text: 1.4x10−3 2.5x 10−3 1.2 2 1 0.8 1.5 0.6 1 0.4 0.5 0.2 0 50 100 150 200 250 300 350 0 10 20 30 40 50 60 70 (a) T = 1 minute bin (b) T = 5 minute bin Figure 4: U-Pattern Volatilities: Stock average of single stock volatility (cid:104) (cid:105) [σ (k)] (black), time average of the cross sectional volatility σ (k, t) (red) α d(cid:104)| |(cid:105) and the average absolute value of the equi-weighted index return µ (blue) d for the CAC 40. Similar results were obtained for the S&P 500. × (cid:98) (cid:98) (cid:98) (cid:98) The N N correlation matrix for a given bin k would be given by (cid:104) (cid:105) − (cid:104) (cid:105) (cid:104) (cid:105) x (k, t)x (k, t) x (k, t) x (k, t) α β α β C (k) = (14) αβ σ (k)σ (k) α β In ﬁgure 5(a) we show the average correlation between stocks (blue) and top eigenvalue λ /N (green) for the CAC 40. As can be seen the largest 1 eigenvalue is a measure of the average correlation between stocks [7, 13, 14, 15, 16]. This average correlation increases during the day from a value around 0.35 to a value around 0.45 when the market closes. For the case of smaller eigenvalues, what we can see is that the amplitude of risk factors decreases during the day (ﬁgure 5(b)), as more and more risk is carried by the market factor (ﬁgure 5(a)) [7]. In order to simplify the computation of the N2 correlation matrices for each bin k in the case of the S&P 500, we computed the correlation matrix C for 4 diﬀerent sets of stocks: r : composed by the 100 ﬁrst stocks of the αβ 0 S&P 500; r : composed by 100 stocks randomly picked; and r : composed 1,2 3 by 200 stocks randomly picked. Figure 6(a) shows λ1 as function of the bins. N Although the values of the eigenvalues seem to be out of scale, it can be seen clearly that the average correlation increases during the day. This scale conﬂict is solved by normalizing the value of the top eigenvalue not by N but by the sample size N (i.e. 100 or 200) (ﬁgure 6(b)). As can be seen 0 the average correlation of the index can be computed by taking a subset of it which means that actually just the more capitalized stocks in the index 8 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: CHEVALLEY-WEIL FORMULA FOR HYPERSURFACES IN PN-BUNDLES OVER CURVES AND MORDELL-WEIL RANKS IN FUNCTION FIELD TOWERS REMKE KLOOSTERMAN Abstract. Let X be a complex hypersurface in a Pn-bundle over a (cid:48) → curve C. Let C C be a Galois cover with group G. In this paper we × (cid:48) × (cid:48) describe the C[G]-structure of Hp,q(X C ) provided that X C × (cid:48) C C is either smooth or n = 3 and X C has at most ADE singularities. C As an application we obtain a geometric proof for an upper bound by Pa´l for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. 1. Introduction Let k be a ﬁeld of characteristic zero, C/k a smooth, geometrically integral (cid:48) → curve, and let f : C C be a (ramiﬁed) Galois cover with Galois group ∈ \ G. Let E/k(C) be a non-isotrivial elliptic curve, i.e., with j(E) k(C) k → and let π : X C be the associated relatively minimal elliptic surface with ⊂ section. Let R C be the set of points over which f is ramiﬁed and let s \ be the number of points in R. Let e be the Euler characteristic of C R, − − i.e., e = 2 2g(C) s. Assume that the discriminant of π does not vanish at any point in R. Let c and d be the degree of the conductor of E/k(C) and the degree E E of the minimal discriminant of E, respectively. P´al showed in [12] using equivariant Grothendieck-Ogg-Shafarevich theory that (cid:48) ≤ − − (1.1) rank E(k(C )) (cid:15)(G, k)(c d /6 e) E E where (cid:15)(G, k) is the Ellenberg constant of (G, k), for a deﬁnition see [3]. This constant depends only on the group G and the ﬁeld K, but not on E. In this paper we will gibe an alternative proof for this bound This paper grew out of an attempt to generalize this simpliﬁed approach (cid:48) to the case where g(C ) > 0. Date: May 27, 2020. 1991 Mathematics Subject Classiﬁcation. 14J27. The author would like to thank Gavril Farkas for pointing out the existence of [1], Mike Roth for various discussions on the best way to calculate the cohomology of Ωi E (kX) for ⊂ E P( ) a hypersurface X P( ) and Orsola Tommasi for various remarks on a previous version of this paper. 1 ##LTLine##

text: CHEVALLEY-WEIL

label: paragraph

text: FORMULA

text: FOR

text: HYPERSURFACES

text: IN

text: PN-BUNDLES

text: OVER

text: CURVES

text: AND

text: MORDELL-WEIL

text: RANKS

text: IN

text: FUNCTION

text: FIELD

text: TOWERS

text: REMKE

text: KLOOSTERMAN

text: Abstract.

text: Let

label: abstract

text: X

text: be

text: a

text: complex

text: hypersurface

text: in

text: a

text: Pn-bundle

text: over

text: a

text: curve

text: C.

text: Let

text: C

text: C

text: be

text: a

text: Galois

text: cover

text: with

text: group

text: G.

text: In

text: this

text: paper

text: we

text: describe

text: the

text: C[G]-structure

text: of

text: Hp,q(X

text: C

text: provided

text: that

text: X

text: C

text: C

text: C

text: is

text: either

text: smooth

text: or

text: n

text: and

text: X

text: C

text: has

text: at

text: most

text: ADE

text: singularities.

text: C

text: As

text: an

text: application

text: we

text: obtain

text: a

text: geometric

text: proof

text: for

text: an

text: upper

text: bound

text: by

text: Pa´l

text: for

text: the

text: Mordell-Weil

text: rank

text: of

text: an

text: elliptic

text: surface

text: obtained

text: by

text: a

text: Galois

text: base

text: change

text: of

text: another

text: elliptic

text: surface.

text: Introduction

text: Let

text: k

text: be

text: a

text: ﬁeld

text: of

text: characteristic

text: zero,

text: C/k

text: a

text: smooth,

text: geometrically

text: integral

text: curve,

text: and

text: let

text: f

text: C

text: C

text: be

text: a

text: (ramiﬁed)

text: Galois

text: cover

text: with

text: Galois

text: group

text: G.

text: Let

text: E/k(C)

text: be

text: a

text: non-isotrivial

text: elliptic

text: curve,

text: i.e.,

text: with

text: j(E)

text: k(C)

text: k

text: and

text: let

text: π

text: X

text: C

text: be

text: the

text: associated

text: relatively

text: minimal

text: elliptic

text: surface

text: with

text: section.

text: Let

text: R

text: C

text: be

text: the

text: set

text: of

text: points

text: over

text: which

text: f

text: is

text: ramiﬁed

text: and

text: let

text: s

text: be

text: the

text: number

text: of

text: points

text: in

text: R.

text: Let

text: e

text: be

text: the

text: Euler

text: characteristic

text: of

text: C

text: R,

text: i.e.,

text: e

text: s.

text: Assume

text: that

text: the

text: discriminant

text: of

text: π

text: does

text: not

text: vanish

text: at

text: any

text: point

text: in

text: R.

text: Let

text: c

text: and

text: d

text: be

text: the

text: degree

text: of

text: the

text: conductor

text: of

text: E/k(C)

text: and

text: the

text: degree

text: E

text: E

text: of

text: the

text: minimal

text: discriminant

text: of

text: E,

text: respectively.

text: P´al

text: showed

text: in

text: using

text: equivariant

text: Grothendieck-Ogg-Shafarevich

text: theory

text: that

text: rank

text: E(k(C

text: k)(c

text: d

text: e)

text: E

text: E

text: where

text: k)

text: is

text: the

text: Ellenberg

text: constant

text: of

text: (G,

text: k),

text: for

text: a

text: deﬁnition

text: see

text: This

text: constant

text: depends

text: only

text: on

text: the

text: group

text: G

text: and

text: the

text: ﬁeld

text: K,

text: but

text: not

text: on

text: E.

text: In

text: this

text: paper

text: we

text: will

text: gibe

text: an

text: alternative

text: proof

text: for

text: this

text: bound

text: This

text: paper

text: grew

text: out

text: of

text: an

text: attempt

text: to

text: generalize

text: this

text: simpliﬁed

text: approach

text: to

text: the

text: case

text: where

text: g(C

text: Date:

text: May

text: Mathematics

text: Subject

text: Classiﬁcation.

text: The

text: author

text: would

text: like

text: to

text: thank

text: Gavril

text: Farkas

text: for

text: pointing

text: out

text: the

text: existence

text: of

text: Mike

text: Roth

text: for

text: various

text: discussions

text: on

text: the

text: best

text: way

text: to

text: calculate

text: the

text: cohomology

text: of

text: Ωi

text: E

text: (kX)

text: for

text: E

text: P(

text: a

text: hypersurface

text: X

text: P(

text: and

text: Orsola

text: Tommasi

text: for

text: various

text: remarks

text: on

text: a

text: previous

text: version

text: of

text: this

text: paper.

