Keywords: MCI subtypes, neuropsychological profile, Grey matter, CSF biomarker, ADNI, Machine learning

We demonstrated that mild cognitive impairment (MCI) participants of the ADNI database (N=640) can be discriminated into 3 coherent and neuropsychologically-defined subgroups. Our clustering approach revealed an amnestic MCI, a mixed MCI and a false positive subgroup. Furthermore, we investigated the neurobiological foundation of these automatically extracted MCI subgroups. Classification modelling exposed that specific predictive features can be used to differentiate amnestic and mixed MCI from healthy controls: CSF Aβ 1-42 concentration for the former and CSF Aβ 1-42 concentration, tau concentration as well as cortical atrophies (especially in the temporal and occipital lobes) for the latter. In contrast, false positive participants exhibited an identical profile to healthy participants in terms of cognitive performance, brain structure and CSF biomarker levels. Our comprehensive data-analytics strategy provide further evidence that multimodal neuropsychological subtyping is both clinically and neurobiologically meaningful. HIGHLIGHTS  Our clustering approach revealed an amnestic MCI, a mixed MCI and a false positive subgroup within the ADNI-defined MCI individuals.  We investigated the neurobiological foundation of these neuropsychologically-defined MCI subtypes.  Classification modelling exposed that specific predictive features can be used to differentiate amnestic and mixed MCI from healthy controls.  However, a subgroup of ADNI-defined MCI individuals displayed a profile similar to healthy participants in terms of brain structure, CSF biomarker levels, and cognitive performance.  These results suggest that neuropsychologically-defined MCI subtypes are neurobiologically grounded.

INTRODUCTION

Mild cognitive impairment (MCI) is considered a transitional stage between normal aging and Alzheimer's disease (AD). There is evidence that around 10-15 percent of MCI patients progress to AD each year, compared to 1-2 percent in the healthy older adult population (Alzheimer's Association 2019; Anderson 2019). However, there is considerable heterogeneity among the MCIdiagnosed individuals and not all of them are at risk for developing AD dementia later in life. Some patients develop non-AD dementia or other neuropsychiatric diseases [START_REF] Slot | Subjective Cognitive Decline and Rates of Incident Alzheimer's Disease and Non-Alzheimer's Disease Dementia[END_REF]. Others remain stable with respect to neuropsychological performance [START_REF] Overton | Diagnostic Stability of Mild Cognitive Impairment, and Predictors of Reversion to Normal Cognitive Functioning[END_REF], or even revert to normal cognitive functioning (Thomas, Edmonds, et al. 2019). There is also a high rate of misdiagnosis using conventional diagnostic criteria based on the DSM-5, with many 'false-positive' MCI cases [START_REF] Edmonds | Early versus Late MCI: Improved MCI Staging Using a Neuropsychological Approach[END_REF]. This heterogeneity of MCI have led the researchers to place great emphasis on subtyping or risk stratification of MCI patients to identify those at increased risk of developing AD and who constitute the optimal target population for therapeutic interventions [START_REF] Dams | Alzheimer's Disease-Related Dementias Summit 2019: National Research Priorities for the Investigation of Traumatic Brain Injury as a Risk Factor for Alzheimer's Disease and Related Dementias[END_REF][START_REF] Winblad | Defeating Alzheimer's Disease and Other Dementias: A Priority for European Science and Society[END_REF].

A common subtyping approach is to classify MCI individuals based on their neuropsychological test scores. Early on, MCI were staged into early and late MCI based on their level of impairment on one memory measure, with the latter being more impaired than the former. This "classical criteria" approach can be seen in the North American Alzheimer's Disease Neuroimaging Initiative (ADNI) and in other samples (eg., [START_REF] Jessen | AD Dementia Risk in Late MCI, in Early MCI, and in Subjective Memory Impairment[END_REF]. This approach has proven to be useful for staging MCI severity by demonstrating a higher risk of conversion to AD in individuals with late MCI compared to those with early MCI. However, there are also a number of limits with this approach including the unreliability of using a single neuropsychological test score to form subgroups, resulting in false positive MCI cases [START_REF] Edmonds | Early versus Late MCI: Improved MCI Staging Using a Neuropsychological Approach[END_REF]Thomas, Eppig, et al. 2019), as well as the low sensitivity for detecting non-amnestic forms of MCI [START_REF] Jak | Quantification of Five Neuropsychological Approaches to Defining Mild Cognitive Impairment[END_REF]). Researchers then developed a "comprehensive criteria" from which multiple subtypes of MCI were identified based on performance on several tests covering a number of cognitive domains (eg., Clark et al. 2013;[START_REF] Jak | Quantification of Five Neuropsychological Approaches to Defining Mild Cognitive Impairment[END_REF][START_REF] Bondi | Neuropsychological Criteria for Mild Cognitive Impairment Improves Diagnostic Precision, Biomarker Associations, and Progression Rates[END_REF]. They consistently revealed an amnestic subtype (impaired memory), a language or dysnomic subtype (impaired language), and a mixed subtype (impaired memory, executive function, attention, verbal fluency, and visuospatial function). It should be noted that, in some studies, the dysexecutive subtype is distinguished from the mixed subtype, with memory being affected only in the latter one; while, in other studies, the mixed subtype is alternately labelled 'dysexecutive' or 'mixed' depending on the authors, even when referring to a subgroup with substantial impairment in overall cognitive performance, including memory. In the present study, this specific group will be referred to as 'mixed MCI'. Interestingly, the mixed MCI subtype has been repeatedly reported to have a higher rate of progression to AD dementia than the other subtypes.

More recently, this finding has been consolidated by studies that empirically derived the exact same subtypes (i.e, amnestic and mixed) using cluster analysis performed on neuropsychological test data [START_REF] Machulda | Neuropsychological Subtypes of Incident Mild Cognitive Impairment in the Mayo Clinic Study of Aging[END_REF][START_REF] Junquera | Clustering Executive Functions Yields MCI Profiles That Significantly Predict Conversion to AD Dementia[END_REF][START_REF] Blanken | Distilling Heterogeneity of Mild Cognitive Impairment in the National Alzheimer Coordinating Center Database Using Latent Profile Analysis[END_REF][START_REF] Edmonds | Heterogeneous Cortical Atrophy Patterns in MCI Not Captured by Conventional Diagnostic Criteria[END_REF].

Several studies further characterized the above neuropsychologically-defined MCI subtypes in terms of their underlying AT(N) biomarkers, namely cerebrospinal fluid (CSF) beta amyloid deposition ('A') and pathologic tau ('T'), and neurodegeneration ('N') as assessed from structural MRI. The objective using the AT(N) framework for AD research [START_REF] Jack | A/T/N: An Unbiased Descriptive Classification Scheme for Alzheimer Disease Biomarkers[END_REF] was to better understand the potential etiologic distinctions underlying the MCI subtypes. Overall, patterns of grey matter atrophy among the MCI subtypes were found to correspond to their profiles of cognitive impairment. Amnestic MCI individuals were reported to have smaller hippocampi [START_REF] He | Differences in Brain Volume, Hippocampal Volume, Cerebrovascular Risk Factors, and Apolipoprotein E4 Among Mild Cognitive Impairment Subtypes[END_REF]. Medial temporal lobe thinning was found in both the amnestic and dysnomic subtypes [START_REF] Whitwell | Patterns of Atrophy Differ Among Specific Subtypes of Mild Cognitive Impairment[END_REF]Edmonds et al. 2016). Lateral temporal lobe atrophy was also found in the dysnomic subtype. A widespread pattern of grey matter atrophy spanning parietal, temporal, and frontal regions was reported in the mixed MCI subtype [START_REF] Dickerson | Dysexecutive versus Amnesic Phenotypes of Very Mild Alzheimer's Disease Are Associated with Distinct Clinical, Genetic and Cortical Thinning Characteristics[END_REF][START_REF] Edmonds | Heterogeneous Cortical Atrophy Patterns in MCI Not Captured by Conventional Diagnostic Criteria[END_REF]). Regarding CSF biomarkers (i.e., p-tau and Aβ 1-42 level), the mixed subtype showed a greater proportion of individuals with positive CSF AD biomarkers than the dysnomic and amnestic subtypes [START_REF] Edmonds | Susceptibility of the Conventional Criteria for Mild Cognitive Impairment to False-Positive Diagnostic Errors[END_REF]. In sum, these results tend to support the idea that MCI subtypes are rather homogeneous in terms of their biological and cerebral injury biomarkers.

A recent study by Kwak and colleagues (2021) addressed the opposite question as to whether heterogeneity in brain atrophy patterns of MCI individuals could allow identification of biologically and clinically meaningful subgroups. They reported one MCI subgroup in which the pattern of brain atrophy resembled that of AD patients (MCI-AD) and another MCI subgroup in which grey matter was similar to that of healthy individuals (MCI-CN). The rate of progression to AD for the MCI-AD subgroup was higher than for the MCI-CN. In terms of biological features, they reported marked differences between MCI-AD and MCI-CN subgroups, including especially more elevated tau and beta-amyloid burden in MCI-AD compared to MCI-CN. On the other hand, they found only a limited degree of overlap between these two MRI-derived (atrophy-centered) subgroups and those empirically derived from neuropsychological test scores, including the amnestic, dysnomic and mixed ones. Thus, whether or not neuropsychological profiles of patients with MCI correspond to real distinct biological subtypes is still an open question.

