Trennung von mehrdimensionalen Mischverteilungen bei ...

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F
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TL =(5, 5) T
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π1

∼
≈
≥p ≤p p
≫
∀
#M M
γ
B(C; r) C r
COR(X, Y ) X Y
det(A) A
f(x; θ) θ
F (x; θ) θ
Femp
Ip
(p × p)
L()
l()

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

Np(μ, Σ) p μ
Σ
Rp×p (p × p)
rg(A) A
TL
tr(A) A
TU
vecA A ∈ Rm×n
vecA ∈ Rmn×1
x X
X p

X X ⊂ R f : X →
R f
f(x) :=
m
πjfj(x)
j=1
fj πj j
∀ j ∈{1,...,m} m j=1 πj =1
fj
θj
f(x; Θ) :=
m
πjfj(x; θj)
j=1
Θ := (π1,...,πm; θ1,...,θm)
fj
X X ⊂ Rp p>1
f : X → R
fj
f

0.0 0.1 0.2 0.3 0.4
y 1 = 0.75 N(2, 1.2)
y 2 = 0.25 N(3.7, 0.8)
f(x) = y 1 + y2
−2 0 2 4 6
x
0.0 0.1 0.2 0.3 0.4
y 1 = 0.75 N(0.5, 1.2)
y 2 = 0.25 N(4.2, 0.8)
f(x) = y 1 + y2
−2 0 2 4 6
X =(X1,...,Xp) p
μ ∈ Rp Σ ∈ Rp×p
X ∼ Np(μ, Σ) X θ := (μ, Σ)
1
f(x; θ) = exp −
det(Σ) 1
2 (x − μ)T Σ −1
(x − μ) ,
F (x; θ) =
(2π) p
2
x1
−∞
...
xp
−∞
(2π) p
2
1
exp −
det(Σ) 1
2 (t − μ)T Σ −1
(t − μ)
x
dtp ... dt1.
p =2
ρ
1
f(x; θ) =
2πσ1σ2 1 − ρ2 ×
1
exp −
2(1 − ρ2 x1 − μ1
)
σ1
2
− 2ρ x1 − μ1
σ1
x2 − μ2
σ2
2
x2 − μ2
+
σ2

p
1
TL∈ Rp TU ∈ Rp
2p K(p)
K(p) :=K(p, TL, TU):={κ =(κ1,...,κp) T | κi ∈{TLi,TUi}, ∀ i =1,...,p}.
κ ∈ K(p)
g(κ) :=♯{κi | κi = TLi, ∀ i =1,...,p}.
X =(X1,...,Xp) p X ∼ Np(μ, Σ) μ ∈
Rp Σ ∈ Rp×p p ≥ 2 f(x; μ, Σ) F (x; μ, Σ)
TL∈ Rp TU ∈
Rp x X
TLi ≤ xi ≤ TUi ∀ i =1,...,p

⎧
⎪⎨
f(x;μ,Σ)
(−1)
fT (x; μ, Σ) := κ∈K(p)
⎪⎩
g(κ) , TL≤
F (κ;μ,Σ)
p x ≤p TU
0 , .
X ∼ MTG(μ, Σ, TL, TU)
fT K(p)
p
p
F p ≥ 2
TL∈ Rp TU ∈ Rp TL≤p TU
F (TL≤ p TU)=
(2π) p
2
=
κ∈K(p)
1
det(Σ)
TUp
TLp
...
TU1
TL1
exp
p
− 1
2 (t − μ)T Σ −1
(t − μ)
dt1 ... dtp
(−1) g(κ) F (κ; μ, Σ)
TUp
TLp
...
TU1
TL1
=
exp
κ∈K(p)
G1(x1; t2,...,tp) =
− 1
2 (t − μ)T Σ −1
(t − μ)
(−1) g(κ) Gp(κ; μ, Σ)
x1
−∞
Gk(x1,...,xk,tk+1,...,tp) =
dt1 ... dtp
exp − 1
2 (t − μ)T Σ −1
(t − μ)
xk
−∞
dt1,
Gk−1(x1,...,xk−1,tk,...,tp) dtk,
p k ∈{1,...,p}

• p =2
TU2
TU1
TL2 TL1
exp
− 1
2 (t − μ)T Σ −1
(t − μ)
dt1 dt2 =
TU2
TL2
(G1(TU1,t2) − G1(TL1,t2)) dt2.
TU2
TL2
(G1(TU1,t2) − G1(TL1,t2)) dt2
= G2(TU1,TU2) − G2(TU1,TL2) − (G2(TL1,TU2) − G2(TL1,TL2))
=
(−1) g(κ) G2(κ; μ, Σ).
κ∈K(2)
•
TUp
TLp
...
TU1
TL1
=
exp
κ∈K(p)
− 1
2 (t − μ)T Σ −1
(t − μ)
(−1) g(κ) Gp(κ; μ, Σ)
dt1 ... dtp
• p +1
TUp+1
TLp+1
...
=
=
TU1
TL1
TUp+1
TLp+1
TUp+1
TLp+1
−
exp
TUp+1
TLp+1
− 1
2 (t − μ)T Σ −1
(t − μ)
...
TU2
TL2
...
TU2
TL2
...
dt1 ... dtp+1
(G1(TU1,t2,...,tp+1) − G1(TL1,t2,...,tp+1)) dt2 ... dtp+1
G1(TU1,t2,...,tp+1) dt2 ... dtp+1
TU2
TL2
G1(TL1,t2,...,tp+1) dt2 ... dtp+1.

TUp+1
TLp+1
...
TU2
TL2
−
G1(TU1,t2,...,tp+1) dt2 ... dtp+1
TUp+1
TLp+1
=
κ∈K(p)
=
κ∈K(p+1)
...
TU2
TL2
G1(TL1,t2,...,tp+1) dt2 ... dtp+1
(−1) g(κ) Gp(TU1, κ; μ, Σ) −
(−1) g(κ) Gp+1(κ; μ, Σ)
p +1
(2π) p
2
κ∈K(p)
1
Gp(κ; μ, Σ) = F (κ; μ, Σ)
det(Σ)
Dichte
0.08
0.06
0.04
0.02
6
y
4
2
Lower and upper Truncation
μ 1 = 140, μ2 = 4.2, Σ11 = 10, Σ12 = 2, Σ21 = 2, Σ22 = 1
0
130
135
140
x
145
(−1) g(κ) Gp(TL1, κ; μ, Σ)
x (i) ∈ R p i =1,...,n X
150

θ =(μ, Σ)
l(θ; x (1) ,...,x (n) )=
=
n
log(fT (x (i) ; θ))
i=1
n
i=1
= − np
2
log
κ∈K(p)
f(x (i) ; θ)
(−1) g(κ)
F (κ; θ)
n
1
log(2π) − log(det(Σ)) −
2 2
− n · log(
κ∈K(p)
n
(x (i) − μ) T Σ −1 (x (i) − μ)
i=1
(−1) g(κ) F (κ; θ)).
−∞ ∞
fT
TUi = ∞ TLi ∈ R \{−∞, ∞} i ∈{1,...,p}
⎧
⎪⎨
f(x;μ,Σ)
(−1)
fTL(x; μ, Σ) := κ∈K(p)
⎪⎩
g(κ) , x ≥
F (κ;μ,Σ)
p TL
0 , .
x (i) ∈ R p i =1,...,n X
θ =(μ, Σ)
l(θ; x (1) ,...,x (n) )=
TUi = ∞ i ∈{1,...,p}
=
n
log(fTL(x (i) ; θ))
i=1
n
log(fT (x (i) ; θ))
i=1