text: ##LTLine##


text: 14 R. KLOOSTERMAN • Since X has ADE singularities it follows from [16, Section 4] that Gr1 Ω˜ F X is concentrated in degree one, and in this degree it is isomorphic to Ω˜1 . X Moreover, in the same section∼Steenbrink shows that for a surface with ADE singularities we have Ω˜1 = ω1 . This implies Hi(ω1 ) = Gr1 H1+i(X) X X X F and hence − χ (ω1 ) = [H1,0(X)] [H1,1(X)] + [H1,2(X)]. G X The deﬁnition of ω1 yields the equality X − χ (ω1 ) = χ (Ω2 (log X)) χ (Ω2 ). G X G P G P Since P is a smooth threefold we ﬁnd that − − χ (Ω2 ) = [H2,0(P )] [H2,1(P )] + [H2,2(P )] [H2,3(P )]. G P Using the deﬁnition of Ω2 (log X) we ﬁnd P − T χ (Ω2 (log X)) = χ (Ω2 (X)) χ (K (2X)) + χ (K (X)) + χ ( ). G P G P G P G P G (cid:3) ∼ Remark 3.11. If Hi(X) = Hi(P ) holds for i = 1 and i = 3 then [H1,0(X)] + [H1,2(X)] = [H2,1(P )] + [H2,3(P )] If, moreover, H2,0(P ) = 0 we have further simpliﬁcations in the formula from Proposition 3.10. O ⊕ ∗L− ⊕ ∗L− In case P = P( f 2 f 3) and X a Weierstrass model all these cancellations happen, and, moreover, [H2,2(P )] = 2[C] in K(C[G]). E Corollary 3.12. Let be a direct sum of at least three line bundles on a ⊂ E (cid:48) → smooth projective curve C. Let X P( ) be a hypersurface. Let f : C (cid:48) × (cid:48) ⊂ ∗E C be a Galois cover. Let X = X C P(f ) be the base-changed C ∗ E → (cid:48) hypersurface. Assume that the natural map Hi(P(f ( ))) Hi(X ) is an ≤ (cid:48) (cid:48) isomorphism for 0 i < dim X and for i = dim X this map is injective. (cid:48) ∈ If X is smooth then for each p, q Z there exist integers a, b, c, depending E on p, q, the degrees of the direct summands of and the ﬁber degree of X, (cid:48) O such that [Hp,q(X )] = a[C] + bχ ( ) + c[C[G]]. (cid:48) G ∈ If X is surface with at most ADE singularities for each p, q Z there exist integers a, b, c, depending on p, q, the degrees of the direct summand of E (cid:48) O and the ﬁber degree of X, such that [Hp,q(X )] = aC+bχ ( )+c[C[G]]+ T (cid:54) G δ[H0( )], where δ = 0 for (p, q) = (1, 1) and δ = 1 for (p, q) = (1, 1). E ⊂ E Corollary 3.13. Let be a direct sum of three line bundles. Let W P( ) (cid:48) → (cid:48) × (cid:48) be a surface. Let C C be a Galois base change such that W :(cid:48)=∼W C C is a surface with at most ADE singularities and such that H1(W ) = H1(P). (cid:48) (cid:48) (cid:48) Let X be the desingularization of W . Then [H1,1(W )] equals − (cid:48) (cid:48) − (cid:48) − T 2[C] χ (Ω2(W )) + χ (K ∗E (2W )) χ (K ∗E (W )) χ ( ) G G P(f ) G P(f ) G and (cid:48) − (cid:48) (cid:48) − (cid:48) [H1,1(X )] = 2[C] χ (Ω2(W )) + χ (K ∗E (2W )) χ (K ∗E (W )) G G P(f ) G P(f )

text: R.

text: KLOOSTERMAN

text: Since

text: X

text: has

text: ADE

text: singularities

text: it

text: follows

text: from

text: Section

text: that

text: Ω˜

text: F

text: X

text: is

text: concentrated

text: in

text: degree

text: one,

text: and

text: in

text: this

text: degree

text: it

text: is

text: isomorphic

text: to

text: Ω˜1

text: X

text: Moreover,

text: in

text: the

text: same

text: section∼Steenbrink

text: shows

text: that

text: for

text: a

text: surface

text: with

text: ADE

text: singularities

text: we

text: have

text: Ω˜1

text: ω1

text: This

text: implies

text: X

text: X

text: X

text: F

text: and

text: hence

text: χ

text: G

text: X

text: The

text: deﬁnition

text: of

text: ω1

text: yields

text: the

text: equality

text: X

text: χ

text: χ

text: (log

text: X))

text: χ

text: G

text: X

text: G

text: P

text: G

text: P

text: Since

text: P

text: is

text: a

text: smooth

text: threefold

text: we

text: ﬁnd

text: that

text: χ

text: G

text: P

text: Using

text: the

text: deﬁnition

text: of

text: Ω2

text: (log

text: X)

text: we

text: ﬁnd

text: P

text: T

text: χ

text: (log

text: X))

text: χ

text: (X))

text: χ

text: (K

text: χ

text: (K

text: (X))

text: χ

text: G

text: P

text: G

text: P

text: G

text: P

text: G

text: P

text: G

text: Remark

text: If

text: Hi(X)

text: Hi(P

text: holds

text: for

text: i

text: and

text: i

text: then

text: If,

text: moreover,

text: we

text: have

text: further

text: simpliﬁcations

text: in

text: the

text: formula

text: from

text: Proposition

text: O

text: ∗L−

text: ∗L−

text: In

text: case

text: P

text: P(

text: f

text: f

text: and

text: X

text: a

text: Weierstrass

text: model

text: all

text: these

text: cancellations

text: happen,

text: and,

text: moreover,

text: in

text: K(C[G]).

text: E

text: Corollary

text: Let

text: be

text: a

text: direct

text: sum

text: of

text: at

text: least

text: three

text: line

text: bundles

text: on

text: a

text: E

text: smooth

text: projective

text: curve

text: C.

text: Let

text: X

text: P(

text: be

text: a

text: hypersurface.

text: Let

text: f

text: C

text: ∗E

text: C

text: be

text: a

text: Galois

text: cover.

text: Let

text: X

text: X

text: C

text: P(f

text: be

text: the

text: base-changed

text: C

text: E

text: hypersurface.

text: Assume

text: that

text: the

text: natural

text: map

text: Hi(P(f

text: Hi(X

text: is

text: an

text: isomorphism

text: for

text: i

text: dim

text: X

text: and

text: for

text: i

text: dim

text: X

text: this

text: map

text: is

text: injective.

text: If

text: X

text: is

text: smooth

text: then

text: for

text: each

text: p,

text: q

text: Z

text: there

text: exist

text: integers

text: a,

text: b,

text: c,

text: depending

text: E

text: on

text: p,

text: q,

text: the

text: degrees

text: of

text: the

text: direct

text: summands

text: of

text: and

text: the

text: ﬁber

text: degree

text: of

text: X,

text: O

text: such

text: that

text: [Hp,q(X

text: a[C]

text: bχ

text: c[C[G]].

text: G

text: If

text: X

text: is

text: surface

text: with

text: at

text: most

text: ADE

text: singularities

text: for

text: each

text: p,

text: q

text: Z

text: there

text: exist

text: integers

text: a,

text: b,

text: c,

text: depending

text: on

text: p,

text: q,

text: the

text: degrees

text: of

text: the

text: direct

text: summand

text: of

text: E

text: O

text: and

text: the

text: ﬁber

text: degree

text: of

text: X,

text: such

text: that

text: [Hp,q(X

text: aC+bχ

text: )+c[C[G]]+

text: T

text: G

text: where

text: δ

text: for

text: (p,

text: q)

text: and

text: δ

text: for

text: (p,

text: q)

text: E

text: E

text: Corollary

text: Let

text: be

text: a

text: direct

text: sum

text: of

text: three

text: line

text: bundles.

text: Let

text: W

text: P(

text: be

text: a

text: surface.