In the present study, we pursue the question of the correspondence between MCI subtypes derived from neuropsychological assessment and their underlying patterns of neurodegeneration and CSF biomarker composition. For this purpose, using the ADNI data (640 MCI individuals and 326 healthy controls), we investigated the accuracy with which brain (i.e. grey matter) atrophy on the one hand and CSF beta amyloid and tau levels on the other hand, can predict neuropsychological subtypes of MCI. If predictive models derived from AT/N biomarkers perform well in classifying neuropsychological profiles of MCI, then such findings will provide compelling evidence of concordance between neuropsychological and neurobiological subtypes. More broadly, the study will provide valuable information about the neuropsychological and neurobiological fingerprintings of MCI, and, by extension, about the need (or not) to profile patients on the basis of multi-modal assessments.

METHODS

Participants

Data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). Written informed consent was obtained from all participants or authorized representatives participating in the study. For more information, including criteria for eligibility, see http://www.adni-info.org. To be included in this work, each participant must have a status of mild cognitive impairment (MCI) or cognitively normal (CN). CN participants showed no signs of depression, mild cognitive impairment, or dementia. ADNI criteria for MCI were : i) subjective memory concern as reported by the subject, their study-partner or clinician, ii) abnormal memory function documented by scoring within education-adjusted ranges on delayed free recall of Story A from the WMS-R Logical Memory II subtest, iii) Mini-Mental State Examination (MMSE) score between 24 and 30, iv) global Clinical Dementia Rating (CDR) score of 0.5, with a Memory Box score of at least 0.5, and v) general cognition and functional performance sufficiently preserved so that a diagnosis of AD could not be made. Included participants must also have an exploitable T1 scan (i.e., the image successfully passed the preprocessing steps as well as visual quality assessment), an exploitable level of CSF biomarkers (no missing or not-a-number quantity), as well as an exploitable score on each questionnaire used in our study (no missing or not-a-number scores). A total of 966 participants met these conditions and thus were included in our study. See table 1 for more information.

Neuropsychological assessments

All MCI subjects in ADNI underwent a neuropsychological assessment at baseline (visit at one month from the screening in the ADNI protocol). The ADNI database provided the raw results of this assessment. For our study, we selected a list of neuropsychological tests according to two criteria : i) the test scores must not be missing and be a valid value, and ii) the test scores must have been used in previous studies using clustering (L. Q. [START_REF] Park | Confirmatory Factor Analysis of the ADNI Neuropsychological Battery[END_REF][START_REF] Edmonds | Early versus Late MCI: Improved MCI Staging Using a Neuropsychological Approach[END_REF]) in order to allow comparison of results. Neuropsychological test scores meeting these two criteria were included in our analysis. These tests included three measures of language : Animal Fluency Test, Boston Naming Test, Naming Object and Fingers Task of the Alzheimer's Disease Assessment Scale-Cognitive Subscale / ADAS-Cog [START_REF] Rosen | A New Rating Scale for Alzheimer's Disease[END_REF], two measures of executive function : Trial Making Test: score A and score B minus A, two measures of visuo-spatial ability : Constructional Praxis Task and Ideational Praxis Task of the ADAS-Cog, and seven measures of memory : Word Recognition Task of the ADAS-Cog, Logical Memory Scale II [START_REF] Chelune | The Wechsler Memory Scale-Revised[END_REF], and short delayed recall, long delayed recall, recognition, learning and forgetting items of the Rey Auditory Verbal Learning Test -RAVLT- [START_REF] Rey | L'examen Clinique En Psychologie[END_REF]. Neuropsychological test scores, for which a lower score represents better performance, were multiplied by minus one, so that a higher score represents better performance. All scores were then transformed into z-scores by mean centering and unit-variance scaling.

Image acquisition

Processing: The structural brain image was acquired for all participants (n=966) with an anatomical 3D T1-weighted MPRAGE sequence. The sequence specifications of ADNI 1 session were TR = 3000 ms, TE = 3.6 ms, FoV = 192 × 192 mm2, flip angle = 8°, voxel resolution = 1.3 × 1.3 ×1.3 mm 3 , and for ADNI 2 session TR = 2300 ms, TE = 3 ms, FoV = 256 × 256 mm 2 , flip angle = 9°, voxel resolution = 1 × 1 × 1 mm 3 . The brain tissue was segmented into grey matter, white matter and cerebrospinal fluid. (Pedregosa et al., 2011, using the argument 'cortmaxprob-thr25-2mm'), quantitative measures of grey matter volume were extracted within the 96 macroscopic brain structures labeled in this atlas in every participant. For the extraction of relevant signal from the structural brain data, the total of 96 regions served as topographic masks to sum the volume information across the voxels belonging to a given region. All region-wise structural grey matter volumes were transformed into z-scores by mean centering and unit-variance scaling.

Structural

Variance explained by total intracranial volume (TIV) and age were regressed out based on a glm approach [START_REF] Friston | Statistical Parametric Maps in Functional Imaging: A General Linear Approach[END_REF].

Cerebrospinal fluid (CSF) biomarkers collection

The ADNI database provided the raw CSF levels of amyloid ß plaques (Aß 1-42 ), total tau (tau) and tau phosphorylated at threonine 181 (ptau). In this work, these 3 biomarkers levels were recorded for each included participant. They were selected according to the AT(N) framework which was proposed to differentially assess the likelihood of progression to AD dementia at the MCI stage. "A" refers to βamyloid deposition (Aβ 1-42 ), "T" refers to pathologic tau, and "N" to neurodegeneration [START_REF] Jack | A/T/N: An Unbiased Descriptive Classification Scheme for Alzheimer Disease Biomarkers[END_REF]. More details of the CSF collection and measurements in the ADNI can be found in Shaw and colleagues (2009). All biomarkers were transformed into z-scores by mean centering and unitvariance scaling.

Identifying hidden group structure: hierarchical clustering

We applied a hierarchical clustering algorithm (agglomerative) to automatically partition patient neuropsychological profiles into homogeneous groups using the standardized (z-scored)

neuropsychological scores from all MCI participants (n = 640). Hierarchical clustering is a general family of clustering algorithms that build nested clusters by merging or splitting them successively [START_REF] Kärkkäinen | Structural Brain Imaging Phenotypes of Mild Cognitive Impairment (MCI) and Alzheimer's Disease (AD) Found by Hierarchical Clustering[END_REF]. This hierarchy of clusters is represented as a tree (or dendrogram, see Figure 1). The root of the tree is the unique cluster that gathers all the samples, the leaves being the clusters with only one sample. Here, agglomerative clustering was performed using a bottom up approach: each observation starts in its own cluster, and clusters are successively merged together.

The metric used for the merge strategy was the sum of squared differences within all clusters (Ward's method), which here was minimized. It is a variance-minimizing approach and in this sense is similar to the k-means objective function but tackled with an agglomerative hierarchical approach. In contrast to previous approach, agglomerative clustering is a method identifying one-to-many mappings [START_REF] Bzdok | Inference in the Age of Big Data: Future Perspectives on Neuroscience[END_REF]: each patient is a member of exactly one group. We used "NbClust" [START_REF] Charrad | NbClust: An R Package for Determining the Relevant Number of Clusters in a Data Set[END_REF], an established R package that simultaneously applied 30 cluster validity metrics. This approach provided complementary indications on the number of groups most supported by the patient data. That is, several clustering schemes were evaluated while varying the number of clusters, to help determining the most appropriate number of clusters for our dataset. These metrics included for example the Duda index, the C-index, and the Gamma index. Please, see the reference above for the full list of metrics. Among the 30 metrics and according to the majority rule, the best number of clusters was 3. Therefore, three groups of patients with distinct neuropsychological profiles were automatically extracted as it provided a useful fit to our clinical sample.

Risk ratio of developing Alzheimer

Risk ratio (RR) was used to assess the risk of developing Alzheimer disease in each extracted MCI subgroups compared to controls. RR was defined as : RR = Cle / Clu where Cle is the cumulative incidence in the exposed group (i.e., each MCI subgroup), and Clu is the cumulative incidence in the unexposed goup (i.e., the control group).

Machine learning prediction of cluster membership from structural brain measures

The relative importance of grey matter volumes to predict membership in each MCI cluster versus control group was analyzed capitalizing on a pattern-learning algorithm L2-penalized logistic regression [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF]. Unlike the common logistic regression, the L2-penalized logistic regression variant has an additional constraint used to reduce the chances of overfitting, which can render the models' prediction of future observation unreliable. The L2-penalized logistic regression estimated the separating hyperplane (i.e., a linear function) yielding out-of-sample accuracies for distinguishing between MCI patients of each cluster and healthy participants. Model-fit and accuracy estimation were carried out as a 5-fold cross-validation procedure. Class imbalance, if present, was handled by changing the class-weight of the scikit-learn logistic regression API. The "balanced" mode uses the class membership to automatically adjust weights inversely proportional to class frequencies. The outcome to be predicted was defined by being healthy (0) or being an MCI patient from one of the three extracted clusters (1). In other words, three models were adjusted using grey matter volumes as input with a first model predicting cluster-derived normal versus controls, a second model predicting amnestic MCI versus controls, and a third one predicting mixed MCI versus controls. This way of engineering transformed a four-class problem into 3 two-class problems. In sum, this quantitative investigation detected if grey matter volume would be predictive of cluster belonging.

