A new test for ARMA models with errors following a general white noise process

Summary The aim of this paper is to present a new test for ARMA models. The consistency of this sequence of tests is proved using the asymptotic separation of the two sequences of probability laws defined by each hypothesis to be tested. Furthermore, we illustrate the adequacy of this test for general ARMA models in which the error process is conditionally heteroscedastic white noise. Therefore, beyond its application to classical ARMA processes, this test is also well-adapted to ARMA-GARCH and ARMA-GTARCH models.     

Streamflow rating uncertainty: Characterisation and impacts on model calibration and performance

Common streamflow gauging procedures require assumptions about the stage-discharge relationship (the ‘rating curve’) that can introduce considerable uncertainties in streamflow records. These rating uncertainties are not usually considered fully in hydrological model calibration and evaluation yet can have potentially important impacts. We analysed streamflow gauge data and conducted two modelling experiments to assess rating uncertainty in operational rating curves, its impacts on modelling and possible ways to reduce those impacts. We found clear evidence of variance heterogeneity (heteroscedasticity) in streamflow estimates, with higher residual values at higher stage values. In addition, we confirmed the occurrence of streamflow extrapolation beyond the highest or lowest stage measurement in many operational rating curves, even when these were previously flagged as not extrapolated. The first experiment investigated the impact on regional calibration/evaluation of: (i) using two streamflow data transformations (logarithmic and square-root), compared to using non-transformed streamflow data, in an attempt to reduce heteroscedasticity and; (ii) censoring the extrapolated flows, compared to no censoring. Results of calibration/evaluation showed that using a square-root transformed streamflow (thus, compromising weight on high and low streamflow) performed better than using non-transformed and log-transformed streamflow. Also, surprisingly, censoring extrapolated streamflow reduced rather than improved model performance. The second experiment investigated the impact of rating curve uncertainty on catchment calibration/evaluation and parameter estimation. A Monte-Carlo approach and the nonparametric Weighted Nadaraya-Watson (WNW) estimator were used to derive streamflow uncertainty bounds. These were later used in calibration/evaluation using a standard Nash-Sutcliffe Efficiency (NSE) objective function (OBJ) and a modified NSE OBJ that penalised uncertain flows. Using square-root transformed flows and the modified NSE OBJ considerably improved calibration and predictions, particularly for mid and low flows, and there was an overall reduction in parameter uncertainty.     

QUANTILE PERIODOGRAM AND TIME‐DEPENDENT VARIANCE

This article investigates the statistical properties of the recently introduced quantile periodogram for time series with time‐dependent variance. The asymptotic distribution of the quantile periodogram is derived in the case where the time series consists of i.i.d. random variables multiplied by a time‐dependent scale parameter. It is shown that the time‐dependent variance is represented approximately additively in the mean of the asymptotic distribution of the quantile periodogram. It is also shown that the strength of the representation is proportional to the squared quantile of the i.i.d. random variables, so that a stronger characterization is expected at upper and lower quantile levels if the time series is centred at zero. These properties are further demonstrated by simulation results. The series of daily returns from the Dow Jones Industrial Average, which is known to exhibit heteroscedastic volatility, serves to motivate the investigation.     

Accounting for Heteroscedasticity in Experimental Design

Power transformation weighting has been found to be a powerful technique for accounting for heteroscedasticity in model fitting. In this paper the transformation weighting concept is used in developing sequential design criteria for precise parameter estimation in heteroscedastic situations. Criteria are proposed for precise estimation of the model and transformation parameters together and for precise estimation of the model parameters alone. Implementation of the criteria is illustrated with two examples from chemical kinetics.     

Nonlinear Lp,-Norm Estimation

An understanding of the inherent variability in micro-computed tomography (micro-CT) data is essential to tasks such as statistical process control and the validation of radiographic simulation tools. These data present unique challenges to variability analysis due to the relatively low resolution of radiographs, and also due to minor variations from run to run which can result in misalignment or magnification changes between repeated measurements of a sample. Such positioning changes artificially inflate the variability of the data in ways that mask true physical phenomena. We present a novel Bayesian nonparametric regression model that incorporates both additive and multiplicative measurement error in addition to heteroscedasticity to address this problem. We use this model to assess the effects of sample thickness and sample position on measurement variability for an aluminum specimen. Supplementary materials for this article are available online.     

