Functional Autoregressive Model Research Articles

Forecasting Canadian Age-Specific Mortality Rates: Application of Functional Time Series Analysis

In the insurance and pension industries, as well as in designing social security systems, forecasted mortality rates are of major interest. The current research provides statistical methods based on functional time series analysis to improve mortality rate prediction for the Canadian population. The proposed functional time series-based model was applied to the three-mortality series: total, male and female age-specific Canadian mortality rate over the year 1991 to 2019. Descriptive measures were used to estimate the overall temporal patterns and the functional principal component regression model (fPCA) was used to predict the next ten years mortality rate for each series. Functional autoregressive model (fAR (1)) was used to measure the impact of one year age differences on mortality series. For total series, the mortality rates for children have dropped over the whole data period, while the difference between young adults and those over 40 has only been falling since about 2003 and has leveled off in the last decade of the data. A moderate to strong impact of age differences on Canadian age-specific mortality series was observed over the years. Wider application of fPCA to provide more accurate estimates in public health, demography, and age-related policy studies should be considered.

Geometric ergodicity and conditional self‐weighted M‐estimator of a GRCAR(p) model with heavy‐tailed errors

We establish the geometric ergodicity for general stochastic functional autoregressive (linear and nonlinear) models with heavy‐tailed errors. The stationarity conditions for a generalized random coefficient autoregressive model (GRCAR()) are presented as a corollary. And then, a conditional self‐weighted M‐estimator for parameters in the GRCAR() is proposed. The asymptotic normality of this estimator is discussed by allowing infinite variance innovations. Simulation experiments are carried out to assess the finite‐sample performance of the proposed methodology and theory, and a real heavy‐tailed data example is given as illustration.

On consistency and sparsity for high-dimensional functional time series with application to autoregressions

Modelling a large collection of functional time series arises in a broad spectral of real applications. Under such a scenario, not only the number of functional variables can be diverging with, or even larger than the number of temporally dependent functional observations, but each function itself is an infinite-dimensional object, posing a challenging task. In this paper, we propose a three-step procedure to estimate high-dimensional functional time series models. To provide theoretical guarantees for the three-step procedure, we focus on multivariate stationary processes and propose a novel functional stability measure based on their spectral properties. Such stability measure facilitates the development of some useful concentration bounds on sample (auto)covariance functions, which serve as a fundamental tool for further convergence analysis in high-dimensional settings. As functional principal component analysis (FPCA) is one of the key dimension reduction techniques in the first step, we also investigate the non-asymptotic properties of the relevant estimated terms under a FPCA framework. To illustrate with an important application, we consider vector functional autoregressive models and develop a regularization approach to estimate autoregressive coefficient functions under the sparsity constraint. Using our derived non-asymptotic results, we investigate convergence properties of the regularized estimate under high-dimensional scaling. Finally, the finite-sample performance of the proposed method is examined through both simulations and a public financial dataset.

Forecasting Day-Ahead Traffic Flow Using Functional Time Series Approach

Nowadays, short-term traffic flow forecasting has gained increasing attention from researchers due to traffic congestion in many large and medium-sized cities that pose a serious threat to sustainable urban development. To this end, this research examines the forecasting performance of functional time series modeling to forecast traffic flow in the ultra-short term. An appealing feature of the functional approach is that unlike other methods, it provides information over the whole day, and thus, forecasts can be obtained for any time within a day. Within this approach, a Functional AutoRegressive (FAR) model is used to forecast the next-day traffic flow. For empirical analysis, the traffic flow data of Dublin airport link road, Ireland, collected at a fifteen-minute interval from 1 January 2016 to 30 April 2017, are used. The first twelve months are used for model estimation, while the remaining four months are for the one-day-ahead out-of-sample forecast. For comparison purposes, a widely used model, namely AutoRegressive Integrated Moving Average (ARIMA), is also used to obtain the forecasts. Finally, the models’ performances are compared based on different accuracy statistics. The study results suggested that the functional time series model outperforms the traditional time series models. As the proposed method can produce traffic flow forecasts for the entire next day with satisfactory results, it can be used in decision making by transportation policymakers and city planners.

Bayesian Testing of Granger Causality in Functional Time Series

We develop a multivariate functional autoregressive model (MFAR), which captures the cross-correlation among multiple functional time series and thus improves forecast accuracy. We estimate the parameters under the Bayesian dynamic linear models (DLM) framework. In order to test for Granger causality from one FAR series to another we employ Bayes Factor. Motivated by the broad application of functional data in finance, we investigate the causality between the yield curves of USA and UK. Bayes factor values shows that no causal relation exists among the interest rates of these two countries. Furthermore, we illustrate a climatology example, suggesting that the meteorological factors Granger cause pollutant daily levels in Delhi. The Github repository https://www.Bayesian-Testing-Of-Granger-Causality-In-Functional-Time-Series contains the detailed study of simulation and real data applications.