Machine learning prediction of cluster membership from CSF biomarkers measures

In order to allow for results comparison, the same algorithm was used in the previous setting and this one. This time, the L2-penalized logistic regression used three CSF biomarkers level (Aß 1-42 , t-tau and p-tau p181 ) as feature input to estimate the separating hyperplane for distinguishing between MCI patients of each cluster and healthy participants. Again, we deployed a 5-fold cross-validation procedure and handled class-imbalance if present. The outcome to be predicted were exactly the same as in the previous setting. That is, being healthy (0) or being an MCI patient from one of the three extracted clusters (1). Thus, three models were adjusted using CSF biomarkers level as input with a first model predicting cluster-derived normal versus controls, a second model predicting amnestic MCI versus controls, and a third one predicting mixed MCI versus controls. In sum, this quantitative investigation detected if CSF biomarkers level would be predictive of cluster belonging.

Testing for significance

Three models based on grey matter volume and three other models based on CSF biomarkers level were conducted separately. Statistical significance for weights in each of the 6 final models was assessed based on (family wise error, multiple-comparison corrected) p-values derived through a rigorous non-parametric permutation approach using the model weights as the test statistic [START_REF] Efron | Large-Scale Inference: Empirical Bayes Methods for Estimation[END_REF][START_REF] Nichols | Nonparametric Permutation Tests for Functional Neuroimaging: A Primer with Examples[END_REF]. Relying on minimal modeling assumptions, a valid null distribution was derived for the achieved weights resulting from the logistic regression fit. In 1000 permutation iterations, the input feature matrix (consecutively brain regions volume and CSF biomarkers level)

was held constant, while the class membership (control versus each clusters) underwent participantwise random shuffling. The empirical distribution generated in this manner reflected the null hypothesis of random association between the input features and class membership across participants. The beta coefficients were recorded in each iteration. The p values were obtained given the distance between the original beta values and the mean beta values obtained during the permutation iterations.

Testing for complex relationships among the grey matter volumes L2-penalized logistic regression (cf. above) selected the most predictive ROI volumes for cluster membership. But this predictive algorithm is constrained to estimate additive effects between ROI volumes. To complement the regression model insights, we combined exploration of more sophisticated ROI-ROI relationships with the evaluation of prediction performance. In this way, we tested the hypothesis of existing non-linear relationships between the ROI volumes and their usefulness for prediction.

The goal here was to assess that a non-linear model would not reach a higher accuracy than the linear models in predicting mixed MCI cluster compared to controls. Note that we reported findings of this analysis only in one setting. That is, input features defined as grey matter volumes, and class defined as whether controls or MCI subjects belonging to the mixed MCI subgroup. given that in the main analysis (using logistic regression), only the accuracy between mixed MCI and controls was significant. However, all the other results can be found in the additional (Supplementary Figure 1).

We thus compared the performance of linear models to the performance of models able to exploit non-linear structure in the ROI volumes for predicting mixed MCI versus control. Model-fit and accuracy estimation were carried out using a 5-fold cross-validation procedure as implemented in the previous analysis. Class imbalance was again handled by changing the class-weight of the scikit-learn model API. Three linear models (ridge regression, logistic regression, and support vector machine)

were benchmarked against three models allowing looking for non-linear interactions (k nearest neighbor, random forest and adaptive boosting). Note that we used the default regularization parameter for each algorithm.

Among the linear predictive pattern-learning algorithms, the ridge regression is commonly used as a shrinkage method. This model encourages small absolute weights on each ROI volume which emphasized the most predictive ROI volumes, while linear support vector machines mapped the ROI volumes as points in space so that volumes of separate categories (predictive of a mixed MCI or a control participant) were divided by a maximal gap between the individuals. Given that similar predictive ROIs were expected to emerge from these linear models, we also compared the obtained coefficients in a supplementary analysis.

Regarding the non-linear models, the k nearest neighbor estimator uses the k closest training examples (in our case, the closest participant brain structure) in the feature space, while the output is determined by a majority vote across these most similar training examples. In other words, the guessed class belonging (mixed MCI or control) for a given new participant can thus be derived from the brain structure of the k closest participants in the training set. Further, the random forest algorithm is an ensemble learning method that operates by constructing a multitude of decision trees and outputs its prediction estimate that is the committee decision across all trees. The cluster belonging for a given patient was thus derived based on the most consistently predicted outcome of the built decision trees. As the last non-linear prediction algorithm, the adaptive boosting algorithm starts by fitting a model on the dataset and then fits additional copies of that model on the same dataset but where the weights of incorrectly judged instances are adjusted such that subsequent fine-tuning of models focusses more on difficult cases.

RESULTS

Identifying hidden group structure: hierarchical clustering

To explore distinct subgroups related to cognitive test assessment patterns among MCI patients, each patient was automatically assigned to one dominant symptom constellation based on a number of cognitive tests. This data-driven exploration exposed 3 distinct symptom clusters (see Figure 1a and 1).

Repartition of MCI patients developing Alzheimer

We examined the association between cluster memberships and AD. Compared to controls, only the amnestic MCI and mixed MCI subtypes exhibited a higher propensity to develop Alzheimer with twice as much risk for the mixed MCI subgroup (See Figure 2). The risk ratios for the amnestic and mixed MCI subgroup were respectivelly 4.38 (p < 0.01) and 7.52 (p < 0.01).

MCI cluster prediction based on grey matter volume

We explored the hypothesis that grey matter volume may predict affiliation to MCI subgroups. A regularized logistic regression was used to automatically identify regions of interest (ROI) with a high discriminant value for distinguishing controls from each MCI subgroup (see Figure 3a and3c). Our analysis strategy revealed that only the mixed MCI subgroup was distinguishable from controls using grey matter volume. The mean accuracy of the averaged GMV models, incorporating only structural MRI data, was 72.13% with a standard error of 3.66%. There were 8 ROIs (p<0. parahippocampal gyrus posterior (see Figure 4a).

MCI cluster prediction based on CSF biomarkers level

We then analyzed the relative importance of the level of Amyloid-β 1 to 42 peptide (Aβ 1-42 ), total tau (Tau), and tau phosphorylated (PTau) for distinguishing controls from each MCI subgroup (see Figure 3a and 3b). Our findings indicated a significant prediction accuracy for discriminating both the mixed MCI (71.60% +/-4.67%) and amnestic MCI (63.42% +/-5.19%) subgroup from controls. However, our model did not perform better than chance to distinguish controls from the cluster-derived normal subgroup. Only the weight associated with the level of Aβ 1-42 (coefficient = 0.40) was significant in predicting amnestic MCI patients while both the level of Aβ 1-42 (coefficient = 0.69) and Tau (coefficient = -0.75) were significant in predicting mixed MCI patients (see Figure 4b).

Testing for complex relationships among ROI volumes to distinguish mixed MCI from controls

Here, we wanted to make sure that a regularized logistic regression was more appropriate to our research setting than a model looking for non-linear effects. To test the hypothesis of existing nonlinear interaction between the ROI volumes, we compared the prediction performance of mixed MCI compared to controls of different linear models (ridge regression, logistic regression, and support vector machine) to the prediction performance of different non-linear ones (k nearest neighbor, random forest, and adaptive boosting) (Fig. 5). Furthermore, we directly compared significant ROI weights obtained with the three linear models.

The three linear models -support vector machine, logistic regression, and ridge regressionobtained on average a similar performance with respectively 71.12% (+/-3.24), 70.96% (+/-3.64), and 72.42% (+/-3.28) accuracy. Note that the tiny difference between our reported mean accuracy with logistic regression in the main analysis (72.13%) and the mean accuracy of the logistic regression in the benchmark analysis (70.96%) might be explained by the randomness in the cross validation split procedure (training-testing sets). On the other hand, the models looking for non-linear interactions-k nearest neighbor, random forest and adaptive boosting-obtained on average 69.84% (+/-4.34), 70% (+/-3.24), and 67.74% (+/-4.53).

As a general observation, the mean accuracy for the linear models (71.5%) was on average higher than the mean accuracy for the non-linear models (69.19%) in our sample. Furthermore, the variance was higher for the non-linear model performances (average standard deviation: 4.21%) than within the linear model performances (average standard deviation: 3.45%). These results suggest that the ROI volumes are predominantly predictive for mixed MCI based on their additive effects. However, it is important to note that our claim might be limited to the size of our sample. Indeed, non-linear models such as adaptive boosting might keep learning and thus may predict better with more data involved in the fitting.

