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# Robustness of the Quadratic Discriminant Function to correlated and uncorrelated normal training samples

- Atinuke Adebanji
^{1, 2}, - Michael Asamoah-Boaheng
^{2}Email author and - Olivia Osei-Tutu
^{1}

**Received:**5 October 2015**Accepted:**13 January 2016**Published:**1 February 2016

## Abstract

This study investigates the asymptotic performance of the Quadratic Discriminant Function (QDF) under correlated and uncorrelated normal training samples. This paper specifically examines the effect of correlation, uncorrelation considering different sample size ratios, number of variables and varying group centroid separators (\(\delta\), \(\delta = 1; 2; 3; 4; 5\)) on classification accuracy of the QDF using simulated data from three populations (\(\pi _{i}, i=1,2,3\)). The three populations differs with respect to their mean vector and covariance matrices. The results show the correlated normal distribution exhibits high coefficient of variation as \(\delta\) increased. The QDF performed better when the training samples were correlated than when they were under uncorrelated normal distribution. The QDF performed better resulting in the reduction in misclassification error rates as group centroid separator increases with non increasing sample size under correlated training samples.

## Keywords

- QDF
- Correlated normal
- Uncorrelated normal
- Group centroid

## Background

Discriminant analysis (DA) as a topic in Multivariate Statistical Analysis has attracted much research interest over the years, with the evaluation of discriminant functions when the covariances matrices are unequal with moderate sizes being well explained by Wahl and Kronmal (1977). Linear discriminant function (LDF) is commonly used by researchers because of its simplicity of form and concept. In spite of theoretical evidence supporting the use of the Quadratic Discriminant Function (QDF) when the covariance matrices are heterogeneous, its actual employment has been sporadic because there are unanswered questions regarding its performance in the practical situation where the discriminant function must be constructed using training samples that do not satisfy the classical assumption of the model. The pioneering work on quadratic discrimination was by Smith (1947) using Fisher’s Iris data. He provided a full expression for the QDF and his results showed the QDF outperforming the LDF when the homogeneity of variance covariance structure was violated.

Marks and Dunn (1974) approached the problem of discrimination by comparing the asymptotic and small sample performance of the QDF, best linear and Fisher’s LDF for both proportional and non-proportional covariance differences under the assumption of normality and unequal covariance matrices. Two populations were used and sample sizes were chosen from 10 to 100. The number of variables selected were 2 and 10. They employed the application of Monte Carlo simulation. Their results indicated that for small samples the QDF performed worse than the LDF when covariances were nearly equal with large dimensions (ie LDF was satisfactory when the covariance matrices were not too different).

Lawoko (1988) studied the performance of the LDF and QDF under the assumption of correlated training samples. The researcher aimed at allocating an object to one of two groups on the basis of measurements on the object. He found that the discriminant functions formed under the model did not perform better than *W* and *Z* formed under the assumption of independent training observation. Asymptotic expected error rate for *W* under the model (\(W_m\)) and *W* were equal when the training observations followed an autoregressive process but there was a slight improvement in the overall error rate when \(W_m\) was used instead of *W* for numerical evaluations of the asymptotic expansions. He concluded that the efficiency of the discriminant analysis estimator is generally lowered by positively correlated training observations.
Mardia et al. (1995) reported that it might be thought that a linear combination of two variables would provide a better discriminator if they were correlated than when they were uncorrelated. However, this is not necessarily so. To show this they considered two bivariate populations \(\pi _{1}\) and \(\pi _{2}\). Supposing \(\pi _{1}\) is \(N_{2}(0,\Sigma )\) and \(\pi _{2}\) is \(N_{2}(\mu ,\Sigma )\) where \(\mu =(\mu _{1},\mu _{2})'\) and with known \(\Sigma\). They indicated that discrimination is improved unless \(\rho\) lies between zero and \(2f/{(1 + f^{2}})\) but a small value of \(\rho\) can actually harm discrimination.

Adebanji and Nokoe (2004) have considered evaluating the quadratic classifier. They restricted their attention to two multivariate normal populations of independent variables. In addition to some theoretical result, with known parameters, they conducted a Monte Carlo simulation in order to investigate the error rates. Results indicated that the total error rate computed showed that there was an increase in the error rate with re-substitution estimator for all *K* values. On the other hand, there was a decline across *K*. The cross-validation estimator showed a steady decline for and across all values *K* and the recorded value showed a substantially low error rate estimates than re-substitution estimator for \(K=4\) and \(K=8\).

Kakaï and Pelz (2010) studied the asymptotic error rates of linear, quadratic rules and conducted a Monte Carlo study in 2, 3 and 5-group discriminant analysis. Hyodo and Kubokawa (2014) studied a variable selection criterion for linear discriminant rule and its optimality in high dimensional data where a new variable procedure was developed for selecting the true variable set.