text: Let

text: C

text: C

text: be

text: a

text: Galois

text: base

text: change

text: such

text: that

text: W

text: C

text: C

text: is

text: a

text: surface

text: with

text: at

text: most

text: ADE

text: singularities

text: and

text: such

text: that

text: Let

text: X

text: be

text: the

text: desingularization

text: of

text: W

text: Then

text: equals

text: T

text: χ

text: χ

text: (K

text: ∗E

text: χ

text: (K

text: ∗E

text: (W

text: χ

text: G

text: G

text: P(f

text: G

text: P(f

text: G

text: and

text: χ

text: χ

text: (K

text: ∗E

text: χ

text: (K

text: ∗E

text: (W

text: G

text: G

text: P(f

text: G

text: P(f


text: CHEVALLEY-WEIL FOR ELLIPTIC SURFACES 7 − then H0(E) = Cs and H2(E) = C( 1)µ and all other cohomology groups vanish. Let Σ = W . From [13, Corollary-Deﬁnition 5.37] it follows that we sing have a long exact sequence of MHS · · · → → ⊕ → → → (2.1) Hi(W ) Hi(X) Hi(Σ) Hi(E) Hi+1(W ) . . . (cid:54) → Note that hi(Σ) = 0 for i = 0. Moreover, the map H0(Σ) H0(E) is clearly an isomorphism, combi∼ning this with the fact that Hi(E) = 0 for (cid:54) (cid:54) i = 0, 2 we obtain that Hi(X) = Hi(W ) for i = 2, 3. → To prove the proposition it suﬃces to show that the map H2(E) − H3(W ) is zero. As H2(E) = C( 1)µ has a pure Hodge structure of weight 2 it suﬃces to show that all the nontrivally Hodge weights of H3(W ) are at least 3. If W is smooth then this is trivially true, so suppose that W is singular. Consider the long exact sequence of the pair (W, W ). Since W smooth has only ADE singularities and the dimension of W is even it follows that (cid:54) − Hi (W ) = 0 for i = 4, and H4(W ) = C( 2)s. The long∼exact sequence of Σ (cid:48) (cid:48) Σ the pair (W , W ) now yields isomorphisms Hi(W ) = Hi(W ) for (cid:54) smooth smooth i = 3, 4 and an exact sequence → → → − → (cid:48) → 0 H3(W ) H3(W ) C( 2)#Σ H4(W ) 0 = H4(W ) smooth smooth Since W is smooth we have that the Hodge weights of H3(W ) smooth smooth(cid:3) are at least 3, and hence the same statement holds true for H3(W ). → ∗ → Lemma 2.9. Consider the inclusion i : W P. Then i : Hk(P) Hk(W ) is an isomorphism for k = 0, 1, 3, is injective for k = 2 and is surjective for k = 4. Proof. For k = 0 the statement is trivial. The case k = 1 can be shown → ◦ → as follows: Consider σ : C W and i σ : C P. Combining these 0→ 0 → morphisms with π : W C, respectively ψ : P C, yield the identity on ∗ ◦ ∗ ∗ ◦ ◦ ∗ C. This implies that π σ and ψ (i σ ) are isomorphisms and that ∗ → 0 0 σ : Hk(C) Hk(W ) is injective. 0 From [9, Lemma IV.1.1] it follows that h1(C) = h1(X) and by the pre- ∗ ∗ vious proposition we have h1(W ) = h1(X). In particular σ and (iσ ) are ∗ 0 0 isomorphisms and therefore i is an isomorphism. For k = 2 note that H2(P) is generated by the ﬁrst Chern classes of a ﬁber O of ϕ and (1). Their images in H2(X) are clearly independent, hence the P → → composition H2(P) H2(W ) H2(X) is injective. For k = 4 note that O ∈ the selﬁntersection of c ( (1)) H4(P) is mapped to a nonzero element 1 P → → in the one-dimensional vector spac∼e H4(X). Hence H4(P) H4(W ) H4(X) is surjective. Since H4(W ) = H4(X) this case follows also. The case k = 3 is slightly more complicated. By successively blowing up points in P we ﬁnd a variety P˜ such that the strict transform of W is

text: CHEVALLEY-WEIL

text: FOR

text: ELLIPTIC

text: SURFACES

text: then

text: Cs

text: and

text: C(

text: and

text: all

text: other

text: cohomology

text: groups

text: vanish.

text: Let

text: Σ

text: W

text: From

text: Corollary-Deﬁnition

text: it

text: follows

text: that

text: we

text: sing

text: have

text: a

text: long

text: exact

text: sequence

text: of

text: MHS

label: equation

text: Hi(W

text: Hi(X)

text: Hi(Σ)

text: Hi(E)

font: default

text: Note

text: that

text: hi(Σ)

text: for

text: i

text: Moreover,

text: the

text: map

text: is

text: clearly

text: an

text: isomorphism,

text: combi∼ning

text: this

text: with

text: the

text: fact

text: that

text: Hi(E)

text: for

text: i

text: we

text: obtain

text: that

text: Hi(X)

text: Hi(W

text: for

text: i

text: To

text: prove

text: the

text: proposition

text: it

text: suﬃces

text: to

text: show

text: that

text: the

text: map

text: is

text: zero.

text: As

text: C(

text: has

text: a

text: pure

text: Hodge

text: structure

text: of

text: weight

text: it

text: suﬃces

text: to

text: show

text: that

text: all

text: the

text: nontrivally

text: Hodge

text: weights

text: of

text: are

text: at

text: least

text: If

text: W

text: is

text: smooth

text: then

text: this

text: is

text: trivially

text: true,

text: so

text: suppose

text: that

text: W

text: is

text: singular.

text: Consider

text: the

text: long

text: exact

text: sequence

text: of

text: the

text: pair

text: (W,

text: W

text: Since

text: W

text: smooth

text: has

text: only

text: ADE

text: singularities

text: and

text: the

text: dimension

text: of

text: W

text: is

text: even

text: it

text: follows

text: that

text: Hi

text: (W

text: for

text: i

text: and

text: C(

text: The

text: long∼exact

text: sequence

text: of

text: Σ

text: Σ

text: the

text: pair

text: (W

text: W

text: now

text: yields

text: isomorphisms

text: Hi(W

text: Hi(W

text: for

text: smooth

text: smooth

text: i

text: and

text: an

text: exact

text: sequence

text: C(

text: smooth

text: smooth

text: Since

text: W

text: is

text: smooth

text: we

text: have

text: that

text: the

text: Hodge

text: weights

text: of

text: smooth

text: are

text: at

text: least

text: and

text: hence

text: the

text: same

text: statement

text: holds

text: true

text: for

text: Lemma

text: Consider

text: the

text: inclusion

text: i

text: W

text: P.

text: Then

text: i

text: Hk(P)

text: Hk(W

text: is

text: an

text: isomorphism

text: for

text: k

text: is

text: injective

text: for

text: k

text: and

text: is

text: surjective

text: for

text: k

text: Proof.

text: For

text: k

text: the

text: statement

text: is

text: trivial.

text: The

text: case

text: k

text: can

text: be

text: shown

text: as

text: follows:

text: Consider

text: σ

text: C

text: W

text: and

text: i

text: σ

text: C

text: P.

text: Combining

text: these

text: 0→

text: morphisms

text: with

text: π

text: W

text: C,

text: respectively

text: ψ

text: P

text: C,

text: yield

text: the

text: identity

text: on

text: C.

text: This

text: implies

text: that

text: π

text: σ

text: and

text: ψ

text: (i

text: σ

text: are

text: isomorphisms

text: and

text: that

text: σ

text: Hk(C)

text: Hk(W

text: is

text: injective.

text: From

text: Lemma

text: it

text: follows

text: that

text: and

text: by

text: the

text: pre-

text: vious

text: proposition

text: we

text: have

text: In

text: particular

text: σ

text: and

text: (iσ

text: are

text: isomorphisms

text: and

text: therefore

text: i

text: is

text: an

text: isomorphism.

text: For

text: k

text: note

text: that

text: is

text: generated

text: by

text: the

text: ﬁrst

text: Chern

text: classes

text: of

text: a

text: ﬁber

text: O

text: of

text: ϕ

text: and

text: Their

text: images

text: in

text: are

text: clearly

text: independent,

text: hence

text: the

text: P

text: composition

text: is

text: injective.

text: For

text: k

text: note

text: that

text: O

text: the

text: selﬁntersection

text: of

text: c

text: is

text: mapped

text: to

text: a

text: nonzero

text: element

text: P

text: in

text: the

text: one-dimensional

text: vector

text: spac∼e

text: Hence

text: is

text: surjective.

text: Since

text: this

text: case

text: follows

text: also.

text: The

text: case

text: k

text: is

text: slightly

text: more

text: complicated.

text: By

text: successively

text: blowing

text: up

text: points

text: in

text: P

text: we

text: ﬁnd

text: a

text: variety

text: P˜

text: such

text: that

text: the

text: strict

text: transform

text: of

text: W

text: is


text: outcome of standard procedures. On this application, normalized mu- tual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering. Keywords: agglomerative hierarchical clustering; Bayesian model comparison; BIC; mutual information; multivariate normal distribu- tions; normalized mutual information. 1 Introduction Cluster analysis aims at uncovering natural groups of objects in a multi- variate dataset [see Jain (2010) for a review]. In the vast variety of methods used in cluster analysis, an agglomerative hierarchical clustering (AHC) is a generic procedure that sequentially merges pairs of clusters that are most similar according to an arbitrary function called similarity measure, thereby generating a nested set of partitions, also called hierarchy (Duda et al., 2000). The choice of the similarity measure indirectly deﬁnes the shape of the clusters, and thus plays a critical role in the clustering process. While this choice is guided by the features of the problem at hand, it is also of- ten restricted to a limited number of commonly used measures, such as the Euclidean distance or Pearson correlation coeﬃcient (D’haeseleer, 2005). In the present work, we focus on the clustering of random variables based on their mutual information, which has recently gained in popularity in clus- ter analysis, notably in the ﬁeld of genomics (Butte and Kohane, 2000; Zhou et al., 2004; Dawy et al., 2006; Priness et al., 2007) and in functional magnetic resonance imaging (fMRI) data analysis (Stausberg et al., 2009; Benjaminsson et al., 2010; Kolchinsky et al., 2014). Mutual information is a general measure of statistical dependency derived from information theory (Shannon, 1948; Kullback, 1968; Cover and Thomas, 1991). A key feature of mutual information is its ability to capture nonlinear interactions for any type of random variables (Steuer et al., 2002); also of interest, it indiﬀerently applies to univariate or multivariate variables and can thus be applied to clusters of arbitrary size. Yet, mutual information is an extensive measure that increases with variable dimensionality. In addition, Iˆ, the ﬁnite-sample estimator of mutual information, suﬀers from a dimensionality-dependent bias (see Appendix ). Several authors have proposed to correct mutual information for dimensionality by using a “normalized” version of mutual information (Li et al., 2001; Kraskov et al., 2005; Kraskov and Grassberger, 2009). In the clustering literature, normalized mutual information is rou- tinely used. However, the impact of such correction procedure has not been extensively evaluated so far. 2

text: outcome

text: of

text: standard

text: procedures.