All examined linear models showed virtually identical prediction performance. As a next step, we also wanted to evaluate the similarity of results (i.e, significant ROI weights) obtained between these three benchmarked linear models. Three ROI weights were systematically significant (Table 2). These ROI included the left posterior middle temporal gyrus; the left anterior parahippocampal gyrus, and the left occipital pole. Four ROI weights were reported as significant by the logistic regression and the support vector machine including the right cuneal cortex, the right anterior parahippocampal gyrus, the right posterior parahippocampal gyrus, and the left occipital fusiform gyrus. In other words, the three linear models capitalized on virtually similar ROIs to discriminate mixed MCI from control individuals.

DISCUSSION

Our study uncovered three partitions of discrete neuropsychologically-based MCI profiles. The first extracted MCI profile was similar to controls in terms of grey matter volumes, CSF biomarker levels, neuropsychological tests scores, as well as risk of developing Alzheimer's disease. The two other extracted MCI profiles showed regional grey matter volume reductions and abnormal CSF biomarker levels, allowing their discrimination from healthy individuals, and were also more at risk of developing Alzheimer's disease. These results support the conclusion that MCI subtypes derived from neuropsychological test scores have relatively clear biological -grey matter volume and CSF features -boundaries.

Subtyping of the MCI individuals using neuropsychological test scores

Our clustering method revealed two distinct, clinically meaningful, subgroups of MCI patients: a mixed MCI profile with low performance on memory, language, executive functioning, and visuospatial function, and an amnestic MCI profile with memory being the only impaired domain. A third profile also came out, with a neuropsychological profile similar to healthy participants. In general, these latent profiles are consistent with those reported in a number of previous studies that also applied clustering methods on a standardized set of neuropsychological tests measuring multiple domains of cognitive functioning [START_REF] Eppig | Statistically-Derived Subtypes and Associations with Cerebrospinal Fluid and Genetic Biomarkers in Mild Cognitive Impairment: A Latent Profile Analysis[END_REF][START_REF] Bondi | Neuropsychological Criteria for Mild Cognitive Impairment Improves Diagnostic Precision, Biomarker Associations, and Progression Rates[END_REF][START_REF] Edmonds | Susceptibility of the Conventional Criteria for Mild Cognitive Impairment to False-Positive Diagnostic Errors[END_REF][START_REF] Blanken | Distilling Heterogeneity of Mild Cognitive Impairment in the National Alzheimer Coordinating Center Database Using Latent Profile Analysis[END_REF]. However, there are also studies that revealed additional profiles to the above-mentioned core MCI profiles, including dysexecutive, visuo-spatial or dysnomic profiles (Clark et al. 2013;[START_REF] Edmonds | Subtle Cognitive Decline and Biomarker Staging in Preclinical Alzheimer's Disease[END_REF][START_REF] Edmonds | Heterogeneous Cortical Atrophy Patterns in MCI Not Captured by Conventional Diagnostic Criteria[END_REF][START_REF] Kwak | Subtyping of Mild Cognitive Impairment Using a Deep Learning Model Based on Brain Atrophy Patterns[END_REF]. Factors that can explain such variations in the profiles are the criteria used to define MCI (prior to the clustering analysis) as well as the set of neuropsychological test scores included in the cluster analysis. For instance, in addition to the amnestic, mixed and cluster derived normal profiles, Clark and colleagues (2013) also reported dysexecutive and visuo-spatial subtypes. However, in their study, to be included as MCI was not based on the conventional diagnostic criterion (as in our present study), but instead on a specific criterion that required low performances on at least two measures within a cognitive domain. In addition, they used items from the Wechsler Intelligence Scale and Wechsler Memory Scale while we used items from the ADAS-cog for assessing visuospatial functioning. Likewise, studies that reported dysnomic MCI subtype assessed language from animal fluency and 30-items Boston Naming Test [START_REF] Kwak | Subtyping of Mild Cognitive Impairment Using a Deep Learning Model Based on Brain Atrophy Patterns[END_REF][START_REF] Edmonds | Susceptibility of the Conventional Criteria for Mild Cognitive Impairment to False-Positive Diagnostic Errors[END_REF], while we further included the Naming Object and Fingers Task of the ADAS-cog. An additional factor that may explain discrepancies between studies is the stability of the chosen clusters. We used multiple distance metrics (n=30, through the nbclust R package) to assess the most stable number of clusters in our sample while a single metric is usually chosen in other studies. Accordingly, we are confident that the choice of three clusters was the most consistent and optimal solution to get nonoverlapping homogeneous groups. It is noteworthy that the higher risk of Alzheimer's dementia observed in the mixed MCI subgroup^p compared to the amnestic MCI subgroup and the normal risk level of the cluster-derived normal subgroup provided clinical validity to this clustering scheme. From a the clinical standpoint, the existence of these MCI subtypes illustrates the problem of diagnosing individuals on the basis of a single test in the memory domain, here the WMS-R Logical Memory Test in the ADNI study. First, it places side by side individuals with memory deficits only and individuals with multi-domain cognitive deficits, who are at different risk of progression to dementia. Second, it leads to false positive MCI diagnoses. Accordingly, and in line with previous recommendations (eg., Thomas, Eppig, et al. 2019;Edmonds, Delano-Wood, Clark, et al. 2015a;[START_REF] Jak | Quantification of Five Neuropsychological Approaches to Defining Mild Cognitive Impairment[END_REF], MCI diagnosis should include a multi-domain neuropsychological assessment and avoid the 'one test equals one domain' methodology.

Predictive value of regional grey matter volume to distinguish MCI patients from healthy individuals

We automatically assessed the extent to which each MCI subgroup could be differentiated from healthy participants based on regional grey matter volumes. Significant accuracy (72%) was obtained only for predicting the mixed MCI subgroup compared with the healthy participants. This finding suggests that the amnestic MCI subgroup and the cluster-derived normal subgroup have a brain structure more similar to healthy participants. Whereas the similarity of regional grey matter volumes in the cluster-derived normal MCI subtype and in the cognitively normal group confirms the conclusion of previous studies drawn from cortical thickness [START_REF] Edmonds | Heterogeneous Cortical Atrophy Patterns in MCI Not Captured by Conventional Diagnostic Criteria[END_REF]2020;[START_REF] Blanken | Disentangling Heterogeneity in Alzheimer's Disease: Two Empirically-Derived Subtypes[END_REF]L. R. Clark et al. 2013), that between the amnestic MCI subgroup and healthy participants may appear surprising. Edmonds and colleagues (2016;2020) found cortical differences between these two populations (i.e., amnestic MCI and controls) in the medial and lateral temporal lobe regions bilaterally as well as in some parietal and frontal regions. Machulda and colleagues (2020) also found differences in the medial temporal regions. Sun and colleagues (2019) reported decreased cortical thickness in medial orbitofrontal, parahippocampal and precuneus in amnestic MCI individuals. The discrepancy between these findings and ours is presumably due to difference in the methodology. Indeed, previous research focused on differences in brain structure in an explanatory fashion (i.e., modeling for inference using statistical significance) whereas in our study, we sought to find predictive patterns (i.e., modeling for prediction using cross-validation). In particular, there is evidence that successful prediction is often associated with a significant p-value, but not vice versa [START_REF] Bzdok | Inference and Prediction Diverge in Biomedicine[END_REF]. Hence, previous brain structure impairments reported in amnestic MCI individuals may have rather poor predictive performance. Accordingly, brain structure should not be regarded as an indicator of main importance to detect amnestic MCI. This proposal is further supported by other studies, albeit with a rather small sample size (respectively 49 and 29 amnestic MCI), that used an explanatory approach and found no differences in brain structure between amnestic MCI individuals and controls [START_REF] Xue | Structural and Functional Disruption of Salience Network in Distinguishing Subjective Cognitive Decline and Amnestic Mild Cognitive Impairment[END_REF][START_REF] Yang | Study of Brain Morphology Change in Alzheimer's Disease and Amnestic Mild Cognitive Impairment Compared with Normal Controls[END_REF].

