Individual Subject Classification for Alzheimer’s Disease based on Incremental Learning Using a Spatial Frequency Representation of Cortical Thickness Data

1.1. Objectives

Alzheimer’s disease (AD) is the most common form of dementia. The incidence of AD doubles every five years after age of 65 (Bain et al. (2008)). As life expectancy increases, the number of AD patients increases accordingly, which causes a heavy socioeconomic burden. Currently used treatments offer a small symptomatic benefit in mild to moderate AD but cannot delay or halt the progression of AD. Recently, Jack et al. (2010) reported that a structural change was observed in human brains a few years before any symptomatic awareness. Thus, the structural change could provide a clue for early detection of AD.

Amnestic mild cognitive impairment (MCI) is known as a prodromal state of AD and has received much attention for early detection of AD. While only 2% of the healthy elders were converted to AD every year, 10 – 15% of amnestic MCI patients were converted to AD annually (Petersen et al. (1999)). However, AD cannot be predicted with only MCI diagnosis, because other forms of dementia can also be preceded by the MCI state (Dubois and Albert (2004)). Some MCI patients were converted to AD and others remained stable or even reversed to a normal status. The former is called MCI converters (MCIc) and the latter is MCI non-converters (MCInc). The classification between MCInc and MCIc was dealt with for early detection of AD (Querbes et al. (2009); Misra et al. (2009)). Since AD is known to reduce brain volumes prior to clinical symptoms of dementia, brain atrophy has also been utilized for this purpose.

However, the classification between MCInc and MCIc is challenging. According to the benchmark results in (Cuingnet et al. (2010)), classification methods in this category showed low accuracies. Cuingnet et al. (2010) stated that low accuracies can be caused certainly by heterogeneity of cortical thinning patterns over MCInc subjects. One can overcome the difficulty in two ways: increasing training data so as to cover all complex patterns, and choosing good features so as to reflect the difference between two groups.

A large volume of data is necessary in order to handle complex patterns of data. Moreover, simultaneous acquisition of such a data set is practically difficult. Because new data are obtained usually from physical examination or during disease diagnosis, the volume of data is increasing steadily. The continuing data acquisition makes conventional classification methods ineffective. Whenever a new data set is obtained, a classification method should train its classifier with an entire data set including the new data set in order to additively reflect the latter of which the volume is small in general. In this study, we employed an incremental learning approach in order to address this issue effectively.

For accurate classification, it is essential to choose eligible features which clearly represent group differences. Subcortical structures such as hippocampi and certain regions of gray matter were substantially more vulnerable in AD; especially, they showed atrophy in an early stage of AD. Accordingly, most of AD classi-fication methods based on brain morphometry utilized one of the following three neuroanatomical features: hippocampal features, tissue probability maps, and cortical thickness data. Among these features, we used cortical thickness data rather than hippocampal features and probability maps. Hippocampal features were known to be difficult to capture exactly (Qiu and Miller (2008); Khan et al. (2008)). The previous study by Cuingnet et al. (2010) also showed that classification methods using hippocampal features were less accurate compared to other methods. The methods based on tissue probability maps are dependent on volume registration, while the methods based on cortical thickness data are dependent on surface registration. In general, surface registration provided better correspondences than volume registration (Desai et al. (2005); Anticevic et al. (2008)). Therefore, we adopt cortical thickness data for AD classification.

Recently, several classification methods based on cortical thickness data have been proposed. Desikan et al. (2009) and Querbes et al. (2009) classified HC, MCI, and AD using the mean values of cortical thickness for neuroanatomically segmented regions as a feature vector. However, the region-wise data cannot reflect local characteristics of smaller regions than the segmented ones. Cuingnet et al. (2010) constructed a classifier using the cortical thickness at every vertex as a feature vector. Although the vertex-wise data can reflect local deformity of a small region, they are sensitive to noise and registration errors. In this study, we overcome the difficulties with the both types of features by adopting the noise-filtered vertex-wise cortical thickness data based on spatial frequency analysis