Dichte
0.06
0.04
0.02
6
y
4
2
0
130
Lower Truncation
μ 1 = 140, μ2 = 4.2, Σ11 = 10, Σ12 = 2, Σ21 = 2, Σ22 = 1
135
fT
TLi = −∞ TUi ∈ R \{−∞, ∞} i ∈{1,...,p}
⎧
⎨ f(x;μ,Σ)
F (TU;μ,Σ) , x ≤
fTU(x; μ, Σ) :=
⎩
p TU
0 , .
140
x
145
150
x (i) ∈ R p i =1,...,n X
θ =(μ, Σ)
l(θ; x (1) ,...,x (n) )=
=
=
n
log(fTU(x (i) ; θ))
i=1
n
f(x (i) ; θ)
log
F (TU; θ)
n
log(f(x (i) ; θ)) − n log(F (TU; θ)))
i=1
i=1

=
n
i=1
p
1
− −
log (2π) 2 det(Σ) 2 exp − 1
2 (x(i) − μ) T Σ −1 (x (i)
− μ)
− n · log(F (TU; θ))
= − np
2
n
1
log(2π) − log(det(Σ)) −
2 2
− n · log(F (TU; θ)).
0.05
0.04
0.03
0.02
0.01
0.00
8
Dichte
6
y
4
2
0
130
Upper Truncation
μ 1 = 140, μ2 = 4.2, Σ11 = 10, Σ12 = 2, Σ21 = 2, Σ22 = 1
135
140
x
145
n
(x (i) − μ) T Σ −1 (x (i) − μ)
i=1
150

p
l(θ; x (1) ,...,x (n) )=− np
2
n
1
log(2π) − log(det(Σ)) −
2 2
− n · log(
κ∈K(p)
(−1) g(κ) F (κ; θ))
n
(x (i) − μ) T Σ −1 (x (i) − μ)
θ
l(θ; x (1) ,...,x (n) )=− np
2
n
1
log(2π) − log(det(Σ)) −
2 2
i=1
n
(x (i) − μ) T Σ −1 (x (i) − μ)
i=1
p
− n · log(
(−1) g(κ) F (κ; θ)) .
κ∈K(p)
p

Z := n
i=1 (x(i) − μ)(x (i) − μ) T θ
d(l(θ; x (1) ,...,x (n) )) = − n
1
d(log(det(Σ))) −
2 2 tr(d(Σ−1 )Z) − 1
2 tr(Σ−1dZ)
− nd log (−1) g(κ)
F (κ; θ)
κ∈K(p)
= − n
2 tr(Σ−1 (dΣ)) − 1
2 tr((−Σ−1 (dΣ)Σ −1 )Z)
− 1
2 tr
Σ −1
n
− (dμ) (x
i=1
(i) n
− μ) − (x
i=1
(i) − μ)(dμ) T
1
− n
(−1) g(κ) F (κ; θ) d
(−1) g(κ)
F (κ; θ)
κ∈K(p)
κ∈K(p)
= − 1
2 tr(Σ−1 (dΣ)Σ −1 nΣ) + 1
2 tr(Σ−1 (dΣ)Σ −1 Z)
+ 1
2 · 2(dμ)T Σ −1
n
(x (i) − μ)
− n
= 1
κ∈K(p)
i=1
1
(−1) g(κ) F (κ; θ) d
κ∈K(p)
2 tr((dΣ)Σ−1 (Z − nΣ)Σ −1 )+(dμ) T Σ −1
− n
κ∈K(p)
1
(−1) g(κ) F (κ; θ) d
κ∈K(p)
(−1) g(κ)
F (κ; θ)
n
(x (i) − μ)
i=1
(−1) g(κ)
F (κ; θ) .
p
F (TU; θ)
F (κ) κ ∈ K(p) TL TU
θ
R R

θ
∂l(θ)
∂θ =
∂l(θ) ∂l(θ)
, = 0.
∂μ ∂Σ
θ (0) θ (0)
∂l(θ)
∂θ
∂l(θ)
∂θ =
∂l(θ)
∂θ
θ=θ (0)
+
∂2l(θ) ∂θ 2
θ=θ (0)
(θ − θ (0) )+R(θ (0) ),
R(θ (0) )
∂2l(θ)
∂θ2
θ=θ (0) θ(0)
θ (0)
∂l(θ)
∂θ ≈
∂l(θ)
∂θ
θ=θ (0)
+
∂2l(θ) ∂θ 2
θ=θ (0)
(θ − θ (0) ).
θ
θ (1) := θ = θ (0) −
θ (k+1) = θ (k) −
∂2l(θ) ∂θ 2
−1 θ=θ (0)
∂2l(θ) ∂θ 2
−1 θ=θ (k)
∂l(θ)
∂θ
∂l(θ)
∂θ
θ=θ (k)
θ=θ (0)
.
k ∈ N0

ε>0
θ (k+1) − θ (k)
θ (k) ≤ ε
l(θ
(k+1) ) − l(θ (k) )
l(θ (k)
≤ ε
)
θ ∗ h
θ (k+1) − θ ∗ ≤h θ (k) − θ ∗ 2 , ∀ k ∈ N0.
R
nlm

−n · log(F (TU; θ)) = −n · log ×
⎛
TU1 TU2
⎝
−∞ −∞
(2π) p
2
1
exp −
det(Σ) 1
2 (t − μ)T Σ −1
(t − μ)
⎞
dt2 dt1⎠
R
θ
nx ·ny
Δx · Δy
Laborwert y
0 2 4 6 8
c(0, T y)
Δy
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c(Tx, Ty) ● ●
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0 5 10 15 20 25 30 35
Laborwert x
●
●
c(T x, 0)
ϕ
θ
F (TU; θ) ≈ 1
2
nx ny
ΔxΔy ϕ
i=1 j=1
(iΔx, jΔy) T ; θ
Obersumme
+ 1
2
nx−1
ny−1
ΔxΔy ϕ
i=0 j=0
(iΔx, jΔy) T ; θ
.
Untersumme

l(θ) ≈− np
n
n
1
log(2π) − log(det(Σ)) − (x
2 2 2
i=1
(i) − μ) T Σ −1 (x (i) − μ)
nx ny
1
− n · log
ΔxΔy ϕ
2
(iΔx, jΔy) T ; θ
+
nx−1
i=0
ny−1
j=0
Δx = TUx
nx
i=1 j=1
ΔxΔy ϕ (iΔx, jΔy) T
; θ
, Δy = TUy
, TU =(TUx,TUy)
ny
T .
R nlm
nlm
θ
F (TU; θ)
nlm
l(θ)
F (TU; θ)
⎧
⎨
exp1{} := exp
⎩ −1
T
iΔx
− μ Σ
2 jΔy
−1
iΔx
− μ
jΔy
⎫ ⎬
⎭ ,
⎧
⎨
exp2{} := exp
⎩ −1
T
(i − 1)Δx
− μ Σ
2 (j − 1)Δy
−1
(i − 1)Δx
− μ
(j − 1)Δy
⎫ ⎬
⎭ ,

Δ1 :=
Δ2 :=
iΔx
− μ,
jΔy
(i − 1)Δx
− μ
(j − 1)Δy
d( − n log(F (TU; θ)))
n
= − d(F (TU; θ))
F (TU; θ)
n ΔxΔy
≈−
F (TU; θ) 2π det(Σ) ×
− 1
det(Σ)
2
3
det(Σ) tr(Σ−1 nx ny
(dΣ)) (exp1{} +exp2{}) +
1
det(Σ)
nx ny
i=1 j=1
i=1 j=1
exp1{} − 1
2 tr(Σ−1 (−(dμ)Δ1 − Δ1(dμ) T ))
− 1
2 tr((−Σ−1 (dΣ)Σ −1 ) · Δ1 · Δ T 1 )
+exp2{} − 1
2 tr(Σ−1 (−(dμ)Δ2 − Δ2(dμ) T ))
.
− 1
2 tr((−Σ−1 (dΣ)Σ −1 ) · Δ2 · Δ T 2 )
W1 := Δ1Δ T 1
iΔx
iΔx
= − μ · − μ
jΔy
jΔy
W2 := Δ2Δ T 2
(i − 1)Δx
(i − 1)Δx
=
− μ ·
− μ
(j − 1)Δy
(j − 1)Δy
T
T