Regarding mixed MCI, a total of 8 ROIs with decreased grey matter volume significantly contributed to the prediction performance. These ROIs included 3 regions from the occipital lobe, namely the left occipital fusiform gyrus, the left occipital pole, and the right cuneal cortex, and 5 regions from the temporal lobe including the right and left, anterior and posterior parahippocampal gyrus, and the left middle posterior temporal gyrus. Note that the weights of 3 of these 8 ROIs (the left middle posterior temporal gyrus, the left anterior parahippocampal gyrus, and the left occipital pole) were systematically significant across the linear model benchmark analysis, suggesting a more robust predictive value for these 3 ROIs. Hence, atrophy in temporal and occipital regions had predictive value for delineating mixed MCI individuals from healthy participants. While widespread atrophy of temporal regions is a typical finding in mixed MCI [START_REF] Edmonds | Patterns of Longitudinal Cortical Atrophy over 3 Years in Empirically Derived MCI Subtypes[END_REF]2016;[START_REF] Machulda | Cortical Atrophy Patterns of Incident MCI Subtypes in the Mayo Clinic Study of Aging[END_REF][START_REF] Kwak | Subtyping of Mild Cognitive Impairment Using a Deep Learning Model Based on Brain Atrophy Patterns[END_REF][START_REF] Johnson | Baseline Predictors of Clinical Progression among Patients with Dysexecutive Mild Cognitive Impairment[END_REF][START_REF] Fernández | Patterns of Brain Atrophy in Dysexecutive Amnestic Mild Cognitive Impairment Raise Confidence about Prodromal AD Dementia[END_REF][START_REF] Ghosh | Mild Cognitive Impairment: A Brief Review and Suggested Clinical Algorithm[END_REF], occipital regions are usually only marginally affected in these individuals. Indeed, it is generally accepted so far that atrophy of the occipital cortex is characteristic of the later stages of Alzheimer's disease (Braak & Braak, 1991). Furthermore, impaired perfusion of the occipital lobe was proposed as a determining marker of dementia with Lewy bodies but not really of Alzheimer's disease [START_REF] Hanyu | Differentiation of Dementia with Lewy Bodies from Alzheimer's Disease Using Mini-Mental State Examination and Brain Perfusion SPECT[END_REF][START_REF] Prosser | Occipital Lobe and Posterior Cingulate Perfusion in the Prediction of Dementia with Lewy Body Pathology in a Clinical Sample[END_REF]. Hence, a striking and novel result of our study is that grey matter volume in occipital cortex is affected as early as the MCI stage. Interestingly, our findings go well with a recent conclusion that loss of grey matter integrity in the lateral and medial temporal lobes as well as in the occipital lobe is responsible of cognitive decline in vulnerable individuals that suffer the deleterious effects of elevated brain amyloid and poor vascular health [START_REF] Saboo | Deep Learning Identifies Brain Structures That Predict Cognition and Explain Heterogeneity in Cognitive Aging[END_REF]. Hence, atrophy of the temporal and occipital lobes may be very valuable marker of cognitively vulnerable individuals. On the other hand, the above-mentioned studies on mixed MCI reported significant grey matter loss in parietal and frontal regions, which were not found to be particularly predictively relevant in our study. Note that we emphasized our discussion on ROIs with the highest and most robust weights as automatically optimized by the model (i.e., L2-penalized logistic regression). However, it is important to keep in mind that the model chose to shrink a ROI coefficient because it brings little or no additional information on top of the other ROIs. Therefore, ROIs with small weights may still be related to the outcome. Nevertheless, the 8 ROIs found to have a significant weight in our study carried a substantial impact on distinguishing mixed MCI subjects from controls.

Predictive value of CSF biomarkers to distinguish MCI patients from healthy individuals

CSF biomarkers were useful to significantly differentiate (72% accuracy) between mixed MCI patients and healthy participants. In particular, the weights associated to the concentration of Aβ 1-42 and concentration of total tau were significant. That is, these two features were repeatedly informative for telling apart both groups. In patients, the concentration of Aβ 1-42 was lower while the concentration of total tau was higher compared to controls. CSF biomarkers were also effective to significantly distinguish amnestic MCI from healthy individuals (63% accuracy). This time, only the weight associated with the concentration of Aβ 1-42 was significant, suggesting that the concentration of Aβ 1-42 was the most contributing feature for the prediction. Overall, these results are in line with several previous studies that have examined biomarker characteristics in empirically derived subtypes of MCI and concluded that MCI patients with amnestic or executive symptoms have amyloid brain pathology and neuronal injury [START_REF] Bangen | Cortical Amyloid Burden Differences Across Empirically-Derived Mild Cognitive Impairment Subtypes and Interaction with APOE Ε4 Genotype[END_REF][START_REF] Edmonds | Heterogeneous Cortical Atrophy Patterns in MCI Not Captured by Conventional Diagnostic Criteria[END_REF]2021;[START_REF] Eppig | Statistically-Derived Subtypes and Associations with Cerebrospinal Fluid and Genetic Biomarkers in Mild Cognitive Impairment: A Latent Profile Analysis[END_REF]Thomas, Eppig, et al. 2019;[START_REF] Edmonds | Susceptibility of the Conventional Criteria for Mild Cognitive Impairment to False-Positive Diagnostic Errors[END_REF]. Indeed, low CSF Aβ 1- 42 level and high CSF tau level are strong predictors of the presence of pathologic amyloid plaques and neurofibrillary abnormalities in the brain [START_REF] Tapiola | Cerebrospinal Fluid β-Amyloid 42 and Tau Proteins as Biomarkers of Alzheimer-Type Pathologic Changes in the Brain[END_REF]). An important outcome of our work is that total tau was found to be a significantly informative feature to separate mixed MCI, but not amnestic MCI, from controls. Accordingly, both amnestic and mixed MCI subtypes would exhibit amyloid pathology while only the mixed subtype would have disrupted neuronal activity. This conclusion is also supported by a clinical interpretation of the concentrations of CSF Aβ 1-42 and total tau observed in our sample. In both subgroups, CSF Aβ 1-42 concentration was less than the cutoff of 192 pg/ml that is commonly used to identify the presence of amyloid pathology [START_REF] Shaw | Cerebrospinal Fluid Biomarker Signature in Alzheimer's Disease Neuroimaging Initiative Subjects[END_REF]).

On the other hand, the cutoff of 93 pg/ml, which identifies disruption of neuronal activity [START_REF] Shaw | Cerebrospinal Fluid Biomarker Signature in Alzheimer's Disease Neuroimaging Initiative Subjects[END_REF], was exceeded in the mixed subgroup only. Finally, it is also important to draw attention to the fact that above and beyond the above-mentioned impaired levels of CSF Aβ 1-42 and total tau in the MCI subgroups, both MCI subgroups were at higher risk to develop AD. This suggests a link between CSF biomarkers and conversion to AD, as pointed out in earlier studies [START_REF] Hansson | Association between CSF Biomarkers and Incipient Alzheimer's Disease in Patients with Mild Cognitive Impairment: A Follow-up Study[END_REF][START_REF] Mattsson | CSF Biomarkers and Incipient Alzheimer Disease in Patients With Mild Cognitive Impairment[END_REF][START_REF] Insel | Amyloid Pathology in the Progression to Mild Cognitive Impairment[END_REF][START_REF] Ortega | Usefulness of CSF Biomarkers in Predicting the Progression of Amnesic and Nonamnesic Mild Cognitive Impairment to Alzheimer's Disease[END_REF][START_REF] Park | Cerebrospinal Fluid Biomarkers for the Diagnosis of Prodromal Alzheimer's Disease in Amnestic Mild Cognitive Impairment[END_REF].

MCI subgroups as distinct MCI phenotypes or distinct stages along the course of Alzheimer?

The mixed MCI subgroup was distinguished from healthy individuals through structural brain atrophy as well as CSF Aβ 1-42 and total tau abnormal levels, while the amnestic MCI subgroup was only separated from healthy controls through CSF Aβ 1-42 abnormal level. Importantly, mixed MCI individuals were at higher risk of conversion to Alzheimer disease than amnestic MCI individuals.

Overall, these findings do concur with the amyloid cascade model of AD progression in which Aβ pathology (as measured by CSF Aβ 1-42 or amyloid PeT) appears first, followed by tau pathology (measured by CSF tau), then neuronal loss (measured by MRI) and then clinical symptoms (Jack Jr et al. 2010;[START_REF] Jack | Amyloid-First and Neurodegeneration-First Profiles Characterize Incident Amyloid PET Positivity[END_REF]. This model has received strong support over the years [START_REF] Balsis | Statistical Model of Dynamic Markers of the Alzheimer's Pathological Cascade[END_REF][START_REF] Yasuno | An Evaluation of the Amyloid Cascade Model Using in Vivo Positron Emission Tomographic Imaging[END_REF][START_REF] Jack | Brain Beta-Amyloid Measures and Magnetic Resonance Imaging Atrophy Both Predict Time-to-Progression from Mild Cognitive Impairment to Alzheimer's Disease[END_REF][START_REF] Rossum | Injury Markers but Not Amyloid Markers Are Associated with Rapid Progression from Mild Cognitive Impairment to Dementia in Alzheimer's Disease[END_REF][START_REF] Broadhouse | Fronto-Temporal Functional Disconnection Precedes Hippocampal Atrophy in Clinically Confirmed Multi-Domain Amnestic Mild Cognitive Impairment[END_REF][START_REF] Han | A Theoretical Analysis of the Synergy of Amyloid and Tau in Alzheimer's Disease[END_REF][START_REF] Weiner | Impact of the Alzheimer's Disease Neuroimaging Initiative, 2004 to 2014[END_REF]X. Yang, Tan, and Qiu 2012;[START_REF] Nettiksimmons | Biological Heterogeneity in ADNI Amnestic Mild Cognitive Impairment[END_REF],

although not all findings align with it and alternative scenarios have emerged where Aβ deposition, tau pathology, neuronal degeneration and cognitive loss aligned in a narrow time sequence [START_REF] Edmonds | Subtle Cognitive Decline and Biomarker Staging in Preclinical Alzheimer's Disease[END_REF][START_REF] Braak | Intraneuronal Tau Aggregation Precedes Diffuse Plaque Deposition, but Amyloid-β Changes Occur before Increases of Tau in Cerebrospinal Fluid[END_REF]. Hence, through the prism of the amyloid cascade model, our MCI subgroups would rather represent distinct stages along the course of AD, the disease progressing from the amnestic stage to the mixed stage. However, the question of whether amnestic and mixed MCI subgroups merely reflect different stages along the course of AD or correspond to distinct MCI phenotypes could only receive a definite answer by examining longitudinal data from the two subgroups. This could hopefully be achieved in future research.

