•Investigation of robustness of a Generalized Quadratic Discriminant Analysis (GQDA) under the presence of Noise in data.•The GQDA is a novel approach integrating linear & quadratic discriminant ...analyses, but is extremely sensitive under mild contamination.•Development of roust versions of GQDA classier by using robust estimators of the mean vector and the dispersion matrix.•Detailed empirical comparison of robust GQDA proposals with 6 robust estimators, 3 classes of model distribution and 4 real data examples.
Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis distance (MMD) classifiers to discriminate between populations with underlying elliptically symmetric distributions competes quite favorably with the QDA classifier when it is optimal and performs much better when QDA fails under non-Normal underlying distributions with heavy tail, e.g. Cauchy distribution. However, the classification rule in GQDA is still based on the sample mean vector and the sample dispersion matrix of a training set, which are extremely non-robust under data contamination. In real world, however, it is quite common to face data which are highly vulnerable to outliers and so the lack of robustness of the classical estimators of the mean vector and the dispersion matrix reduces the efficiency of the GQDA classifier significantly, increasing the misclassification errors. The present paper investigates the performance of the GQDA classifier when the classical estimators of the mean vector and the dispersion matrix used therein are replaced by various robust counterparts. Applications to various real data sets as well as simulation studies reveal far better performance of the proposed robust versions of the GQDA classifier. A comparative study has been made to advocate the appropriate choice of the robust estimators to be used in a specific situation.
In this article, we take up the experimental situation of a heteroscedastic one-way layout model in the presence of a set of controllable covariates. For the joint estimation of the elementary ...contrasts of a set of test treatments with a control and the effects of covariates, sufficient conditions for the existence of an A-optimal design are identified. When these sufficient conditions are not met, we propose highly A-efficient designs. The methods of construction of A-optimal and highly A-efficient designs are discussed. For different values of the parameters of the design, A-efficiency of the proposed designs are tabulated for a comparative study.
In linear discriminant analysis, the assumption of equality of the dispersion matrices of different classes leads to a classification rule based on minimum Mahalanobis distance from the class ...centres. However, without this assumption, the resulting quadratic discriminant classifier involves, in addition to the Mahalanobis distances, the ratio of the determinants of the dispersion matrices as a factor. In fact, it has been observed that, for discriminating between populations with underlying elliptically symmetric distributions, such classifiers also incorporate similar factors, apart from the Mahalanobis distances.
In this paper, a nonparametric classification technique which generalizes discriminant analysis has been proposed. The method of cross-validation is used to make the technique adaptive to a given dataset. An extensive simulation study is presented to illustrate the potential of the method. Finally, through implementation on a number of real-life data sets, it has been demonstrated that the proposed generalized quadratic discriminant analysis (GQDA) compares very favourably with other nonparametric methods, and is computationally cost-effective.
•A nonparametric classification technique has been proposed.•It generalizes Fisher׳s discriminant analysis.•Cross-validation is used to make the technique adaptive to given data.•Found empirically to compare very favourably with other nonparametric methods.•Computationally cost-effective.
In this article experimental situations are considered where a main effects plan is to be used to study m two-level factors using n runs, partitioned into b blocks, ith block having size ki, where ...∑i=1bki=n. For n odd, D- and E-optimal designs are characterized. Simple construction methods for these optimal designs are given.
In this paper we consider screening experiments where
m
factors are to be studied using
n
experimental runs and where
m
is relatively large compared to
n
. In such experimental settings unreplicated ...fractional factorial (FF) designs are often used. Unreplicated FF designs have received a good deal of study over the past 15 years because they present problems in terms of use and analysis. One major problem with unreplicated FF design is the lack of reliable estimates for experimental error. Because of this a number of methods have been developed for identifying significant effects which include both graphical and more objective methods. However, there does not seem to be a clear winner among the methods suggested. In this paper we suggest as an alternative the use of partially replicated FF designs. The use of partial replication provides a non-model dependent estimate for pure error which allows for simpler and more standard methods for identifying significant effects through the use of
F
-and
T
-tests for lack of fit and individual contrasts. To illustrate the possibilities we focus on 16 run designs and provide some partially replicated designs for 4 ≤
m
≤ 11. We compare our proposed designs in terms of the A-optimality criterion and Estimation Capacity to regular unreplicated FF and Generalized Minimum Aberration non-regular FF designs. Our results indicate that partially replicated FF are competitive in terms of the criteria mentioned.