d( − n log(F (TU; θ)))
≈ n · tr(Σ −1 n ΔxΔy
(dΣ)) −
F (TU; μ) 2π det(Σ) ×
nx ny
exp1{} − (dμ)
i=1 j=1
T Σ −1 Δ1 − 1
2 tr((−Σ−1 (dΣ)Σ −1
)W1)
+exp2{} =
1
2
+
nx
i=1 j=1
nx
− (dμ) T Σ −1 Δ2 − 1
2 tr((−Σ−1 (dΣ)Σ −1 )W2)
n
ny
(exp1{} +exp2{}) i=1 j=1
(dμ) T Σ −1
nx ny
i=1 j=1
exp1{}Δ1 +exp2{}Δ2 nx ny
n
exp ny
1{}tr((−Σ
(exp1{} +exp i=1 j=1
2{})
−1 (dΣ)Σ −1 )W1)
+exp 2{}tr((−Σ −1 (dΣ)Σ −1 )W2)
+ n · tr(Σ −1 (dΣ)) .
(dμ) (dΣ) d(lnorm(θ; x (1) , ..., x (n) ))
d(lnorm(θ; x (1) , ..., x (n) )) = 1
2 tr((dΣ)Σ−1 (Z − nΣ)Σ −1 )+(dμ) T Σ −1
n
(x (i) − μ)
d(l(θ; x (1) , ..., x (n) )=d(lnorm(θ; x (1) , ..., x (n) )+d(−n log(F (TU; θ))).
i=1

d(l(θ; x (1) , ..., x (n) ))
≈ 1
2 tr((dΣ)Σ−1 (Z − nΣ)Σ −1 )+(dμ) T Σ −1
+
+
1
2
nx
i=1 j=1
nx
i=1 j=1
n
ny
(exp1{} +exp2{}) i=1
(dμ) T Σ −1
nx ny
n
(x (i) − μ)
i=1 j=1
exp1{}Δ1 +exp2{}Δ2 nx ny
n
exp ny
1{}tr((−Σ
(exp1{} +exp i=1 j=1
2{})
−1 (dΣ)Σ −1 )W1)
+exp 2{}tr((−Σ −1 (dΣ)Σ −1 )W2)
= 1
2 tr((dΣ)Σ−1 (Z + nΣ)Σ −1 )+
nx ny
i=1 j=1
+(dμ) T Σ −1
nx ny
i=1 j=1
+ n · tr(Σ −1 (dΣ))
nx
i=1 j=1
n
×
ny
(exp1{} +exp2{})
exp1{}tr((−Σ −1 (dΣ)Σ −1 )W1)+exp2{}tr((−Σ −1 (dΣ)Σ −1
)W2)
n
x (k) − nμ +
k=1
1
2
nx
i=1 j=1
n
×
ny
(exp1{} +exp2{})
iΔx
(i − 1)Δx
exp1{} +exp2{} jΔy
(j − 1)Δy
= 1
2 tr((dΣ)Σ−1 (Z + nΣ)Σ −1 )+
nx ny
i=1 j=1
+(dμ) T Σ −1
nx ny
i=1 j=1
nx
i=1 j=1
n
×
ny
(exp1{} +exp2{})
− (exp1{} +exp2{})μ
exp1{}tr((−Σ −1 (dΣ)Σ −1 )W1)+exp2{}tr((−Σ −1 (dΣ)Σ −1
)W2)
n
x (k) − 3nμ +
k=1
nx
i=1 j=1
2n
×
ny
(exp1{} +exp2{})
iΔx
(i − 1)Δx
exp1{} +exp2{} .
jΔy
(j − 1)Δy

d(l(θ; x (1) , ..., x (n) )) = 0
(dμ) (dΣ) μ Σ
(dμ) μ
μ Σ (dΣ)
Σ
d(l(θ; x (1) , ..., x (n) ))
R nlm
R
R nlm
n
n
Δx1 · ...· Δxn
n

p
X ∼ Np(μ;Σ) rg(Σ) = p B ∈ Rp×p Σ=BBT
U := B−1 (X − μ) U ∼ Np(0; Ip) Ip
p Σ
Σ=BBT
B −1

Σ
Σij i, j ∈{1, ..., p} Σ
Σij =
j
k=1
bikbjk i ≥ j
⎧
0 ij .
bjj
bikbjk
k=1
B
C := B−1
B
C = B−1 p
φp(x) Φp(x)
f(X; θ) =
1
det(Σ−1 ) φp(B −1 (X − μ)),
F (X; θ) = Φp(B −1 (X − μ)).
d
κ∈K(p)
(−1) g(κ)
F (κ; θ)
d
κ∈K(p)
(−1) g(κ)
F (κ; θ)
=d
=
κ∈K(p)
=
κ∈K(p)
κ∈K(p)
(−1) g(κ) Φp(B −1
(κ − μ); θ)
(−1) g(κ) φp(B −1 (κ − μ); θ)d(B −1 (κ − μ))
(−1) g(κ) φp(B −1 (κ − μ); θ)(−B −1 +(d(B −1 ))(κ − μ)).

d(l(θ)) := d(l(θ; x (1) , ..., x (n) ))
d(l(θ)) = 1
2 tr((dΣ)Σ−1 (Z − nΣ)Σ −1 )+(dμ) T Σ −1
−
κ∈K(p)
κ∈K(p)
n
(−1) g(κ) F (κ; θ) ×
n
(x (i) − μ)
i=1
(−1) g(κ) φp(B −1 (κ − μ); θ)(−B −1 +(d(B −1 ))(κ − μ))
= 1
2 tr((dΣ)Σ−1 (Z − nΣ)Σ −1 )+(dμ) T Σ −1
−
κ∈K(p)
κ∈K(p)
n
(−1) g(κ) F (κ; θ) ×
n
(x (i) − μ)
(−1) g(κ) φp(B −1 (κ − μ); θ)(−B −1 − B −1 (dB)B −1 (κ − μ))
l(θ; x (1) , ..., x (n) ) θ
d 2 (l(θ)) = 1
2 tr((dΣ)(d(Σ−1 ))(Z − nΣ)Σ −1 )+ 1
2 tr((dΣ)Σ−1 (Z − nΣ)(d(Σ −1 )))
+ 1
n
(x (i) − μ)
i=1
2 tr((dΣ)Σ−1 (dZ − n(dΣ))Σ −1 )+(dμ) T (d(Σ −1 ))
− (dμ) T Σ −1
n(dμ) −
(
n
(−1)
κ∈K(p)
g(κ) · (−1) ×
F (κ; θ)) 2
(−1) g(κ) φp(B −1 (κ − μ); θ)(−B −1 − B −1 (dB)B −1 (κ − μ)) 2 κ∈K(p)
+
κ∈K(p)
κ∈K(p)
+
κ∈K(p)
n
(−1) g(κ) F (κ; θ) ×
(−1) g(κ) dφp(B −1 (κ − μ); θ)(−B −1 − B −1 (dB)B −1 (κ − μ))
(−1) g(κ) φp(B −1 (κ − μ); θ) − (d(B −1 )) + (d((d(B −1 ))(κ − μ)))
= 1
2 tr((dΣ)(d(Σ−1 ))(Z − nΣ)Σ −1 )+ 1
2 tr((dΣ)Σ−1 (Z − nΣ)(d(Σ −1 )))
+ 1
2 tr((dΣ)Σ−1 (dZ − n(dΣ))Σ −1 )+(dμ) T (d(Σ −1 ))
i=1
n
(x (i) − μ)
i=1
.