Short-Term Electricity Prices Forecasting Using Functional Time Series Analysis

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.

Bayesian Adaptive Lasso for the Partial Functional Linear Spatial Autoregressive Model

This study introduces a partial functional linear spatial autoregressive model which can explore the relationship between a scalar spatially dependent response variable and predictive variables containing both multiple scalar covariates and a functional covariate. With approximating to the functional coefficient by Karhunen–Loève representation, we propose a Bayesian adaptive Lasso method to simultaneously estimate unknown parameters and select important covariates in the model, which can be performed by combining the Gibbs sampler and the Metropolis–Hastings algorithm. Some simulation studies are conducted and the results show that the proposed Bayesian method behaves well.

Modeling Functional Time Series and Mixed-Type Predictors With Partially Functional Autoregressions

In many business and economics studies, researchers have sought to measure the dynamic dependence of curves with high-dimensional mixed-type predictors. We propose a partially functional autoregressive model (pFAR) where the serial dependence of curves is controlled by coefficient operators that are defined on a two-dimensional surface, and the individual and group effects of mixed-type predictors are estimated with a two-layer regularization. We develop an efficient estimation with the proven asymptotic properties of consistency and sparsity. We show how to choose the sieve and tuning parameters in regularization based on a forward-looking criterion. In addition to the asymptotic properties, numerical validation suggests that the dependence structure is accurately detected. The implementation of the pFAR within a real-world analysis of dependence in German daily natural gas flow curves, with seven lagged curves and 85 scalar predictors, produces superior forecast accuracy and an insightful understanding of the dynamics of natural gas supply and demand for the municipal, industry, and border nodes, respectively.

Forecasting of high-resolution electricity consumption with stochastic climatic covariates via a functional time series approach

This paper proposes a functional autoregressive model with stochastic functional covariates, denoted by FARSX, to depict high-resolution data dynamics. An easy-to-implement procedure is proposed to estimate the model parameters under the frameworks of an expansion of multiresolution B-spline basis functions and an adaptive lasso criterion with a two-layer sparsity assumption. We derive the consistency of the proposed estimators under mild conditions. The effectiveness of the estimation procedure allows us to further construct a FARSX model with time-varying parameters under a rolling window framework to capture stochastic effects of functional covariates timely and enhance the prediction accuracy. In the empirical study, the FARSX method with time-varying parameters is applied to the high-resolution electricity consumption and intraday temperatures in Belgium and the U.S. separately. The investigation results reveal that the FARSX model with time-varying parameters provides more reliable day-ahead predictions than several existing models.

Statistical inference of varying-coefficient partial functional spatial autoregressive model

In this paper, based on spatial autoregression and partial functional linear regression, we introduce a new varying-coefficient partial functional spatial autoregressive model. The functional principal component analysis and B-spline are adopted to approximate the slope function and varying-coefficient functions respectively. Then, the instrumental variable method gives final estimators. Under some regular conditions, we further study the asymptotic normality of the parameter and the convergence rates of slope function and coefficient functions. Lastly, the finite sample performance of the proposed methodology is evaluated by simulation studies and a practical data example.

Seasonal functional autoregressive models

Functional autoregressive models are popular for functional time series analysis, but the standard formulation fails to address seasonal behaviour in functional time series data. To overcome this shortcoming, we introduce seasonal functional autoregressive time series models. For the model of order one, we derive sufficient stationarity conditions and limiting behaviour, and provide estimation and prediction methods. Moreover, we consider a portmanteau test for testing the adequacy of this model, and we derive its asymptotic distribution. The merits of this model are demonstrated using simulation studies and via an application to hourly pedestrian counts.

A functional autoregressive model based on exogenous hydrometeorological variables for river flow prediction

In this research, a functional time series model was introduced to predict future realizations of river flow time series. The proposed model was constructed based on a functional time series’s correlated lags and the essential exogenous climate variables. Rainfall, temperature, and evaporation variables were hypothesized to have substantial functionality in river flow simulation. Because an actual time series model is unspecified and the input variables’ significance for the learning process is unknown in practice, it was employed a variable selection procedure to determine only the significant variables for the model. A nonparametric bootstrap model was also proposed to investigate predictions’ uncertainty and construct pointwise prediction intervals for the river flow curve time series. Historical datasets at three meteorological stations (Mosul, Baghdad, and Kut) located in the semi-arid region, Iraq, were used for model development. The prediction performance of the proposed model was validated against existing functional and traditional time series models. The numerical analyses revealed that the proposed model provides competitive or even better performance than the benchmark models. Also, the incorporated exogenous climate variables have substantially improved the modeling predictability performance. Overall, the proposed model indicated a reliable methodology for modeling river flow within the semi-arid region.