Nonlinear hierarchical models for predicting cover crop biomass using Normalized Difference Vegetation Index

Incorporating cover crops into agricultural systems can improve soil structural properties, increase nutrient availability, reduce erosion and loss of agrochemicals, and suppress weeds. These benefits are a function of the amount of cover crop biomass that enters the soil. The ability to easily and inexpensively quantify the spatial variability of cover crop biomass is needed to better understand and predict its potential as an input to agricultural systems. Here, we explore the use of Normalized Difference Vegetation Index (NDVI) as a source of information for improving accuracy and precision of cover crop biomass prediction. We focus on developing models that account for biomass variability within and among fields. These models are used to produce digital data layers of predicted biomass and associated uncertainty. We propose hierarchical nonlinear models with field random effects and a residual variance function to accommodate strong heteroscedasticity. These models are motivated using aboveground biomass of red clover (Trifolium pratense L.) measured on three different dates in five fields in southwest Michigan. Model adequacy was assessed using the Deviance Information Criterion. Given this criterion, the “best” fitting model included field effects and a polynomial function to account for non-constant residual variance. Importantly, we demonstrate that accounting for heteroscedasticity in the model fitting is critical for capturing uncertainty in subsequent biomass prediction.     

Laplace random effects models for interlaboratory studies

A model is introduced for measurements obtained in collaborative interlaboratory studies, comprising measurement errors and random laboratory effects that have Laplace distributions, possibly with heterogeneous, laboratory-specific variances. Estimators are suggested for the common median and for its standard deviation. We provide predictors of the laboratory effects, and of their pairwise differences, along with the standard errors of these predictors. Explicit formulas are given for all estimators, whose sampling performance is assessed in a Monte Carlo simulation study.     

Robust diagnostics for the heteroscedastic regression model

The assumption of equal variance in the normal regression model is not always appropriate. Cook and Weisberg (1983) provide a score test to detect heteroscedasticity, while Patterson and Thompson (1971) propose the residual maximum likelihood (REML) estimation to estimate variance components in the context of an unbalanced incomplete-block design. REML is often preferred to the maximum likelihood estimation as a method of estimating covariance parameters in a linear model. However, outliers may have some effect on the estimate of the variance function. This paper incorporates the maximum trimming likelihood estimation ( [Hadi and Luce?o, 1997] and [Vandev and Neykov, 1998]) in REML to obtain a robust estimation of modelling variance heterogeneity. Both the forward search algorithm of Atkinson (1994) and the fast algorithm of Neykov et al. (2007) are employed to find the resulting estimator. Simulation and real data examples are used to illustrate the performance of the proposed approach.     

Cumulative continuous predictions for bole and aboveground woody biomass in Eucalyptus globulus plantations in northwestern Spain

At present there is increasing interest in modelling biomass to estimate carbon sequestration or the availability of forest products for use as bioenergy. The biomass of different tree components can be estimated to provide more detailed information. However, the different components have not been clearly defined. Moreover, the greater the number of components considered, the more difficult it is to fit the system of equations with any guarantee of statistical robustness. To overcome these limitations, we developed a continuous function that predicts cumulative biomass from the stump until any top diameter (including the biomass of branches). We also used two different methods to predict bole biomass: a cumulative continuous biomass function and conversion from volume to biomass by use of a taper equation and average wood density. We used a mixed-effects modelling framework to account for correlated errors in developing the taper equation. We developed a separate equation to estimate the foliar biomass for use in estimating total aboveground tree biomass. The cumulative aboveground woody biomass equation is implicitly additive, and no heteroscedasticity was observed, thereby resolving two of the main modelling goals in the development of biomass equations. For predicting cumulative bole biomass, estimation from volume generated less error, after bias correction, than direct estimation. Moreover, the indirect method also yields useful variables such as volume and height limits. Other variables such as carbon and nutrient contents, calorific power, ash content, etc. can be estimated by multiplying the mean contents by the predicted biomass or, for more accurate predictions, by using equations based on the relative diameter.     

Further Consideration of Heteroscedasticity in Fitting Kinetic Models

Box and Hill [6] recently proposed a method for using power transformation weighting in least squares analysis to account for changing variance. Such an approach can be useful when the original data are heteroscedastic but adequate weight estimates are not available, and when the original data are homoscedastic but heteroscedasticity is induced by the data analyst in linearising a nonlinear model.

Several aspects of their proposal are examined for practical implications in fitting chemical kinetic models and a more direct algorithm is recommended for fitting nonlinear models to heteroscedastic data. Methods for testing model adequacy and assessing parameter precision in such situations are also discussed.