− (dμ) T Σ −1
n(dμ) −
(
n
(−1)
κ∈K(p)
g(κ) · (−1) ×
F (κ; θ)) 2
(−1) g(κ) φp(B −1 (κ − μ); θ)(−B −1 − B −1 (dB)B −1 (κ − μ)) 2 κ∈K(p)
+
κ∈K(p)
n
(−1) g(κ) F (κ; θ)
κ∈K(p)
(−B −1 − B −1 (dB)B −1 (κ − μ)) +
(−1) g(κ) φp(B −1 (κ − μ); θ)(B −1 (κ − μ)) ×
κ∈K(p)
(−1) g(κ) φp(B −1 (κ − μ); θ) ×
− (d(B −1 )) + (d 2 (B −1 ))(κ − μ)+(d(B −1 )) · (−1)(dμ)
.
dμ dΣ dB
dB
B
Σ
Σ
vec
F : S → Rm×p S ⊆ Rn×q C
S B(C; r) ⊂ S C r U Rn×q U

B Σ
⎛
⎜
dB(Σ; U) = ⎝
1
2 √ Σ11 u11
− Σ21 √
Σ11 u21
2 √ Σ11 3 u11 + 1
Σ2 21
Σ2
11
Σ22− Σ2 Σ21
u11 −
21
Σ22−
Σ11 Σ2 21
Σ11 0
u21 +
1
Σ22− Σ2 21
Σ 11
u22
⎞
⎟
⎠ .
Σ
U ∼ Np(0; Ip) C := B −1
TP =(t1, ..., tp)
ui = p j=1 cij(tj − μj) i ∈{1, ..., p}
∂φp ∂ 1
(u) =
∂μ ∂μ
=
1
(2π) p
2
(2π) p
2
exp
exp
− 1
− 1
2 uT u
2 uT
u
·
− 1
· 2u
2
· (−C T )
= φp(u) u C T ∂φp
(u)
∂Σ
=
,
∂ 1
∂Σ (2π)
p
exp −
2
1
2 uT =
u
1
(2π) p
exp −
2
1
2 uT
u · − 1
· 2u ·
2 ∂u
∂Σ
= φp(u)(−u) · ∂u
.
∂Σ
Σ ∂u
∂Σ
ui = p j=1 cij(tj − μj) i ∈{1, ..., p} Σ C
u = C(x − μ)

p
p θ
p = 2 B
Σ=
⎛
B = ⎝
C := B −1 ⎛
⎜
= ⎝−
∂φ2
∂μ (u) = φ2(u) u C T
∂φ2
∂Σ (u) = φ2(u)(−u) · ∂u
∂Σ
Σ11 Σ12
Σ21 Σ22
√ Σ11
Σ21 √
Σ11
= BB T
0
Σ22 − Σ2 21
Σ11
√ 1
Σ11
Σ21
Σ11 Σ22− Σ2 21
Σ11
⎞
⎠
0
1
Σ22− Σ2 21
Σ 11
TP =(t1,t2) ui = 2
j=1 cij(tj − μj) i ∈{1, 2}
⎛
⎜
u = ⎝−
Σ21
Σ11 Σ22− Σ2 21
Σ11
1
√ Σ11 (t1 − μ1)
(t1 − μ1)+
1
Σ22− Σ2 21
Σ 11
⎞
⎟
⎠ .
⎞
⎟
(t2 − μ2) ⎠ .
B

p =2
l(θ; x (1) ,...,x (n) ) l(θ
∂l(θ)
∇l(θ) = ,
∂μ1
∂l(θ)
,
∂μ2
∂l(θ)
,
∂σ1
∂l(θ)
,
∂σ2
∂l(θ)
T ∂ρ
l(θ)
∂l(θ)
= −n
∂μ1
∂
log(
∂μ1
(−1)
κ∈K(p)
g(κ)
F (κ; θ))
1
−
2(1 − ρ2 n
− 2
)
i=1
x(i) 1 − μ1
σ2 − 2ρ
1
−(x(i)
2 − μ2)
σ1σ2
n
= −
(−1)
κ∈K(p)
g(κ)
∂
(
F (κ; θ) ∂μ1
(−1)
κ∈K(p)
g(κ)
F (κ; θ))
+ 1
1 − ρ2 n
(i)
x 1 − μ1
σ
i=1
2 − ρ
1
x(i)
2 − μ2
,
σ1σ2
∂l(θ)
n
= −
∂μ2
(−1) g(κ)
∂
(
F (κ; θ) ∂μ2
(−1)
κ∈K(p)
g(κ)
F (κ; θ))
κ∈K(p)
n
x (i)
2
− μ2
+ 1
1 − ρ2 σ
i=1
2 2
∂l(θ)
= −n
∂σ1
1
− n
σ1
∂
log(
∂σ1
∂l(θ)
∂σ2
∂l(θ)
∂ρ
−
1
2(1 − ρ 2 )
= − n
−
σ1
+ 1
1 − ρ 2
= − n
−
σ2
+ 1
1 − ρ 2
n
i=1
κ∈K(p)
n
i=1
κ∈K(p)
n
i=1
κ∈K(p)
− 2 (x(i) 1
− ρ x(i) 1
− μ1
σ1σ2
,
(−1) g(κ)
F (κ; θ))
− μ1) 2
σ 3 1
+2ρ (x(i) 1
− μ1)(x (i)
2
σ 2 1 σ2
.
− μ2)
n
(−1) g(κ)
∂
(
F (κ; θ) ∂σ1
(−1)
κ∈K(p)
g(κ)
F (κ; θ))
(x (i)
1
− μ1) 2
σ 3 1
− ρ (x(i) 1
− μ1)(x (i)
2
σ 2 1 σ2
− μ2)
,
n
(−1) g(κ)
∂
(
F (κ; θ) ∂σ2
(−1)
κ∈K(p)
g(κ)
F (κ; θ))
(x (i)
2
− μ2) 2
σ 3 2
1
∂
= −n (−2)ρ − n
1 − ρ2 ∂ρ
− ρ (x(i) 1
log(
κ∈K(p)
− μ1)(x (i)
2
σ1σ 2 2
− μ2)
,
(−1) g(κ)
F (κ; θ))

+
+
ρ
(1 − ρ 2 ) 2
1
2(1 − ρ 2 ) 2
= 2ρn
−
1 − ρ2 +
ρ
(1 − ρ 2 ) 2
+ 1
1 − ρ 2
n
i=1
n
i=1
κ∈K(p)
n
i=1
n
i=1
x (i)
1
(x (i)
1
2 − μ1
σ1
− μ1)(x (i)
2
σ1σ2
(i)
x 2 +
2 − μ2
σ2
− μ2)
− 2ρ (x(i) 1
− μ1)(x (i)
2
σ1σ2
n
(−1) g(κ)
∂
( (−1)
F (κ; θ) ∂ρ
κ∈K(p)
g(κ)
F (κ; θ))
x (i)
1
(x (i)
1
2 − μ1
σ1
− μ1)(x (i)
2
σ1σ2
(i)
x 2 +
− μ2)
.
2 − μ2
σ2
− 2ρ (x(i) 1
− μ1)(x (i)
2
σ1σ2
− μ2)
− μ2)
F κ ∈ K(p) μ1,μ2,σ1,σ2 ρ
μ1
x1
x2
∂
1
F (x; θ) =
∂μ1
2πσ1σ2 1 − ρ2 −∞ −∞
×
1
exp −
2(1 − ρ2 t1 − μ1
) σ1
− 1
1 − ρ2
t1 − μ1
− ρ t2 − μ2
σ1σ2
=
x1
x2
−∞ −∞
σ 2 1
f(t) − 1
1 − ρ2 2
t1 − μ1
∂l(θ)
∂μ1
∂l(θ)
∂μ1
= 1
1 − ρ 2
κ∈K(p)
n
i=1
x (i)
1
− μ1
σ 2 1
(−1) g(κ)
κ1
σ 2 1
− μ2
− ρ x(i) 2
σ1σ2
κ2
−∞ −∞
− 2ρ t1 − μ1
σ1
dt2 dt1
− ρ t2
− μ2
σ1σ2
−
f(t) − 1
1 − ρ2
κ∈K(p)
t1 − μ1
σ 2 1
· t2 − μ2
σ2
dt2 dt1 .
2
t2 − μ2
+
×
σ2
n
(−1) g(κ) F (κ; θ) ×
− ρ t2
− μ2
σ1σ2
dt2 dt1 .
l(θ) μ2 σ1 σ2 ρ

