Regression Models with Time Series Errors

In models of the form Y_t = r ( X_t ) + Z_t , where r is an unknown function and { X_t } is a covariate process independent of the stationary error { Z_t }, we give conditions under which estimators based on residuals Z ₁, ..., Z _n obtained from linear smoothers are asymptotically equivalent to those based on the actual errors Z ₁, ..., Z_n .     

Cointegration in frequency domain

Existence of a cointegration relationship between two time series in the time domain imposes restrictions on the series zero-frequency behaviour in terms of their squared coherence, phase and gain, in the frequency domain. I derive these restrictions by studying cross-spectral properties of a cointegrated bivariate system. Specifically, I demonstrate that if two difference stationary series, X_t and Y_t , are cointegrated with a cointegrating vector [1   b ] and thus share a common stochastic trend, then at the zero frequency, the squared coherence of (1- L ) X_t and (1- L ) Y_t will equal one, their phase will equal zero, and their gain will equal | b |.     

Estimation of the Dominating Frequency for Stationary and Nonstationary Fractional Autoregressive Models

This paper was motivated by the investigation of certain physiological series for premature infants. The question was whether the series exhibit periodic fluctuations with a certain dominating period. The observed series are nonstationary and/or have long-range dependence. The assumed model is a Gaussian process X _t whose m th difference Y_t = (1 − B ) ^m X_t is stationary with a spectral density f that may have a pole (or a zero) at the origin. the problem addressed in this paper is the estimation of the frequency ω_max where f achieves the largest local maximum in the open interval (0, π). The process X_t is assumed to belong to a class of parametric models, characterized by a parameter vector θ, defined in Beran (1995). An estimator of ω_max is proposed and its asymptotic distribution is derived, with θ being estimated by maximum likelihood. In particular, m and a fractional differencing parameter that models long memory are estimated from the data. Model choice is also incorporated. Thus, within the proposed framework, a data driven procedure is obtained that can be applied in situations where the primary interest is in estimating a dominating frequency. A simulation study illustrates the finite sample properties of the method. In particular, for short series, estimation of ω_max is difficult, if the local maximum occurs close to the origin. The results are illustrated by two of the data examples that motivated this research.     

Some Results Concerning the Asymptotic Distribution of Sample Fourier Transforms and Periodograms for a Discrete-time Stationary Process with a Continuous Spectrum

We consider a stationary process ( X_t , t = 0, ±1, ...) with a continuous spectrum. Denote by D_n (λ) a tapered Fourier transform of ( X ₀, X ₁, ..., X _{n −1}) at (angular) frequency λ. We obtain the asymptotic distribution of D_n (λ) and the joint asymptotic distribution of { D_n (λ _j ), 1 ≤ j ≤ k } with continuity of the spectral density f (.) at the relevant frequencies as the only assumption concerning the second-order structure of ( X_t ); all other assumptions required are easily stated. The results are extended to processes for which f (.) is continuous except at λ = 0, with lim_λ←0 f (λ)λ^{2 d} = K , a constant, where 0 < d < ½, as is typical of certain types of processes with long-range dependence. Results for the sample periodogram, proportional to | D_n (λ)|², follow immediately.     

A TIME SERIES MODEL WITH SUDDENLY CHANGING PARAMETERS

Abstract. A process X_t = θ_t + e , is investigated where e_t is a strict white noise and θ_t is a Markov chain with two real states. A realization of X_t fluctuates around two levels which correspond to the two states of θ_t . Formulae for the extrapolation of the process X_t and for the mean square error of the extrapolation are derived. The moments of X_t and its covariance function are calculated. The results are used to derive moment estimators for the parameters of the model.     

ON STATIONARITY OF THE SOLUTION OF A DOUBLY STOCHASTIC MODEL

Abstract. Consider the discrete parameter process {X_I} satisfying the doubly stochastic model X_t =ø_tX_t-1+ε_t where {ø} and {ε_t} are also stochastic processes. Necessary and sufficient conditions on {ø} are given for { X₁ } to be a second order process. When {ø_t} is a strictly stationary process, some sufficient conditions in terms of {ø} are given which guarantee the wide sense stationarity of {X_t} . It turns out that for these problems the distribution and dependence structure of the process {log |ø|} play an important role.     

DISCRIMINANT ANALYSIS FOR STATIONARY VECTOR TIME SERIES

Abstract. In this paper, we shall consider the case where a stationary vector process { X_t } belongs to one of two categories described by two hypotheses π ₁ and π ₂. These hypotheses specify that { X_t } has spectral density matrices f (Λ) and g (Λ) under π ₁ and π ₂, respectively. Although Gaussianity of { X_t } is not assumed, we can formally make the Gaussian likelihood ratio (GLR) based on X (1),… X ( T ). Then an approximation I ( f : g ) of the GLR is given in terms of f (Λ) and g (Λ). If f (Λ) and g (Λ) are known, we can use I ( f : g ) as a classification statistic. It is shown that I ( f : g ) is a consistent classification criterion in the sense that the misclassification probabilities converge to zero as T →∝. When g is contiguous to f , we discuss non-Gaussian robustness of I ( f : g ). A sufficient condition for the non-Gaussian robustness will be given. Also a numerical example will be given.     

