Simultaneous Measurement Imputation and Rehabilitation Outcome Prediction for Achilles Tendon Rupture

IN

DEGREE PROJECT COMPUTER SCIENCE AND ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2018,

Simultaneous Measurement Imputation and Rehabilitation Outcome Prediction for Achilles Tendon Rupture

CHARLES HAMESSE

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

P

P S

P

P

⇥

⇥ ⇥ ⇥

k

. . . . . .

. . . . . .

⇥ . . . . . . ⇥

⇥ ⇥ . . . ⇥ ⇥ ⇥ . . .

⇥ . . . . . .

⇥ . . . . . .

⇥

D = 0 BB BB BB B@

D1,1 D1,2 · · · D^1,M D1,M +1 · · · D^{1,M +P} D2,1 D2,2 · · · D^1,M D2,M +1 · · · D^{2,M +P}

| {z }

M

DN,1 DN,2 · · · DN,M

| {z }

P

DN,M +1 · · · DN,M +P

1 CC CC CC CA

9>

>>

>>

>>

=

>>

>>

>>

>; N

M = 297

P = 63 N = 442 N ⇥ M

P N ⇥ P S

D = (P|S)

N⇥ M N⇥ P

N M

•

D_ij ⇠ N (0, 1) i, j 2 [1, N] ⇥ [1, M]

•

Dij 2 [0, 1] i, j 2 [1, N] ⇥ [1, M]

P

Anm

Pnm

Inm

udn

vdm

D

M

N

A U

V

p(U| U²) = YN i=1

N (uⁱ|µ^U, _U²1),

p(V| V²) = YM j=1

N (v^j|µ^V, ²_V1) .

I

I(n, m) = 8<

:

1 Pn,m

0

P

p(P|U, V, P²) = YN n=1

YM m=1

"

N (Pnm|u^Tnvm, _P²)

#I(n,m)

,

N (x|µ, ²)

µ ²

A

Pnm

P

p(P|U, V, ²P) = YN n=1

YM m=1

"

N (Pnm|u^Tnvm+ bu,n+ bv,m, _P²)

#I(n,m)

,

b_u,n n b_v,m

m

U V R

x y

xy = ( x)( y) (x, y) ( x, y)

x y

Rij

u v

u^Tv A 2 R^D⇥D

u^Tv = (Au)^T(A ^Tv)

U 2 R^D⇥N A

U⁰ = AU D⇥ N

V 2 R^D⇥M A ^T R

A

U⁰ D = 2

"

a₁₁ a₁₂ a21 a22

#

| {z }

A

"

u₁₁ u₁₂ x₁₃ . . . u_1N u21 x22 u23 . . . u2N

#

| {z }

U

=

"

1 0 u₁₃ . . . u_1N 0 1 u23 . . . u2N

#

| {z }

U⁰

Anm

Pnm

Inm

udn

Xnm

vdm

Bnp

Snp

I_np⁰

bp

wmp

M N

P D

Pˆ

Anm

Pnm

Inm

u_dn

Xnm

v_dm

B_np

Snp

I_np⁰

b_p wdp

D

M P

N Uˆ

D⇥D

S

p(S| W, b, X) = YN n=1

YP p=1

"

N (Snp| xnwp+ bp, _S²)

#I⁰np

, p(W) = N (W | 0, ²w1),

p(b) = N (b | 0, ²b).

X W b

S

B I⁰

X = ˆP = I⇤ P + (1 I) ⇤ A I N ⇥ M

X = ˆU

Pˆ Uˆ

Pˆ Uˆ

p(S| ✓, X) = YN n=1

YP p=1

"

N (Snp | (xn; ✓), _S²)

#I⁰np

.

NN ✓

L Hl = tanh⇣

Hl 1Wl+ bl

⌘

l = 1, ..., L

Hl l H0 = X Wl

l 1 l bl l

w

j l

nlj, nlj,

w_lij l 1 i l j

p(wlij) =N (wlij | 0, w²lij),

2

wlij = 2

nlj, + nlj,

.

W

b ✓

S P

P3 S3

P6 S6

z x

p(z|x) = p(z, x)

p(x) = p(z, x) R p(z, x)dz

p(x, z) x z p(z | x )

q(z)

q(z)⇡ p(z | x )

q(z; ) q(z)

p(z| x)

d(q, p) q(z)

d(q, p)

P Q

D_KL(q||p) =X

z

q(z) log q(z) p(z| x).

P S

Pˆ Uˆ

Sˆ

P

P S

N = 100 M = 30 P = 10 D = 10

p(U) = YD d=1

YN n=1

N (udn|0.5, 0.5), p(V) = YD d=1

YM m=1

N (vdm|0.5, 0.5).

U V

V D⇥D

P P

p(W) = YM m=1

YP p=1

N (w^mp|0, 0.5), p(b) = YP p=1

N (b^p|0, 0.5).

S p(S) = N (S|PW + b, 0.1).

0 10 20 30 40 50 60 70 80 90 2

1 0 1 2

0 10 20 30 40 50 60 70 80 90 2

1 0 1 2

0 10 20 30 40 50 60 70 80 90

2 1 0 1 2

0 10 20 30 40 50 60 70 80 90 2

1 0 1 2

0 10 20 30 40 50 60 70 80 90

2 1 0 1 2

0 10 20 30 40 50 60 70 80 90 2

1 0 1 2

P S

P

P

D

0.2 0.4

0.6 1 2 3 4 5 6 7

0.1 0.15 0.2 0.25 0.3

P

D = 5

D

10²

10³

10⁴ 1 2 3 4 5 6 7 8

0.1 0.15 0.2 0.25 0.3

D 3

P

S

S

S

D 2 [1, 20] ²U 2

V

D

P S P S Pˆ

Pˆ

Uˆ Pˆ

Uˆ

P

S

[0, 1]

10³ 10⁴

P

S Uˆ

Pˆ

Pˆ

P S

Pˆ Sˆ

P P3

P6

P P3 P6

S3 S6 S12

Pˆ Pˆ3

Pˆ₆

S

0 0.2 0.4

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 0

200 400

P

P

0 0.2 0.4

0 5 10 15 20 25 30 35 40 45 50 55 60

0 200 400

S

S

www.kth.se