p>2
p =2
F (x; θ) =
x1
x2
−∞ −∞
1
2π det(Σ)
θ W := (t − μ)(t − μ) T
· exp − 1
2 (t − μ)T Σ −1
(t − μ)
dt2 dt1
x1
x2
1
dF = d
2π
−∞ −∞
exp −
det(Σ)
1
2 (t − μ)T Σ −1
(t − μ)
1
+
2π det(Σ) d
exp − 1
2 (t − μ)T Σ −1
(t − μ)
dt2 dt1
x1
x2
1
=
· (−1
2π 2
−∞ −∞
)
1
3
det(Σ) det(Σ) · tr(Σ−1
(dΣ)) exp − 1
2 (t − μ)T Σ −1
(t − μ)
1
+
2π det(Σ) exp
− 1
2 (t − μ)T Σ −1
(t − μ) ×
− 1
2 tr((−Σ−1 (dΣ)Σ −1 )W ) − 1
2 tr(Σ−1 (t − μ)(−dμ) T − (dμ)(t − μ) T
)
dt2 dt1
x1
x2
1
= −
4π
−∞ −∞
· tr(Σ
det(Σ)
−1
(dΣ)) exp − 1
2 (t − μ)T Σ −1
(t − μ)
1
+
2π det(Σ) exp
− 1
2 (t − μ)T Σ −1
(t − μ) ×
1
2 tr((Σ−1 (dΣ)Σ −1 )W )+(dμ) T Σ −1
(t − μ)
dt2 dt1
x1
x2
1
=
2π
−∞ −∞
exp −
det(Σ)
1
2 (t − μ)T Σ −1
(t − μ) ×
− 1
2 tr(Σ−1 (dΣ)) + 1
2 tr((Σ−1 (dΣ)Σ −1 )W )+(dμ) T Σ −1
(t − μ) dt2 dt1

=
x1
x2
1
2π
exp −
det(Σ)
1
2 (t − μ)T Σ −1
(t − μ) ×
1
2 tr((dΣ)Σ−1 (W − Σ)Σ −1 )+(dμ) T Σ −1
(t − μ) dt2 dt1.
−∞ −∞
d(l(θ; x1, ..., xN)) = 0
⇔ 0= 1
2 tr((dΣ)Σ−1 (Z − NΣ)Σ −1 )+(dμ) T Σ −1
⎛
− N ⎝
−∞ −∞
TU1
TU2
−∞ −∞
d 2 (l(θ; x1, ..., xN))
1
2π det(Σ)
N
(xi − μ)
i=1
· exp − 1
2 (t − μ)T Σ −1
(t − μ)
TU1 TU2
1
2π
exp −
det(Σ)
1
2 (t − μ)T Σ −1
(t − μ) ×
1
2 tr((dΣ)Σ−1 (W − Σ)Σ −1 )+(dμ) T Σ −1
(t − μ) dt2 dt1.
= 1
2 tr((dΣ)(d(Σ−1 ))(Z − NΣ)Σ −1 )+ 1
2 tr((dΣ)Σ−1 (Z − NΣ)(d(Σ −1 )))
+ 1
2 tr((dΣ)Σ−1 (dZ − N(dΣ))Σ −1 )+(dμ) T (d(Σ −1 ))
− (dμ) T Σ −1 N(dμ)
⎛
TU1
− N · (−1) ⎝
TU1
−∞ −∞
1
TU2
TU2
−∞ −∞
1
2π det(Σ)
1
2π det(Σ)
N
(xi − μ)
i=1
· exp − 1
2 (t − μ)T Σ −1
(t − μ)
· exp − 1
2 (t − μ)T Σ −1
(t − μ) ×
2 tr((dΣ)Σ−1 (W − Σ)Σ −1 )+(dμ) T Σ −1 (t − μ)
⎛
+ ⎝
TU1
TU2
−∞ −∞
1
2π det(Σ)
dt2 dt1
· exp − 1
2 (t − μ)T Σ −1
(t − μ)
2
⎞
dt2 dt1⎠
dt2 dt1⎠
−1
⎞
dt2 dt1⎠
⎞
−2
×
−1
×

+
1
TU1
TU2
1
2π det(Σ)
· exp − 1
2 (t − μ)T Σ −1
(t − μ) ×
−∞ −∞
2 tr((dΣ)Σ−1 (W − Σ)Σ −1 )+(dμ) T Σ −1 2 (t − μ)
1
+
2π
· exp −
det(Σ) 1
2 (t − μ)T Σ −1
(t − μ) ×
1
2 tr((dΣ)(d(Σ−1 ))(W − NΣ)Σ −1 )+ 1
2 tr((dΣ)Σ−1 (W − NΣ)(d(Σ −1 )))
+ 1
2 tr((dΣ)Σ−1 (dW − N(dΣ))Σ −1 )+(dμ) T (d(Σ −1 ))(t − μ)
− (dμ) T Σ −1
(dμ) dt2 dt1 .
μ Σ
F
x = μ
F (x; θ)
x = μ
F (x) =
|j|≤n
DjF (μ)
(x − μ)
j!
j +
|j|=n+1
DjF (z)
(x − μ)
j!
j

z ∈ {μ + t(x − μ) | t ∈ [0, 1]} ,
⎛ ⎞
j1
⎜
j = ⎜ ⎟
⎟
⎝ ⎠ ∈ Nm0 ,
jm
|j| := j1 + ...+ jm ( j) ,
m
j! := ji! ,
D j F :=
(x − μ) j =
i=1
∂ |j|
∂x j1
1 · ...· ∂xjm F,
m
m
(xi − μi) ji .
i=1
x1
x2
1
F (x; θ) =
2πσ1σ2 1 − ρ2 −∞ −∞
×
1
exp −
2(1 − ρ2 t1 − μ1
)
=
x1
x2
−∞ −∞
1
2π det(Σ)
σ1
2
− 2ρ t1 − μ1
σ1
· t2 − μ2
σ2
· exp − 1
2 (t − μ)T Σ −1
(t − μ)
2
t2 − μ2
+
dt2 dt1
dt2 dt1
x2
∂F
1
(x) =
∂x1 2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
·
σ1
t2
2
− μ2 t2 − μ2
+
dt2 ,
σ2 σ2
μ2
∂F
1
(μ) =
∂x1 2πσ1σ2 1 − ρ2 exp
1
−
2(1 − ρ2 t2
2
− μ2
dt2 ,
) σ2
−∞
x1
∂F
1
(x) =
∂x2 2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 t1 − μ1
)
σ1
2
− 2ρ t1 − μ1
σ1
· x2 − μ2
σ2
σ2
2
x2 − μ2
+
dt1 ,
σ2

∂F
(μ) =
∂x2
μ1
−∞
1
2πσ1σ2 1 − ρ2 exp
1
−
2(1 − ρ2 t1
2
− μ1
dt1 ,
) σ1
∂2F ∂x2 (x) =
1
∂2F ∂x2 (μ) =
1
x2
1
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
σ1
1
−
2(1 − ρ2
2
)
x1 − μ1
− 2ρ
σ1
t2
− μ2 1
dt2
σ2 σ1
x2
∂2F ∂x2 x1
(x) =
2
−∞
∂2F ∂x2 (μ) =
2
· t2 − μ2
σ2
2
t2 − μ2
+
×
1
=
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
·
σ1
t2
2
− μ2 t2 − μ2
+
×
σ2 σ2
− 1
1 − ρ2
x1 − μ1
− ρ
σ1
t2
− μ2 1
dt2 ,
σ2 σ1
μ2
1
2πσ1σ2 1 − ρ2 −∞
exp
1
−
2(1 − ρ2
2
t2 − μ2 ρ
) σ2 1 − ρ2
t2 − μ2
dt2 ,
σ1σ2
1
2πσ1σ2 1 − ρ2 ×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
x2
2
− μ2 x2 − μ2
+
×
σ2 σ2
1
−
2(1 − ρ2
2
)
x2 − μ2
− 2ρ
σ2
t1
− μ1 1
dt1
σ1 σ2
x1
1
=
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
x2
2
− μ2 x2 − μ2
+
×
σ2 σ2
− 1
1 − ρ2
x2 − μ2
− ρ
σ2
t1
− μ1 1
dt1 ,
σ1 σ2
μ1
1
2πσ1σ2 1 − ρ2 exp
1
−
2(1 − ρ2
2
t1 − μ1 ρ
) σ1 1 − ρ2
t1 − μ1
dt1 ,
σ1σ2
−∞
σ2

