Riemannian tangent space mapping and elastic net regularization for cost-effective EEG markers of brain atrophy in Alzheimers disease

Riemanni an tangent space mapping and elastic net r egulariza tion f or cost-effec tive EEG markers of brain atr oph y in Alzheimer’ s disease W olfgang Fruehwirt Medical University of V ienna & University of Oxford Matthias Gerstgrasser University of Oxford Pengfei Zhang University of Oxford Leonard W eydemann Medical Uni versity of V ien na Markus W aser T echnical Univ ersity o f Den m ark Reinhold Schmidt Medical University of Graz Thomas Benk e Medical University of Innsbru ck Peter Dal-Bianco Medical University of V ienna Gerhard Ransmayr Linz General Hospital Dieter Grossegger Dr . Grossegger & Dr bal GmbH Heinrich Garn Austrian Institute of T echnolo gy Gareth W . P eters University College London Stephen Roberts University of Oxford Georg Dorffner Medical University of V ienna Abstract The diagnosis of Alzheim e r’ s disease (AD) in ro utine clinical practice is mo st common ly based on subjecti ve clinical interpretations. Quantitative electroen- cephalog raphy (QEEG) measures have been shown to reflect neurodegenerative processes in AD and migh t qualify as afforda b le and thereby widely available markers to facilitate th e ob jectivization of AD assessment. Here, we present a novel framew ork combining Riemann ia n ta n gent space m apping and elastic n et regression for the development o f brain atro p hy markers. While m o st AD QEEG studies are based on small samp le sizes a n d psycholog ical test scores as o utcome measures, here we train an d test ou r mo dels using data of one o f the largest prospective EE G AD trials ev er co nducted , in cluding MRI biomarkers of brain atrophy . 1 Intr oduction Having been successfully app lied in domains such as computer vision [3 0], radar sign al processing [5], and d iffusion tensor imag ing [23] for year s, the introd uction o f a Riemann ian manifold of sym- metric p ositiv e-definite (SPD) matrice s to brain signal analysis rep r esents a powerful altern ativ e to more traditional inf ormation extraction pro tocols. Only recently has it been shown that Riemannian Brain-Compu ter Interface (BCI) metho ds outpe r form state-of-the- a rt Euclidian spatial filtering and machine learnin g techniques [9]. Five recent internation al BCI competition s – includin g la st year ’ s Microsoft Cortana brain decoding challenge – h av e been won using Riemann ian geometry [ 9 , 3]. Sev eral reasons for this su ccess have been p roposed in the literatu re. First, in the form of cov ariance matrices, SPD matr ices are under stood to b e excellent r epresentation s of the raw electroph ysiolog- ical br ain signal, wh ile redu cing its unwanted variations [ 20]. They h av e therefo re beco me f un- 31st Conference on Neural Information Processing Systems (N I P S 2017), L ong Beach, CA, USA. damental elements in me th ods such as com mon spa tial pattern and c a nonical correlation analysis. Second, SPD matrice s are tr aditionally treated within Euclidian framew orks, ignor ing their intrinsic non-E u clidian structure. Neglecting this fu ndamen tal characteristic m ay lead to deficient results [1 ] . These p oints not o n ly have be en foun d advantageous in BCI design but also make a stron g case fo r the use of a Rieman nian SPD matr ix m anifold in the assessment o f n euron a l degeneratio n a s can be found in Alzheimer’ s disease (AD). AD is the most com m on fo rm of dementia and ultimately fatal. The combin ation of its se verity an d looming global epidemic scale – c a used by the ageing o f our so c ie ty – m akes AD a majo r pub lic health co ncern [2]. Due to its degener ativ e natu re, early accur a te diagno sis a nd effective clinical monitorin g are cruc ial. Howe ver , when it comes to r o utine clinical practice, AD assessment is most common ly don e by subjective clinical interpretatio ns at an alr eady prog ressed stage of th e disease. So far, n o cost-effecti ve, wid ely-used biom arkers h av e been established to facilitate the ob jectiviza- tion of diagnosis and disease pr ogression assessment. T o pro mote th e screening an d mon itoring of as many individuals as possible, such m arkers sho uld n o t be depe n dent on costly equ ipment, such as