Modeling of water consumption in Saudi Arabia using classical and modern time series methods

Overpopulation, industrialization, urbanization, and the spreading out of irrigated agricultural lands are the driving forces to increase the demand of water in the Kingdom of Saudi Arabia. Improvement, production, and management of various commodities are one of the objectives of vision 2030 of the Kingdom. In this paper, we seek to find the paramount model for water consumption in Saudi Arabia using classical and modern tools of time series analysis. From the classical time series analysis, we employed multiplicative seasonal autoregressive integrated moving average (SARIMA) model, while functional autoregressive (FAR) model was adopted under modern time series analysis. A data of water consumption from January 2001 to December 2016 were analyzed, and the results were compared using these two methods. This study is the first ever application of FAR method in Saudi Arabia using water consumption data. Statistical analysis exhibits that the best model for water consumption data under classical approach is SARIMA (1,0,2)(2,1,1) [12] while FAR (1) under modern paradigm. A comparison between these methods reveals that FAR modeling is better than SARIMA in training as well as in forecasting of water consumption data in the Kingdom. This kind of modeling is beneficial for future policy implications.

A review study of functional autoregressive models with application to energy forecasting

AbstractIn this data‐rich era, it is essential to develop advanced techniques to analyze and understand large amounts of data and extract the underlying information in a flexible way. We provide a review study on the state‐of‐the‐art statistical time series models for univariate and multivariate functional data with serial dependence. In particular, we review functional autoregressive (FAR) models and their variations under different scenarios. The models include the classic FAR model under stationarity; the FARX and pFAR model dealing with multiple exogenous functional variables and large‐scale mixed‐type exogenous variables; the vector FAR model and common functional principal component technique to handle multiple dimensional functional time series; and the warping FAR, varying coefficient‐FAR and adaptive FAR models to handle seasonal variations, slow varying effects and the more challenging cases of structural changes or breaks respectively. We present the models’ setup and detail the estimation procedure. We discuss the models’ applicability and illustrate the numerical performance using real‐world data of high‐resolution natural gas flows in the high‐pressure gas pipeline network of Germany. We conduct 1‐day and 14‐days‐ahead out‐of‐sample forecasts of the daily gas flow curves. We observe that the functional time series models generally produce stable out‐of‐sample forecast accuracy.This article is categorized under: Statistical Models > Semiparametric Models Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Day-ahead high-resolution forecasting of natural gas demand and supply in Germany with a hybrid model

As the natural gas market is moving towards short-term planning, accurate and robust short-term forecasts of the demand and supply of natural gas is of fundamental importance for a stable energy supply, a natural gas control schedule, and transport operation on a daily basis. We propose a hybrid forecast model, Functional AutoRegressive and Convolutional Neural Network model, based on state-of-the-art statistical modeling and artificial neural networks. We conduct short-term forecasting of the hourly natural gas flows of 92 distribution nodes in the German high-pressure gas pipeline network, showing that the proposed model provides nice and stable accuracy for different types of nodes. It outperforms all the alternative models, with an improved relative accuracy up to twofold for plant nodes and up to fourfold for municipal nodes. For the border nodes with rather flat gas flows, it has an accuracy that is comparable to the best performing alternative model.

Forecasting One-Day-Ahead Electricity Prices for Italian Electricity Market Using Parametric and Nonparametric Approaches

Over the last three decades, accurate modeling and forecasting of electricity prices has become a key issue in competitive electricity markets. As electricity price series usually exhibit several complex features, such as high volatility, seasonality, calendar effect, non-stationarity, non-linearity and mean reversion, price forecasting is not a trivial task. However, participants of electricity market need price forecast to make decisions in their daily activity in the market, such as trading, risk management or future planning. In this study we consider linear and nonlinear models for one-day-ahead forecast of electricity prices using components estimation techniques. This approach requires to filter out the structural, deterministic components from the original time series and to model the residual component by means of some stochastic process. The final forecast is obtained by combining the predictions of both these components. In this work, linear and non-linear models are applied to both, deterministic and stochastic, components. In the case of stochastic component, AutoRegressive, Nonparametric AutoRegressive, Functional AutoRegressive, and Nonparametric Functional AutoRegressive have been considered. Furthermore, two naive benchmarks are applied directly to the price time series and their results are compared with our proposed models. An application of the proposed methodology is presented for the Italian electricity market (IPEX). Our analysis suggests that, in terms of Mean Absolute Error, Mean Absolute Percentage Error, and Pearson correlation coefficient, best results are obtained when deterministic component is estimated by using parametric approach. Further, Functional AutoRegressive model performs relatively better than the rest while Nonparametric AutoRegressive is highly competitive.