∂2F (x) =φ(x) ,
∂x1∂x2
∂2F 1
(μ) =φ(μ) =
∂x1∂x2
2πσ1σ2 1 − ρ2 ,
∂3F ∂x3 (x) =
1
∂3F ∂x3 (x) =
2
x2
1
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
σ1
− 1
1 − ρ2
x1 − μ1
− ρ
σ1
t2
2 − μ2 1
σ2 σ1
+ − 1
1 − ρ2
1
σ2 x2
1
1 2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
σ1
x2
1
=
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 x1 2 − μ1
) σ1
− 1
1 − ρ2
x1 − μ1
− ρ
σ1
t2 − μ2
σ2
+ − 1
1 − ρ2
1 ∂F
(x) ,
∂x1
x1
σ 2 1
− 2ρ x1 − μ1
σ1
1
σ1
2
· t2 − μ2
σ2
dt2 ×
dt2
· t2 − μ2
σ2
· t2 − μ2
σ2
1
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
x2 − μ2
σ2
− 1
1 − ρ2
x2 − μ2
− ρ
σ2
t1
2 − μ1 1
dt1
σ1 σ2
+ − 1
1 − ρ2
1
σ2 x1
1
1 2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
x2 − μ2
σ2
2
t2 − μ2
+
σ2
2
t2 − μ2
+
dt2
σ2
2
t2 − μ2
+
×
σ2
2
x2 − μ2
+
×
σ2
2
x2 − μ2
+
dt1
σ2

∂x 2 1 ∂x2
∂x1∂x 2 2
x1
1
=
2πσ1σ2 1 − ρ2 −∞
×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
x2
2
− μ2 x2 − μ2
+
×
σ2 σ2
− 1
1 − ρ2
x2 − μ2
− ρ
σ2
t1
2 − μ1 1
σ1 σ2
+ −
dt1
1
1 − ρ2
1
σ2 ∂F
(x) ,
2 ∂x2
∂3F 1
(x) =
2πσ1σ2 1 − ρ2 ×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
·
σ1
x2
2
− μ2 x2 − μ2
+
×
σ2 σ2
1
−
2(1 − ρ2
2
)
x1 − μ1
− 2ρ
σ1
x2
− μ2 1
σ2 σ1
= φ(x) − 1
1 − ρ2
x1 − μ1
− ρ
σ1
x2
− μ2 1
,
σ2 σ1
∂3F 1
(x) =
2πσ1σ2 1 − ρ2 ×
1
exp −
2(1 − ρ2 x1 2 − μ1
− 2ρ
) σ1
x1 − μ1
·
σ1
x2
2
− μ2 x2 − μ2
+
×
σ2 σ2
1
−
2(1 − ρ2
2
)
x2 − μ2
− 2ρ
σ2
x1
− μ1 1
σ1 σ2
= φ(x) − 1
1 − ρ2
x2 − μ2
− ρ
σ2
x1
− μ1 1
.
σ1 σ2
∂3 F
∂x 3 1
∂3F ∂x3
2
∂3F ∂x2 1∂x2 ∂3F ∂x1∂x2 2
z ∈{μ + t(x − μ) | t ∈ [0, 1]}

−n · log(
(−1) g(κ) F (κ; θ)).
κ∈K(p)
μ Σ

p
p
p
X ∼ MTG(μ, Σ, TL, TU) fT (x; μ, Σ) f(x; μ, Σ)
p
p
TL ∈ Rp TU ∈ Rp
E(X) Var(X)
M
X =(X1,...,Xp) ∼ MTG(μ, Σ, TL, TU)
E(X)
Var(X)
⎡
⎢
Var(X) =Σ+Σ⎢
⎣
E(X) =μ − Σ
κ∈K(p)
κ∈K(p)
κ∈K(p)
κ∈K(p)
(−1) g(κ) ∇2 κF (κ)
(−1) g(κ) F (κ) −
⎛
⎜
⎝
(−1) g(κ) ∇κF (κ)
(−1) g(κ) F (κ)
κ∈K(p)
κ∈K(p)
,
(−1) g(κ) ∇κF (κ)
(−1) g(κ) ⎞
⎟ ⎥
⎠ ⎥
F (κ) ⎦ Σ,
∇x x ∇ 2 x = ∇x∇ T x
M
2 ⎤

M(t) =
=
TU
TL
κ∈K(p)
TU
TL
exp{t T x}fT (x) dx
exp
1
(−1) g(κ) F (κ)
(2π) p
2
1
×
det(Σ)
− 1
2 [(x − μ)T Σ −1 (x − μ) − 2t T
x]
μ =0 μ
exp − 1
2 [xT Σ −1 x − 2t T
x] =exp − 1
2 [xT Σ −1 x − 2t T x + t T
1
Σt] exp
2 tT
Σt
=exp − 1
2 (x − Σt)T Σ −1
1
(x − Σt) exp
2 tT
Σt
M(t) =
=
=
κ∈K(p)
TU
TL
exp
κ∈K(p)
TU−Σt
TL−Σt
κ∈K(p)
κ∈K(p)
1
(−1) g(κ) F (κ)
(2π) p
2
1
×
det(Σ)
− 1
2 (x − Σt)T Σ −1
(x − Σt)
1
(−1) g(κ) F (κ)
exp
(2π) p
2
− 1
2 xT Σ −1
x
(−1) g(κ) F (κ − Σt)
(−1) g(κ) F (κ)
1
×
det(Σ)
1
dx exp
2 tT
Σt
dx.
1
dx exp
2 tT
Σt
1
exp
2 tT
Σt .
∇tM
∇2 tM
1
∇tM(t) =ΣtM(t) −
(−1) g(κ) F (κ) exp
1
2 tT
Σt Σ ×
κ∈K(p)

p
⎛
⎝
κ∈K(p)
∇ 2 tM(t) =M(t)Σ T + ∇tM(t)(Σt) T −
1
Σt exp
2 tT
Σt
κ∈K(p)
1
− Σexp
2 tT
Σt
⎞
(−1) g(κ) ∇κ−ΣtF (κ − Σt) ⎠
κ∈K(p)
κ∈K(p)
t =0
∇tM(0) = −Σ
∇ 2 tM(0) = Σ + Σ
μ =0
E(Xμ=0) =−Σ
κ∈K(p)
1
(−1) g(κ) F (κ) ×
(−1) g(κ) ∇κ−ΣtF (κ − Σt)
κ∈K(p)
κ∈K(p)
κ∈K(p)
T
Σ
(−1) g(κ) ∇ 2
κ−ΣtF (κ − Σt) Σ
κ∈K(p)
(−1) g(κ) ∇κF (κ)
(−1) g(κ) F (κ)
(−1) g(κ) ∇ 2 κF (κ)
κ∈K(p)
(−1) g(κ) F (κ)
(−1) g(κ) ∇κF (κ)
(−1) g(κ) F (κ)
Σ.
.
μ
(−1)
κ∈K(p)
E(X) =μ − Σ
g(κ) ∇κF (κ)
κ∈K(p)
(−1) g(κ) F (κ)
.
μ =0
Var(Xμ=0) =E(Xμ=0X T μ=0) − E(Xμ=0)E(Xμ=0) T

