A total of 30 AD patients (46.7% female; mean age of 75 at the standard deviation of 7), and 30 age-matched healthy normal controls (50% female; mean age of 77 at the standard deviation of 5) are selected from the ADNI public database http://www.loni.ucla.edu/ADNI/Data/. ADNI is a large five-year study launched in 2004 by the National Institute on Aging (NIA), the National Institute of Biomedical Imaging and Bioengineering (NIBIB), the Food and Drug Administration (FDA), private pharmaceutical companies and nonprofit organizations, as a $60 million public-private partnership. The primary goal of ADNI has been to test whether serial MRI, PET, other biological markers, and clinical and neuropsychological assessments acquired at multiple sites (as in a typical clinical trial), can replicate results from smaller single site studies measuring the progression of MCI and early AD. Determination of sensitive and definite markers of very early AD progression is destined to aid researchers and clinicians to monitor the effectiveness of new treatments, and diminish the time and cost of clinical trials. The Principal Investigator of this initiative is Michael W. Weiner, M.D., VA Medical Center and University of California, San Francisco.

All the AD and NC subjects in this study had successfully undergone MRI scanning, cognitive tests and clinical evaluation at baseline, 6^th months and 2^nd year follow up.

Some demographic parameters such as age, sex and years of education have remarkable impact on brain atrophic measures and to avoid their influence on the study, subjects of two groups must be matched regarding them. Difference in gender among the two groups is tested with the Chi-square test and matched (p = 0.796). Independent two sample student t-test is used to test inter-group differences in age and years of education. As there are no significant differences in age (p = 0.188) and years of education (p = 0.554) among the two groups, they were ignored in diagnosing AD in this study. Baseline MMSE and PBVC in all three time intervals of baseline to the 6^th month follow up (PbvcSc-6), 6^th month to 24^th month follow up (Pbvc6-24) and baseline to 24^th month follow up (PbvcSc-24) indicate significant differences between the two groups (Table 1).

These results approve that the two groups are disparate based on longitudinal volume changes, but it does not specify the way of classifying one individual subject into one of these groups based on above features.

DA is a statistical technique used to differentiate groups when the underlying features are quantitative and normally distributed 31. It is an appropriate method for classifying patterns of subjects into two desired separated groups, AD and NC.

The aim of DA is to analyze group separation power for a set of normally distributed features or pattern of features. Test of normality for all three atrophic measures imply their normal distribution through both groups (Table 2).

The simplest and first way to this is using total means of features as threshold values. Patterns with feature values above it will be assigned to one group and the ones bellow it to the other.

Referring to the total means of Table 1, results of classification will be as shown in Table 3. It is obvious that long-term atrophy rates yield higher accuracy.

These values may not be the optimal threshold values and for comprehensive evaluation, Receiver Operating Characteristic (ROC) curve analysis is carried out. ROC curve plots for all of the three features and associated parameters are shown in Figure 3.

The highest diagnostic accuracy of 90% is achieved by using PbvcSc-24 and a specific threshold value. To evaluate generalization capacity of this feature, leave-one-out-cross-validation is conducted. Finding discloses lower generalization accuracy besides the memorization (Table 4).

After that, two other features are included in DA to see whether the accuracy is enhanced or not. A key assumption of DA is that the features should not be highly correlated, but these three features are highly correlated (Table 5).

It is clear that PbvcSc-24 has high correlation with PbvcSc-6 and Pbvc6-24 and this violates the terms of analysis. To overcome this we use principal component analysis (PCM) to convert them to uncorrelated features. There are two main steps in conducting PCA:

• Step 1: Assessment of data suitability

Sample size or factorability of data, and the strength of the relationship among the features are two main issues to consider in determining whether a particular data set is suitable for PCA or not. A sample size over feature space dimension ratio of 10/1 has been recommended 32. To put it in other words, at least 10 samples for each feature are needed to be PC analyzed. This criterion is passed in the study. Moreover, two statistical measures are also available for analyzing suitability of the sample size. Bartlett's test of sphericity 33, and the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy 34. The Bartlett's test of sphericity should be significant (p < 0.05) and the KMO index which ranges from 0 to 1, should be greater than 0.6 for the PCA to be considered appropriate. These two measures for our dataset are shown in Table 6.

Factorability of data samples are also confirmed according to these measures. In order for feature relationship to be strong, correlation between features should be at least 0.3 which is at this rate in our case (Table 5).