We present an individual subject classification method based on incremental learning for AD diagnosis and AD prediction using the cortical thickness data. These data are mapped onto a spatial frequency domain from the surface of a cortex by using the manifold harmonic transform. The basis functions for this transformation are smooth and periodic with different frequencies. Since high frequency components are sensitive to noise and registration errors rather than group differences, we cut off these components to filter out noise, which also effectively reduces the dimension of data as observed in lossy data compression. We construct our classifier based on principal component analysis (PCA) and linear discriminant analysis (LDA) (Zhao et al. (2003)), which enables incremental learning for AD diagnosis and AD prediction. The efficacy of the proposed classification method was demonstrated using several experiments, including comparison with ten other well-known classification methods and also validation of incremental classification.

1.2. Previous Work

Numerous studies on brain morphometry in AD have been conducted in the past two decades. It turns out that AD tends to deform brain regions. For example, volume reduction of temporal lobes was observed in brains of AD patients, according to studies based on voxel-based morphometry (Good et al. (2002); Karas et al. (2003); Chételat et al. (2005)). Reduction of cortical thickness was also observed on cortical surfaces, in particular, medial temporal lobes and entorhinal cortices (Thompson et al. (2003); Lerch et al. (July 2005); Dickerson et al. (2009)). In (Wang et al. (2006)), volume reduction and shape deformity of hippocampi were shown in mild AD. Although these studies reported new findings on AD, they did not provide automatic tools for AD diagnosis and AD prediction. However, these findings were significant in studies on AD classification since vulnerable structures to AD can be excellent features for these tasks.

Based on the results of such morphometry studies, many classification methods have been proposed for AD diagnosis and AD prediction recently. Colliot et al. (2008) and Gerardin et al. (2009) utilized hippocampal volumes and hippocampal shapes for AD classification, respectively. Colliot et al. (2008) normalized hippocampal volumes with respect to the total intracranial volume, and the average of the two normalized hippocampal volumes for both hemispheres was used to classify HC, MCI and AD. In (Gerardin et al. (2009)), hippocampal shapes were modeled via the spherical harmonics transform, and then the support vector machine (SVM) was employed to find a hyperplane to maximally separate HC and AD. A technical limitation of these methods was that extracted subcortical structures from MR images were often unacceptable due to their small shapes and ill-defined boundaries (Chupin et al. (2007)). Klöppel et al. (2008) applied the SVM to voxels of a tissue probability map which indicates the probability of different tissue classes (grey matter, white matter, and cerebrospinal fluid). The approach can be adapted easily to other anatomical features due to its methodological simplicity. Specifically, Cuingnet et al. (2010) applied it to cortical thickness data for AD classification. However, the approach was sensitive to noise and registration errors because the spatial coherence of features was not considered. Unlike the method in (Klöppel et al. (2008)), other methods parcellated MR volumes or cortical surfaces, and extracted a representative value such as the average cortical thickness from each region. Ye et al. (2008) and Magnin et al. (2009) classified AD and HC by using the tissue probability of each parcellated region. In (Desikan et al. (2009); Querbes et al. (2009)), the cortical thickness value of each segmented region was extracted to build a feature vector. Ye et al. (2008), Desikan et al. (2009), Magnin et al. (2009), and Querbes et al. (2009) parcellated MR volumes based on neuroanatomical knowledge, and Fan et al. (2007) adaptively parcellated MR volumes to define discriminative regions. However, based on region segmentations, such methods poorly reflected detailed spatial variation of features. Furthermore, none of the above methods addressed the incremental learning issue in classification.

