# Overview of Algorithms for Using Particle Morphology in Pre-Detonation Nuclear Forensics
## 4 Lda  
The multivariate T2 statistic uses the statistical distance between two sample means and leads to a simple and reasonably robust test, for example, of whether the ADU group mean is different from the AUC mean. Linear discriminant analysis (LDA) uses the statistical distance from a test observation to each of the ADU and AUC group means to assign the test observation to the ADU or AUC group. LDA is one of the oldest [12; 13; 15] and simplest pattern recognition methods (\"supervised learning\" in ML jargon) used to assign a group membership to a test case based on training data that has known group assignments. Figure 16 plots simulated group 1 and group 2 data for \(p=2\), with the true and estimated (from 100 observations per group) linear decision boundaries. The correlation is 0.5, and the true mean shift is 3 units in the direction of x1 only (a 3-unit mean shift to the east, and the standard deviation in both groups is 1). The mean vector of group 1 is (0,0) and the mean vector of group 2 is (3,0). By appealing to likelihood ratio theory, or Bayes rule, the true linear decision boundary is computed using \((\mu_{1}-\mu_{2})^{T}\Sigma^{-1}X\) and the estimated linear decision boundary is computed using \((\overline{x}_{1}-\overline{x}_{2})^{T}\Sigma^{-1}X\) where \(X=(x_{1},x_{2})\)[13; 14; 16]. Figure 14: The pdf of T2 for normal data (with correlation 0.60) and for lognormal data, also with a correlation 0.60, \(p=15\), and \(n_{1}=n_{2}=20\). Figure 15: The same as Figure 14, but for \(n_{1}=n_{2}=200\). Similarly to using the non-centrality parameter in Section 3 to quantify the effect of a mean shift from \(\mu_{1}\) to \(\mu_{2}\), the total probability of misclassification (TPM) by LDA (the probability that a group 1 item is classified as group 2, or a group 2 item is classified as group 1, TPM = 1\(-\)CCR), is given by the cumulative normal probability \(\text{TPM}=\Phi\left(\frac{-8}{2}\right)\), where \(\delta^{2}=(\mu_{1}-\mu_{2})^{T}\Sigma^{-1}(\mu_{1}-\mu_{2})\)[16]. For the data in Figure 16, the true TPM = 0.042 and the estimated TPM is 0.045. Figure 16 plots the pdf for the estimated TPM based on 105 sets of 100 simulated observations per group, both for normal data and log normal data. Of course, real data are rarely so close in distribution to multivariate normal (MVN). For example, Figure 17 plots 25 simulated observations from each of 4 clusters, where clusters 1 and 2 belong to group 1 and clusters 3 and 4 belong to group 2. ML methods such as learning vector quantization (LVQ [14]) or mixture discriminant analysis (MDA [14]) training are effective for this type of data. However, with more than \(p=2\) predictors, it is rarely so evident that clusters are present, so LDA, LVQ, and MDA are compared. MDA assigns a test case to the nearest cluster and corresponding group. Although real data rarely closely follow the LDA assumptions that the data are MVN with the same covariance matrix and possibly different means, LDA sometimes has a competitive CCR compared to other ML options such as MDA and LVQ. Note that due to processing, aging, and/or storage conditions, clustering, such as in Figure 17, could occur and performance claims regarding how distinct group 1 is from group 2 depend on the relative frequency of each cluster. For example, cluster 4 in group 2  
Figure 16: The estimated pdf for TPM based on 105 sets of 100 simulated observations from each of group 1 and group 2, with the true TPM = 0.042, \(n_{1}=n_{2}=20\), 0.6 correlation between \(x_{1}\) and \(x_{2}\). Figure 17: Mixture data with clusters 1 and 2 in group 1 and clusters 3 and 4 in group 2.  
is quite distinct from all of group 1, but cluster 3 from group 2 is not distinct from cluster 2 of group 1. Therefore, performance claims regarding group separability could depend on the relative frequencies of the clusters in the training and testing data. Applying leave-one-out cross validation (CV) to 103 simulated observations in each of the four clusters in each group, the estimated TPM using either LDA or MDA is 0.