LIMITATIONS

It is important to note that data from the ADNI 1 were acquired from a 1.5 T scanner while data from the ADNI 2 were acquired from a 3T scanner. Finally, our results suggested that AD progression may start affecting memory and CSF biomarkers, followed by an alteration of brain structure and of the other cognitive functions. As such, multimodal neuropsychological subtyping, in addition to being clinically meaningful, is also biologically and neurologically meaningful. Furthermore, our results suggested that AD progression may start affecting memory and CSF biomarkers, followed by an alteration of brain structure and of the other cognitive functions. The brains (c) indicates the grey matter quantity average difference between controls and each MCI subgroups. The redder the area, the higher the atrophy compared to controls. As a general observation, a better performance was achieved when dissociating mixed MCI from the controls using grey matter volume, as well as when using CSF biomarkers level. However, the models could not segregate cluster-derived normal from controls using these modalities. The amnestic MCI subgroup was distinguishable from the controls based on CSF biomarkers level but not on grey matter volume. In sum, 8 rois passed the (Bonferroni corrected) threshold and thus had a significant coefficient in predicting mixed MCI versus controls based solely on grey matter volume. These rois are located within the temporal and occipital lobes. Controls had on average more grey matter than mixed MCI subjects in all these significant roi. We explored the hypothesis that more complex patterns may explain relationships between different ROI volumes and thus be helpful to distinguish mixed MCI from controls. We thus compared the predicting performance of models looking for additive effects (left side) to the prediction performance of models looking for non-linear effects (right side). The transparent orange violin plots display the in-sample accuracies (train set) while the plain orange plots display the generalization performance (test set). The width of the violins illustrates the density of the obtained performances.

For instance, the shape of the first plain orange violin plot on the left side (skinny on each end and wide in the middle) indicates that the obtained accuracies are highly concentrated around the median. The height of the violins indicates the variability (i.e., range of the obtained accuracies). Short violins represent a slight while long violins represent a substantial variability. Linear models including the ridge regression (Ridge L2), the logistic regression (LogReg L2), and the support vector machine (SVM L2) are plotted on the left side of the dashed bar. Non-linear models including the k nearest neighbor (kNN), the random forest (RandForest) and the adaptive boosting (AdaBoost) are plotted on the right side. As a general observation, the plain orange violin plots of the linear models indicate on average a better performance with less variance thus appear to be more adapted to this setting. These results suggest that the ROI volumes are perhaps mostly individually predictive as much as this evidence is supported by dataset

  05) that consistently contributed to predicting mixed MCI. These ROIs included the left occipital fusiform gyrus (weight = 0.73), the right (weight = 0.93) and left (weight = 1.11) parahippocampal gyrus anterior, the right cuneal cortex (weight = 0.58), the left middle posterior temporal gyrus (weight = 1.05), the left occipital pole (weight = -1.10), and the right (weight = -0.84) and left (weight = 0.58)

Figure 1 :

 1 Figure 1: automatic extraction of 3 MCI clusters Three subgroups were extracted from a cohort of 640 MCI patients. Polar plot (a) shows the Z score for each neuropsychological test included in the clustering procedure. The grey lines represent each extracted cluster (from darker to lighter: mixed MCI, amnestic MCI and cluster-derived normal) while the dotted black line represent the Z score of the controls. A higher score represents a greater performance. Dendrogram (b) shows the best clustering scheme, 3 subgroups according to 30 metrics, extracted from a hierarchical clustering based on a cohort of 640 MCI participants. In sum, within the MCI diagnosed participants, we could extract three very specific subtypes, the mixed MCI subtype scoring low at almost every test, the amnestic MCI subtype, scoring low mostly on test assessing memory, and the cluster-derived normal subtype, scoring mostly like controls except for the logical memory scale II.

Figure 2 :

 2 Figure 2: Risk ratio of MCI patients developing AlzheimerScatter plot (a) displays the participant first and second component of a PCA analysis on the neuropsychological tests included in our clustering analysis. Note that this PCA analysis was computed for the sake of visualization only, however, more details about this analysis can be found in the supplementary. The grey bar graphs (b) display the risk ratio of developing Alzheimer disease in each extracted subgroup compared to controls. The red dotted line represents a risk ratio similar to the control group risk ratio. These results exhibit a higher propensity to develop Alzheimer for MCI patients in the mixed MCI and amnestic MCI subgroups with twice as much risk for the mixed MCI subgroup.

Figure 3 :

 3 Figure 3: MCI cluster prediction based on grey matter or CSF levelWe explored the hypothesis that grey matter volume on the one hand, and level of Amyloid-β 1 to 42 peptide (Aβ42), total tau (tau), and tau phosphorylated (ptau) on the other hand may predict affiliation to MCI subgroups. Note that all subgroups were extracted based on neuropsychological test scores only, thus the CSF biomarkers and grey matter information did not take part into the cluster extraction procedure. Violin plots (a) display the generalization performance (test set) of the prediction using grey matter volume (blue) and CSF biomarker level (orange) between controls and each MCI subgroup. The width of the violins illustrates the density of the obtained performances. For instance, the shape of the first blue violin plot on the left side (skinny on each end and wide in the middle) indicates that the obtained accuracies are highly concentrated around the median. The height of the violins indicates the variability (i.e., range of the obtained accuracies). Short violins represent a slight while long violins represent a substantial variability. A non-parametric test was applied to assess the (Bonferroni corrected) significance of the accuracy, that is, to evaluate if such an accuracy could be obtained by chance alone. The significant accuracies are represented with a black star. The line graph (b) displays the mean (with the standard deviation) CSF biomarkers level (Aβ42, tau, and ptau) per MCI subgroups as well as in controls. The brains (c) indicates the grey matter quantity average difference between controls and each MCI subgroups. The redder the area, the higher the atrophy compared to controls. As a general observation, a better performance was achieved when dissociating mixed MCI from the controls using grey matter volume, as well as when using CSF biomarkers level. However, the models could not segregate cluster-derived normal from controls using these modalities. The amnestic MCI subgroup was distinguishable from the controls based on CSF biomarkers level but not on grey matter volume.

Figure 4 :

 4 Figure 4: Map of coefficient for the predictions of controls versus each MCI clusterPrediction of controls versus mixed MCI subgroup membership was assessed using grey matter volume (a) or CSF biomarkers level (b) and using regularized logistic regression models. The colormap on each glass brain (a) depicts the final coefficient value for each region of interest (roi). A nonparametric test was computed to assess significance of the coefficients. That is, to evaluate if a high coefficient was high only by chance or not. Significant rois are outlined in yellow. For each significant roi, boxplots of the distribution of grey matter volume per subjects for controls (pink) and mixed MCI (light green) are displayed. The heatmap (b) displays the final coefficient value for each CSF biomarker, with significant biomarkers outlined in yellow. In sum, 8 rois passed the (Bonferroni corrected) threshold and thus had a significant coefficient in predicting mixed MCI versus controls based solely on grey matter volume. These rois are located within the temporal and occipital lobes. Controls had on average more grey matter than mixed MCI subjects in all these significant roi.

Figure 5 :

 5 Figure 5: Probing complex relationships among the ROI volumes.We explored the hypothesis that more complex patterns may explain relationships between different ROI volumes and thus be helpful to distinguish mixed MCI from controls. We thus compared the predicting performance of models looking for additive effects (left side) to the prediction performance of models looking for non-linear effects (right side). The transparent orange violin plots display the in-sample accuracies (train set) while the plain orange plots display the generalization performance (test set). The width of the violins illustrates the density of the obtained performances. For instance, the shape of the first plain orange violin plot on the left side (skinny on each end and wide in the middle) indicates that the obtained accuracies are highly concentrated around the median. The height of the violins indicates the variability (i.e., range of the obtained accuracies). Short violins represent a slight while long violins represent a substantial variability. Linear models including the ridge regression (Ridge L2), the logistic regression (LogReg L2), and the support vector machine (SVM L2) are plotted on the left side of the dashed bar. Non-linear models including the k nearest neighbor (kNN), the random forest (RandForest) and the adaptive boosting (AdaBoost) are plotted on the right side. As a general observation, the plain orange violin plots of the linear models indicate on average a better performance with less variance thus appear to be more adapted to this setting. These results suggest that the ROI volumes are perhaps mostly individually predictive as much as this evidence is supported by dataset

  1b) grouping the MCI patients: a mixed MCI subgroup (294 MCI patients) harbored low scores at almost every test (maximum 0.6 points on average), an amnestic MCI subgroup (207 MCI patients) scored low only on test assessing memory (maximum 1 point on average), and a cluster-derived normal subgroup (139 MCI patients) included MCI patients with a scoring profile virtually identical to controls (at least 1 point on average) except for one test, the logical memory scale II. The three subgroups were homogeneous in terms of sex and age (Supplementary Table

  It is an ongoing debate if scans acquired from different scanners can be merged. Many studies reported highly reproducible correspondence between volumes(Roche et al, 2013; Ho et al, 2010) while other studies suggested different methods to increase consistency across field strengths(Keihaninejad et al 2010). Here, we have made the decision to similarly preprocess all scans. Additionally, we only used grey matter volume as neuroimaging data, other measure can be of interest such as tractography, white matter, cortical thickness, and so on. Another limitation of our study includes the use of one dataset (i.e., ADNI). ADNI is not a population-based study and there are strict inclusion and exclusion criteria for selection of participants, which can affect generalizability of our findings. Therefore, validating our models and outcomes in other population-based studies and clinical trials' data would be an important next step.