MRI, or PET scann ers. Theref ore, we focus on inexpensive ap paratuses, namely electro en- cephalog raphy (EEG) devices. T heir no n-inv asi veness a nd low noise level adds to their suitability for large-scale use in irritable patients such as tho se fo und with in the spectrum of AD. Addition ally , research suggests that qu antitative electr oenceph alograp hy (QEEG) reflects neurod egenerative pro - cesses in AD (for re views , see [32, 12, 1 0]) . Therefo re, we aim to dev elop a Riem a nnian framework for QEEG mar kers of neuronal d egen eration in AD and empir ically in vestigate its usefulness. T o be able to combine the me r its of Riemannian geometry with th e advantages of sop h isticated Euclidea n regularization and variable selection tech- niques like the elastic n et (see 2 .3), we map SPD matrices into th e tangen t space (see 2.2). Sustaining the distance relationship of elements, this projectio n and a subsequ ent vector iz a tion allows to treat SPD matrices as Euclidean entities. All existing Riemann ian brain signal analysis methods use covariance as measure of depend ence, thereby implicitly assuming a multiv ariate Gaussian distribution o f data and linear associations be- tween the activities of b rain regio n s. Howe ver , bo th pro perties mig ht no t b e fulfilled [2 9, 33]. Hence, we examine the usefu lness of ran k correlation fo r constructing SPD ma tr ices – capturing no n-linear relationships in da ta that is often far from normally distributed. For model training an d testing, we use one of the la rgest A D EEG data sets ever collecte d in a p rospective manner, including MRI biomarkers o f brain atrop hy . As frequ e ncy-specific QEEG inform ation ha s been proven useful in the AD domain [33], we fu rthermo re analyze the ef fectiv eness o f a special type of spatiofrequen tial SPD matrix. Finally , to ev aluate th e real adde d value of tangent space mapping, we c o mpare results achieved b y this m e th od with those a chieved by using regular E uclidean procedu res. 2 Materials and Methods 2.1 Experimental data AD patients we r e prospectively recruited at f our tertiary memo ry clinics (Medical Universities of Graz, Innsbr uck, V ienna, and the Gen eral Hospital Lin z, PR ODEM coho r t study by th e Austrian Alzheimer Society [2 7], suppo rted by the Au strian Research Pro motion Agen cy FFG, p roject n o. 82746 2). W e exclusively analyz ed participa nts (N = 110 ) who h a d a stru ctural MRI scan within 60 days o f the ba selin e EEG measur ement, a max im al MMSE score [14] o f 28, and a CDR [19] fro m 0.5 to 1. Acquisition of structu ral T 1-weighte d image s was acco mplished on 1.5 and 3 T esla MR scanne r s (Siemens). W e used FreeSurfer volumetric analyses [13] to build two MRI biom arkers o f b rain atrophy , i.e., cerebral volume an d hippo campal volume divided by the total intracranial volume (ratios referred to as BrainV ol an d HippV ol). Continuou s EEG (alpha trace EEG recorder, 10-20 ele c trode placement) was analy zed f or a n eyes- closed r esting co ndition (EC, 180 sec; prediction of BrainV ol) and the encoding per iod of a paired- associate word list task (WL T , adapted version of [25], 140 sec; predictio n of HippV ol). Research has repeated ly sh own (i) the importance of hippocam pal activity during the WL T [ 7, 8], and (ii) the sensiti vity of paired-associative m emory to early AD-related changes [15, 24]. For details o n th e entire PR ODEM exper imental pr otocol an d preprocessing p ip eline, see [33, 17, 16]. 2 2.2 Feature g eneration T o op timize the nu mber of time points av ailable, we determ ined the maximal signal length with gu ar- anteed quasi-station ary prop erties using an augmen ted Dickey-Fuller test [11]. The de-ar tifacted signal was partition ed into 4-sec segments according ly . SPD matrices wer e estimated using sample covariance (SCM, see 1) and Kendall rank correlatio n (KE N, see [22]). In the following a ll proce- dures will be described by taking the example of sample covariance. EEG se gments were considered as n by t matrices X s ∈ R nxt , ( s = 1 . . . S ) , n being th e n umber of electrodes, t being th e numb er of time samples. For each segment s a spatial covariance matrix C s was estimated: C s = 1 t − 1 X s X ⊺ s ∈ R n × n (1) T o take d