text: On

text: this

text: application,

text: normalized

text: mu-

text: tual

text: information

text: had

text: a

text: highly

text: atypical

text: behavior,

text: in

text: the

text: sense

text: that

text: it

text: systematically

text: favored

text: very

text: large

text: clusters.

text: These

text: initial

text: experiments

text: suggest

text: that

text: the

text: proposed

text: Bayesian

text: alternatives

text: to

text: mutual

text: information

text: are

text: a

text: useful

text: new

text: tool

text: for

text: hierarchical

text: clustering.

text: Keywords:

text: agglomerative

text: hierarchical

text: clustering;

text: Bayesian

text: model

text: comparison;

text: BIC;

text: mutual

text: information;

text: multivariate

text: normal

text: distribu-

text: tions;

text: normalized

text: mutual

text: information.

label: section

text: Introduction

text: Cluster

text: analysis

text: aims

text: at

text: uncovering

text: natural

text: groups

text: of

text: objects

text: in

text: a

text: multi-

text: variate

text: dataset

text: [see

text: Jain

text: for

text: a

text: review].

text: In

text: the

text: vast

text: variety

text: of

text: methods

text: used

text: in

text: cluster

text: analysis,

text: an

text: agglomerative

text: hierarchical

text: clustering

text: (AHC)

text: is

text: a

text: generic

text: procedure

text: that

text: sequentially

text: merges

text: pairs

text: of

text: clusters

text: that

text: are

text: most

text: similar

text: according

text: to

text: an

text: arbitrary

text: function

text: called

text: similarity

text: measure,

text: thereby

text: generating

text: a

text: nested

text: set

text: of

text: partitions,

text: also

text: called

text: hierarchy

text: (Duda

text: et

text: al.,

text: The

text: choice

text: of

text: the

text: similarity

text: measure

text: indirectly

text: deﬁnes

text: the

text: shape

text: of

text: the

text: clusters,

text: and

text: thus

text: plays

text: a

text: critical

text: role

text: in

text: the

text: clustering

text: process.

text: While

text: this

text: choice

text: is

text: guided

text: by

text: the

text: features

text: of

text: the

text: problem

text: at

text: hand,

text: it

text: is

text: also

text: of-

text: ten

text: restricted

text: to

text: a

text: limited

text: number

text: of

text: commonly

text: used

text: measures,

text: such

text: as

text: the

text: Euclidean

text: distance

text: or

text: Pearson

text: correlation

text: coeﬃcient

text: (D’haeseleer,

text: In

text: the

text: present

text: work,

text: we

text: focus

text: on

text: the

text: clustering

text: of

text: random

text: variables

text: based

text: on

text: their

text: mutual

text: information,

text: which

text: has

text: recently

text: gained

text: in

text: popularity

text: in

text: clus-

text: ter

text: analysis,

text: notably

text: in

text: the

text: ﬁeld

text: of

text: genomics

text: (Butte

text: and

text: Kohane,

text: Zhou

text: et

text: al.,

text: Dawy

text: et

text: al.,

text: Priness

text: et

text: al.,

text: and

text: in

text: functional

text: magnetic

text: resonance

text: imaging

text: (fMRI)

text: data

text: analysis

text: (Stausberg

text: et

text: al.,

text: Benjaminsson

text: et

text: al.,

text: Kolchinsky

text: et

text: al.,

text: Mutual

text: information

text: is

text: a

text: general

text: measure

text: of

text: statistical

text: dependency

text: derived

text: from

text: information

text: theory

text: (Shannon,

text: Kullback,

text: Cover

text: and

text: Thomas,

text: A

text: key

text: feature

text: of

text: mutual

text: information

text: is

text: its

text: ability

text: to

text: capture

text: nonlinear

text: interactions

text: for

text: any

text: type

text: of

text: random

text: variables

text: (Steuer

text: et

text: al.,

text: also

text: of

text: interest,

text: it

text: indiﬀerently

text: applies

text: to

text: univariate

text: or

text: multivariate

text: variables

text: and

text: can

text: thus

text: be

text: applied

text: to

text: clusters

text: of

text: arbitrary

text: size.

text: Yet,

text: mutual

text: information

text: is

text: an

text: extensive

text: measure

text: that

text: increases

text: with

text: variable

text: dimensionality.

text: In

text: addition,

text: Iˆ,

text: the

text: ﬁnite-sample

text: estimator

text: of

text: mutual

text: information,

text: suﬀers

text: from

text: a

text: dimensionality-dependent

text: bias

text: (see

text: Appendix

text: Several

text: authors

text: have

text: proposed

text: to

text: correct

text: mutual

text: information

text: for

text: dimensionality

text: by

text: using

text: a

text: “normalized”

text: version

text: of

text: mutual

text: information

text: (Li

text: et

text: al.,

text: Kraskov

text: et

text: al.,

text: Kraskov

text: and

text: Grassberger,

text: In

text: the

text: clustering

text: literature,

text: normalized

text: mutual

text: information

text: is

text: rou-

text: tinely

text: used.

text: However,

text: the

text: impact

text: of

text: such

text: correction

text: procedure

text: has

text: not

text: been

text: extensively

text: evaluated

text: so

text: far.


text: Figure 2: Simulation study. Computational time (top), adjusted Rand index (middle) and proportion of correct classiﬁcations (bottom) for D = 20 (left) and D = 40 (right). 17 ##LTFigure##

text: Figure

text: Simulation

label: caption

text: study.

text: Computational

text: time

text: (top),

text: adjusted

text: Rand

text: index

text: (middle)

text: and

text: proportion

text: of

text: correct

text: classiﬁcations

text: (bottom)

text: for

text: D

text: (left)

text: and

text: D

text: (right).

text: ##LTFigure##

label: figure


text: This data set was ﬁrst used in Roverato (1999) and later re-analyzed in Marrelec and Benali (2006) in the context of conditional independence graphs. It originates from a study investigating early diagnosis of HIV in- fection in children from HIV positive mothers. The variables are related to various measures on blood and its components: X and X immunoglobin 1 2 G and A, respectively; X the platelet count; X , X lymphocyte B and 4 3 5 T4, respectively; and X the T4/T8 lymphocyte ratio. The sample sum- 6 mary statistics are given in Table 1. Roverato (1999) found that the cor- relations between X and any other variable had strong probability around 4 zero and hypothesized that the model was overparametrized. Based on the simulation study, we performed clustering of the data with the fol- lowing methods: BayesCov(Auto), BayesCorr(Auto), Bic(Auto), Infomut, InfomutNorm, SingleAbs, AverageAbs, CompleteAbs, WardAbs, SpecCorrAbs and SpecCorrparAbs. For spectral clustering, we used either 1 or 10 repe- titions of the k-means step; since the results obtained for 1 step of k-means were highly variable for 3, 4, and 5 clusters, we discarded these results. Table 1: Toy example. Summary statistics for the HIV data: sample vari- ances (main diagonal), correlations (lower triangle) and partial correlations (upper triangle) [from Roverato (1999)]. − − − x 8.8374 0.479 0.043 0.033 0.356 0.236 1 − − − x 0.483 0.1919 0.068 0.084 0.224 0.110 2 − x 0.220 0.057 8924231.9 0.085 0.552 0.330 3 − − x 0.040 0.133 0.149 20392.4 0.091 0.013 4 − x 0.253 0.124 0.523 0.179 1952795.2 0.384 5 − − − x 0.276 0.314 0.183 0.064 0.213 1.378 6 x x x x x x 1 2 3 4 5 6 The resulting clusterings are given in Fig 4 and Table 2. All hierar- chical clusterings started by clustering X and X (lymphocite). This was 3 5 conﬁrmed by SpecCorrAbs-10 when required to provide 5 clusters. All hier- archical clustering methods then clustered X and X (immunoglobin). This 1 2 result was in agreement with both SpecCorrAbs-10 and SpecCorrparAbs-10 when required to provide 4 clusters. After the second step, we observed four behaviors for the AHC algorithms and classiﬁed them accordingly: (G1) BayesCov, BayesCorr, Infomut and InfomutNorm; (G2) SingleAbs and AverageAbs; (G3) Bic; (G4) CompleteAbs and WardAbs. While not a hierarchical clustering, SpecCorrAbs-10 provided successive clusterings that were in agreement with methods in (G2). Algorithms in 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: data

text: set

text: was

text: ﬁrst

text: used

text: in

text: Roverato

text: and

text: later

text: re-analyzed

text: in

text: Marrelec

text: and

text: Benali

text: in

text: the

text: context

text: of

text: conditional

text: independence

text: graphs.

text: It

text: originates

text: from

text: a

text: study

text: investigating

text: early

text: diagnosis

text: of

text: HIV

text: in-

text: fection

text: in

text: children

text: from

text: HIV

text: positive

text: mothers.

