ROBUST REGRESSION AND INTERPOLATION FOR TIME SERIES

Abstract. In this paper we shall consider the interpolation problem under the condition that the spectral density of a stationary process concerned is vaguely known (i.e., Huber's ε -contaminated model). Then we can get a minimax robust interpolator for the class of spectral densities S ={ g:g(x)=(1-ε)f(x)+εh(x)ε Ar D_o, 0<ε<1}, where f(x) is a known spectral density and D ₀ is a certain class of spectral densities. Also we shall consider the time series regression problem under the condition that the residual spectral density is vaguely known. Then we can get a minimax robust regression coefficient estimate for the class of the residual spectral densities S .     

ON THE ROBUST PREDICTION AND INTERPOLATION OF TIME SERIES IN THE PRESENCE OF CORRELATED NOISE

Abstract. We consider the problem of predicting and interpolating linearly a time series which can be represented as the sum of a model process with known spectral density and a noise process. The spectral density of the noise process is unknown with the exception of an upper bound for its integral. Some partial information of quite general kind about the cross spectral density of model and noise is available. We prove the existence of a robust predictor which minimizes the maximal mean-square error, where the maximum is taken over all spectral densities which may arise from the circumstances described above as spectral density of the predicted time series. An analogous result holds for the related interpolation problem. We describe how to derive the minimax robust predictor and interpolator in concrete situations. The method is illustrated by determining the robust predictor explicitly for three examples where model and noise may be arbitrarily, only causally or not at all correlated.     

Testing non‐parametric hypotheses for stationary processes by estimating minimal distances

In this article, new tests for non‐parametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L²‐distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.     

Robust Hypothesis Testing and Robust Time Series Interpolation And Regression

Abstract. Recent results on minimax robust time series interpolation and regression coefficient estimation are generalized and extended through a relationship with robust hypothesis testing. The spectral uncertainty classes in the time series problems are assumed to be convex and to satisfy an integral constraint such as on the variance of the process. It is shown that robust solutions in such cases can always be obtained from the least-favourable probability density functions for corresponding hypothesis testing problems. A specific class, the bounded spectral densities from the band model, is considered to illustrate the results.     

LINEAR INTERPOLATORS AND THE INVERSE CORRELATION FUNCTION OF NON-STATIONARY TIME SERIES

Abstract. The inverse correlation function of a stationary time series was introduced by Cleveland (The inverse autocorrelations of a time series and their applications. Technometrics 14 (1972), 277–93). In this paper inverse correlations are defined for non-stationary time series {x_t, integer t} such that y_t= (1 —B^s)^dx_t is second-order stationary. The linear interpolator and the inverse process of {x_t} are also defined:their weights are shown to be time invariant and proportional to the inverse correlations. The interpolation method for the estimation of the inverse correlation function of a stationary time series is extended to the non-stationary series {x_t} and the asymptotic properties of the estimates are found to be similar to those in the stationary case.     

A wavelet‐Fisz approach to spectrum estimation

Abstract. We propose a new approach to wavelet threshold estimation of spectral densities of stationary time series. Our proposal addresses the problem of heteroscedasticity and non‐normality of the (tapered) periodogram. We estimate thresholds for the empirical wavelet coefficients of the periodogram as appropriate linear combinations of the periodogram values similar to empirical scaling coefficients. Our solution introduces ‘asymptotically noise‐free reconstruction thresholds’ which parallels classical wavelet theory for nonparametric regression with independently and identically distributed Gaussian errors. Our simulations show promising results that clearly improve on existing approaches. In addition, we derive theoretical results on the near‐optimal rate of convergence of the minimax mean‐square risk for a class of spectral densities, including those of low regularity.     

Artificial and Hybrid Fuzzy Linear Neural Network‐Based Estimation of Seed Oil Content of Safflower

Inexpensive and rapid methods for measurement of seed oil content by near infrared reflectance spectroscopy (NIRS) are useful for developing new oil seed cultivars. Adopting default multiple linear regression (MLR), the predictions of safflower oil content were made by 20–140 samples using a Perten Inframatic 8620 NIR spectrometer. Although the obtained interpolation results of MLR had desired accuracy, the extrapolation was extremely poor. The extrapolation determination coefficient (R²) and standard error (SE) of cross validation for MLR models were 0.63–0.78 and 3.71–4.44, respectively. In order to overcome the accuracy limitation of linear MLR models, a common suggestion is to use a nonlinear artificial neural network (ANN); however, it needs a large number of data to yield significant accurate results. We developed a novel robust hybrid fuzzy linear neural (HFLN) network to capture simultaneously linear and nonlinear patterns of data with a limited number of safflower samples. Empirical extrapolation results showed that the HFLN had higher R² (=0.85) and lower SE (=1.83) compared to those obtained by MLR and ANN models. It is concluded that hybrid methodologies could be used to construct efficient and appropriate models for estimation of seed oil content set up on NIR system.     