To the best of our knowledge, incremental learning has not been attempted in neuroimaging. However, since most of training data for classification are sequentially obtained, studies on incremental learning have received steady attentions in artificial intelligence and computer vision. Incremental learning-based versions of statistical techniques such as PCA and LDA have been reported. Hall et al. (1998) proposed an incremental PCA method which updates the PCA transformation matrix sequentially with each of additional training data. Hall et al. (2002) extended it to deal with a set of new training data simultaneously. Levy and Lindenbaum (2000) and Lim et al. (2004) later enhanced the computational efficiency and reduced the computational error during the update. The method of Lim et al. (2004) was used in a visual tracking method (Ross et al. (2008)) in order to adapt to changes in the appearance of the target. Pang et al. (2005) proposed an incremental LDA (ILDA) scheme which can handle large training data. They showed that the classification accuracy increased incrementally with additional data.

1.3. Materials

Data used in this paper were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (www.loni.ucla.edu/ADNI). The ADNI was launched in 2003 by the National Institute on Aging (NIA), the National Institute of Biomedical Imaging and Bioengineering (NIBIB), the Food and Drug Administration (FDA), non-profit organizations and private pharmaceutical companies. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of MCI and early AD. Determination of sensitive and specific markers in an early stage of AD progression is intended to aid researchers and clinicians to develop new treatments and monitor their effectiveness, as well as to reduce the time and cost of clinical trials. For more details, we refer the readers to www.adni-info.org.

In this paper, MRIs of 491 subjects who belong to one of HC, MCI, and AD groups were analyzed. The eligibility criteria of subjects applied in ADNI are described at http://www.adni-info.org/Scientists/ADNIGrant/ProtocolSummary.aspx. Briefly, enrolled subjects in ADNI were between ages of 55–90 (inclusive) and spoke either English or Spanish. All subjects must be willing and able to undergo all test procedures including neuroimaging and agree to longitudinal follow-up. Specific psychoactive medications were excluded. General inclusion/exclusion criteria are as follows:

1. HC subjects: Mini-Mental State Examination (MMSE) (Folstein et al. (1975)) scores between 24–30 (inclusive), a Clinical Dementia Rating (CDR) of 0, non-depressed, non-MCI, and nondemented. The age range of normal subjects was roughly matched to that of MCI and AD subjects. Therefore, there should be minimal enrollment of normals under the age of 70.

2. MCI subjects: MMSE scores between 24–30 (inclusive), a memory complaint, objective memory loss measured by education-adjusted scores on Wechsler Memory Scale Logical Memory II, a CDR of 0.5, absence of significant levels of impairment in other cognitive domains, essentially preserved activities of daily living, and an absence of dementia.

3. Mild AD subjects: MMSE scores between 20–26 (inclusive), CDR of 0.5 or 1.0, and meeting NINCDS/ADRDA criteria for probable AD.

In the ADNI procedure, all subjects received the baseline clinical/cognitive examinations including 1.5 T structural MRI, and were re-evaluated at specified intervals (6 or 12 month) for 2–3 years. The baseline MRI scans were downloaded from the ADNI database, which were used as the input data in our experiments. We also utilized the follow-up examination results in order to separate MCI into MCIc and MCInc. The separation was performed by following the same policy as that in (Cuingnet et al. (2010)). MCI subjects who were converted to AD within 18 months were classified into MCIc, and those not converted into AD within the same period were classified into MCInc. Further, we excluded MCI subjects who did not follow up the examinations more than 18 months. Table 1 shows the demographic characteristics of the participants.

2.1. Overview

In this section, we present an incremental classification method for AD diagnosis and AD prediction using cortical thickness data. Given an MR volume of a subject, AD diagnosis determines whether she/he is in HC or AD. On the other hand, AD prediction discriminates MCIc from MCInc, provided with an MR volume of an MCI subject. We also deal with classification between HC and MCIc for comparison with the benchmark results in (Cuingnet et al. (2010)). Our classification method consists of two steps: group classifier training and individual subject classification. Figure 1 shows an overall structure of the method. The former step trains a group classifier with labeled MR volumes. This step first filters out high frequency components from the cortical thickness data at vertices, which has been extracted from the MR volumes, in order to remove noise, and then trains the group classifier with resulting data. The latter step classifies unlabeled subjects by using an individual subject classifier. This classifier is initialized with the group classifier trained in the previous step and incrementally updated. Given the MR volume of a subject, its feature vector representing the noise-filtered cortical thickness data is acquired as in group classifier training. The classifier performs AD diagnosis or AD prediction using the feature vector. Unlike existing methods for AD classification, our individual subject classifier not only classifies an unlabeled subject but also enhances the classifier itself based on incremental learning after labeling the subject.