text: The

text: variables

text: are

text: related

text: to

text: various

text: measures

text: on

text: blood

text: and

text: its

text: components:

text: X

text: and

text: X

text: immunoglobin

text: G

text: and

text: A,

text: respectively;

text: X

text: the

text: platelet

text: count;

text: X

text: X

text: lymphocyte

text: B

text: and

text: respectively;

text: and

text: X

text: the

text: lymphocyte

text: ratio.

text: The

text: sample

text: sum-

text: mary

text: statistics

text: are

text: given

text: in

text: Table

text: Roverato

text: found

text: that

text: the

text: cor-

text: relations

text: between

text: X

text: and

text: any

text: other

text: variable

text: had

text: strong

text: probability

text: around

text: zero

text: and

text: hypothesized

text: that

text: the

text: model

text: was

text: overparametrized.

text: Based

text: on

text: the

text: simulation

text: study,

text: we

text: performed

text: clustering

text: of

text: the

text: data

text: with

text: the

text: fol-

text: lowing

text: methods:

text: BayesCov(Auto),

text: BayesCorr(Auto),

text: Bic(Auto),

text: Infomut,

text: InfomutNorm,

text: SingleAbs,

text: AverageAbs,

text: CompleteAbs,

text: WardAbs,

text: SpecCorrAbs

text: and

text: SpecCorrparAbs.

text: For

text: spectral

text: clustering,

text: we

text: used

text: either

text: or

text: repe-

text: titions

text: of

text: the

text: k-means

text: step;

text: since

text: the

text: results

text: obtained

text: for

text: step

text: of

text: k-means

text: were

text: highly

text: variable

text: for

text: and

text: clusters,

text: we

text: discarded

text: these

text: results.

text: Table

text: Toy

text: example.

text: Summary

text: statistics

text: for

text: the

text: HIV

text: data:

text: sample

text: vari-

text: ances

text: (main

text: diagonal),

text: correlations

text: (lower

text: triangle)

text: and

text: partial

text: correlations

text: (upper

text: triangle)

text: [from

text: Roverato

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: The

text: resulting

text: clusterings

text: are

text: given

text: in

text: Fig

text: and

text: Table

text: All

text: hierar-

text: chical

text: clusterings

text: started

text: by

text: clustering

text: X

text: and

text: X

text: (lymphocite).

text: This

text: was

text: conﬁrmed

text: by

text: when

text: required

text: to

text: provide

text: clusters.

text: All

text: hier-

text: archical

text: clustering

text: methods

text: then

text: clustered

text: X

text: and

text: X

text: (immunoglobin).

text: This

text: result

text: was

text: in

text: agreement

text: with

text: both

text: and

text: when

text: required

text: to

text: provide

text: clusters.

text: After

text: the

text: second

text: step,

text: we

text: observed

text: four

text: behaviors

text: for

text: the

text: AHC

text: algorithms

text: and

text: classiﬁed

text: them

text: accordingly:

text: BayesCov,

text: BayesCorr,

text: Infomut

text: and

text: InfomutNorm;

text: SingleAbs

text: and

text: AverageAbs;

text: Bic;

text: CompleteAbs

text: and

text: WardAbs.

text: While

text: not

text: a

text: hierarchical

text: clustering,

text: provided

text: successive

text: clusterings

text: that

text: were

text: in

text: agreement

text: with

text: methods

text: in

text: Algorithms

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 5: Toy example. Result of clustering for BayesCorr, BayesCov and Bic. The values on the y axis correspond to the log Bayes factor in 10 favor of the global clustering obtained at each step compared to a model where all variables are independent (step 0 of hierarchical clustering). The dotted lines correspond to clustering steps that were not performed with the automatic stopping rule. young healthy subjects. See Appendix for details on data collection and preparation. The data are available online7. We ﬁrst aimed at establishing which clustering algorithms yielded sim- ilar results on these datasets. We more speciﬁcally investigated a 7-cluster solution, as this level of decomposition has been examined several times in the literature (Bellec et al., 2010; Power et al., 2011; Yeo et al., 2011). Each clustering algorithm was applied to the time series of each subject independently. For a given pair of methods, the similarity between the cluster solutions generated by both methods on the same subject was eval- uated with the Rand. Note that the raw, unnormalized Rand index was used here, as we did not compare cluster solutions with diﬀerent numbers of clusters, which is the main motivation of the normalization. The unnor- malized Rand index has a more intuitive interpretation than its adjusted counterpart. The Rand indices were averaged across subjects, hence re- sulting into a method-by-method matrix capturing the (average) similarity of clustering outputs for each pair of methods (see Fig 6). An AHC with Ward’s criterion was applied to this matrix in order to identify clusters of methods with similar cluster outputs. We visually identiﬁed ﬁve clusters of methods that had high (> 0.7) average within-cluster Rand index. The largest cluster included classical AHCs such as CompleteAbs, AverageAbs, WardAbs, as well as the Bic and BayesCov methods proposed here. It should be noted that this class of algorithms generated solutions for this prob- lem that were very close to one another (average within-cluster Rand index 7http://figshare.com/articles/Atlanta resting state fMRI time series preprocessed using the AAL template/1521155 22 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Toy

text: example.

text: Result

text: of

text: clustering

text: for

text: BayesCorr,

text: BayesCov

text: and

text: Bic.

text: The

text: values

text: on

text: the

text: y

text: axis

text: correspond

text: to

text: the

text: log

text: Bayes

text: factor

text: in

text: favor

text: of

text: the

text: global

text: clustering

text: obtained

text: at

text: each

text: step

text: compared

text: to

text: a

text: model

text: where

text: all

text: variables

text: are

text: independent

text: (step

text: of

text: hierarchical

text: clustering).

text: The

text: dotted

text: lines

text: correspond

text: to

text: clustering

text: steps

text: that

text: were

text: not

text: performed

text: with

text: the

text: automatic

text: stopping

text: rule.

text: young

text: healthy

text: subjects.

text: See

text: Appendix

text: for

text: details

text: on

text: data

text: collection

text: and

text: preparation.

text: The

text: data

text: are

text: available

text: We

text: ﬁrst

text: aimed

text: at

text: establishing

text: which

text: clustering

text: algorithms

text: yielded

text: sim-

text: ilar

text: results

text: on

text: these

text: datasets.

text: We

text: more

text: speciﬁcally

text: investigated

text: a

text: solution,

text: as

text: this

text: level

text: of

text: decomposition

text: has

text: been

text: examined

text: several

text: times

text: in

text: the

text: literature

text: (Bellec

text: et

text: al.,

text: Power

text: et

text: al.,

text: Yeo

text: et

text: al.,

text: Each

text: clustering

text: algorithm

text: was

text: applied

text: to

text: the

text: time

text: series

text: of

text: each

text: subject

text: independently.

text: For

text: a

text: given

text: pair

text: of

text: methods,

text: the

text: similarity

text: between

text: the

text: cluster

text: solutions

text: generated

text: by

text: both

text: methods

text: on

text: the

text: same

text: subject

text: was

text: eval-

text: uated

text: with

text: the

text: Rand.

text: Note

text: that

text: the

text: raw,

text: unnormalized

text: Rand

text: index

text: was

text: used

text: here,

text: as

text: we

text: did

text: not

text: compare

text: cluster

text: solutions

text: with

text: diﬀerent

text: numbers

text: of

text: clusters,

text: which

text: is

text: the

text: main

text: motivation

text: of

text: the

text: normalization.

text: The

text: unnor-

text: malized

text: Rand

text: index

text: has

text: a

text: more

text: intuitive

text: interpretation

text: than

text: its

text: adjusted

text: counterpart.

text: The

text: Rand

text: indices

text: were

text: averaged

text: across

text: subjects,

text: hence

text: re-

text: sulting

text: into

text: a

text: method-by-method

text: matrix

text: capturing

text: the

text: (average)

text: similarity

text: of

text: clustering

text: outputs

text: for

text: each

text: pair

text: of

text: methods

text: (see

text: Fig

text: An

text: AHC

text: with

text: Ward’s

text: criterion

text: was

text: applied

text: to

text: this

text: matrix

text: in

text: order

text: to

text: identify

text: clusters

text: of

text: methods

text: with

text: similar

text: cluster

text: outputs.

text: We

text: visually

text: identiﬁed

text: ﬁve

text: clusters

text: of

text: methods

text: that

text: had

text: high

text: average

text: within-cluster

text: Rand

text: index.

text: The

text: largest

text: cluster

text: included

text: classical

text: AHCs

text: such

text: as

text: CompleteAbs,

text: AverageAbs,

text: WardAbs,

text: as

text: well

text: as

text: the

text: Bic

text: and

text: BayesCov

text: methods

text: proposed

text: here.

