strengths of these two approaches. Existing representation learning models which rely on latent topological structure such as self-organising maps, exhibit markedly lower clustering performance compared to recent deep clustering methods. To close this performance gap, we (a) present a novel way to fit self-organizing maps with probabilistic cluster …
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 …
In this paper we introduce a micro-clustering strategy for Functional Boxplots. The aim is to summarize a set of streaming time series splitted in non overlapping windows. It is a two step strategy which performs at first, an on-line summarization by means of functional data structures, named Functional Boxplot micro-clusters; then it reveals the final summarization by processing, off-line, the functional data structures. Our main contribute consists in providing a new definition of micro-cluster based on Functional Boxplots and, in defining a proximity measure which allows to compare and upda…
In economics we often face a system, which intrinsically imposes a structure of hierarchy of its components, i.e., in modelling trade accounts related to foreign exchange or in optimization of regional air protection policy. A problem of reconciliation of forecasts obtained on different levels of hierarchy has been addressed in the statistical and econometric literature for many times and concerns bringing together forecasts obtained independently at different levels of hierarchy. This paper deals with this issue in case of a hierarchical functional time series. We present and critically discu…
In several environmental applications data are functions of time, essentially con- tinuous, observed and recorded discretely, and spatially correlated. Most of the methods for analyzing such data are extensions of spatial statistical tools which deal with spatially dependent functional data. In such framework, this paper introduces a new clustering method. The main features are that it finds groups of functions that are similar to each other in terms of their spatial functional variability and that it locates a set of centers which summarize the spatial functional variability of each cluster. …
IBM Research（IBM研究院）；Sandia National Lab（桑迪亚国家实验室）；Texas A&M（德克萨斯A&M大学）；IBM Corporate Social Responsibility（IBM企业社会责任部）
Climate change, population growth, and water scarcity present unprecedented challenges for agriculture. This project aims to forecast soil moisture using domain knowledge and machine learning for crop management decisions that enable sustainable farming. Traditional methods for predicting hydrological response features require significant computational time and expertise. Recent work has implemented machine learning models as a tool for forecasting hydrological response features, but these models neglect a crucial component of traditional hydrological modeling that spatially close units can ha…
In the context of capacity planning, forecasting the evolution of informatics servers usage enables companies to better manage their computational resources. We address this problem by collecting key indicator time series and propose to forecast their evolution a day-ahead. Our method assumes that data is structured by a daily seasonality, but also that there is typical evolution of indicators within a day. Then, it uses the combination of a clustering algorithm and Markov Models to produce day-ahead forecasts. Our experiments on real datasets show that the data satisfies our assumption and th…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Alibaba Cloud Intelligence Business Group（阿里云智能事业群）
Topological Data Analysis (TDA) is a recent approach to analyze data sets from the perspective of their topological structure. Its use for time series data has been limited. In this work, a system developed for a leading provider of cloud computing combining both user segmentation and demand forecasting is presented. It consists of a TDA-based clustering method for time series inspired by a popular managerial framework for customer segmentation and extended to the case of clusterwise regression using matrix factorization methods to forecast demand. Increasing customer loyalty and producing acc…
In this work a robust clustering algorithm for stationary time series is proposed. The algorithm is based on the use of estimated spectral densities, which are considered as functional data, as the basic characteristic of stationary time series for clustering purposes. A robust algorithm for functional data is then applied to the set of spectral densities. Trimming techniques and restrictions on the scatter within groups reduce the effect of noise in the data and help to prevent the identification of spurious clusters. The procedure is tested in a simulation study, and is also applied to a rea…
Central University of Finance and Economics（中央财经大学）；Beijing Wuzi University（北京物资学院）
Analyzing both temporal and spatial patterns for an accurate forecasting model for financial time series forecasting is a challenge due to the complex nature of temporal-spatial dynamics: time series from different locations often have distinct patterns; and for the same time series, patterns may vary as time goes by. Inspired by the successful applications of deep learning, we propose a new model to resolve the issues of forecasting household leverage in China. Our solution consists of multiple RNN-based layers and an attention layer: each RNN-based layer automatically learns the temporal pat…
There is an increasingly rich literature about Bayesian nonparametric models for clustering functional observations. However, most of the recent proposals rely on infinite-dimensional characterizations that might lead to overly complex cluster solutions. In addition, while prior knowledge about the functional shapes is typically available, its practical exploitation might be a difficult modeling task. Motivated by an application in e-commerce, we propose a novel enriched Dirichlet mixture model for functional data. Our proposal accommodates the incorporation of functional constraints while bou…
This paper presents a new statistical method for clustering step data, a popular form of health record data easily obtained from wearable devices. Since step data are high-dimensional and zero-inflated, classical methods such as K-means and partitioning around medoid (PAM) cannot be applied directly. The proposed method is a novel combination of newly constructed variables that reflect the inherent features of step data, such as quantity, strength, and pattern, and a multivariate functional principal component analysis that can integrate all the features of the step data for clustering. The pr…