The contributions of our approach are two-fold: First, we represent the cortical thickness data of a subject in terms of their frequency components, employing the manifold harmonic transform (Levy (2006); Qiu et al. (2006); Vallet and Lévy (2008)). The basis functions for this transform are obtained from the eigenvectors of the Laplace-Beltrami (LB) operator, which is dependent only on the geometry of a cortical surface but not on the cortical thickness function defined on it. This facilitates individual subject classification based on incremental learning. Second, our classifier showed high accuracy in both AD diagnosis and AD prediction. According to the benchmark data in (Cuingnet et al. (2010)), no AD classifiers produced good results in the both categories. Intuitively, methods based on region-wise features poorly reflect the detailed spatial variation of cortical thickness, and those based on vertex-wise features are sensitive to noise. Adopting a vertex-wise cortical thickness representation scheme, we can still achieve robust classification to noise by filtering out high frequency components of cortical thickness data while reflecting their spatial variation. We believe that this compromise enabled both AD diagnosis and AD prediction with high accuracy.

2.2. Group Classifier Training

2.3. Individual Subject Classification

In this section, we describe how to classify individual subjects using the group classifier trained in the previous section. We also discuss how to enhance the performance of the classifier incrementally with new training data. Given an MR volume of a unlabeled subject, its feature vector is first acquired as described in Section 2.2.1. This process performs PCA and LDA in sequence to transform the feature vector x of the subject to a point z in the LDA space:

where W_P and W_L are the PCA and LDA transformation matrices defined in Section 2.2.2, respectively. Given z in the LDA space, our classifier maps the subject onto one of the groups {G₁, G₂, …, G_g}. That is, the subject is classified to group G_i if z is closer to the projection of the mean x̄_i of G_i onto the LDA space than those of the others. After the unlabeled subject is classified, its label is validated by a clinician to be used for training. It is time-consuming to train the classifier with the entire training data whenever new training data are added. Therefore, the group classifier training scheme is slightly modified for training an individual subject classifier. In other words, the individual subject classifier is trained incrementally without using the previously used training data so that the time efficiency is dependent only on the size of new data.

Suppose that the group classifier has been built with a training data set, X₁ = {x₁, …, x_N}. The classifier can be modeled with a set of parameters, Ω = {W_P, W_L, x̄₁, …, x̄_g}, where x̄_i, 1 ≤ i ≤ g is the mean of feature vectors x_ik, 1 ≤ k ≤ N_i belonging to group G_i. Given an additional data set, X₂ = {x_N+1, …, x_N+M } for small M ≤ 1, our objective is to update Ω incrementally with X₂ in order to obtain a new model ${\mathrm{\Omega }}^{\prime }=\{W_P^{\prime },W_L^{\prime },{\overline{x}}_1^{\prime },\dots ,{\overline{x}}_g^{\prime }\}$.

In order to update W_P, we first modify the mean vector x̄ and the covariance matrix V with X₂, and then derive the new PCA matrix $W_P^{\prime }$ from these, by employing an incremental PCA method (Lim et al. (2004)). For clarity of explanation, suppose that M = 1, that is, X₂ = {x_N+1} is a singleton set. The new mean x̄′ is computed from the old mean x̄ as follows:

The new covariance matrix V′ is

For details in deriving the transformation matrix W_P′, we refer the readers to (Hall et al. (1998)). Hall et al. (2002) proposed an incremental PCA method which update a classifier with a set of feature vectors rather than a single feature vector, and Lim et al. (2004) enhanced the computational efficiency for incremental PCA. We adopt the method in (Lim et al. (2004)), the source code of which is available at http://www.cs.toronto.edu/dross/code/.