istinct fr equential aspects into accou nt, we created a special type of matrix b y band -pass filtering X s multiple times ( δ = 2 to 4 H z , θ = 4 to 8 Hz α = 8 to 13 Hz, β 1 = 13 to 15 Hz) and vertically co n catenating the resulting signal to X S F s = [ X ( δ ) ; X ( θ ) ; X ( α ) ; X ( β 1 ) ] ⊺ ∈ R 4 n × t . Then, the sample cov ariance ma tr ix C S F s was estimated: C S F s = 1 t − 1 X S F s X S F ⊺ s ∈ R 4 n × 4 n (2) The com mon Eu clidean distance ( d euc ) between two matrices C 1 , C 2 and the correspo nding mean ( M euc ) of se veral m atrices C 1 , . . . , C N can be defined as d euc ( C 1 , C 2 ) = k C 1 − C 2 k F (3) M euc ( C 1 . . . C N ) = 1 N N X i =1 C i (4) where k . k F denotes th e Fro benius n orm. Howev er , the Euclid ean space suffers fro m several dis- advantages, as – for instan ce – th e av eraging of SPD matrices may lead to a swelling effect (the determinan t o f the Eu clidean mean can be strictly larger tha n the or iginal determin ants [ 1]). T o av oid su ch artifacts fr om geometry , a more natural metric fo r SPD matrices, the Lo g-Euclid ean distance d log , with the correspond ing mean M log , can be used (e.g., [34]): d log ( C 1 , C 2 ) = k log( C 1 ) − log( C 2 ) k F (5) M log ( C 1 . . . C N ) = exp 1 N N X i =1 log ( C i ) ! (6) Further, SPD matrices can be tre a ted in their n ativ e Riemannian space usin g geodesic distance d r ie and the Riem annian geometr ic m e an, of ten referr e d to as Karc her mean [2 1], M r ie , wh ich minimizes the sum of squared d r ie : d r ie ( C 1 , C 2 ) = k log  C − 1 2 1 C 2 C − 1 2 1  k F = " N X i =1 log 2 λ i # 1 2 (7) M r ie ( C 1 . . . C N ) = argmin C N X i =1 d r ie 2 ( C i , C ) (8) where λ i are the eigen values of C − 1 2 1 C 2 C − 1 2 1 . As M r ie has no closed -form solution fo r N > 2 , we optimized it using the relaxed Richardson iteration [ 6]. For a revie w on the advantages of Rieman- nian geometry in brain signal processing and d etailed formal definitions, see [9, 30]. Sets of C s and C S F s were separa tely av eraged on patient level u sing the aforemention ed mean calcu- lation metho ds (4, 6, 8) fo r bo th EEG paradig ms (EC and WL T), and both measur es of depend e nce (CO V and KEN), resulting in sub ject-specific matrices M z ( z bein g a patient) for all variants. For Riemannian (T AN r ie ) and Log-Eu clidean (T AN log ) tangent space-based features, M z of th e corre- sponding mean type was mapped into the tang e nt sp a c e F z = upper  M − 1 2 G Log M G ( M z ) M − 1 2 G  (9) 3 where M G was computed altern ativ ely using M r ie (8) for T AN r ie or M log (6) for T AN log . F or a formal definition of the Riemann ian tangent space , see [4, 3 0]. For the Euclid ean control cond ition (EUC) both, the av eraging on subject as well as gro up le vel, was done using M euc (4). Applying u pper(. ) as a n oper ator vector iz in g the upp er triangular part o f a SPD matr ix , th e fe a- ture vectors F z of C s ∈ R n ( n +1) / 2 , g iv en n = 19 resulting in 19 0 d imensions, and F z of C S F s ∈ R 4 n (4 n +1) / 2 , given n = 19 resulting in 2926 dimensions, were created. 2.3 Elastic net regression and repeated nested cr oss-validation The elastic net [3 5] was used as a regularization an d variable selectio n techn ique to estimates a sparse regression m odel based on F z . It imposes a combin ation of the ℓ 1 (lasso, [28]) and ℓ 2 (ridge, [18]) penalties on r egression coef ficients. While enjoying a similar sparsity of representatio n as the lasso, th e elastic net enco urages a grouping effect, wh ere stron gly correlate d predictors – as p resum- ably pr esent in our data set du e to spatially ad ja c ent electro d e placement – tend to be in or out of the model together [35]. W e used a 10 × 10 two-lev el nested cross-validation to determine gen eralization perfo rmance. Th e inner loop was includ ed to sensibly choose a value for the regulariza tio n parameter λ with minimal expected g e neralization er ror [ 31]. The λ value that resu lted in th e lowest m ean squared erro r (MSE) in the inner loop was used to fit models in the outer loop. The parameter α , represen ting the we ig ht of lasso ( ℓ 1 ) versus ridge ( ℓ 2 ) optim iza tion, was set at 0.5. Age and g ender were in tr oduced in Brain - V ol models, whereas th e ma gnetic field strength (varying values o f T esla between centers mig ht influence the analysis of smaller structur es) was additionally introduced in th e Hip pV ol m odels. All variables wer e n ormalized bef ore mo del fitting. T o reduc e the variability o f p rediction ou tcome resulting from ran d om training– test set splitting, we rep eated the entire ne sted cross-validation pro - cedure 1 00 times. This allowed us to average out variability and report the range o f resu lts of multiple permu ta tio ns [26]. 