A survey is given of papers which have influenced or have been influenced by the Growth Curve Model due to Potthoff & Roy (1964). The review covers, among others, methods of estimating parameters, the canonical version of the model, tests, extensions, incomplete data, Bayesian approaches and covariance structures.

The growth curve model: a review

In this paper a muIt,ivariat.e EWMA chart for time series is introduced. In principle, it is a generalization of the control scheme of Lowry et al. (I992) for multivarite indendent observations. The autocovariances of the EWMA recursion are derived for stationary multivariate time series. IYsing tllese reslllts a co11t.rol chart hased or1 t11 illt.ivariate EWMA recursion is proposed. For a multivariate autoregressive process of order 1, a sufficient. condition is given such that the in-control average run length (ARL) is invariant, withrespect to the covariance of the White Noise process. This scheme is compared with the MEWMA control chart of Lowry et al. (1992) applied to the residuals. By an extensive Monte carlo study the ARL of both charts are determined for several multivariate autoregressive processses.

Ewma charts for multivariate time series

Landslides are among the great destructive factors which cause lots of fatalities and financial losses all over the world every year. Studying of the factors affecting occurrence of landslides in a region and zoning the resulting damages will certainly play a crucial role in mitigating such phenomena. In this research, through geological maps and field studies, we primarily prepared a map for landslide distributions in Zab basin—an area of 520 km2 in the southwest mountainsides of West Azerbaijan Province. By applying other source of information such as the existing thematic maps, we studied and defined the factors (slope, slope aspect, distance to road, distance to drainage network, distance to fault, land use and land cover, geological factors, horizontal gravity acceleration of earthquakes, and climatic condition of the studied area) that affect occurrence of the landslides. To get better precision and higher speed and facility in our analysis, all descriptive and spatial information were entered into geographic information system (GIS) system and Ilwis software. We also used Satellite images (Landsat ETM+ and SPOT 5), producing land cover and landslide-inventory maps, respectively. After preparation of the influential parameters on landslides, we drew the zoning maps of slide hazard via four different statistical methods and then evaluated and compared them. By analyzing the obtained index and by comparing landslide distribution map and zoning map of landslide susceptibility prepared by each of the methods in GIS environment, we found that bivariate method of information value analysis, bivariate method of density-area, multivariate method with linear regression analysis, and multivariate method of discriminate analysis take priority, respectively. Finally, as this research shows, despite their simplicity, bivariate statistical methods have more acceptable precision than multivariate methods, and consequently, they are more compatible with landslide susceptibility of the region. From the results, lithology, slope, annual rainfall, land cover, slope aspect, distance to waterway, distance to road, horizontal gravity acceleration, and distance to fault are very influential to landslides in the region.

Evaluation and comparison of bivariate and multivariate statistical methods for landslide susceptibility mapping (case study: Zab basin)

Publisher Summary This chapter focuses on the distribution aspects of quadratic forms in normal variables only. In the theory of least squares, in variance component analysis, in an estimation including MINQUE theory and the testing of hypothesis, and in some problems in time series analysis, quadratic forms play an important role. Because the basic assumption of normality is used in the study of quadratic forms, knowledge about the extent of the validity of this basic assumption is essential when it is already known that a quadratic form is distributed as a Chi-square variate, or a linear function and a quadratic form are independently distributed, or the regression of a quadratic form on a linear function is constant. The chapter discusses the necessary and sufficient conditions for Chi-squaredness and independence, exact distribution of quadratic forms, asymptotic distribution of quadratic forms, the characterization of the distributions, and the estimation of fixed effects.

14 Quadratic forms in normal variables

Testing the hypothesis of multivariate white noise is seen as the selection of the order of a multivariate autoregressive model for the observed time series. Therefore, multivariate white noise tests can be carried out by applying autoregressive order-determination criteria such as AIC, BIC, etc. It is known, for example, that the BIC criterion estimates consistently the order of an autoregression. An order-determination criterion with this property leads to a white noise test with a significance level approaching zero as n, the number of observations, increases. The order of an autoregressive moving-average model is proposed to be determined by applying this kind of white noise test. The resulting model building procedure is a generalization of the procedure proposed by G.E.P. Box and G.M. Jenkins (1970). >

On the use of autoregressive order determination criteria in multivariate white noise tests

A note on the joint distribution of correlated quadratic forms

A note on variance balanced designs

Contemporary Textbooks on Multivariate Statistical Analysis: A Panoramic Appraisal and Critique@@@An Introduction to Multivariate Statistical Analysis (2nd ed.).@@@The Theory of Linear Models and Multivariate Analysis.@@@Multivariate Statistics: A Vector Space Approach.@@@Multivariate Statistical Inference.@@@Methods for Statistical Data Analysis of Multivariate Observations.@@@Multivariate Analysis. (2nd ed.).@@@Multivariate Analysis.@@@Multivariate Statistical Methods (2nd ed.).@@@Aspects of Multivariate Statistical Theory.

C.G. Khatri

Papers

A note on the joint distribution of correlated quadratic forms

A note on variance balanced designs

Unified treatment of Cramér-Rao bound for the nonregular density functions

Minimum variance quadratic unbiased estimate of a linear function of variances and covariances under Manova model