With W_p, x̄, x̄₁, …, x̄_g computed, we employ an incremental LDA method (Pang et al. (2005)) to compute the LDA transformation matrix W_L. Unlike the group classifier described in Section 2.2, this method incrementally computes the between- and within-class scatter matrices in the feature space (rather than in the PCA subspace) and then maps the results onto the PCA subspace to finally compute W_L. Let $S_B^X$ and $S_W^X$ be the between- and within-class scatter matrices in the feature space, respectively. These matrices can be computed with a small number of parameters including group means x̄, x̄₁, …, x̄_g and group covariance matrices V₁, …, V_g in the feature space, which can efficiently be updated in an incremental manner. We describe how to convert $S_W^X$ in the feature space to S_W in the PCA subspace using the PCA transformation matrix W_P:

where x_ik is the k^th feature vector in the group G_i. Similarly, $S_B^X$ can be converted to S_B, that is, $S_B=W_P^T{S}_B^X{W}_P$. Finally, W_L are constructed with eigenvectors of $S_B^{-1}{S}_W$.

3.1. Noise Removal

The first experiment was intended to validate our noise removal scheme described in Section 2.2.1. This scheme removes noise from cortical thickness data by filtering out their high frequency components after determining the cut-off frequency F. We first performed noise removal using the scheme and then performed statistical analysis to compare the cortical thickness data before and after noise removal.

To perform noise filtering, we determined the cut-off dimension F using the training data set consisting of 80 HC, 65 MCInc, 37 MCIc, and 62 AD subjects as given in Table 2. Therefore, the resulting dimension F is unbiased to any specific test data for validation. The cut-off dimension F was chosen by setting the goodness of fit G = 0.025. With G = 0.025, the cut-off frequency F was set to 2,400. Figure 4 plots accuracy of each of group classifiers with respect to the cut-off dimension F. Notice that we did not try to choose the optimal value of the dimension F in a classification-specific manner as observed in the figure. Rather, we chose a cut-off dimension F for all classifications by conservatively setting G = 0.025. This cut-off frequency F was used for all experiments described in Sections 3.1, 3.2 and 3.3.

In order to compare cortical thickness data before and after noise removal, we separately performed group analysis for the original cortical thickness and its noise-filtered one using the test data of each data subset in Table 2. Since the group analysis results for all three data subsets were similar to each other, we only present the results for Dataset 1 (HC vs. AD) for conciseness of explanation. We first performed noise removal for the cortical thickness data of each test subject by cutting off their high frequency components using the previously-determined F = 2, 400. We then measured the mean difference between AD and HC groups for the corresponding noise-filtered data subset as well as the original one. The first row in Figure 5 shows the mean difference in the original data subset, and the second row shows the mean difference in the noise-filtered data subset. One can visually verify that the mean differences are similar to each other. The third row visualizes the difference between the first and second rows: For more than 90% of the whole surface region, the original and noise-filtered cortical thickness data are the same within 0.15 mm error. We also performed a t-test at each vertex in order to validate the hypothesis that the cortical thickness can discriminate AD subjects from HC subjects. The first and the second rows in Figure 6 show the resulting t-statistics maps for the original and noise-filtered data subsets, respectively. Statistically significant regions in the noise-filtered data subset were similar to those in the original one. The third row shows the difference of absolute t-statistics values at every vertex between the original and noise-filtered data subsets. A warm color represents that the noise-filtered data subset is statistically more significant than the original one, while a cold color represents the opposite case. For more than 63% of the whole surface region, the noise-filtered data subset has greater absolute t-statistics values, which verifies that our noise removal scheme improves the statistical analysis results similarly to the work in (Chung et al. (2007)).