A note on L1 density estimation for linear processes

In this paper, we consider the L ₁ performance of a kernel estimator, f^_n of the density of a linear process X_t ∑^∞ _{k =0} a _k Z _t−k , a ₀ = 1, where { Z _t } is a sequence of independent and identically distributed (i.i.d.) random variables with E | Z ₁|^ε< ∞, for some ε > 1, and { a_k } is a sequence of reals converging to zero at a certain rate. Asymptotic minimizations of the integrated L ₁ risk of f_n and its upper bounds are considered. This paper extends the earlier results for the i.i.d. case by Devroye and Gyorfi ( Nonparametric Density Estimation: The L₁ View. New York: Wiley, 1985) and by Hall and Wand (Minimizing the L ₁ distance in nonparametric density estimation, J. Multivariate Anal. 26 (1988), 59–88) to the linear process case. Numerical examples to illustrate the performance of f_n are also presented.     

Anisotropic Thermal Expansion of β-Ca2SiO4 Monoclinic Crystal

Crystals of β-Ca₂SiO₄ (space group P 12₁/ n 1) were examined by high-temperature powder X-ray diffractometry to determine the change in unit-cell dimensions with temperature up to 645°C. The temperature dependence of the principal expansion coefficients (α_i) found from the matrix algebra analysis was as follows: α₁= 20.492 × 10⁻⁶+ 16.490 × 10⁻⁹ ( T - 25)°C⁻¹, α₂= 7.494 × 10⁻⁶+ 5.168 × 10⁻⁹( T - 25)°C⁻¹, α₃=−0.842 × 10⁻⁶− 1.497 × 10⁻⁹( T - 25)°C⁻¹. The expansion coefficient α₁, nearly along [302] was approximately 3 times α₂ along the b -axis. Very small contraction (α₃) occurred nearly along [     01]. The volume changes upon martensitic transformations of β↔α_L' were very small, and the strain accommodation would be almost complete. This is consistent with the thermoelasticity.     

Effect of Phosphorus Impurity on Tricalcium Silicate T1: From Synthesis to Structural Characterization

Alite is the major compound of anhydrous Portland cement: it is composed of tricalcium silicate Ca₃SiO₅ (C₃S) modified in composition and crystal structure by ionic substitutions. Alite is also the main hydraulic phase of cement and the most important for subsequent strength development. Using raw meals (rich in Ca₃P₂O₈) as alternative fuels in cement plants raises the question about the effect of phosphorus on C₃S and its consequences on reactivity with water. This paper deals with a systematic study of C₃S triclinic T₁ polymorph doped with P₂O₅ in the range 0–0.9 wt%. All the samples were characterized by X-ray powder diffraction (XRD), scanning electron microscopy (SEM), and electron-microprobe analysis. The appearance of a phase rich in phosphorus is shown. It displays a structure derivative of the α'_H–Ca₂SiO₄ polymorph, noted α'_H–C₂S(P). As phosphorus content increases, C₃S is more and more decomposed into free lime and α'_H–C₂S(P). The α'_H phase was detected from 0.1 wt% P₂O₅ and located at the interfaces of C₃S grains. Two identification keys are proposed in order to highlight the α'_H–C₂S(P) phase: the XRD angular window at 2θ_Cu=32.8°–33.2° and a smooth aspect on SEM micrographs.     

STATIONARITY OF THE SOLUTION OF Xt= AtXt-1+εt AND ANALYSIS OF NON-GAUSSIAN DEPENDENT RANDOM VARIABLES

Abstract. We give general and concrete conditions in terms of the coefficient (stochastic) process {A_t} so that the (doubly) stochastic difference equation X_t= A_tX_t-1+ε_t has a second-order strictly stationary solution. It turns out that by choosing {A_t} and the "innovation" process {ε_t} properly, a host of stationary processes with non-Gaussian marginals and long-range dependence can be generated using this difference equation. Examples of such nowGaussian marginals include exponential, mixed exponential, gamma, geometric, etc. When {A_t} is a binary time series, the conditional least-squares estimator of the parameters of this model is the same as those of the parameters of a Galton-Watson branching process with immigration.     