text: It

text: should

text: be

text: noted

text: that

text: this

text: class

text: of

text: algorithms

text: generated

text: solutions

text: for

text: this

text: prob-

text: lem

text: that

text: were

text: very

text: close

text: to

text: one

text: another

text: (average

text: within-cluster

text: Rand

text: index

text: resting

text: state

text: fMRI

text: time

text: series

text: preprocessed

text: using

text: the

text: AAL

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: estimated than the whole covariance matrix, while the same does not hold for a precision matrix. From there, we could expect the covariance-based approach to perform better than the precision-based approach, and the dif- ference to increase with increasing D. This was conﬁrmed on our synthetic data, where the precision-based approach behaved as well as BayesCov and BayesCorr for D = 6 but had worse results than these two approaches for ∈ { } D 10, 20, 40 . Besides, performance of the automated stopping rule was much more eﬃcient with BayesCov and BayesCorr than it was with the precision-based approach. As a ﬁnal note, basing the method on concen- tration matrices yielded a slower algorithm, arguably because of the matrix inversions that are required. Sample covariance matrix vs. full dataset. Intuitively, the structure of dependence of a multivarite normal distribution is included in its covari- ance matrix. All existing algorithms that we used here do not need the full dataset but only the sample covariance (or correlation) matrix. This is why we started with a likelihood model that only considers the covariance matrix [see Equations (3) and (7)]. Rigorously, this model is only valid for ≥ N D; for N < D, one should resort to a model of the full data as being multivariate normal with a mean µ and covariance matrix Σ. Nonetheless, we kept the ’intuitive’ approach as it has fewer hyperparameters, is easier to deal with, and leads to formulas that are simpler to interpret. Also, the ≥ resulting similarity measure [Equation (11)] is not restricted to N D, but is well deﬁned for N < D as well. From a practical perspective, the ’intu- itive’ algorithm only requires the sample covariance matrix as input, while a full model would also require the sample mean. Finally, this simpler model exhibited good behavior on our synthetic data, even for small datasets. 4.3 Directions for future work Computational costs. Regarding the computational cost, measures de- rived from mutual information or a Bayesian approach are more demanding than standard methods such as Average or Ward. The derivation of the similarity matrix and the search for the most similar pairs of clusters are the two critical operations that can be optimized to decrease computation time. It is always possible to speed up these two steps by taking advan- tage of the fact that the similarity matrices of two successive iterations of the algorithm have many elements in common, as all but one element of a partition at a given iteration are identical to the elements of the parti- tion of the previous iteration. Critically, in the case of Average and Ward methods, it is in addition possible to derive the similarity matrix at every iteration only based on the similarity matrix at initialization through succes- sive updates using the Lance-Jambu-Williams recursive formula (Batagelj, 1988). By contrast, other measures, including BayesCov, BayesCorr, Bic, 30

text: estimated

text: than

text: the

text: whole

text: covariance

text: matrix,

text: while

text: the

text: same

text: does

text: not

text: hold

text: for

text: a

text: precision

text: matrix.

text: From

text: there,

text: we

text: could

text: expect

text: the

text: covariance-based

text: approach

text: to

text: perform

text: better

text: than

text: the

text: precision-based

text: approach,

text: and

text: the

text: dif-

text: ference

text: to

text: increase

text: with

text: increasing

text: D.

text: This

text: was

text: conﬁrmed

text: on

text: our

text: synthetic

text: data,

text: where

text: the

text: precision-based

text: approach

text: behaved

text: as

text: well

text: as

text: BayesCov

text: and

text: BayesCorr

text: for

text: D

text: but

text: had

text: worse

text: results

text: than

text: these

text: two

text: approaches

text: for

text: D

text: Besides,

text: performance

text: of

text: the

text: automated

text: stopping

text: rule

text: was

text: much

text: more

text: eﬃcient

text: with

text: BayesCov

text: and

text: BayesCorr

text: than

text: it

text: was

text: with

text: the

text: precision-based

text: approach.

text: As

text: a

text: ﬁnal

text: note,

text: basing

text: the

text: method

text: on

text: concen-

text: tration

text: matrices

text: yielded

text: a

text: slower

text: algorithm,

text: arguably

text: because

text: of

text: the

text: matrix

text: inversions

text: that

text: are

text: required.

text: Sample

text: covariance

text: matrix

text: vs.

text: full

text: dataset.

text: Intuitively,

text: the

text: structure

text: of

text: dependence

text: of

text: a

text: multivarite

text: normal

text: distribution

text: is

text: included

text: in

text: its

text: covari-

text: ance

text: matrix.

text: All

text: existing

text: algorithms

text: that

text: we

text: used

text: here

text: do

text: not

text: need

text: the

text: full

text: dataset

text: but

text: only

text: the

text: sample

text: covariance

text: (or

text: correlation)

text: matrix.

text: This

text: is

text: why

text: we

text: started

text: with

text: a

text: likelihood

text: model

text: that

text: only

text: considers

text: the

text: covariance

text: matrix

text: [see

text: Equations

text: and

text: Rigorously,

text: this

text: model

text: is

text: only

text: valid

text: for

text: N

text: D;

text: for

text: N

text: D,

text: one

text: should

text: resort

text: to

text: a

text: model

text: of

text: the

text: full

text: data

text: as

text: being

text: multivariate

text: normal

text: with

text: a

text: mean

text: µ

text: and

text: covariance

text: matrix

text: Σ.

text: Nonetheless,

text: we

text: kept

text: the

text: ’intuitive’

text: approach

text: as

text: it

text: has

text: fewer

text: hyperparameters,

text: is

text: easier

text: to

text: deal

text: with,

text: and

text: leads

text: to

text: formulas

text: that

text: are

text: simpler

text: to

text: interpret.

text: Also,

text: the

text: resulting

text: similarity

text: measure

text: [Equation

text: is

text: not

text: restricted

text: to

text: N

text: D,

text: but

text: is

text: well

text: deﬁned

text: for

text: N

text: D

text: as

text: well.

text: From

text: a

text: practical

text: perspective,

text: the

text: ’intu-

text: itive’

text: algorithm

text: only

text: requires

text: the

text: sample

text: covariance

text: matrix

text: as

text: input,

text: while

text: a

text: full

text: model

text: would

text: also

text: require

text: the

text: sample

text: mean.

text: Finally,

text: this

text: simpler

text: model

text: exhibited

text: good

text: behavior

text: on

text: our

text: synthetic

text: data,

text: even

text: for

text: small

text: datasets.

text: Directions

text: for

text: future

text: work

text: Computational

text: costs.

text: Regarding

text: the

text: computational

text: cost,

text: measures

text: de-

text: rived

text: from

text: mutual

text: information

text: or

text: a

text: Bayesian

text: approach

text: are

text: more

text: demanding

text: than

text: standard

text: methods

text: such

text: as

text: Average

text: or

text: Ward.

text: The

text: derivation

text: of

text: the

text: similarity

text: matrix

text: and

text: the

text: search

text: for

text: the

text: most

text: similar

text: pairs

text: of

text: clusters

text: are

text: the

text: two

text: critical

text: operations

text: that

text: can

text: be

text: optimized

text: to

text: decrease

text: computation

text: time.

text: It

text: is

text: always

text: possible

text: to

text: speed

text: up

text: these

text: two

text: steps

text: by

text: taking

text: advan-

text: tage

text: of

text: the

text: fact

text: that

text: the

text: similarity

text: matrices

text: of

text: two

text: successive

text: iterations

text: of

text: the

text: algorithm

text: have

text: many

text: elements

text: in

text: common,

text: as

text: all

text: but

text: one

text: element

text: of

text: a

text: partition

text: at

text: a

text: given

text: iteration

text: are

text: identical

text: to

text: the

text: elements

text: of

text: the

text: parti-

text: tion

text: of

text: the

text: previous

text: iteration.

text: Critically,

text: in

text: the

text: case

text: of

text: Average

text: and

text: Ward

text: methods,

text: it

text: is

text: in

text: addition

text: possible

text: to

text: derive

text: the

text: similarity

text: matrix

text: at

text: every

text: iteration

text: only

text: based

text: on

text: the

text: similarity

text: matrix

text: at

text: initialization

text: through

text: succes-

text: sive

text: updates

text: using

text: the

text: Lance-Jambu-Williams

text: recursive

text: formula

text: (Batagelj,

text: By

text: contrast,

text: other

text: measures,

text: including

text: BayesCov,

text: BayesCorr,

text: Bic,


text: framework can be conveniently used to compare nested partitions, that is, diﬀerent levels of a hierarchy. We also deal with the issue of setting the hyperparameters. Finally, we show how the Bayesian solution can naturally provide for an automatic stopping rule. 2.1 Bayesian model comparison Let X be a D-dimensional multivariate normal variable with known mean2 µ and unknown covariance matrix Σ. Deﬁne X and X as two i j disjoint subvectors of X (of size D and D , respectively), and X ∪ as their i j i j union (of size D ∪ = D + D ). Assume that we have to decide whether i j i j we should cluster X and X based on their level of dependence. To this i j M M M end, consider two competing models and . In , X and X are I D I i j independent and the distribution of X ∪ can be decomposed as the product i j of the marginal distributions of X and X . Under such condition, Σ ∪ , i j i j the restriction of Σ to X ∪ , is block diagonal with blocks Σ and Σ , the i j M i j restrictions of Σ to X and X , respectively. In , by contrast, X and i j { } D i X are dependent. Given a dataset x , . . . , x of N independent and j 1 N identically distributed (i.i.d.) realizations of X and S the corresponding sample sum-of-square matrix(cid:88) N − − S = (x µ)(x µ)t, n n n=1 we propose to quantify the similarity between X and X as the log Bayes i j M factor, that is, the log ratio of the marginal model likelihoods of versus M D I |M p(S ∪ ) s(X , X ) = ln i j |MD . (1) i j p(S ∪ ) i j I Each marginal model likelihood can be expressed as an integral over the model parameters as described below. 2.2 Marginal model likelihood under the hypothesis of de- pendence 2For the sake of simplicity, we assumed a known mean in the following theoretical development. If the mean is unknown (as will be the case in the simulation and real data − sections), this development is sti(cid:88)ll valid, with N replaced by N 1 and S by N − − (x m)(x m)t, n n n=1 (cid:88) where m is the sample mean N 1 m = x . N n n=1 4 ##LTLine## ##LTLine## ##LTLine##

text: framework

text: can

text: be

text: conveniently

text: used

text: to

text: compare

text: nested

text: partitions,

text: that

text: is,

text: diﬀerent

text: levels

text: of

text: a

text: hierarchy.

