Appendix Alzheimer's Disease Modelling and Staging through Independent Gaussian Process Analysis of Spatio-Temporal Brain Changes

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Appendix Alzheimer’s Disease Modelling and Staging

through Independent Gaussian Process Analysis of

Spatio-Temporal Brain Changes

Clement Nader, Nicholas Ayache, Philippe Robert, Marco Lorenzi

To cite this version:

Clement Nader, Nicholas Ayache, Philippe Robert, Marco Lorenzi. Appendix Alzheimer’s Disease

Modelling and Staging through Independent Gaussian Process Analysis of Spatio-Temporal Brain

Changes. 2018. �hal-01849180�

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Appendix

Alzheimer’s Disease Modelling and Staging through

Inde-pendent Gaussian Process Analysis of Spatio-Temporal Brain

Changes

Clement Abi Nader1, Nicholas Ayache1, Philippe Robert2,3, and Marco Lorenzi1

1 _{UCA, Inria Sophia Antipolis, Epione Research Project} 2 _{UCA, CoBTeK}

3 _{Centre Memoire, CHU de Nice}

A. Lower bound derivation

In this section we detail the derivation of the lower bound : log(p(Y, C|σ, λ)) = log[ Z A Z S Z S0

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S, S0|λ)dAdSdS0]

= log[ Z A Z S Z S0

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S0|S, λ)p(S)dAdSdS0] (1) If we know S this completely determines S’, thus we have R p(S0_{|S, λ)dS}0 _{= 1}

which gives us : log(p(Y, C|σ, λ)) = log[ Z A Z S

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S)dAdS]

= log[ Z

A

Z

S

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S)q1(A)q2(S) q1(A)q2(S)

dAdS]

= log[EA∼q1,S∼q2[

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S) q1(A)q2(S)

]]

> EA∼q1,S∼q2[log[

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S) q1(A)q2(S)

]]

(2) This is obtained thanks to Jensen’s inequality. Finally this leads us to :

EA∼q1,S∼q2[log[

p(Y |A, S, σ)p(C|S0, λ)p(A)p(S) q1(A)q2(S) ]] = E

A∼q1,S∼q2[log[p(Y |A, S, σ)]]

+ ES∼q2[log(P (C|S0, λ))]

− D[q1(A|Y )||p(A)]

− D[q2(S|Y )||p(S)]

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In the Method section we introduced the approximation q1(A) = N s

Y

n=1

N (µn, Σ(α, β)).

The covariance matrix is shared by all the spatial processes which gives us the set of spatial parameters :

ψ = {µn, n ∈ [1, N s], α, β} (4)

Following [5] we introduce for each GP two vectors, Ωn with a prior p(Ωn) =

N (0, 1

lnI) for each element and Wn with a prior p(Wn) = N (0, I), such that

Sn(t) = Φ(tΩn)Wn. Where Φ is chosen to obtain a RBF kernel as explained in

[5]. We define the approximated distributions q3(Wn) = QjN (mn,j, s2n,j) and

q4(Ωn) =QjN (αn,j, βn,j2 ) of p(Wn) and p(Ωn). Using these approximations and

following [5], we can derive a lower bound for S with the same technique than above. We have the set of temporal parameters :

θ = {mn, sn, αn, βn, ln, n ∈ [1, N s]} (5)

Now we can obtain every term of (3). The Kullback-Leibler of a multivariate Gaussian has a closed-from :

D[q1(A|X)||p(A)] = 1 2 N s X n=1 T r(Σ) + µT_nµn− F − log[det(Σ)] (6)

Using the factorized form of q2and the fact that the different Gaussian processes

are independent from each other we can write :

D[q2(S|X)||p(S)] = N s

X

n=1

D[q3(Wn)|p(Wn)] + D[q4(Ωn)|p(Ωn)] (7)

Since the approximations q3 and q4 and their respective priors are normally

distributed we have an analytic formula for both Kullback-Leibler divergences. D[q3(Wn)|p(Wn)] = 1 2 X j s2_n,j+ µ2_n,j− 1 − log(s2n,j) (8) D[q4(Ωn)|p(Ωn)] = 1 2 X j βn,j2 ln+ α2n,jln− 1 − log(βn,j2 ln) (9)

As in [10] we employ the reparameterization trick to have an efficient way of sampling the expectations of (3). Thus we have :

– Wn,j= mn,j+ sn,j∗ n,j – Ωn,j= αn,j+ βn,j∗ ζn,j – An= µn+ Σ 1 2 n ∗ κn Which gives us :

EA∼q1,S∼q2[log(p(Y |A, S, σ))] = E,ζ,κ[log(p(Y |m, s, α, β, µ, Σ, σ))] (10)

ES∼q2[log(p(C|S 0

, λ))] = E,ζ[log(p(C|m, s, α, β, λ))] (11)

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B. Kronecker factorization

Here we detail how to split the covariance matrix in a Kronecker product of three matrices along each spatial dimensions. We have :

Σi,j(α, β) = α exp(−

||ui− uj||2

2β ) (12)

We can use the separability properties of the exponential to decompose the covariance between two locations ui= (xi, yi, zi) and uj= (xj, yj, zj) :

Σi,j(α, β) = α exp(− (xi− xj)2 2β ) exp(− (yi− yj)2 2β ) exp(− (zi− zj)2 2β ) (13) So Σ can be decomposed into the Kronecker product of 1D processes :

Σ = Σx⊗ Σy⊗ Σz (14)

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C. Comparison with ICA

We performed a comparison of our algorithm with ICA on a similar example than in 3.1. However the data was generated in a simplified setting since ICA can’t be applied when the time associated to each image is unknown. To do so we assigned the ground truth parameter tpbeforehand. The goal was to compare the

separation performances of both our algorithm and ICA, on data generated with three latent spatio-temporal processes. In Figure 1 we observe that the sources estimated by ICA are more noisy and uncertain than the ones estimated by our method, highlighting the performances of our algorithm in terms of sources separation. 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Source 0 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Source 1 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Source 2 Time Amplitude (a) Map 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Map 1 0.0 0.5 1.0 1.5 2.0 Map 2 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 (b) 0.0 0.5 1.0 Original Time 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 Original Time 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 Original Time 0.0 0.2 0.4 0.6 0.8 1.0 0.250.00 0.25

Estimated Time Estimated Time_{0.250.00 0.25} Estimated Time_{0.250.00 0.25}

Amplitude (c) Map 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Map 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Map 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (d) 0.0 0.5 1.0 2 4 6 8 10 12 14 Source 0 0.0 0.5 1.0 0 5 10 15 20 Source 1 0.0 0.5 1.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Source 2 Time Amplitude (e) Map 0 0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 Map 1 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10 Map 2 0.15 0.10 0.05 0.00 (f)

Fig. 1: First Row : Raw sources. Second Row : Sources estimated by our method. Third Row : Sources estimated by ICA.

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D. ADNI

Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and DOD ADNI. ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imag-ing and BioengineerImag-ing, and through generous contributions from the followImag-ing: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Ara-clon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company;CereSpir, Inc.;Cogstate;Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; Eu-roImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Re-search & Development, LLC.; Johnson & Johnson Pharmaceutical ReRe-search & Development LLC.;Lumosity;Lundbeck;Merck & Co., Inc.; Meso Scale Diagnos-tics, LLC.;NeuroRx Research; Neurotrack Technologies;Novartis Pharmaceuti-cals Corporation; Pfizer Inc.; Piramal Imaging;Servier; Takeda Pharmaceutical Company; and Transition Therapeutics.The Canadian Institutes of Health Re-search is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern Califor-nia Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern Cal-ifornia. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.