3.2. Group Classification Performance

The next experiment was to show the group classification performance of our method against other classification methods. Many high-dimensional classification methods for automatic AD classification have recently been presented. Fan et al. (2007) and Querbes et al. (2009) used different data sets from the ADNI database. In order to compare different methods in their classification performances, one would have to implement all these methods. Fortunately, Cuingnet et al. (2010) presented benchmark results for AD classification methods using the ADNI database. The same data from the ADNI database were used to compare the performances of ten classification methods. Since the subject identifications for the data were provided as a supplement, we were able to repeat an identical experiment in (Cuingnet et al. (2010)) with our method in order to exploit their benchmark results for performance comparison. Now, we briefly explain how the benchmark results were generated.

Cuingnet et al. (2010) performed three classification experiments: HC vs. AD, HC vs. MCIc, and MCIc vs. MCInc. They compared ten classification methods: Voxel-Direct, Voxel-Direct VOI, Voxel-Atlas, Voxel-STAND, Voxel-COMPARE, Thickness-Direct, Thickness-Atlas, Thickness-ROI, Hippo-Volume and Hippo-Shape. The first five methods employed voxel-based segmented tissue probability maps. The next three methods used cortical thickness data. The last two methods were based on hippocampal features. For details of the classification methods, we refer the readers to (Cuingnet et al. (2010)). For assessment of classification performances, we used data subsets, Datasets 1, 2, and 3 (See Table 2 for composition of each data subset), which are the same data sets as those in (Cuingnet et al. (2010)).

We trained three group classifiers for HC vs. AD classification, MCInc vs. MCIc classification, and HC vs. MCIc classification with the training data in their respective data subsets. Specifically, for training a group classifier, we first performed noise removal for the training data in the corresponding data subset using the cut-off frequency F = 2, 400 determined in the previous section. We then employed PCA to reduce the dimension of the feature space, which prevents the singularity problem in performing LDA. As shown in Section 2.2.2, we empirically decided this dimension k by setting the percentage of the total variance to 70%. Finally, the group classifier was obtained by performing LDA with the transformed training data in the PCA space.

After training the classifiers, we assessed the sensitivity and the specificity of each classification with the test data in the respective data subset as follows:

The sensitivity and the specificity were 82% and 93% for HC vs. AD classification, 63% and 76% for MCIc classification, and 66% and 89% for MCIc vs. HC classification, respectively. We compared the classification performance of our method with those of the other ten methods used in (Cuingnet et al. (2010)). Table 3 summarizes the classification performances of the ten classification methods together with that of ours. The sum of the sensitivity and specificity for each method was used as the measure of classification performance. Figure 7 depicts the performances of the methods in their descending order. Our method received good evaluations in all classifications: It showed the highest performance in MCInc vs. MCIc classification and the second highest performance in HC vs. AD classification. In HC vs. MCIc classification, it was ranked in the fourth position. In HC vs. AD classification, the performance of ours was similar to that of Voxel-Direct showing the highest performance. In HC vs. MCIc classification, three methods, Voxel-Atlas, Thickness-ROI, and Voxel-STAND showed higher performances than ours. However, these four methods showed inferior performances to ours in MCInc and MCIc classification. Voxel-STAND showed a similar performance to ours in both HC vs. MCIc classification and MCInc and MCIc classification, but our method is superior to Voxel-STAND in HC vs. AD classification. The benchmark results show that our classification method exhibits its high performance compared to the other methods.