We propose a novel semi-parametric classifier based on Mahalanobis distances of an observation from the competing classes. Our tool is a generalized additive model with the logistic link function ...that uses these distances as features to estimate the posterior probabilities of the different classes. While popular parametric classifiers like linear and quadratic discriminant analyses are mainly motivated by the normality of the underlying distributions, the proposed classifier is more flexible and free from such parametric assumptions. Since the densities of elliptic distributions are functions of Mahalanobis distances, this classifier works well when the competing classes are (nearly) elliptic. In such cases, it often outperforms popular nonparametric classifiers, especially when the sample size is small compared to the dimension of the data. To cope with non-elliptic and possibly multimodal distributions, we propose a local version of the Mahalanobis distance. Subsequently, we propose another classifier based on a generalized additive model that uses the local Mahalanobis distances as features. This nonparametric classifier usually performs like the Mahalanobis distance based semiparametric classifier when the underlying distributions are elliptic, but outperforms it for several non-elliptic and multimodal distributions. We also investigate the behaviour of these two classifiers in high dimension, low sample size situations. A thorough numerical study involving several simulated and real datasets demonstrate the usefulness of the proposed classifiers in comparison to many state-of-the-art methods.
Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) ...classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis distance (MMD) classifiers to discriminate between populations with underlying elliptically symmetric distributions competes quite favorably with the QDA classifier when it is optimal and performs much better when QDA fails under non-Normal underlying distributions, e.g. Cauchy distribution. However, the classification rule in GQDA is based on the sample mean vector and the sample dispersion matrix of a training sample, which are extremely non-robust under data contamination. In real world, since it is quite common to face data highly vulnerable to outliers, the lack of robustness of the classical estimators of the mean vector and the dispersion matrix reduces the efficiency of the GQDA classifier significantly, increasing the misclassification errors. The present paper investigates the performance of the GQDA classifier when the classical estimators of the mean vector and the dispersion matrix used therein are replaced by various robust counterparts. Applications to various real data sets as well as simulation studies reveal far better performance of the proposed robust versions of the GQDA classifier. A Comparative study has been made to advocate the appropriate choice of the robust estimators to be used in a specific situation of the degree of contamination of the data sets.
In this paper we consider a variant of the traditional noncircular model for crossover designs. Instead of assuming that ea.Ch treatment - applied to an experimental unit imparts the same carryover ...effect regardless of the treatment applied to the next period on the same unit, we consider the model which assumes two types of carryover effects that extend from a period to the next period. One type is called self carryover effect when a treatment is followed by itself in the next period on the same unit and the other type is called mixed carryover effect when a treatment is followed by any other treatment in the next period. Such models are useful in sensory trials. Efficient estimation and testing of the direct treatment effects (imparted by the treatment itself on the experimental unit of application) as well as the carryover effects under different models are of interest to the practitioners from application and model building point of view and have been addressed by many researchers. In the present article the problem of identification of optimal designs for the estimation of the mixed carryover effects has been taken up. It is shown that under the self and mixed carryover model generalised Patterson's balanced designs (termed also totally balanced designs in the literature}, which are known to be universally optimal for the estimation of the direct treatment effects are also universally optimal for the estimation of the mixed carryover effects provided that the number of periods exceeds two.
AMS (2000} Subject Classification : 62K05.
Critical Sets in Orthogonal Arrays SahaRay, Rita; Seberry, Jennifer
Electronic notes in discrete mathematics,
05/2003, Letnik:
15
Journal Article
A critical set consists of the minimum amount of information needed to recreate combinatorial structures uniquely. A
minimal critical set is a critical set of minimum cardinality. In this paper we ...consider orthogonal arrays constructed from Mutually Orthogonal Latin Squares (MOLS) and obtain bounds on the possible sizes of the
minimal critical sets. If there exists a set of
k MOLS of order
n and each Latin Square has a
minimal critical set of size atmost c
l, then we show that there is a minimal critical set of size
COA, of the corresponding
OA(n
2, k +
2,
n 2) satisfying
CQA ≤
kcl. This bound is shown to be exact for
n = 3 by actual construction of a
minimal critical set. However, for
n ≥ 4 this bound can be improved upon. Using the fact that for
n = 4 the Latin Square representing the elementary abelian 2-group has a
minimal critical set of size 5, and only this form or its isotopic has orthogonal mates, we exhibit a
minimal critical set of size 11 for an
OA(16,5,4,2). For
n > 4,
n odd, we consider
OA(
n
2, k + 2,n,2
) made from a back circulant Latin Square which is a particular Latin Square having initial row in the standard form and subsequent rows formed by translating the previous row one element to the left. We prove by actual construction, that if the minimal critical set for a back circulant latin square is of size cl, then any of its orthogonal mate can be uniquely reconstructed given information on c
l − 1 cells only. Interestingly enough, for
n = 5, starting with a minimal critical set of size 6, we construct a minimal critical set of size 18 for a complete set of MOLS and the corresponding orthogonal array. Identification of a critical set for
n = 7 is also discussed.
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