=Σ+Σ
⎢
=Σ+Σ⎢
⎣
(−1) g(κ) ∇2 κF (κ)
(−1) g(κ) ⎛
⎜
Σ − ⎝Σ
F (κ)
κ∈K(p)
⎡
κ∈K(p)
(−1) g(κ) ∇2 κF (κ)
(−1) g(κ) F (κ) −
⎛
⎜
⎝
κ∈K(p)
κ∈K(p)
(−1) g(κ) ∇κF (κ)
(−1) g(κ) ⎞
⎟
⎠
F (κ)
κ∈K(p)
κ∈K(p)
(−1) g(κ) ∇κF (κ)
(−1) g(κ) ⎞
⎟ ⎥
⎠ ⎥
F (κ) ⎦ Σ.
κ∈K(p)
κ∈K(p)
2
2 ⎤
μ
Var(X) =Var(Xμ=0)
F (−a; R)
κ∈K(p) (−1)g(κ) F (κ)
p
F (x; μ, Σ) =
x1
−∞
...
xp
−∞
φ(u; μ, Σ)dup ...du1
1

p
x 2
0 2 4 6 8 10
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●TL
0 2 4 6 8 10
TL=(5, 5) T
R
R tmvtnorm
x 1

x =(x (1) ,...,x (n) )
Θ := (π1,...,π3; θ1,...,θ3) f
f(x; Θ) :=π1f1(x; θ1)+π2f2(x; θ2)+(1− π1 − π2)f3(x; θ3),
fr θr ∈ Rpr
πr r ∈{1, 2, 3}
0 ≤ πr ≤ 1 ∀ r ∈{1, 2, 3}
3
πr =1.
[TL, TU]
TL∈ Rp TU ∈ Rp r=1
f(x; Θ) ≈ π2f2(x; θ2)
f2
f1 f3

θ2 π2 Femp
f
θ2 π2
TL TU
ˆ θ := ˆ θ2
θ2
π2 ˆ θ
TU1
TL1
TU1
TL1
...
TUp
TLp
...
TUp
TLp
f(x; Θ) dx ≈ π2
TU1
TL1
...
TUp
TLp
f2(x; ˆ θ)dx
f(x; Θ) dx = F (TL≤ p x ≤ p TU)
≈ Femp(TL≤ p xi ≤ p TU)
= m
⇒ π2 ≈
[TL, TU]
=
n
m
⎛
TU1 TUp
⎜
⎝ ... f2(x;
n
ˆ ⎞−1
⎟
θ)dx⎠
.
TL1
ˆ θ
γ
TLp

f2
F2
F2

χ2
β
´
13

D()
D(Np(0,Ip),Q)=0 ⇔ Q = Np(0,Ip),
Np(0,Ip) p Q
D
Dβ(P, Q) = | ˆ P (t) − ˆ Q(t)| 2 ϕβ(t)dt
R p
ˆ P (t) ˆ Q(t) P
Q ϕβ(t)
Np(0,β2Id)
ϕ(t) =
1
(2πβ2 ) p
2
−|t| 2
exp
2β2
t ∈ R p
||
β n
p
βp(n) = 1
√ 2
2p +1
4
1
p+4
· n 1
p+4 .

HZβ(p)
HZβ(p) = 1
n 2
−
n
n
j=1 k=1
2
(1 + β2 ) p
2
exp − β2
2 |Yj − Yk| 2
1
n
n
β
exp −
2
2(1 + β2
2
|Yj| +
)
j=1
1
(1 + 2β2 ) p
2
|Yj − Yk| 2 = (Xj − Xk) T S −1 (Xj − Xk)
|Yj| 2 = (Xj − ¯ X) T S −1 (Xj − ¯ X),
¯ X S
p
HZ
p
HZ ˆ θ
ˆ θ
HZk
p
HZk HZ
p
x1,...,xn xi ∈ Rp fT (x, θ) =:fT (θ)
p θ := (μ, Σ)

ˆθ
HZ := HZ( ˆ θ; x1,...,xn) =HZ( ˆ θ)
n1 n fT ( ˆ θ)
HZk := HZ( ˆ θ k ; x k 1 ,...,xk n)
p−
{x k i }1≤i≤n ∀ k ∈{1,...,n1}
p = #(HZk ≥ HZ)
n1

TU
TL
π = 1

p
i
si
xi ∈ R2 i ∈{1,...,1000}
μ =(24, 4.2) T
10 2
Σ= .
2 1
s =(2.5, 0.5) T
TU0
TU0 = μ + s =(26.5, 4.7) T
TU1
TU1 = TU0 +(0,s2) T =(26.5, 5.2) T .
μ

Laborwert y
2 4 6 8
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15 20 25 30 35
Laborwert x
Laborwert y
3 4 5 6 7 8 9
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24 26 28 30 32 34 36
Laborwert x
●
●
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●
●
δ
TL p
θ
p
p
p
●
●
●
●
●
●

p
p
p pfix
M
M =
j ∈{1, ..., J}|pj ≥ pfix
p
θ
{x1,...,xn} xi ∈ Rp TL ∈ Rp
TL≤x1 ≤x2 ≤...≤xn TL≤ p xi ∀ i ∈{1,...,n}.
ˆ θ0 =(ˆμ 0, ˆ Σ0)
j := 1 xi
i
xi ≥ p ˆμ 0 xi ≥ˆμ 0
TU1 = xi
TUj = xn j = j +1 i = i+1
TUj = xi
θ
ˆθj =(ˆμ j, ˆ Σj).

HZ
M := {x∗ 1 ,...,x∗mj } = {xi | xi ∈ [TL, TUj]} mj := #M
M
HZ := HZ( ˆ θj; x ∗ 1 ,...,x∗ mj )=HZ(ˆ θj)
p pj
μHZ σHZ HZ
pj =1− F (HZ; μHZ,σHZ),
F
M = j ∈{1, ..., J}|pj =max{p1,...,pJ}
j0 := arg j∈M{min mj},
J TUj0
j0

{x1,...,xn} xi ∈ Rp TL ∈ Rp
TL≤x1 ≤x2 ≤...≤xn TL≤ p xi ∀ i ∈{1,...,n}.
a =(a1,...,ap) T := (0,...,0) T j := 1
s =(s1,...,sp) T δ>0 ˆ θ0 =(ˆμ 0, ˆ Σ0)
TUj
l := p
al = al +1 TUj = TU0 +(a1 · s1,...,ap · sp) T sl l
TU0 = ˆμ 0 + s
xi δ TUj
xi ∈ B(TUj; δ)
xi ∈ B(TUj; δ)
l>1 al = ...= ap =0 l = l −1
θ
ˆθj =(ˆμ j, ˆ Σj).
HZ
M := {x∗ 1 ,...,x∗mj } = {xi | xi ∈ [TL, TUj]} mj := #M
M
HZ := HZ( ˆ θj; x ∗ 1 ,...,x∗ mj )=HZ(ˆ θj)
p pj
μHZ σHZ HZ
pj =1− F (HZ; μHZ,σHZ),

F j = j +1
M = j ∈{1, ..., J}|pj =max{p1,...,pJ}
j0 := arg j∈M{min mj},
J TUj0
j0
ˆμ
p
p
μ
S =(S1,...,Sp)
μi i ∈{1,...,p}
√ Σii

μ
μ
S
Si = μi + Σii
= μi +
⎛ p p
⎜ l=2
⎜ p
⎝
p
|Σlm|
m=1
m

i ∈{1,...,p}
⎧
p √
Σll
⎪⎨ μi +Σii |Σli| |Σli| ≥0.3 l ∈{1,...,p}\i
l=1
Si =
l=i
⎪⎩
μi +Σii .
f1
μ1
F1

TU
TL p
θ
ˆ θ
p
p
p πj := π1 j F1
πj
[TL, TU]
πj πj > 1
p
p

{x1,...,xn} xi ∈ Rp TL ∈ Rp
TL≤x1 ≤x2 ≤...≤xn TL≤ p xi ∀ i ∈{1,...,n}
fr(x, θ r ) r ∈{1,...,R}
p
f(x) =
R
r=1
π r fr(x; θ r ).
f1
(ˆπ 1 ,...,ˆπ R ; ˆ θ 1
0,..., ˆ θ R
0 )
p
j := 1 xi i
ˆμ 1 0 ≤ p xi ˆμ 1 0≤xi
TUj = xi
TUj = xn j = j +1 i = i+1
TUj = xi
θ 1 j
ˆθ 1
j =(ˆμ 1 j, ˆ Σ 1 j) .
π1 j
M := {x∗ 1 ,...,x∗m} = {xi | xi ∈ [TL, TUj]} m := #M