CONCLUSION

In summary, our research revealed 3 latent subgroups underlying MCI participants of the ADNI database: an amnestic MCI, a mixed MCI and a false positive subgroup. Leveraging on machine learning, our findings further suggest that MCI subtypes, extracted from multimodal neuropsychological approach, have proper biological and neurological characteristics. As such, our findings pave the way to fine-grained, biologically and neurologically meaningful MCI diagnosis.

Table 1 :

 1 information about the included ADNI subjects

	Diagnosis	N total	N male	N female Age mean (SD) N ADNI 1 N ADNI 2
	MCI	640	376	264	73.42(7,66)	374	266
	CN	326	162	164	75,15(5.57)	211	115

Table 2 :

 2 Significant ROI weights for the benchmarked linear models

	LogReg L2	SVM L2	Ridge L2

Acknowledgements

The study was funded by the CARSAT (Caisse Régionale d'Assurance Retraite et de la Santé au Travail) of Normandie (France) and from donation to the Caen Normandie Santé foundation (Crédit Agricole Normandie and Normandie-Seine, Harmonie Mutuelle, société SAMMed), and co-funded by the European Union and the Normandie Region as part of the Operationnal FEDER program 2014-2020 Normandie. Data collection and sharing for this project was funded by the Alzheimer's Disease Neuroimaging Initiative (ADNI). See the supplementary for more information.

Code availability

Python was selected as the scientific computing engine. Scikit-learn (Pedregosa et al., 2011) provided efficient, unit-tested implementations of state-of-the-art statistical learning algorithms (http://scikitlearn.org). All analysis scripts of the present study are readily accessible to the reader online (https://github.com/JLefortBesnard/MCI_cluster_prediction).

Declaration of Competing Interest

The authors have no actual or potential conflicts of interest.

Credit author statement

JLB: Study concept, plan of analysis, data management, image analysis, statistical analysis, literature search and review, write-up of parts of the manuscript, revision of the manuscript; MN: Study concept, plan of analysis, assistance with image and statistical analysis, literature search and review, revision of the manuscript; ND: Study concept, plan of analysis, assistance with image and statistical analysis, revision of the manuscript; FC: Study concept, plan of analysis, assistance with image and statistical analysis, literature search and review, write-up of parts of the manuscript, revision of the manuscript; LD: Study concept, plan of analysis, assistance with statistical analysis, literature search and review, revision of the manuscript.


Keywords: Blind source separation, time series clustering, energy sustainability, bond graph

Unsupervised manufacturing process identification using non-intrusive sensors R. DELABEYE a , M. GHIENNE a , A. KOSECKI a , J.-L. DION a , a. Quartz Laboratory, EA7393, ISAE-Supméca, (romain.delabeye@isae-supmeca.fr, martin.ghienne@isae-supmeca.fr, arkadiusz.kosecki@isae-supmeca.fr, jean-luc.dion@isae-supmeca.fr)

Résumé :

Le suivi de la durabilité énergétique dans l'industrie manufacturière se heurte à un problème d'échelle. La surveillance d'indicateurs de performance est essentielle, en utilisant cependant le moins de capteurs possible et en limitant leur intrusivité vis-à-vis des systèmes existants. Les capteurs non-intrusifs sont particulièrement adaptés à de telles applications, en cela qu'ils captent de nombreuses sources depuis un lieu distant de celles-ci. La reconstitution des indicateurs-cibles nécessite toutefois davantage de traitement du signal. Les méthodes présentées dans cet article visent avant tout à reconstruire, de manière non-supervisée, le processus de production d'une machine à partir de données issues de capteurs. Une série de mesures est ainsi séquencée temporellement en opérations distinctes. Leur contenu en termes d'actionneurs actifs est ensuite estimé par décomposition. Ces méthodes sont toutes particulièrement adaptées aux signaux apparaissant comme stationnaires par morceaux dans la représentation temps-fréquence.

Abstract :

Energy sustainability in the manufacturing industry faces a scalability issue. Monitoring appropriate performance indicators is essential, yet as few sensors as possible should be used, and with limited intrusiveness (software-or hardware-wise). Non-intrusive sensors are well suited to such applications, as multiple sources can be sensed at once. Recovering the desired indicators requires additional signal processing though. This paper focuses on recovering a machine's process from sensor data in an unsupervised fashion, and unveiling which actuators are active within each operation. The proposed method is particularly well suited to mixed signals which appear as stationary in the time-frequency domain within each operation.

Introduction

The manufacturing industry is energy-intensive by construction. It is therefore essential to improve its energy sustainability, starting with the monitoring of energy key performance indicators (EnPI). These are tedious to estimate and forecast due to the lack of high-fidelity models for the industrial machines making up manufacturing plants. Moreover, their identification requires much intrusive data which are either lacking or unavailable (e.g., from sensors located inside the machines). For this reason, nonintrusive sensors are more convenient. For instance, accelerometers, temperature and current sensors sense much information, at the expense of a greater difficulty to link these observations to the measurand of interest through physical models.

Instead, models can be entirely built up from data through model synthesis, without a priori knowledge.

There are two critical aspects to these models. First, the underlying physics semantics must be sufficiently rich to allow for EnPI to be inferred. Second, the constitutive elements of the production process should be inferred as well. A process is defined here as a sequence of operations performed by the same physical components. Associating different models to specific operations and actuators enables their respective energy sustainability analysis, thus allowing for process enhancement or component replacement.

This paper focuses on the second aspect, namely breaking down a process into distinct operations and finding the actuators used in each of these operations. Non-intrusive sensors are considered here, these have the particularity of yielding mixed signals originated from different actuators. The process identification algorithm shall operate in a completely unsupervised fashion. The control inputs in particular are assumed to be unknown. This is a special case of under-determined Blind Source Separation (BSS), where restrictions apply on the mixing matrices. Additionally, actuator activation statuses are sought rather than the source signals themselves. The signals are assumed to have short transient responses (with respect to an operation's duration), stationarity being a common assumption in such problems [START_REF] Sadhu | A review of output-only structural mode identification literature employing blind source separation methods[END_REF].

Despite similarities with a soft clustering problem [START_REF] Baadel | Overlapping clustering : A review[END_REF], where the measurement at a certain time step can belong to multiple clusters, the related techniques tend to soften the transitions from a cluster to another. Fuzzy Clustering is used in [START_REF] Honda | Extraction of local independent components using fuzzy clustering[END_REF] for time-varying mixing matrices in speech signal reconstruction, and in [START_REF] Hathaway | Switching regression models and fuzzy clustering[END_REF] for switching regression models. In BSS problems, source signals are reconstructed from mixed measurement signals. The number of sources is often assumed to be known a priori [START_REF] Hyvärinen | Independent component analysis : algorithms and applications[END_REF]. This assumption can be lifted through subspace methods [START_REF] Dong | Blind separation of sound sources from the principle of least spatial entropy[END_REF][START_REF] Antoni | Second order blind source separation techniques (so-bss) and their relation to stochastic subspace identification (ssi) algorithm[END_REF], typically by thresholding the eigenvalue spectrum for the high Signal-to-Noise-Ratio (SNR) scenario, or using alternative problem-specific criteria [START_REF] Loesch | Source number estimation and clustering for underdetermined blind source separation[END_REF].

Along with problem formulation, a dynamical systems representation is described in Section 2.1. The proposed method consists in a two-step process : clustering into operations and decomposing into actuator activation times, detailed in Section 2.2 and Section 2.3 respectively. Simulation and experimental results for this algorithm are detailed in Section 3.

Method

Problem formulation

As the machine evolves in time and in its process, from an operation to another, sudden changes in sensor signals are encountered (assuming the transient responses are short-enough). Several operations can use the same actuators though. Splitting the signal in terms of operations is helpful for prioritizing process enhancements, whereas decomposing the operations into a minimal set of atoms, representing machine actuators if encountered alone at least once, is essential to isolate an actuator's performance and deterioration.

Bond graph theory [START_REF] Borutzky | Bond graph modelling of engineering systems[END_REF] is well-suited to energy sustainability modeling. Indeed, such a graph represents the power distribution, towards elements classified either as energy dissipation (R) or storage (I or C).

Energy efficiency and consumption can thus be inferred from a minimal set of variables. 0-and 1junctions, as well as multi-port components (TF and GY ) represent power transfers without loss (this is essentially a generalization of Kirchhoff's laws). The discrete-event aspect is modeled as switched power junctions [START_REF] Borutzky | Bond graph model-based fault diagnosis of hybrid systems[END_REF], represented as 1 δ i in Figure 1, where the power is forced to zero when the actuator is inactive.

Figure 1 -Overview of the proposed method, illustrated upon a machine composed of a pump, a motor and a heating coil (synthetic signal).