An enormous deal of study has been made since Fisher’s (1936) original work on discriminant analysis as well as several other researchers tackling similar problem. Some estimation methods have been proposed and some sampling properties derived. However, there is little investigation done on large sample properties of these functions. Also a considerable number of studies had been carried out on discriminant analysis but not much is done on the effect or the performance of the QDF under correlated and uncorrelated data with varying sample size ratios, different variable selections and with different centroid separators for three populations.

In this study we therefore investigate the performance of classification functions (i.e Quadratic Discriminant Functions) when the covariance matrices are heterogeneous with the data of interest being correlated, sample size ratios being unequal, considering different number of variables and varying values of group centroid separator (\(\delta\)).

## Methods

### Simulation design

To evaluate the performance of QDF for correlated and uncorrelated training samples of distributions, we considered a Monte Carlo study with multivariate normally correlated random data generated for three populations with their mean vector \(\mu _1=(0, \ldots , 0)\), \(\mu _2=(0, \ldots , \delta )\) and \(\mu _3=(0, \ldots , 2\delta )\) respectively. The covariance matrices, \(\Sigma _i\) (i = 1, 2, 3). Where \(k\ne l,\) \(\sigma _{kl}=0.7\) for all groups except the diagonal entries given as \(\sigma _{k}^2 =i\), for \(i=1,2,3\). The covariance matrices were transformed to be uncorrelated to generate the uncorrelated data. The QDF was then performed in each case and the leave-one-out method was used to estimate the proportion of observations misclassified.

- 1.
Mean vector separator which is set at \(\delta\) from 1 to 5 where \(\delta\) is determined by the difference between the mean vectors.

- 2.
Sample sizes which are also specified. Here 14 values of \(n_1\) set at 30, 60, 100, 150, 200, 250, 300, 400, 500, 600, 700, 800, 1000, 2000 and the sample size of \(n_2\) and \(n_3\) are determined by the sample ratios at 1:1:1, 1:2:2 and 1:2:3 and these ratios also determine the prior probabilities to be considered.

- 3.
The number of variables for this study is also specified. The number of variables are set at 4, 6 and 8 following Murray (1977) who considered this in selection of variables in discriminant analysis.

- 4.
The size of population 1 \((n_1)\) is fixed throughout the study and the sizes of population 2 and population 3, \(n_2\) and \(n_3\) respectively are determined by the sample size ratio under consideration.

### Subroutine for QDF

Series of subroutines were written in MatLab to perform the simulation and discrimination procedures on QDF. Below are the important ones.

### Classification into several populations

Generalization of classification procedure for more than two discriminating groups (ie from 2 to \(g\ge 2\)) is straight forward. However, not much is known about the properties corresponding sample classification function, and in particular, their error rates have not been fully investigated. Therefore, we focus only on the Minimum ECM classification with equal misclassification cost and Minimum TPM for multivariate normal population with unequal covariance matrices (quadratic discriminant analysis).

#### Minimum ECM classification with equal misclassification cost

#### Minimum TPM rule for unequal-covariance normal populations

*ith*population is defined as

The quadratic score \(d_{i}^{Q}({\mathbf {x}})\) is composed of contributions from the generalized variance \(|{\Sigma }_i|\), the prior probability \(p_i\), and the square of the distance from *x* to the population mean \(\mu _i\).

*x*to \(\Pi _k\) if the quadratic score

### The quadratic classifier (\(\Sigma _1\ne \Sigma _2\))

This function is easily extended to the 3 group classification where 2 cut off points are required for assigning observations to the 3 groups (Johnson and Wichern 2007).

## Results

This section presents the performance of QDF when the training data are correlated and then when they are uncorrelated.

### Effects of sample size on QDF under correlated and uncorrelated normal distribution

Effects of sample size on QDF for correlated normal based on error rates, CV and SD