Increased Rhamnolipid Concentration and Productivity Achieved with Advanced Process Design

Rhamnolipids are biosurfactants having several applications. A major limitation in rhamnolipid production is low productivity, which decreases significantly during stationary-phase production. In this study, fermentations were first made with nitrogen-limited stationary phase. Long-term rhamnolipid production (up to 505 h) could be maintained with low-rate N-source addition but the intermittent cell growth led to lower productivity (q_p), particularly apparent at the highest addition rate. Four fermentations were next made under non-N-limited stationary phase without and with N-source supplementation; q_p could be much higher at 24–26 mg g⁻¹h⁻¹. Three final fermentations were designed to build the maximum cell concentration to 30 g L⁻¹ in two growth phases where the growth rate in the second phase was regulated by N-addition to control foaming. Cultures then entered non-N-limited stationary phase and were N-supplemented. At an optimal rate of 15% growth-N per 24 h to maintain cell activity, a highest rhamnolipid concentration of 120 g L⁻¹ was obtained after 144 h with overall productivity of 839 mg L⁻¹h⁻¹.     

A Shrinked Forecast in Stationary Processes Favouring Percentage Error

Abstract. In stationary time‐series forecasting, the commonly used criterion for selecting a proper forecast is the mean square error (MSE), which is minimized by the conditional expectation of future observation given the entire past known as a minimum MSE forecast. In this paper, mean square percentage error (MSPE) instead of is used to forecast autoregressive moving average (ARMA)(p,q) series. The suggested forecast takes the form of or (CV_t+1 is the coefficient of variation for one step ahead) times the minimum MSE forecast, which performs better not only in MSPE, but also in mean absolute percentage error (MAPE) than the ordinary MSE forecast in simulation studies. A real data example also supported this result. We conclude that, if percentage error is a prime concern, this shrinked version of MSE forecast performs better than the ordinary forecast in the stationary ARMA(p,q) model.     

Quantifying the uncertainty introduced by discretization and time‐averaging in two‐fluid model predictions

The two‐fluid model (TFM) has become a tool for the design and troubleshooting of industrial fluidized bed reactors. To use TFM for scale up with confidence, the uncertainty in its predictions must be quantified. Here, we study two sources of uncertainty: discretization and time‐averaging. First, we show that successive grid refinement may not yield grid‐independent transient quantities, including cross‐section–averaged quantities. Successive grid refinement would yield grid‐independent time‐averaged quantities on sufficiently fine grids. Then a Richardson extrapolation can be used to estimate the discretization error, and the grid convergence index gives an estimate of the uncertainty. Richardson extrapolation may not work for industrial‐scale simulations that use coarse grids. We present an alternative method for coarse grids and assess its ability to estimate the discretization error. Second, we assess two methods (autocorrelation and binning) and find that the autocorrelation method is more reliable for estimating the uncertainty introduced by time‐averaging TFM data. © 2017 American Institute of Chemical Engineers AIChE J, 63: 5343–5360, 2017     

On limiting spectral distribution of large sample covariance matrices by VARMA(p,q)

We studied the limiting spectral distribution of large‐dimensional sample covariance matrices of a stationary and invertible VARMA(p,q) model. Relationship of the power spectral density and limiting spectral distribution of large population dimensional covariance matrices of ARMA(p,q) is established. The equation about Stieltjes transform of large‐dimensional sample covariance matrices is also derived. As applications, the classical M‐P law, VAR(1) and VMA(1) can be regarded as special examples.     

The restricted likelihood ratio test for autoregressive processes

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (?1, 1] × (?1, 1)^p?1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.     