Ti2ZrO Phases Formed in the Titanium and Zirconia Interface After Reaction at 1550°C

Hot-pressed 3 mol% Y₂O₃ partially stabilized ZrO₂ was reacted with titanium at 1550°C/30 min. The interface was characterized by analytical transmission microscopy (transmission electron microscopy/energy-dispersive spectroscopy). The lamellar and the spherical Ti₂ZrO as well as the orthorhombic β'-Ti were found to exist in the titanium side after cooling down to room temperature. The crystal structures of the lamellar and the spherical Ti₂ZrO were orthorhombic and hexagonal, respectively. On heating, the dissolution of a large amount of zirconium and oxygen into titanium gave rise to the metastably supersaturated disordered α-Ti(Zr, O) solid solution where two different Ti₂ZrO phases subsequently precipitated, while the β-Ti coexisting with α-Ti at high temperatures was transformed to the orthorhombic β'-Ti during cooling. The spherical hexagonal Ti₂ZrO was an ordered structure, with Zr and O occupying substitutional and interstitial sites, respectively. The orientation relations between α-Ti and the lamellae orthorhombic Ti₂ZrO were determined to be [0001]_α-Ti//[110]_Ti2ZrO and (10 1 0)_α-Ti//(1 1 0)_Ti2ZrO; meanwhile, those between the α-Ti and the spherical hexagonal Ti₂ZrO were [0001]_α-Ti//[0001]_Ti2rO and (10 1 0)_α-Ti//(10 1 0)_Ti2ZrO.     

Dielectric, Piezoelectric, and Pyroelectric Properties of Lead Zirconate—Lead Zinc Niobate Ceramics

New piezoelectric and pyroelectric ceramics consisting of antiferroelectric lead zirconate (PZ) and relaxor ferroelectric lead zinc niobate (PZN) are studied from an application view-point of the field-induced antiferroelectric-to-ferroelectric phase transition. An antiferroelectric-ferroelectric phase boundary exists in PbZr_x(Zn_1/3Nb_2/3)_1−xO₃ (PZZN-1000x) close to x = 0.93 to 0.94 at room temperature. A new ferroelectric rhombohedral phase change, F_α–F'_α, at low temperature is found and studied by the temperature dependence of the pyroelectric coefficient. Electrical poling in these ceramics is easy, and the coercive field E_c∼8 to kV/cm is rather low. Samples with compositions in the range PZZN-86 to PZZN-92 have a large electromechanical coupling constant, k (k_t and k₁₅∼50% to 60%), and a low dielectric constant, ɛ_s (ɛ^T₃₃/ɛ₀= 260 to 320, ɛ^T₁₁/ɛ₀= 380). PZZN ceramics appear to be potential candidates for high-frequency ultrasonic transducers used in the thickness shear mode. The pyroelectric figure of merit (F_v) of these ceramics is comparable to the values published for the PZT-based or PbTiO₃-based materials.     

Three Crystalline Polymorphs of KFeSiO4, Potassium Ferrisilicate

Orthorhombic α-KFeSiO₄ ( a =0.5478, b =0.9192, c =0.8580 nm), hexagonal β-KFeSiO₄ ( a =0.5309, c =0.8873 nm), and hexagonal γ-KFeSiO₄ ( a =0.5319, c =0.8815 nm) were synthesized by devitrification of KFeSiO₄ glass. Powder X-ray diffraction data are given for all three polymorphs. Alpha KFeSiO₄, the high-temperature polymorph, melts congruently at 1197°± 2°C. Mössbauer spectroscopy of the α phase indicates that Fe³⁺ occupies two tetrahedral sites in the lattice. Beta KFeSiO₄, the low-temperature polymorph, and γ-KFeSiO₄, a metastable polymorph, appear to be isomorphous with kalsilite, KAISiO₄, and synthetic kaliophilite, KAISiO₄, respectively, and it is proposed that β- and γ -KFeSiO₄ are linked by Si-Fe order-disorder. Beta KFeSiO₄ transforms slowly into α -KFeSi0₄ above 910°C but the transformation was not shown to be reversible in the present dry-heating experiments.     

ESTIMATION FOR THE FIRST-ORDER DIAGONAL BILINEAR TIME SERIES MODEL

Abstract. The problem of estimation of the parameter b in the simple diagonal bilinear model { X _t }, X_t = e_t + be _{t -1} X_{t -1}, is considered, where { e_t } is Gaussian white noise with zero mean and possibly unknown variance ². The asymptotic normality of the moment estimator of b is established for the two cases when ² is known and ² is unknown. It is noted that the limit distribution of the least-squares cannot easily be derived analytically. A bootstrap comparison of the sampling distributions of the least-squares and moment estimates shows that both are asymptotically normal with the least-squares estimate being the more efficient.     