Centroid | Sample size (n) | SD | CV | Mean error rate |
---|---|---|---|---|

\(\delta = 1\) | 90 | 0.0530 | 0.04276 | 0.1240 |

180 | 0.0536 | 0.04656 | 0.1151 | |

300 | 0.0543 | 0.04853 | 0.1120 | |

450 | 0.0547 | 0.04903 | 0.1116 | |

750 | 0.0534 | 0.04908 | 0.1089 | |

900 | 0.0526 | 0.04857 | 0.1082 | |

1200 | 0.0524 | 0.04868 | 0.1077 | |

1500 | 0.0529 | 0.04893 | 0.1081 | |

1800 | 0.0526 | 0.04881 | 0.1077 | |

2100 | 0.0531 | 0.04913 | 0.1080 | |

6000 | 0.0524 | 0.04898 | 0.1069 | |

\(\delta = 2\) | 90 | 0.0447 | 0.05084 | 0.0880 |

180 | 0.0402 | 0.05215 | 0.0771 | |

300 | 0.0384 | 0.05224 | 0.0735 | |

450 | 0.0394 | 0.05439 | 0.0724 | |

750 | 0.0370 | 0.05172 | 0.0715 | |

900 | 0.0370 | 0.05217 | 0.0710 | |

1200 | 0.0369 | 0.05249 | 0.0703 | |

1500 | 0.0367 | 0.05191 | 0.0707 | |

1800 | 0.0372 | 0.05265 | 0.0706 | |

2100 | 0.0367 | 0.05222 | 0.0702 | |

6000 | 0.0365 | 0.05234 | 0.0698 | |

\(\delta = 3\) | 90 | 0.0300 | 0.6047 | 0.0479 |

180 | 0.0264 | 0.5983 | 0.0442 | |

300 | 0.0245 | 0.5940 | 0.0412 | |

450 | 0.0239 | 0.5907 | 0.0495 | |

750 | 0.0232 | 0.5780 | 0.0401 | |

900 | 0.0230 | 0.5763 | 0.0400 | |

1200 | 0.0226 | 0.5708 | 0.0396 | |

1500 | 0.0225 | 0.5727 | 0.0393 | |

1800 | 0.0223 | 0.5682 | 0.0393 | |

2100 | 0.0226 | 0.5728 | 0.0395 | |

6000 | 0.0224 | 0.5698 | 0.0393 | |

\(\delta = 4\) | 90 | 0.0200 | 0.7509 | 0.0266 |

180 | 0.0159 | 0.7195 | 0.0221 | |

300 | 0.0144 | 0.6727 | 0.0215 | |

450 | 0.0139 | 0.6732 | 0.0207 | |

750 | 0.0135 | 0.6574 | 0.0205 | |

900 | 0.0131 | 0.6455 | 0.0203 | |

1200 | 0.0124 | 0.6286 | 0.0198 | |

1500 | 0.0126 | 0.6353 | 0.0199 | |

1800 | 0.0128 | 0.6301 | 0.0203 | |

2100 | 0.0128 | 0.6382 | 0.0201 | |

6000 | 0.0125 | 0.6284 | 0.0200 | |

\(\delta = 5\) | 90 | 0.0123 | 0.9798 | 0.0126 |

180 | 0.0094 | 0.8717 | 0.0108 | |

300 | 0.0084 | 0.8252 | 0.0102 | |

450 | 0.0078 | 0.7838 | 0.0100 | |

750 | 0.0071 | 0.7489 | 0.0095 | |

900 | 0.0069 | 0.7399 | 0.0093 | |

1200 | 0.0070 | 0.7158 | 0.0098 | |

1500 | 0.0066 | 0.7058 | 0.0093 | |

1800 | 0.0066 | 0.7016 | 0.0093 | |

2100 | 0.0065 | 0.7002 | 0.0093 | |

6000 | 0.0064 | 0.6843 | 0.0094 |

From Table 1, the effects of the sample size on the QDF for the various group centroids (\(\delta =1,2,3,4,5\)) for the correlated samples gave an indication that, generally as the sample size increases with increasing group centroids, the mean error rates decreases marginally in that order. The standard deviation of the error rate for the correlated normal distribution reveals that as the sample size increases, standard deviation of the error rate for sample size ratio 1:1:1 exhibit low standard deviations for \(\delta =1\). For a particular \(\delta\), the standard deviation decreases as the number of variables also increases. From \(\delta =2\) to \(\delta =5\), the standard deviations decreases as the sample size increases asymptotically. There is a sharp decrease of the standard deviation of sample size ratio 1:1:1.

Effects of sample size on QDF for uncorrelated normal based on error rates, CV and SD