Regularized Partially Functional Autoregressive Model

Functional time series and high-dimensional scalar predictors frequently arise in a wide range of modern economic and business applications, which require statistical models that can simultaneously handle the temporal and causal dependence that are prevalent in large sets of mixed-type data. We propose a partially functional autoregressive model (pFAR) to describe the dynamic evolution of the serially correlated functional response on its own lagged values and the causal relation with a large amount of exogenous scalar predictors. Our estimation is conducted by facilitating the sieve method and a two-layer sparsity assumption that is imposed on groups and elements. In the high-dimensional setting, the sparse structure is completely unknown and it is identified entirely data-driven with a forward-looking criterion. In addition, asymptotic properties of the estimators are established. Extensive simulation studies show that the pFAR model accurately identifies the sparse structure with a convincing and stable predictive performance. The power of the pFAR model is further confirmed by real data analysis of day-ahead gas demand and supply curve predictions of multiple nodes in the German natural gas transmission network with different functions. Given the historical values of the daily curves and 85 scalar predictors, the model detects several essential categories of mixed-type predictors with insightful economic interpretation. It also provides appealing out-of-sample forecast accuracy when compared to a number of popular alternative models.

Forecasting of electricity price through a functional prediction of sale and purchase curves

AbstractThis work proposes a new approach for the prediction of the electricity price based on forecasting aggregated purchase and sale curves. The basic idea is to model the hourly purchase and the sale curves, to predict them and to find the intersection of the predicted curves in order to obtain the predicted equilibrium market price and volume. Modeling and forecasting of purchase and sale curves is performed by means of functional data analysis methods. More specifically, parametric (FAR) and nonparametric (NPFAR) functional autoregressive models are considered and compared to some benchmarks. An appealing feature of the functional approach is that, unlike other methods, it provides insights into the sale and purchase mechanism connected with the price and demand formation process and can therefore be used for the optimization of bidding strategies. An application to the Italian electricity market (IPEX) is also provided, showing that NPFAR models lead to a statistically significant improvement in the forecasting accuracy.

Modeling seasonality and serial dependence of electricity price curves with warping functional autoregressive dynamics

Electricity prices are high dimensional, serially dependent and have seasonal variations. We propose a Warping Functional AutoRegressive (WFAR) model that simultaneously accounts for the cross time-dependence and seasonal variations of the large dimensional data. In particular, electricity price curves are obtained by smoothing over the $24$ discrete hourly prices on each day. In the functional domain, seasonal phase variations are separated from level amplitude changes in a warping process with the Fisher–Rao distance metric, and the aligned (season-adjusted) electricity price curves are modeled in the functional autoregression framework. In a real application, the WFAR model provides superior out-of-sample forecast accuracy in both a normal functioning market, Nord Pool, and an extreme situation, the California market. The forecast performance as well as the relative accuracy improvement are stable for different markets and different time periods.

Functional regression models for South African economic indicators: a growth curve perspective

AbstractIn this paper, we compare three functional regression models from a growth curve perspective to predict the relationship between two economic variables, specifically we compare a functional concurrent model, a functional historical model and a functional autoregressive model (FAR). The dependent and the independent variables are cumulated over the annual time window for the growth curve analyses. These models are used to predict exports (real) for the South African economy in terms of electricity demand. The data analysed consist of 33 years of exports (in ZAR million) at annual quarterly frequency, and electricity demand (in GwH) at monthly totals. Exploratory analysis included phase‐plane plots for the two series. For the prediction exercise, the baseline concurrent model was evaluated against the other two models, and their performance compared using the root‐mean‐square error (RSME) measure, the relative performance in terms of the ratio of the RMSEs, and a Kolmogorov–Smirnov based hypothesis test to compare the distributions of the RMSEs of the models. Our results show that from the growth curve perspective, for the prediction of exports in terms of electricity for the SA economy, the FAR model of lag(1) outperforms both the concurrent model and the historical model by far.