Structural Disorder and Intracrystalline Microtexture of α'H-(Ba0.24Ca0.76)2SiO4

The structural disorder and twin microtexture of (Ba_0.24Ca_0.76)₂SiO₄ crystal were investigated by X-ray powder diffraction, precession method, and optical microscopy. The crystal at 298 K was orthorhombic (space group Pnma , Z =4) with a =0.70490(5) nm, b =0.55399(4) nm, c =0.95532(7) nm, and V =0.37306(4) nm³. The crystal structure was satisfactorily described by a split-atom model, involving the positional disorder of Ba/Ca and O atoms across the (010) mirror plane. The crystal was found to be isostructural with α'_H-Ca₂SiO₄, thus regarded as the stabilized α'_H phase. The crystal grain was composed of orthorhombic domains in three different orientations. The unit cells within each domain were related to one another by the threefold pseudo-symmetry axis lost during the α-to-α'_H transition. Each domain was most probably made up of sub-domains of the two mirror-related structural configurations with P 2₁/ n symmetry.     

Iron and Spinel Precipitation in Iron-Doped Sapphire

Single crystals of Al₂O₃ containing 0.5 wt% Fe were exposed to low p o₂ atmospheres at 1500°C to produce precipitate phases. Analytical TEM identified the precipitate phases as spinel (hercynite) and iron, with the following orientation relationships: 〈001〉_Fe‖〈2 2 01〉_s with {1 1 0}_Fe‖{11 2 0}_s, 〈111〉_Fe‖〈10 1 0〉_s with {0 1 1}_Fe‖ (0001)_s, and 〈1 1 0〉_H‖〈01 1 0〉_s with {111}_H‖ (0001)_s. The three phase fields observed — (Fe, Al)₂O₃, spinel +α-Al₂O₃, and iron +αAl₂O₃— are in accordance with phase stability diagrams. Precipitation kinetics indicate that oxygen is mobile in the reduced region of the crystal.     

Cell Constants for Dicalcium Silicate-Alkali Fluoride Complexes

X-ray diffraction powder patterns were used to determine the cell constants for the complexes 2CaO·SiO_2·LiF and 2CaO·SiO₂·KF by Lipson's method of indexing, assuming orthorhombic structures. The resulting cell constants are α₀= 18.2, b₀=20.6, and c₀=7.2 Å for the Li complex and α₀=18.7, b₀=20.5, and c₀6.9 Å for the K complex.     

Stacking Faults and Stacking Fault Energy of Hexagonal Barium Titanate

Stacking faults in hot-pressed hexagonal BaTiO₃ (the 6H-polytype) ceramic have been analyzed adopting the 2π g·R _F=0 (or 2n π ) invisibility criteria, and dislocations were examined by the g·b =0 effective invisibility criteria, both using transmission electron microscopy. Perfect basal dislocations with b_B= 1/3〈 1 2 1 0〉 have dissociated into pairs of prism-plane half-partials with b_Pr= 1/3〈01 1 0〉 by glide in the (0002) fault plane of an α-type extended planar stacking fault. Consequently, planar fault F_A divided by half-partials was segmented into the initially faulted regions (F_A1, F_A3, F_A5) of extended planar stacking faults and the complex-faulted regions (F_A2, F_A4) of complex extrinsic stacking faults superposed by the stacking fault ribbon created by half-partials. High-resolution imaging enables unambiguous differentiation of stacking sequence for all fault segments parted by half-partials. Change of the stacking sequence across half-partials reveals the atomic configurations of the faulted regions and the shear directions along 〈01 1 0〉 in (0002). The stacking fault energy of h-BaTiO₃ derived from the separation between half-partials is compared with data from the literature. The discrepancy is attributed to non-equilibrium configuration and to the stacking sequence altered in the faulted regions embedded with half-partials.     

Nucleation and Growth of α'-SiAlON on α-Si3N4

Morphology, composition, and growth defects of α'-SiAION have been studied in a fine-grained material with an overall composition Y_0.33Si₁₀Al₂O₁N₁₅ prepared from α-Si₃N₄, AlN, Al₂O₃, and Y₂O₃ powders. TEM analysis has shown that fully grown α'-SiAloN grains always contain an α-Si₃N₄ core, implicating heterogeneous nucleation operating in the present system. The growth mode is epitaxial, despite the composition and lattice parameter difference between α-Si₃N₄ and α'-SiAlON. The inversion boundary that separates two domains in the seed crystal is seen to continue in the grown α'-SiAION. Lacking a special growth habit, the growth typically proceeds from more than one site on the seed crystal, and the different growth fronts impinge on each other to give an equiaxed appearance of α'-SiAlON. Misfit dislocations on the α/α'interface are identified as [0001] type ( b = 5.62 Å) and 1/3 [1 2 10] type ( b = 7.75 Å). These nucleation and growth characteristics dictate that microstructural control of α'-SiAlON must rest on the size distribution of the starting α-Si₃N₄ powder.