Observable trend-projecting state-space models

Much attention has focused in recent years on the use of state-space models for describing and forecasting industrial time series. However, several state-space models that are proposed for such data series are not observable and do not have a unique representation, particularly in situations where the data history suggests marked seasonal trends. This raises major practical difficulties since it becomes necessary to impose one or more constraints and this implies a complicated error structure on the model. The purpose of this paper is to demonstrate that state-space models are useful for describing time series data for forecasting purposes and that there are trend-projecting state-space components that can be combined to provide observable state-space representations for specified data series. This result is particularly useful for seasonal or pseudo-seasonal time series. A well-known data series is examined in some detail and several observable state-space models are suggested and compared favourably with the constrained observable model.     

Observable trend-projecting state-space models

Much attention has focused in recent years on the use of state-space models for describing and forecasting industrial time series. However, several state-space models that are proposed for such data series are not observable and do not have a unique representation, particularly in situations where the data history suggests marked seasonal trends. This raises major practical difficulties since it becomes necessary to impose one or more constraints and this implies a complicated error structure on the model. The purpose of this paper is to demonstrate that state-space models are useful for describing time series data for forecasting purposes and that there are trend-projecting state-space components that can be combined to provide observable state-space representations for specified data series. This result is particularly useful for seasonal or pseudo-seasonal time series. A well-known data series is examined in some detail and several observable state-space models are suggested and compared favourably with the constrained observable model.     

The Use of Treatment Concurrences to Assess Robustness of Binary Block Designs Against the Loss of Whole Blocks

Criteria are proposed for assessing the robustness of a binary block design against the loss of whole blocks, based on summing entries of selected upper non‐principal sections of the concurrence matrix. These criteria improve on the minimal concurrence concept that has been used previously and provide new conditions for measuring the robustness status of a design. The robustness properties of two‐associate partially balanced designs are considered and it is shown that two categories of group divisible designs are maximally robust. These results expand a classic result in the literature, obtained by Ghosh, which established maximal robustness for the class of balanced block designs.     

Improved conditions for the robustness of binary block designs against the loss of whole blocks

Several authors have investigated conditions for a binary block design, D, to be maximally robust such that every eventual design obtained from D by eliminating r_[υ]−1 blocks is connected, where r_[υ] is the smallest treatment replication. Four new results for the maximal robustness of D with superior properties are given. An extension of these results to widen the assessment of robustness of the planned design is also presented.     

A note on the robustness of PBIBD(2)s against breakdown in the event of observation loss

Robustness against design breakdown following observation loss is investigated for Partially Balanced Incomplete Block Designs with two associate classes (PBIBD(2)s). New results are obtained which add to the body of knowledge on PBIBD(2)s. In particular, using an approach based on the E‐value of a design, all PBIBD(2)s with triangular and Latin square association schemes are established as having optimal block breakdown number. Furthermore, for group divisible designs not covered by existing results in the literature, a sufficient condition for optimal block breakdown number establishes that all members of some design sub‐classes have this property.     

Simple pilot procedures for the avoidance of disconnected experimental designs

Summary.  Two simple pilot procedures are proposed for avoiding the problem of dealing with a disconnected experimental design. Both procedures should be carried out on the selected design before any experimentation is considered. The first procedure is a check that the suggested design is connected with respect to treatments. This makes use of the information matrix for the model and provides feed-back on a disconnected design. The second procedure specifies which observations are influential in causing a connected design to become disconnected, with respect to any set of parameter effects, if these observations are lost during the experimental period. This specification is found by examining the projection matrix for the model. These pilot procedures are illustrated by several examples.     

Uncorrelated residuals and an exact test for two variance components in experimental design

The error contrasts from an experimental design can be constructed from uncorrelated residuals normally associated with the linear model. In this paper uncorrelated residuals are defined for the linear model that has a design matrix which is less than full rank, typical of many experimental design representations. It transpires in this setting, that for certain choices of uncorrelated residuals, corresponding to recursive type residuals, there is a natural partition of information when two variance components are known to be present. Under an assumtion of normality of errors this leads to construction of appropriate F-tests for testing heteroscedasticity. The test, which can be optimal, is applied to two well known data sets to illustrate its usefullness.