We compared the discriminative regions for our method with those in Thickness-Direct and Thickness-Atlas which used, as feature data, the cortical thickness at every vertex and the mean cortical thickness over every parcellated region, respectively. Both of the methods adopted the support vector machine (SVM) which finds a hyperplane that maximally separates groups. In the case of the linear SVM, the value of the i^th component of the vector v that is orthogonal to the separating hyperplane represents the contribution of the component to classification. That is, if the value of the i^th component of v is zero, the i^th element of every feature vector does not affect the classification result. Conversely, if the value is larger than the others, the classification result is more sensitive to the i^th element than the other elements. This vector plays the same role as the separating axis w in LDA since the value of the i^th component in w also represents the contribution of the i^th element of each feature vector in the PCA space to classification. Therefore, the analysis of both v and w gives the discriminative region for classification. In (Cuingnet et al. (2010)), the orthogonal vectors v to the hyperplanes of Thickness-Direct and Thickness-Atlas have been visualized. We also visualized the axis w of LDA by converting it to a pair of vectors on the left and right atlas meshes: w in the PCA space was first transformed to a vector x = W_P w in the feature space. The vector is then divided into two parts: frequency components { $f_1^L,\dots ,f_F^L$} for the left atlas mesh and frequency components { $f_1^R,\dots ,f_F^R$} for the right atlas mesh. These frequency components are finally transformed to two cortical thickness vectors on the left and right atlas meshes using Equation (3), respectively. We divided these vectors by their magnitudes to obtain two unit vectors for visualization. Figure 8 depicts these vectors on the atlas meshs for HC vs. AD classification, MCInc vs. MCIc classification, and HC vs. MCIc classification. The entorhinal cortex was the most discriminative for AD classification, and the lateral temporal lobe and the prefrontal cortex were also discriminative, which is consistent with discriminative regions of Thickness-Direct and Thickness-Atlas.

In general, near-by regions in a cortex have correlated brain functions, and are similarly deformed by brain diseases. By representing the noise-filtered cortical thickness data of a subject in terms of the spatial frequency components, our method reflects spatial coherency of the data, which resulted in the spatial coherency of discriminative regions. Our discriminative regions were therefore smoother than those of Thickness-Direct: The vertex-wise cortical thickness representation adopted by Thickness-Direct poorly reflects spatial relationship of the feature data. On the other hand, Thickness-Atlas is based on a region-wise representation of the thickness data, which poorly reflects detailed spatial variation of the thickness data. Due to this property of Thickness-Atlas, the cortical thickness data at all vertices in the same parcellated region contributed equally to the classification (see Figure 6 in (Cuingnet et al. (2010))).

3.3. Incremental Classification Performance

In this section, we demonstrated the effectiveness of incremental classification in both accuracy and efficiency. We initialized three individual subject classifiers, a HC vs. AD classifier, an MCInc vs. MCIc classifier, a HC vs. MCIc classifier using the training data of data subsets, Datasets 1, 2, and 3, respectively. The resulting classifiers were validated with the test data in their respective data subsets.

We began with demonstrating the accuracy of incremental classification. As the training data to each of three individual subject classifiers, we generated one hundred random permutations of the entire training subjects in the corresponding data subset in order to keep the experiment unbiased to a specific ordering of the subjects in the data subset. For each permutation, the accuracy of the classifier was measured as follows: We applied the classifier to the test data in the data subset while updating the classifier incrementally with the training data. Specifically, we iteratively supplied the training data of ten subjects at a time for incremental learning until all the training data were used up. Whenever the classifier was updated with these data, the entire test data were used to estimate the accuracy of the classifier. By averaging the results over all permutations, we measured the accuracy of the classifier with respect to the number of used training subjects. As plotted in Figure 9, the accuracy of every individual subject classifier tended to converge to that of the respective group classifier trained with the entire training data in the corresponding data subset as the number of used training subjects approached to that of the training subjects in the data subset.

We next validated the time efficiency of incremental classification by employing the group and individual subject classifiers for HC vs. AD classification. Similarly results would be obtained for MCInc vs. MCIc classification and HC vs. MCIc classification. The group classifier was initially trained with the training data of Dataset 1, and the individual subject classifier was initialized with this group classifier. We increased the size of our data set up to (including) 13,000 by cloning the test data of Dataset 1.

We separately measured the computation times for the group and incremental subject classifiers by incrementally supplying test subjects to the both classifiers in the unit of 100 subjects. The both classifiers were updated whenever new training data were added. Figure 10 shows the computation time of each classifier excluding that for the feature vector construction. The computation time for incremental learning was constantly 1.4 seconds regardless of the cumulative size of training data since it is dependent on the size of new training data (or new labeled test data). On the other hand, the computation time of batch learning rapidly growing in the cumulative size of the training data. The experiment of batch learning with more than 13,000 subjects was unable to be performed due to lack of the memory space.