A Nonparametric Prewhitened Covariance Estimator

This paper proposes a new nonparametric spectral density estimator for time series models with general autocorrelation. The conventional nonparametric estimator that uses a positive kernel has mean squared error no better than n^?4/5. We show that the best implementation of our estimator has mean squared error of order n^?8/9, provided there is sufficient smoothness present in the spectral density. This is, of course, achieved by bias reduction; however, unlike most other bias reduction methods, like the kernel method with higher‐order kernels, our procedure ensures a positive definite estimate. Our method is a generalization of the well‐known prewhitening method of spectral estimation; we argue that this can best be interpreted as multiplicative bias reduction. Higher‐order expansions for the proposed estimator are derived, providing an improved bandwidth choice that minimizes the mean squared error to the second order. A simulation study shows that the recommended prewhitened kernel estimator reduces bias and mean squared error in spectral density estimation.     

Selective Extraction of Fe3+ Cation by Calixarene-Based Cyclic Ligands

Abstract

The selective liquid-liquid extraction of Fe³⁺ cation from the aqueous phase to the organic phase was carried out by using p-tert-butylcalix[4]arene [L₁], ca-lix[4]arene [L₂], p-nitro-calix[4]arene [L₃], calix[4]arene p-sulfonic acid [L₄], p-(diethylamino)methylcalixt4]arene [L₅], tetramethyl-p-tert-butylcalix[4]arene tet-raketone [L₆], 25,27-dimethyl-26,28-dihydroxy-p-tert-butylcalix[4]arene diketone [L₇], calix[4]arene-bearing dioxime group on the lower rim [L₈], and a monooxime [L₉]. The effect of varying pH upon the extraction ability of calixarenes substituted with electron-donating and electron-withdrawing groups at their p-position was examined. Observed results were compared with those found for unsubstituted calix[4]arene.     

Concentration dependence,boundary layer resistance,and the “time-lag” diffusion coefficient

Diffusion coefficient measurements have been shown to be strongly influenced by boundary layer resistance and concentration dependence. When half-times for desorption or absorption are used to find diffusion coefficients from the equation for a constant coefficient (T_½ = Dt/L² = 0.049), the values so obtained require correction. Diffusion coefficients found by the “time-lag” method are also influenced strongly by surface resistance and concentration dependence. The usual equation for a constant diffusion coefficient gives a break-through curve with an extrapolation to the time-lag, T_L = Dt/L² = 1/6. This factor can rise to ½ for concentration dependence. Where boundary layer resistance is encountered, (a situation which appears to be quite common), T_L is significantly increased and the slope at steady state is decreased.     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

ON THE SELECTION OF RANDOM SAMPLING SCHEMES FOR THE SPECTRAL ESTIMATION OF CONTINUOUS TIME PROCESSES

Abstract. Let X={X(t), -∝<t<∝} be a continuous time stationary process with spectral density φ_x(Λ) and {φ_k} be a stationary point process independent of X. The problem of selecting the sampling point process {τ_k} for the estimation of φ_x(Λ) based on the discrete time observations {X(τ_k), τ_k} is considered. Sufficient conditions are established for the admittability of a broad class of delayed renewal point processes. Examples are given for delayed renewal point processes {τ_k} which have a mixture of Gamma densities for their inter-arrival times. Guidelines are given for the selection of specific point processes in the estimation of broadband and narrowband spectral density functions.     

New coordination compounds of some rare-earth metal complexes with sulfur and nitrogen Schiff bases and their in vitro antibacterial and antifungal properties

A series of rare-earth metal complexes with Schiff bases have been prepared by the interactions of hydrated lanthanide(III) chloride with the sodium salts of 1-(2-thienyl)ethanone hydrazinecarbothioamide (L¹H) and 1-(2-thienyl)ethanonehydrazinecarboxamide (L²H) in 1:3 molar ratios and characterized by their elemental analyses, molar conductance and IR, NMR (¹H and ¹³C) electronic and EPR spectral studies. The spectral data suggested that the complexes have a hexa-coordinated environment around the central metal atoms. Elemental analyses and NMR spectral data of the ligands and their metal(III) complexes agree with their proposed structures. The synthesized Schiff bases and their new metal complexes have been screened for in vitro antibacterial activity against Gram-negative (Escherichia coli) and Gram-positive (Pseudomonas cepacicola) bacterial strains and for in vitro antifungal activity against Fusarium oxysporum and Macrophomina phaseolina. All compounds showed significant antibacterial and antifungal activities against microbial species.