Han Lin Shang

Rainbow Plots, Bagplots, and Boxplots for Functional Data

We propose new tools for visualizing large amounts of functional data in the form of smooth curves. The proposed tools include functional versions of the bagplot and boxplot, which make use of the first two robust principal component scores, Tukey’s data depth and highest density regions. By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with existing methods for detecting outliers in functional data, and show that our methods are better able to identify outliers. An R-package containing computer code and datasets is available in the online supplements.

Nonparametric time series forecasting with dynamic updating

We present a nonparametric method to forecast a seasonal univariate time series, and propose four dynamic updating methods to improve point forecast accuracy. Our methods consider a seasonal univariate time series as a functional time series. We propose first to reduce the dimensionality by applying functional principal component analysis to the historical observations, and then to use univariate time series forecasting and functional principal component regression techniques. When data in the most recent year are partially observed, we improve point forecast accuracy by using dynamic updating methods. We also introduce a nonparametric approach to construct prediction intervals of updated forecasts, and compare the empirical coverage probability with an existing parametric method. Our approaches are data-driven and computationally fast, and hence they are feasible to be applied in real time high frequency dynamic updating. The methods are demonstrated using monthly sea surface temperatures from 1950 to 2008.

Functional time series approach for forecasting very short-term electricity demand

This empirical paper presents a number of functional modelling and forecasting methods for predicting very short-term (such as minute-by-minute) electricity demand. The proposed functional methods slice a seasonal univariate time series (TS) into a TS of curves; reduce the dimensionality of curves by applying functional principal component analysis before using a univariate TS forecasting method and regression techniques. As data points in the daily electricity demand are sequentially observed, a forecast updating method can greatly improve the accuracy of point forecasts. Moreover, we present a non-parametric bootstrap approach to construct and update prediction intervals, and compare the point and interval forecast accuracy with some naive benchmark methods. The proposed methods are illustrated by the half-hourly electricity demand from Monday to Sunday in South Australia.

Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density

Error density estimation in a nonparametric functional regression model with functional predictor and scalar response is considered. The unknown error density is approximated by a mixture of Gaussian densities with means being the individual residuals, and variance as a constant parameter. This proposed mixture error density has a form of a kernel density estimator of residuals, where the regression function is estimated by the functional Nadaraya-Watson estimator. A Bayesian bandwidth estimation procedure that can simultaneously estimate the bandwidths in the kernel-form error density and the functional Nadaraya-Watson estimator is proposed. A kernel likelihood and posterior for the bandwidth parameters are derived under the kernel-form error density. A series of simulation studies show that the proposed Bayesian estimation method performs on par with the functional cross validation for estimating the regression function, but it performs better than the likelihood cross validation for estimating the regression error density. The proposed Bayesian procedure is also applied to a nonparametric functional regression model, where the functional predictors are spectroscopy wavelengths and the scalar responses are fat/protein/moisture content, respectively.

Methods for Scalar-on-Function Regression

Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images, etc. are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorizing the basic model types as linear, nonlinear and nonparametric. We discuss publicly available software packages, and illustrate some of the procedures by application to a functional magnetic resonance imaging dataset.

A sampling algorithm for bandwidth estimation in a nonparametric regression model with a flexible error density

The unknown error density of a nonparametric regression model is approximated by a mixture of Gaussian densities with means being the individual error realizations and variance a constant parameter. Such a mixture density has the form of a kernel density estimator of error realizations. An approximate likelihood and posterior for bandwidth parameters in the kernel-form error density and the Nadaraya–Watson regression estimator are derived, and a sampling algorithm is developed. A simulation study shows that when the true error density is non-Gaussian, the kernel-form error density is often favored against its parametric counterparts including the correct error density assumption. The proposed approach is demonstrated through a nonparametric regression model of the Australian All Ordinaries daily return on the overnight FTSE and S&P 500 returns. With the estimated bandwidths, the one-day-ahead posterior predictive density of the All Ordinaries return is derived, and a distribution-free value-at-risk is obtained. The proposed algorithm is also applied to a nonparametric regression model involved in state-price density estimation based on S&P 500 options data.

Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density

We investigate the issue of bandwidth estimation in a functional nonparametric regression model with function-valued, continuous real-valued and discrete-valued regressors under the framework of unknown error density. Extending from the recent work of Shang (2013) [‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics & Data Analysis, 67, 185–198], we approximate the unknown error density by a kernel density estimator of residuals, where the regression function is estimated by the functional Nadaraya–Watson estimator that admits mixed types of regressors. We derive a likelihood and posterior density for the bandwidth parameters under the kernel-form error density, and put forward a Bayesian bandwidth estimation approach that can simultaneously estimate the bandwidths. Simulation studies demonstrated the estimation accuracy of the regression function and error density for the proposed Bayesian approach. Illustrated by a spectroscopy data set in the food quality control, we applied the proposed Bayesian approach to select the optimal bandwidths in a functional nonparametric regression model with mixed types of regressors.

Bagplots, Boxplots and Outlier Detection for Functional Data

We propose some new tools for visualizing functional data and for identifying functional outliers. The proposed tools make use of robust principal component analysis, data depth and highest density regions. We compare the proposed outlier detection methods with the existing “functional depth” method, and show that our methods have better performance on identifying outliers in French male age-specific mortality data.

Grouped Functional Time Series Forecasting: An Application to Age-Specific Mortality Rates

ABSTRACT 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. in 2011 to functional time series, where age is considered as a continuum. The grouped functional time series methods are used to produce point forecasts of mortality rates that are aggregated appropriately across different disaggregation factors. For evaluating forecast uncertainty, we propose a bootstrap method for reconciling interval forecasts. Using the regional age-specific mortality rates in Japan, obtained from the Japanese Mortality Database, we investigate the one- to ten-step-ahead point and interval forecast accuracies between the independent and grouped functional time series forecasting methods. The proposed methods are shown to be useful for reconciling forecasts of age-specific mortality rates at the national and sub-national levels. They also enjoy improved forecast accuracy averaged over different disaggregation factors. Supplementary materials for the article are available online.

Bootstrap methods for stationary functional time series

Bootstrap methods for estimating the long-run covariance of stationary functional time series are considered. We introduce a versatile bootstrap method that relies on functional principal component analysis, where principal component scores can be bootstrapped by maximum entropy. Two other bootstrap methods resample error functions, after the dependence structure being modeled linearly by a sieve method or nonlinearly by a functional kernel regression. Through a series of Monte-Carlo simulation, we evaluate and compare the finite-sample performances of these three bootstrap methods for estimating the long-run covariance in a functional time series. Using the intraday particulate matter (