text: We

text: also

text: deal

text: with

text: the

text: issue

text: of

text: setting

text: the

text: hyperparameters.

text: Finally,

text: we

text: show

text: how

text: the

text: Bayesian

text: solution

text: can

text: naturally

text: provide

text: for

text: an

text: automatic

text: stopping

text: rule.

text: Bayesian

text: model

text: comparison

text: Let

text: X

text: be

text: a

text: D-dimensional

text: multivariate

text: normal

text: variable

text: with

text: known

text: µ

text: and

text: unknown

text: covariance

text: matrix

text: Σ.

text: Deﬁne

text: X

text: and

text: X

text: as

text: two

text: i

text: j

text: disjoint

text: subvectors

text: of

text: X

text: (of

text: size

text: D

text: and

text: D

text: respectively),

text: and

text: X

text: as

text: their

text: i

text: j

text: i

text: j

text: union

text: (of

text: size

text: D

text: D

text: D

text: Assume

text: that

text: we

text: have

text: to

text: decide

text: whether

text: i

text: j

text: i

text: j

text: we

text: should

text: cluster

text: X

text: and

text: X

text: based

text: on

text: their

text: level

text: of

text: dependence.

text: To

text: this

text: i

text: j

text: M

text: M

text: M

text: end,

text: consider

text: two

text: competing

text: models

text: and

text: In

text: X

text: and

text: X

text: are

text: I

text: D

text: I

text: i

text: j

text: independent

text: and

text: the

text: distribution

text: of

text: X

text: can

text: be

text: decomposed

text: as

text: the

text: product

text: i

text: j

text: of

text: the

text: marginal

text: distributions

text: of

text: X

text: and

text: X

text: Under

text: such

text: condition,

text: Σ

text: i

text: j

text: i

text: j

text: the

text: restriction

text: of

text: Σ

text: to

text: X

text: is

text: block

text: diagonal

text: with

text: blocks

text: Σ

text: and

text: Σ

text: the

text: i

text: j

text: M

text: i

text: j

text: restrictions

text: of

text: Σ

text: to

text: X

text: and

text: X

text: respectively.

text: In

text: by

text: contrast,

text: X

text: and

text: i

text: j

text: D

text: i

text: X

text: are

text: dependent.

text: Given

text: a

text: dataset

text: x

text: x

text: of

text: N

text: independent

text: and

text: j

text: N

text: identically

text: distributed

text: (i.i.d.)

text: realizations

text: of

text: X

text: and

text: S

text: the

text: corresponding

text: sample

text: sum-of-square

text: N

text: S

text: (x

text: µ)(x

text: µ)t,

text: n

text: n

text: we

text: propose

text: to

text: quantify

text: the

text: similarity

text: between

text: X

text: and

text: X

text: as

text: the

text: log

text: Bayes

text: i

text: j

text: M

text: factor,

text: that

text: is,

text: the

text: log

text: ratio

text: of

text: the

text: marginal

text: model

text: likelihoods

text: of

text: versus

text: M

text: D

text: I

text: |M

text: p(S

text: s(X

text: X

text: ln

text: i

text: j

text: |MD

text: i

text: j

text: p(S

text: i

text: j

text: I

text: Each

text: marginal

text: model

text: likelihood

text: can

text: be

text: expressed

text: as

text: an

text: integral

text: over

text: the

text: model

text: parameters

text: as

text: described

text: below.

text: Marginal

text: model

text: likelihood

text: under

text: the

text: hypothesis

text: of

text: de-

text: pendence

text: the

text: sake

text: of

text: simplicity,

text: we

text: assumed

text: a

text: known

text: mean

text: in

text: the

text: following

text: theoretical

text: development.

text: If

text: the

text: mean

text: is

text: unknown

text: (as

text: will

text: be

text: the

text: case

text: in

text: the

text: simulation

text: and

text: real

text: data

text: sections),

text: this

text: development

text: is

text: valid,

text: with

text: N

text: replaced

text: by

text: N

text: and

text: S

text: by

text: N

text: (x

text: m)(x

text: m)t,

text: n

text: n

text: where

text: m

text: is

text: the

text: sample

text: mean

text: N

text: m

text: x

text: N

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Desrosiers (UNF) for his expertise and suppport running the simulations on the grid engine. Part of this study was performed while G.M. was at the UNF. Author contributions G.M. and P.B. conceived and designed the experiments, performed the experiments, and analyzed the data. G.M., A.M., and P.B. contributed reagents/materials/analysis tools and wrote the manuscript. G.M. conceived the theoretical model. Competing Interests P.B. oﬀers consulting in brain image analysis for two Contract Research Organizations, NeuroRX and Biospective, Montreal, Canada. Funding A.M. is supported by Deutsche Forschungsgemeinschaft (DFG) grant SFB 936/Z3. P.B. is supported by the Fonds de Recherche du Qu´ebec – Sant´e (FRQS). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript A Illustration of the behavior of estimated mutual information In the case of a D-dimensional variable X with a multivariate normal distribution, the mutual information between two subvectors X and X is i j given by | | | | 1 Σ Σ I(X , X ) = ln | i |j , (25) i j 2 Σ ∪ i j ∈ { ∪ } | · | where Σ , k i, j, i j , is the covariance matrix of X and is the k k usual determinant function. Estimation bias. In this case, mutual information is estimated using its plug-in estimator Iˆ, that is, Equation (25) where the model covariance ma- trices have been replaced by their estimators (i.e., the corresponding sample covariance matrices). For large N, this estimator suﬀers from a systematic (cid:18) (cid:19) bias, as it was shown that (Marrelec and Benali, 2011) D D 1 E(Iˆ) = I(X , X ) + i j + O . i j 2N N2 This bias is additive and is a (quadratic) function of the problem dimen- sionality. This is consistent with the fact that 2N I(X , X ) asymptotically i j follows a chi square distribution with D D degrees of freedom (Kullback, § § i j 1968, Chap. 12, 3.6); (Press, 2005, 11.3.2). 33 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Desrosiers

text: (UNF)

text: for

text: his

text: expertise

text: and

text: suppport

text: running

text: the

text: simulations

text: on

text: the

text: grid

text: engine.

text: Part

text: of

text: this

text: study

text: was

text: performed

text: while

text: G.M.

text: was

text: at

text: the

text: UNF.

text: Author

text: contributions

text: G.M.

text: and

text: P.B.

text: conceived

text: and

text: designed

text: the

text: experiments,

text: performed

text: the

text: experiments,

text: and

text: analyzed

text: the

text: data.

text: G.M.,

text: A.M.,

text: and

text: P.B.

text: contributed

text: reagents/materials/analysis

text: tools

text: and

text: wrote

text: the

text: manuscript.

text: G.M.

text: conceived

text: the

text: theoretical

text: model.

text: Competing

text: Interests

text: P.B.

text: oﬀers

text: consulting

text: in

text: brain

text: image

text: analysis

text: for

text: two

text: Contract

text: Research

text: Organizations,

text: NeuroRX

text: and

text: Biospective,

text: Montreal,

text: Canada.

text: Funding

text: A.M.

text: is

text: supported

text: by

text: Deutsche

text: Forschungsgemeinschaft

text: (DFG)

text: grant

text: SFB

text: P.B.

text: is

text: supported

text: by

text: the

text: Fonds

text: de

text: Recherche

text: du

text: Qu´ebec

text: Sant´e

text: (FRQS).

text: The

text: funders

text: had

text: no

text: role

text: in

text: study

text: design,

text: data

text: collection

text: and

text: analysis,

text: decision

text: to

text: publish,

text: or

text: preparation

text: of

text: the

text: manuscript

text: A

text: Illustration

text: of

text: the

text: behavior

text: of

text: estimated

text: mutual

text: information

text: In

text: the

text: case

text: of

text: a

text: D-dimensional

text: variable

text: X

text: with

text: a

text: multivariate

text: normal

text: distribution,

text: the

text: mutual

text: information

text: between

text: two

text: subvectors

text: X

text: and

text: X

text: is

text: i

text: j

text: given

text: by

text: Σ

text: Σ

text: I(X

text: X

text: ln

text: i

text: |j

text: i

text: j

text: Σ

text: i

text: j

text: where

text: Σ

text: k

text: i,

text: j,

text: i

text: j

text: is

text: the

text: covariance

text: matrix

text: of

text: X

text: and

text: is

text: the

text: k

text: k

text: usual

text: determinant

text: function.

text: Estimation

text: bias.

text: In

text: this

text: case,

text: mutual

text: information

text: is

text: estimated

text: using

text: its

text: plug-in

text: estimator

text: Iˆ,

text: that

text: is,

text: Equation

text: where

text: the

text: model

text: covariance

text: ma-

text: trices

text: have

text: been

text: replaced

text: by

text: their

text: estimators

text: (i.e.,

text: the

text: corresponding

text: sample

text: covariance

text: matrices).