3 Results and Discussion Results are dep icted in T a b le 1. For bo th prediction problem s (Brain V ol, Hip pV ol), the best models were of spatiofrequ ential natur e (indicated in bold in the table), highlighting the im p ortance o f frequen cy-specific info rmation for QEEG AD markers. Wh en co mparing spatiofreq uential models, the best tangen t space mappin g mode ls significan tly outp e rforme d the Eu clidean reference m odels (BrainV ol, p = 0 .003; HippV ol, p = 0.0 30). Differences between mo del perfor mances were assessed by statistically comparing squared errors of test set p redictions an d av eraging p -values acro ss r epeated cro ss-validation. Further, we calcu la te d test statistics for ev aluating the stand- alone perform ance of the best mod e ls (BrainV ol, p = 0.003 ; Hippvol, p = 0.0 11). For th e prediction of BrainV ol (measured during EC resting state) CO V yielded lower ro o t-mean- square error s (RMSE) than KEN. Info rmation on th e the mag nitude of the sign al at certain sites – wh ich is present in the diagon al elements of CO V but no t KEN matr ices – seem to be essential. Whereas fo r hippo campus-m ediated m emory enco ding d uring th e WL T , the interaction between brain r egions (neuro nal networks), as measured b y off-diago nal matrix elements, seem to be of predo minant imp ortance – explaining the supe rior results for KEN. Further, the EEG sig nal during an active eyes-open task is presumab ly less n ormally distributed ( even af ter so phisticated pre-pr ocessing) then dur ing a resting EC p eriod, add itionally explaining deviating results f or CO V and KEN. Interestingly , T AN log achieved better r esults than T AN r ie . Barachant [ 3] also in ter alia used tangent space mapping with a Lo g-Euclid ean reference point ( M G ) for winning Micro so ft’ s ’mind reading ’ challeng e. Should futur e stud ies suppor t the sup eriority – or at least equ ality – of T AN log , co mputation al co st could be dra m atically dec r eased due to th e alg orithmic simplicity of Log-Eu clidean as comp ared to Riemannian mean c alculation. T o the best of o u r knowledge, th is is the first article reporting a Riemannian appr oach fo r building QEEG markers o f neurona l degeneration . 4 T able 1: Mean, minim um and maximu m root-m e an-squar e error (RMSE) of 1 00 n ested cross- validation repetitions for pred icting the normalized whole-br ain v olume (BrainV ol) and norma lized hippoca m pus v olume (HippV ol) for various combination s of measures of d epend e n ce (Dep ; covari- ance, CO V ; Kendall r ank co r relation, KEN), geometric app roaches (App roach; Euclidean , E UC; tangent space map ping with Log- Euclidean mean, T AN log , an d Riemann ian mean, T AN r ie ), and spatial (S), or spatiofrequen tial (SF) matrix designs (Design). BrainV ol HippV ol Dep Approach Design RMSE Min Max RMSE Min Max EUC SF 1.70E-03 1.57E-03 2,53E-03 2.09E-07 1.63E-07 4.26E-07 CO V T AN log SF 1.23E-03 1.10E-03 1.37E-03 1.86E-07 1.71E-07 2.04E-07 T AN r ie SF 1.43E-03 1.27E-03 1,63E-03 1.80E-07 1.67E-07 2.04E-07 EUC SF 1.75E-03 1.61E-03 2.11E-03 1.78E-07 1.65E-07 2.05E-07 KEN T AN log SF 1.56E-03 1.43E-03 1.77E-03 1.44E-07 1.30E-07 1.44E-07 T AN r ie SF 1.58E-03 1.41E-03 1.91E-03 1.47E-07 1.31E-07 1.65E-07 EUC S 2.02E-03 1.57E-03 3.80E-03 1.69E-07 1.55E-07 2.17E-07 CO V T AN log S 1.34E-03 1.25E-03 1.50E-03 1.77E-07 1.62E-07 2.01E-07 T AN r ie S 1.35E-03 1.24E-03 1.46E-03 1.75E-07 1.56E-07 2.10E-07 EUC S 1.73E-03 1.59E-03 1.90E-03 1.68E-07 1.51E-07 1.95E-07 KEN T AN log S 1.56E-03 1.43E-03 1.79E-03 1.79E-07 1.63E-07 2.18E-07 T AN r ie S 1.55E-03 1.44E-03 1.73E-03 1.81E-07 1.63E-07 2.56E-07 5 Refer ences [1] Arsigny , V ., Fillard, P ., Pennec , X., & A yach e, N. 2007. 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