• Step 2: Feature extraction

In this step the number of features involved in discriminating groups, should be specified. This involves balancing two contradicting needs which are the need to find a simple solution with as few factors as possible and the need to explain as much of the variance in the original data set as possible. There are a number of techniques that can be used to specify the number of features to be kept. One of them is Kaiser's criterion 35, according to which, only features with an eigenvalue of 1.0 or more are retained. The eigenvalue of a feature represents the amount of the total variance explained by that feature. Extracting features by this method leads to selecting only one feature (Table 7).

The next test is known as Scree test 36. It plots each of the eigenvalues and inspects the plot to find a point at which the shape of the curve changes direction toward horizontal or an elbow. Keeping all factors above the elbow is recommended, as these features contribute the most to the explanation of the variance in the data set. In the case of our study, two of the features settle above the elbow and can be kept (Figure 4).

Other method in determining number of features is parallel analysis 37. Parallel analysis involves comparing the value of the eigenvalues with those obtained from a randomly generated data set of the same size. Only those eigenvalues that exceed the corresponding values from the random data set are kept. According to this analysis, only one of features can be kept (Table 8).

Regarding to the three abovementioned methods, only one of the features must be selected for discriminating subjects. Referring to the Table 7, it carries 79.371% of total variance among data which seems not satisfactory. Indeed, PCA is used as a data exploration technique, so the interpretation and the way we use it is up to our judgment, rather than any hard and fast statistical rules. Here in this article, it is supposed that the algorithm is interested only in components that have an eigenvalue of 0.6 or more. By extracting two uncorrelated features, with which 99.863% of total variance among data will be carried, which is highly satisfactory.

To investigate the contribution degree of initial features in newly extracted ones, refer to Table 9. It can be seen from this table that most of the features load quite strongly (above 0.4) on them (except PbvcSc-6 on PC2).

As expected, the new extracted features are highly uncorrelated (Table 10).

DA can be carried on by these two newly extracted uncorrelated features.

Calculated unstandardized canonical discriminant function is:

d s = ( 0 . 3 4 7 * P C 1 ) - ( 0 . 5 9 2 * P C 2 ) + 2 . 0 6 2

With ds as discriminant score, Table 11 shows the mean of ds for two groups of subjects which are conspicuously far apart each other.

To measures the association between the ds and the groups, Canonical correlation should be considered (Table 12). A high value (near 1) shows that the function discriminates quite well.

With regard to canonical correlation of 0.671 in this study, discrimination power of these extracted features is conceived as moderate. Wilk's Lambda shows the proportion of the total variance (55%) in the ds not explained by differences among groups (Table 13). A small Lambda value (near 0) indicates that the group's mean ds differs. The Sig (p < 0.001) is for the Chi-square test which indicates there is a highly significant difference between the groups' centroids.

To investigate the impact of each extracted feature on the discriminant function, correlation (in order of importance) of each feature with the ds is calculated (Table 14). It is revealed that PC1 has highest impact on discrimination process.

In k-fold cross-validation, the initial data set is randomly partitioned into k non-overlapping subsets or "folds" (D₁, D₂, ... , D_k) each of which with approximately equal size. Training and testing is performed k times. In iteration i, subset D_i is reserved as test set, and the remaining subsets are collectively used to train the model. To put it simple, in the first iteration, subsets D₂, ... , D_k are used as the training set in order to obtain a first model, which is tested on D1; the second iteration is trained on subsets D₁, D₃, ..., D_k and tested on D₂, and so on. For classification, the accuracy estimation is the overall number of correct classifications from the k iterations, divided by the total number of tuples in the initial data.

Leave-one-out is a special case of k-fold cross-validation where k is set to the number of initial tuples. That is, only one sample is left out at a time for the test set.

It is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences 38. The other main advantage of PCA is that once you have found these patterns in the data, you can compress the data by reducing the number of dimension, without much loss of information. This technique is used in feature extraction to reduce feature space dimension and make features more discriminative.

PCA involves the eigenvalue decomposition of data covariance matrix to generate features that are optimally uncorrelated

I ( i 1 , i 2 , i 3 , ⋯ , i m ) = A T ⋅ P ( p 1 , p 2 , p 3 , ⋯ , p n )

Where P is the original pattern of features and I is the pattern of uncorrelated features. A is the eigenvalue of covariance matrix.