Centroid | Sample size (n) | SD | CV | Mean error rate |
---|---|---|---|---|

\(\delta = 1\) | 90 | 0.0681 | 0.3483 | 0.1955 |

180 | 0.0641 | 0.3920 | 0.1636 | |

300 | 0.0603 | 0.3965 | 0.1522 | |

450 | 0.0603 | 0.4092 | 0.1473 | |

750 | 0.0575 | 0.4059 | 0.1417 | |

900 | 0.0573 | 0.4065 | 0.1409 | |

1200 | 0.1387 | 0.4096 | 0.1387 | |

1500 | 0.0571 | 0.4114 | 0.1388 | |

1800 | 0.0567 | 0.4123 | 0.1374 | |

2100 | 0.0567 | 0.4129 | 0.1374 | |

6000 | 0.0559 | 0.4116 | 0.1359 | |

\(\delta = 2\) | 90 | 0.0524 | 0.3968 | 0.1321 |

180 | 0.0469 | 0.4222 | 0.1111 | |

300 | 0.0421 | 0.4062 | 0.1037 | |

450 | 0.0424 | 0.4251 | 0.0997 | |

750 | 0.0409 | 0.4218 | 0.0970 | |

900 | 0.0415 | 0.4313 | 0.0962 | |

1200 | 0.0402 | 0.4209 | 0.0955 | |

1500 | 0.0398 | 0.4219 | 0.0944 | |

1800 | 0.0400 | 0.4230 | 0.0946 | |

2100 | 0.0399 | 0.4239 | 0.0940 | |

6000 | 0.0394 | 0.4223 | 0.0934 | |

\(\delta = 3\) | 90 | 0.0345 | 0.4200 | 0.0823 |

180 | 0.0296 | 0.4341 | 0.0683 | |

300 | 0.0277 | 0.4417 | 0.0028 | |

450 | 0.0263 | 0.4395 | 0.0599 | |

750 | 0.0256 | 0.4372 | 0.0586 | |

900 | 0.0257 | 0.4385 | 0.0586 | |

1200 | 0.0255 | 0.4393 | 0.0581 | |

1500 | 0.0253 | 0.4386 | 0.0578 | |

1800 | 0.0248 | 0.4330 | 0.0573 | |

2100 | 0.0250 | 0.4361 | 0.0574 | |

6000 | 0.0249 | 0.4380 | 0.0568 | |

\(\delta = 4\) | 90 | 0.0236 | 0.4866 | 0.0486 |

180 | 0.0189 | 0.4991 | 0.0379 | |

300 | 0.0173 | 0.4889 | 0.0354 | |

450 | 0.0160 | 0.4802 | 0.0334 | |

750 | 0.0158 | 0.4815 | 0.0327 | |

900 | 0.0153 | 0.4744 | 0.0323 | |

1200 | 0.0151 | 0.4725 | 0.0319 | |

1500 | 0.0151 | 0.4728 | 0.0318 | |

1800 | 0.0147 | 0.4660 | 0.0315 | |

2100 | 0.0147 | 0.4663 | 0.0315 | |

6000 | 0.0146 | 0.4656 | 0.0313 | |

\(\delta = 5\) | 90 | 0.0156 | 0.5989 | 0.0260 |

180 | 0.0121 | 0.5892 | 0.0205 | |

300 | 0.0098 | 0.5613 | 0.0174 | |

450 | 0.0093 | 0.5367 | 0.0173 | |

750 | 0.0090 | 0.5507 | 0.0163 | |

900 | 0.0085 | 0.5262 | 0.0161 | |

1200 | 0.0085 | 0.5260 | 0.0162 | |

1500 | 0.0085 | 0.5272 | 0.0161 | |

1800 | 0.0083 | 0.5182 | 0.0160 | |

2100 | 0.0083 | 0.5221 | 0.0159 | |

6000 | 0.0081 | 0.5109 | 0.0158 |

### Effect of number of variables on QDF (under correlated and uncorrelated normal distribution)

The effect of number of variables on the QDF under the correlated and uncorrelated normal distribution are discussed under this subsection.

### Effect of group centroid separator on QDF under correlated and uncorrelated normal distribution

## Conclusion

The study focussed on the asymptotic performance of the QDF under correlated and uncorrelated normally distributed training samples. Under this distribution, the performance of the QDF under varying sampling ratios, selected number of variables and different group centroid separators were extensively studied. The QDF recorded minimum misclassification error rates and high variability as the sample size increased asymptotically under correlated normal distribution, thereby increasing the accuracy of classification of observations with the function. The performance of the QDF deteriorated when the sample size ratio was 1:2:3 as \(\delta\) increased with increasing sample size. However, the performance of the function was appreciably good under both correlated and uncorrelated normal distributions when their estimated average misclassification error rate decreased with increasing number of variables (from 4 to 8). This results shows some partial conformity with the study of Lawoko (1988) where the researcher found that the efficiency of the QDF and other classifiers are generally lowered by positively correlated training observations. Generally, the study found that, the QDF performed better resulting in the reduction in misclassification error rates as group centroid separator increases with non increasing sample size and under correlated training samples. This results therefore shows some partial conformity with the studies by Marks in 1974. Marks approached the problem of discrimination by comparing the performance of QDF with other classifiers. Although he considered only two populations, the QDF performance was abysmal under small sample size selection when covariance matrices were nearly equal with large dimensions.

## Declarations

### Authors' contributions

AA worked on the literature review and background of the study as well as suggestion of the appropriate method and software used and also supervised the entire research work. MAB prepared the manuscript and discussed part of the studies findings. OOT conducted the data analysis through simulation with the used of MATLAB software and also presented and discussed part of the findings. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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