Forschungsprojekt ::

Stochastic Model Building and Model Checks with Applications

Projektbeschreibung

1. The Problem

A main problem in stochastic model building is to find an adequate stochastic model for the problem and/or for the data under consideration. To examine the model it is important to develop suitable model checks. In practice, however, the chosen model is seldomly checked. In this project we want to develop a suitable model check based on residual partial sums processes (CUSUM technique) for growth curve models.

Especially, we have in mind the growth curve model we developed for the analysis of the desinfection mechanics of waste-water in marine environment, see Bischoff, Huang and Yang (2006). A corresponding growth curve model for binary responses is a second model we are interested in. Such data occur in our context when our original data are judged/fixed as good or not good according
as they are smaller or larger than a given threshold. For the second model a suitable growth curve model has still to be developed. Additionally, it is of interest to compare the results of both models. But such a comparison is only useful if both models are adequate which must be investigated by model checks.

2. The Experiments

The example of a real experiment is taken from Bischoff, Huang and Yang (2006) about the natural desinfection mechanism of the waste-water in marine environment for ocean outfall systems without chlorination. In the study of the desinfection on waste-water in marine environment by Bischoff et al. (2006), two natural factors, consisting of light intensity and salinity, one controllable factor, the volumetric mixing ratio of sea-water to waste-water, and one random effect factor, the existence of predator, were investigated. Experiments were performed based on a rotatable central composite design with different factor level combinations for investigating the die-off rate of E. coli bacteria, which are considered as an important indicator of fecal pollution in marine environment. Methods for establishing a stochastic model and corresponding analysis for the data obtained from the experiments were provided.

In this work, the method developed for model check on growth curve model will apply to the data from the experiments. In the meantime, while monitoring of the water quality at marine bathing sites, E. coli is one of the most important indicators of the water quality. It is quite common to present the water quality as passing or not passing with binary responses and repeated data. The methods developed for the growth curve model for the binary response experiments will be tested by reducing the real data to whether passing or not passing a threshold, to see how the results compared with those with the real data.

3. Current state of art, techniques and objectives

To check homogeneity of data it is common and popular to use CUSUM (Cumulative SUM, partial sums) techniques. Partial sums of least squares residuals are used to check ordinary regression models, see MacNeill (1978a,b), Ploberger and Krämer (1992), Bischoff (1998, 2002), Bischoff and Miller (2000). Cramèr-von Mises or Kolmogorov(-Smirnov) type statistics can be applied to the partial sums of least squares residuals, see MacNeill (1978a,b), Bischoff, Hashorva, Hüsler and Miller (2003, 2004, 2005). Results for more complicated regression models are rare, see MacNeill and Jandhyala (1985) for non-linear regression, MacNeill and Jandhyala (1993) for spatial data and MacNeill and Tang (1993) for certain correlation.

One objective of our project is to develop a model check based on the least squares residuals for our growth curve model developed in Bischoff et al. (2006). A second objective is to develop a suitable growth curve model for our corresponding binary data. Moreover, for this growth curve model we have to develop a model check based on the partial sums of the least squares residuals as well.

It is worth mentioning that our investigations are hindered by correlation of the data. Moreover, for the theoretical results asymptotic techniques are usually applied. It is not obvious, however, whether asymptotic arguments are applicable for our specific model and data. Therefore, the theoretical investigations have to be accompanied by simulations and/or resampling techniques.

Based on the above investigations, we should be able to build up some stochastic models and provide useful information on the understanding of the treatment processes of the waste-water. Later it may be used as basis of the planning of the future sewage treatment plan in Taiwan.

By the results of our study we also intend to increase the level of intuition for the considered type of models to make it more applicable to wider problem areas. This methodological development is of potential usefulness for such diverse applications as experimental economics, biochemistry, clinical trials, industrial production, etc.

References
Bischoff, W. (1998). A functional central limit theorem for regression models. Annals of Statistics 26, 1398 - 1410.

Bischoff, W. and Miller, F. (2000). Asymptotically optimal tests and optimal designs for testing the mean in regression models with applications to change-point problems. Annals of the Institute of Statistical Mathematics 52, 658-679.

Bischoff, W. (2000). Asymptotically optimal tests for some growth curve models under non-normal error structure. Metrika 50, 195-203.

Bischoff, W. (2002). The structure of residual partial sums limit processes of linear regression models. Theory of Stochastic Processes 8(24), N1-2, 23-28.

Bischoff, W., Hashorva, E., Hüsler, J. and Miller, F. (2003). Exact asymptotics for boundary crossings of the Brownian bridge with trend with applications to the Kolmogorov test. Annals of the Institute of Statistical Mathematics 55,
849-864.

Bischoff, W., Hashorva, E., Hüsler, J. and Miller, F. (2004). On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models. Statistics and Probability
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Bischoff, W., Hashorva, E., Hüsler, J. and Miller, F. (2005). Analysis of a change-point regression problem in quality control by partial sums processes and Kolmogorov type tests. Metrika 62, 85-98.

Bischoff, W., Huang, M.-N. L. and Yang, L. (2006). Growth curve models for stochastic modeling and analyzing of natural desinfection of waste-water. Environmetrics 17, 827-847.

Huang, M.-N. L., and Wong, K.-F. (2002). $s^{k-p}$ fractional factorial designs in $s^{b}$ blocks. Metrika 56, 163-170.

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Huang, M.-N. L., Chen, R.-B. , Lin, C.-S. and Wong, W. K. (2006). Optimal designs for parallel models with correlated responses. Statistica Sinica 16, 121-133. (SCIE)

MacNeill, I.B. (1978 a). Properties of sequences of partial sums of polynomial regression residuals with applications to test for change of regression at unknown times. Ann. Statist. 6, 422-433.

MacNeill, I.B. (1978 b). Limit processes for sequences of partial sums of regression residuals. Ann. Prob. 6, 695-698.

MacNeill, I.B. and Jandhyala, V.K. (1985). The residual process for nonlinear regression. J. Appl. Prob. 2}, 957-963.

MacNeill, I.B. and Jandhyala, V.K. (1993). Change-point methods for spatial data. Patil, G. P. (ed.) et al., Multivariate environmental statistics. Papers presented at the 7th international conference on multivariate analysis held at
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MacNeill, I.B. and Tang, S.M. (1993). The effect of serial correlation on tests for parameter change at unknown time. Ann. Statist. 21, 552-575.

Ploberger, W. and Krämer, W. (1992). The CUSUM test with OLS residuals. Econometrica 60, 271-285.