text: For

text: large

text: N,

text: this

text: estimator

text: suﬀers

text: from

text: a

text: systematic

text: bias,

text: as

text: it

text: was

text: shown

text: that

text: (Marrelec

text: and

text: Benali,

text: D

text: D

text: E(Iˆ)

text: I(X

text: X

text: i

text: j

text: O

text: i

text: j

text: This

text: bias

text: is

text: additive

text: and

text: is

text: a

text: (quadratic)

text: function

text: of

text: the

text: problem

text: dimen-

text: sionality.

text: This

text: is

text: consistent

text: with

text: the

text: fact

text: that

text: I(X

text: X

text: asymptotically

text: i

text: j

text: follows

text: a

text: chi

text: square

text: distribution

text: with

text: D

text: D

text: degrees

text: of

text: freedom

text: (Kullback,

text: i

text: j

text: Chap.

text: (Press,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Extensivity of the measure. Besides the above problem of estimation, mutual information itself is an extensive quantity i.e., it mechanically in- creases with an increase in the dimensionality of the problem. To better demonstrate this eﬀect, consider a model of homogeneous matrix for Σ. More speciﬁcally, let A (ρ) be a D-by-D homogeneous matrix with param- D eter ρ, i.e., a matrix with 1s on the diagonal and all oﬀ-diagonal elements − equal to ρ. A (ρ) has two eigenvalues: 1 + (D 1)ρ with multiplicity 1 D − (associated with the vector composed only of 1s) and 1 ρ with multiplicity − D 1 (associated with the subspace of vectors with a zero mean). This ≤ covariance matrix is therefore positive deﬁnite for 0 ρ < 1 and its deter- − − − minant is given by [1 + (D 1)ρ](1 ρ)D 1. Assuming that Σ = A (ρ), D mutual information can be expressed as − − 1 [1 + (D 1)ρ][1 + (D 1)ρ] I(X , X ) = ln − i j− , (26) i j 2 (1 ρ)[1 + (D + D 1)ρ] i j ∈ { ∪ } where D , k i, j, i j , is the dimension of X . This expression can k k lead to counter-intuitive results: the mutual information is about 0.31 for D = D = 5 and ρ = 0.3, while it is about 0.33 for D = D = 7 and i j i j ρ = 0.25. More generally, I(X , X ) is an increasing function of D and i j i D (as can be seen by diﬀerentiation). This means that, for the same value j of ρ, mutual information will be larger for larger values of D and D . It i j also means that mutual information will favor the merging of two variables with smaller marginal correlations if the dimensionality of the variables is large enough, thus systematically favoring larger clusters. To give a feeling (cid:48) − of the amplitude of this behavior, set D = D 1 and assume that we have (cid:48) (cid:29) k k D ρ 1. Mutual information can then be approximated by k (cid:48) (cid:48) (cid:48) (cid:48) ≈ 1 D D ρ 1 ρ 1 D D I(X , X ) ln −i j (cid:48) = ln − + ln i(cid:48) j . i j 2 (1 ρ)D ∪ 2 1 ρ 2 D ∪ i j i j B Asymptotic form of the log Bayes factor → ∞ Assume that N . Starting from the expression φ of Equation (15), we have (cid:18) (cid:19) (cid:88) − D −N + ν | | k N + ν + 1 d k k φ(N + ν , Λ + S ) = ln Λ + S + ln Γ . k k k k k 2 2 d=1 (cid:98) ((cid:98)27) Deﬁning S as the standard sample covariance matrix, i.e., S = NS , k k k the ﬁrst term of the right-hand side can be expanded as 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Extensivity

text: of

text: the

text: measure.

text: Besides

text: the

text: above

text: problem

text: of

text: estimation,

text: mutual

text: information

text: itself

text: is

text: an

text: extensive

text: quantity

text: i.e.,

text: it

text: mechanically

text: in-

text: creases

text: with

text: an

text: increase

text: in

text: the

text: dimensionality

text: of

text: the

text: problem.

text: To

text: better

text: demonstrate

text: this

text: eﬀect,

text: consider

text: a

text: model

text: of

text: homogeneous

text: matrix

text: for

text: Σ.

text: More

text: speciﬁcally,

text: let

text: A

text: (ρ)

text: be

text: a

text: D-by-D

text: homogeneous

text: matrix

text: with

text: param-

text: D

text: eter

text: ρ,

text: i.e.,

text: a

text: matrix

text: with

text: on

text: the

text: diagonal

text: and

text: all

text: oﬀ-diagonal

text: elements

text: equal

text: to

text: ρ.

text: A

text: (ρ)

text: has

text: two

text: eigenvalues:

text: (D

text: with

text: multiplicity

text: D

text: (associated

text: with

text: the

text: vector

text: composed

text: only

text: of

text: and

text: ρ

text: with

text: multiplicity

text: D

text: (associated

text: with

text: the

text: subspace

text: of

text: vectors

text: with

text: a

text: zero

text: mean).

text: This

text: covariance

text: matrix

text: is

text: therefore

text: positive

text: deﬁnite

text: for

text: ρ

text: and

text: its

text: deter-

text: minant

text: is

text: given

text: by

text: (D

text: ρ)D

text: Assuming

text: that

text: Σ

text: A

text: (ρ),

text: D

text: mutual

text: information

text: can

text: be

text: expressed

text: as

text: (D

text: (D

text: I(X

text: X

text: ln

text: i

text: j−

text: i

text: j

text: (D

text: D

text: i

text: j

text: where

text: D

text: k

text: i,

text: j,

text: i

text: j

text: is

text: the

text: dimension

text: of

text: X

text: This

text: expression

text: can

text: k

text: k

text: lead

text: to

text: counter-intuitive

text: results:

text: the

text: mutual

text: information

text: is

text: about

text: for

text: D

text: D

text: and

text: ρ

text: while

text: it

text: is

text: about

text: for

text: D

text: D

text: and

text: i

text: j

text: i

text: j

text: ρ

text: More

text: generally,

text: I(X

text: X

text: is

text: an

text: increasing

text: function

text: of

text: D

text: and

text: i

text: j

text: i

text: D

text: (as

text: can

text: be

text: seen

text: by

text: diﬀerentiation).

text: This

text: means

text: that,

text: for

text: the

text: same

text: value

text: j

text: of

text: ρ,

text: mutual

text: information

text: will

text: be

text: larger

text: for

text: larger

text: values

text: of

text: D

text: and

text: D

text: It

text: i

text: j

text: also

text: means

text: that

text: mutual

text: information

text: will

text: favor

text: the

text: merging

text: of

text: two

text: variables

text: with

text: smaller

text: marginal

text: correlations

text: if

text: the

text: dimensionality

text: of

text: the

text: variables

text: is

text: large

text: enough,

text: thus

text: systematically

text: favoring

text: larger

text: clusters.

text: To

text: give

text: a

text: feeling

text: of

text: the

text: amplitude

text: of

text: this

text: behavior,

text: set

text: D

text: D

text: and

text: assume

text: that

text: we

text: have

text: k

text: k

text: D

text: ρ

text: Mutual

text: information

text: can

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text: where we used the fact that D ∪ = D +(cid:98) D(cid:98), and where we set i j i j (cid:98) | | | | 1 S S Iˆ(Xi, Xj) = ln | i |j . 2 S ∪ i j C Hyperparameter optimization { } Given a clustering X , . . . , X of X, optimizing the marginal likeli- 1 K hood wi(cid:88)th respect to a diagonal Λ(cid:88)leads to the optimization of − − D K ν D + D − N + ν D + D | | kd ln Λ k ln S + Λ , dd k k 2 2 d=1 k=1 where k is the cluster containing X . Diﬀerentiation with respect to Λ d d dd (cid:2) (cid:3) leads to − − ν D + D − N + ν D + D − kd kd (S + Λ ) 1 = 0. (30) 2Λ 2 kd kd dd dd To obtain the solution of this equation, we notice that the equation is (cid:8)(cid:2) (cid:3) (cid:9) equivalent to − ν D + D − − Λ = − kd (S + Λ ) 1 1 . dd N + ν D + D kd kd dd k d Optimizing at the ﬁrst level (i.e., with D clusters and D = 1) yields k d − ν D + 1 Λ = S . dd dd N D Real fMRI data: datasets and preprocessing We used the ’Atlanta’ resting-state fMRI database (Liu et al., 2009). This resource was made publicly available as part of the 1000-connectome project8 (Biswal et al., 2010). The Atlanta sample includes 28 subjects (age ranging from 22 to 57 years, 15 women) with one structural MRI and one fMRI run each (205 volumes, TR = 2 s), acquired on a 3 T scanner. The datasets were preprocessed using the neuroimaging analysis kit9 (NIAK), version 0.6.5c (Bellec et al., 2012). The parameters of a rigid body motion were ﬁrst estimated at each time frame of the fMRI dataset (no correction of inter-slice diﬀerence in acquisition time was applied). The median vol- ume of the fMRI time series was coregistered with a T individual scan 1 8http://www.nitrc.org/projects/fcon 1000/ 9http://wiki.bic.mni.mcgill.ca/index.php/NiakFmriPreprocessing 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: ’Atlanta’

text: resting-state

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text: volumes,

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text: datasets

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text: neuroimaging

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text: et

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text: parameters

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