As illustrated in Figure 1, we propose the following method : a sensor signal, mixing actuator-originated signals, is clustered into distinct operations to begin with ; each operation is then decomposed into atoms (actuators) when possible. These actuators essentially relate to distinct sets of causal paths, which is helpful information for the subsequent bond graph model synthesis.

Formally, a univariate time series X ∈ R 1×T ′ is considered, representing T ′ sensor measurements evenly sampled over time. The Short-Time Fourier Transform (STFT) operator ST F T {X} is applied to the data, the extracted features make up the matrix Z ∈ C f ×T where f denotes the number of features (frequency bins in this case), and T is the new number of samples (time window indices). The underlying system performs n ops distinct operations using n act different actuators, with n ops ≥ n act .

The matrix L ∈ {0, 1} nops×T represents the operation the system is in at each time step t ∈ 1, T , with ∥L t ∥ 0 = 1. Whereas the activation statuses of a system's actuators over time are gathered in a matrix L ∈ {0, 1} nact×T , with ∥ Lt ∥ 0 ≥ 1. Thus the time series L c and La indicate whether the system is in operation c (cluster) and uses actuator a at each time step. Matrix superscripts and subscripts represent vectors lying in the row and column spaces respectively.

Clustering into operations

Data is clustered to begin with. Clustering is a division of data into groups of similar objects [START_REF] Rai | A survey of clustering techniques[END_REF]. Time series (hard) clustering is an unsupervised technique grouping timestamps according to a similarity measure between vectors describing data at each time step : a data point belongs to one cluster only. This topic has been extensively tackled in the literature, the k-means algorithm is considered here, applied to the magnitude of the data's STFT, |Z|. Any relevant descriptor applied to sliding windows could be used though, the first four statistical moments for example. The number of clusters (operations) n ops is chosen manually using a dendrogram. In most cases, this number cannot be known a priori and other clustering techniques such as DBSCAN or its variant OPTICS [START_REF] Wang | A novel underdetermined blind source separation method based on optics and subspace projection[END_REF] are better suited.

The one-hot encoded labels L are thus obtained, indicating in which operation the system is at each timestamp.

Decomposition into actuators

In this section, decomposition refers to expressing an operation as the composition of other operations. Assuming the system's actuators have all been encountered alone at least once, i.e. activated while all others were deactivated, the decomposition yields the actuators' activation sequence. If this assumption is not met, then a minimal set of independent operations is found. Should the data be aperiodic (still under the stationarity assumption), and should the application converting the data into features be linear, a composition would thus correspond to a linear combination in the least-squares sense.

Alternatively, if the data is periodic, these signal superimpositions are slightly more difficult to unveil. Indeed, each time an actuator is activated, its signal's phase changes and so does the mixed signal's.

In order to tackle the latter case, a shifted STFT is applied to the raw data. The STFT is in essence a sliding discrete Fourier transform (DFT) and is hence linear by definition. Due to window sliding, a progressive phase shift occurs. Although the time delay can easily be retrieved, the phase shift is signal-dependent. Exact phase retrieval [START_REF] Magron | Model-based stft phase recovery for audio source separation[END_REF] is computationally expensive and unnecessary in this context. Instead, at each time step, the phase of Z t is measured at the maximum of magnitude and shifted accordingly. This takes out both the phase shift of the windowing process as well as the inherent phase of the signal. To this end, the phase-shift operator is used, defined as :

S ∆ = diag e -j 2πk N ∆ 1≤k≤N ( 1
)
where N is the number of frequency bins and ∆ is the time delay. This shifted STFT makes the phase consistent within a cluster and reduces the intra-cluster variance. The centroid matrix C ∈ C f ×nops can thus be computed, each vector in the column space corresponding to the average of all samples belonging to each cluster.

Because the phase-shift operator is nonlinear with respect to the time delay, the phase of a composite centroid will be different than the sum of its constituents. The sought combination between a centroid C c and all others, {C i } i̸ =c , can be expressed as :

arg min {∆ i } 1≤i≤nops {α i } 1≤i≤nops ∥C c - i̸ =c α i S ∆ i C i ∥ 2 2 (2)
where the coefficients α = {α i } 1≤i≤nops and ∆ = {∆ i } 1≤i≤nops are (ordered) sets of scalars, representing combination coefficients and time-delays respectively. The functional is linear in α and nonlinear with respect to ∆, albeit periodic.

In some cases, this projection can lead to misinterpretations. On one hand, if an actuator operates at different regimes, the centroids of the two operations will be collinear, with a combination coefficient corresponding to the ratio between both regimes. On the other hand, in real case scenarios, multiple actuators can also produce very close signatures. For instance, several electrical devices can induce current consumptions at 50Hz. The decomposition would not apply in such a case. This is also why subspace methods in general would not be well suited.

In order to tackle this aspect, we compute a residual matrix R ∈ R nops×2 nops , with ∀r c g ∈ R :

r c g = min {∆ i } i∈g (2) ∥C c -i∈g (2) S ∆ i C i ∥ 2 2 ∥C c ∥ 2 2 (3)
where ∀g ∈ 1, n ops , g (2) corresponds to a possible group (i.e., a combination) of centroids, with • (2) denoting a number expressed in base 2.

By doing so, we constrain the values of α i to {0, 1} and evaluate all possible combinations. In the case of a composite operation, this residual (or an alternative distance) would fall below a problem-dependent threshold τ . The Broyden-Fletcher-Goldfarb-Shanno (BFGS) [START_REF] Fletcher | Practical methods of optimization[END_REF] algorithm is used to compute these residuals. This approach does not scale due to an obvious combinatorial explosion. For actual systems running only a few distinct operations, this is not limiting.

For clarity, the method is wrapped up as a pseudo-code in Algorithm 1.

for 1 ≤ c ≤ n ops do for 1 ≤ g ≤ 2 nops do r c g ← min {∆ i } i∈g (2) ∥C c -i∈g (2) S ∆ i C i ∥ 2 2 ∥C c ∥ 2 2 end if min 1≤g≤2 nops r c g ≤ τ then
Cluster c is composed of the clusters present in g [START_REF] Baadel | Overlapping clustering : A review[END_REF] , ∀t ∈ {cluster c}, Lt ← g [START_REF] Baadel | Overlapping clustering : A review[END_REF] ; else Cluster c cannot be decomposed, ∀t ∈ {cluster c}, Lt ← L t ; end end Remove empty rows in L (composite operations), yield L

Results and discussion

The proposed methods have been applied to synthetic and real use cases.

The synthetic use cases follow a pattern : three different waves (no transient) successively activated for 10 seconds each, then all three active simultaneously for 10 seconds. For brevity, only one of such experiment has been shown :

x(t) = δ 2 (t) sin (2π50t) + δ 3 (t) 2 sin (2π50t + 2π 3 ) + δ 1 (t) 2 sgn sin (2π70t) + w(t) (4)
where δ 1 (t), δ 2 (t), δ 3 (t) denote the switches statuses at time t. The output signal has been supplemented with a zero-mean Gaussian noise w(t) with standard deviation σ = .8.

The physical use case consists in an automated coffee machine. It is composed of a pump, a heating coil as well as two motors. The process shown in Figure 2b starts with a series of short heating and pumping sequences, followed by a longer pumping operation, and four heating operations superimposed with the pumping ; finally one of the motors displaces the infuser. The experiments run on the synthetic use cases highlight the method's resilience to noise as well as its ability to perform as expected when the signal frequencies are equal or differ, given equal or different signal amplitudes.

As for the coffee making process, all devices produce current signals with a fundamental frequency at 50Hz and different amplitudes. The superimposition of the pumping (switch δ 1 ) and the heating (switch δ 5 ) operations is accurately detected.

Conclusions

In this paper, a decomposition method was proposed in order to detect superimposed operations from non-intrusive sensor data. This approach has been validated upon several synthetic scenarios as well as a laboratory experiment.

Future work will focus on better handling controlled systems capable of operating at different regimes over a production cycle. The approach will be further developed so as to include the transient responses which contain much information regarding the subsystems and could help their differentiation with one another. At last, a bond graph model synthesis algorithm will be designed.

Aknowledgments

This work was funded by the EnerMan project from the European Union's Horizon 2020 research and innovation programme under grant agreement No 958478.

Algorithm 1 :

 1 proposed algorithm Input: Discretized time series data X ∈ R 1×T ′ ; expected number of operations n ops ; Result: Clustering labels L ∈ {0, 1} nops×T and decomposition labels L ∈ {0, 1} nact×T ; Step 1 : extract features and apply clustering Extract features accurately describing the operations Apply K-means algorithm; Yield the (one-hot encoded) clustering labels L; Step 2 : extract features for decomposition Z ∈ C f ×T , Z = STFT{X} for 1 ≤ t ≤ T do measure the phase at the maximum of magnitude ∆ m = ∠ arg max z∈Zt |z|; shift the phase accordingly Z t ← S ∆m Z t ; end Compute C ∈ C f ×nops : ∀c, given N c samples in cluster c, C c ← 1 Nc t∈{cluster c} Z t Step 3 : decomposition L ← L;

  Figure 2 -Decomposition results