M
m
n
Ej(m)
n
⇒ π 1 j ≈
[TL, TUj]
=
= π 1 TUj1 TUjp
j ... f1(x; ˆ θ 1
j) dx
n
TL1
TUj1
TL1
...
TLp
m
TUjp
TLp
f1(x; ˆ θ 1
j) dx
Ej(m) ˆ θj
HZ
HZ := HZ( ˆ θ 1
j; x∗ 1 ,...,x∗m)=HZ( ˆ θ 1
j)
n1 m fT ( ˆ θ 1
j)
{x (k)
i }1≤i≤m ∀ k ∈{1,...,n1} .
fT (θ)
ˆ θ (k) ∀ k ∈{1,...,n1}
HZk := HZ( ˆ θ (k) ; x (k)
1 ,...,x(k) m )
p pj
pj = #(HZk ≥ HZ)
n1
M = j ∈{1, ..., J}|pj =max{p1,...,pJ}
j0 := arg j∈M{min πj},
J TUj0
j0
,

{x1,...,xn} xi ∈ Rp TL ∈ Rp
TL≤xi TL≤ p xi ∀ i ∈{1,...,n}
fr(x, θ r ) r ∈{1,...,R}
p
f(x) =
R
r=1
π r fr(x; θ r ).
f1
a =(a1,...,ap) T := (0,...,0) T j := 1
s = (s1,...,sp) T δ > 0
(ˆπ 1 ,...,ˆπ R ; ˆ θ 1
0,..., ˆ θ R
0 )
p
TUj
l := p
al = al +1 TUj = TU0 +(a1 · s1,...,ap · sp) T sl l
TU0 = ˆμ 1 0 + s
xi δ TUj
xi ∈ B(TUj; δ)
xi ∈ B(TUj; δ)
l>1 al = ...= ap =0 l = l −1
θ 1 j
ˆθ 1
j =(ˆμ 1 j, ˆ Σ 1 j)
π1 j
M := {x∗ 1 ,...,x∗m} = {xi | xi ∈ [TL, TUj]} m := #M

M
m
n
E(m)
n
⇒ π 1 j ≈
[TL, TUj]
=
= π 1 TUj1 TUjp
j ... f1(x; ˆ θ 1
j) dx
n
TL1
TUj1
TL1
...
TLp
m
TUjp
TLp
f1(x; ˆ θ 1
j) dx
HZ
HZ := HZ( ˆ θ 1
j; x ∗ 1 ,...,x∗ m)=HZ( ˆ θ 1
j)
n1 m fT ( ˆ θ 1
j)
{x (k)
i }1≤i≤m ∀ k ∈{1,...,n1} .
fT (θ)
ˆ θ (k) ∀ k ∈{1,...,n1}
HZk := HZ( ˆ θ (k) ; x (k)
1 ,...,x(k) m )
p pj
pj = #(HZk ≥ HZ)
n1
M = j ∈{1, ..., J}|pj =max{p1,...,pJ}
j0 := arg j∈M{min πj},
J TUj0
j0
.

πr r ∈{1,...,R}
h
μ 1 = (140, 4.2) T , μ 2 = (155, 7.8) T , μ 3 = (145, 6.0) T
Σ 1
10 2
= , Σ
2 1
2
25 3
= , Σ
3 4
3
25 3
= .
3 4
F G
f(x) = 0.75 · h(x; μ 1 , Σ 1 )+0.25 · h(x; μ 2 , Σ 2 ),
g(x) = 0.75 · h(x; μ 1 , Σ 1 )+0.25 · h(x; μ 3 , Σ 3 ).
F
G
S = (S1,S2) T
= (μ 1 1 + Σ1 11
Σ 1 22
|Σ1 21 |
= (145, 5.78) T .
,μ 1 2 + Σ1
22 Σ1 11 )
|Σ21|

F
G
μ3
Laborparameter B
2 4 6 8 10 12
●
start
130 140 150 160 170
Laborparameter A
F
Laborparameter B
0 2 4 6 8 10
●
start
130 135 140 145 150 155 160
Laborparameter A
G

S =(S1,...,Sp)
Si
S
Si = ξi + ψi
ξi
ψi
ξi μ1 i i ∈{1,...,p}
μ1 i
π1
⎛
ξi = π 1
⎜
⎝
μ 1 i +
p
Σ1 ii
l=1
l=i
p
l=2
p
m=1
m

μ1
πr r ∈{2,...,R}
ξi ψi
fr
p
l=2
p
m=1
m

Ei
E =(E1,...,Ep)
Ei =
R
π r
⎜
⎝
r=1
⎛
μ r i +
Σr ii
l=1
l=i
p
l=2
p
Σr ll
p
|Σr lm |
⎞
⎟
⎠
m=1
m

EF EG
EF ≈ (151.67, 7.95) T EG ≈ (149.17, 7.50) T .
S = (145, 5.78) T
F
μ1 G
p
Laborparameter B
2 4 6 8 10 12
●
start
stop
130 140 150 160 170
●
Laborparameter A
F
Laborparameter B
0 2 4 6 8 10
●
start
stop
130 135 140 145 150 155 160
●
Laborparameter A
G
μ1 E
p

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140 142 144 146 148
4.5 5.0 5.5 6.0 6.5 7.0 7.5
Laborwert A
Laborwert B
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start
stop
p
G
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p = 0
0 < p

μ1
G
4.17 μ1 2
μ1
F1
μ1
μ1
μ1

θ
ˆ θ
100(1 − α)% p
x (i)
(μ − x (i) ) T Σ −1 (μ − x (i) ) ≤
p(n − 1)
n − p Fp,n−p(α)
Fp,n−p 100(1 − α) F p
n−p Σ
Σ
λi ·
p(n − 1)
n − p Fp,n−p(α)
λi i Σ
100(1 − α)

θ
i simRIi
simRIi =
μi −
p(n − 1)
n − p Fp,n−p(α)
p(n − 1)
Σii ,μi +
n − p Fp,n−p(α)
Σii
i ∈{1,...,p}
p p
z
θ
RIi i ∈{1,...,p}
RIi =
μi − z (1−α/2) Σii ,μi + z (1−α/2) Σii
z (1−α/2) (1 − α/2)
p

p
z
p
z
z
z
z p =2
Z =
(4.24, 1.17)
[3.22, 5.26] [0.62, 1.71]
z
[3.42, 5.05] [0.73, 1.61]

2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Laborparameter A
Laborparameter B
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Laborparameter A
Laborparameter B
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z
z
2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Laborparameter A
Laborparameter B
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p
95%
95% α
100(1 − α)%
95%
95% P1
P4
P1 =(4.00, 2.20), P2 =(4.33, 2.28), P3 =(4.10, 1.75) P4 =(4.55, 2.30)
95%
• P1 65% 95%
• P2 94% 95%
• P3 96% 95%
• P4 99%
95%
95%

Laborparameter B
1.0 1.5 2.0 2.5 3.0
2.0 2.5 3.0 3.5 4.0 4.5 5.0
Laborparameter A
P1 ●
●
85%−Referenzbereich
90%−Referenzbereich
95%−Referenzbereich
97.5%−Referenzbereich
99%−Referenzbereich
P 3
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
P 2
●
P 4
●

p
p
1.0
p
HZ
p
p HZ
p

p
p
p
p
HZ
HZ
HZ
p
p

R
R
R
p
|Yj − Yk| 2 =(Xj − Xk) T S −1 (Xj − Xk)
R

n
n = 4000

n = 4000
q(n) =0.00001n 3 +0.0563n 2 +89.3n
n ≫ 4000
p
n ≫ 4000 n

Zeitverhältnis: Schleife/Vektor
0 500000 1000000 1500000 2000000
●
Regressionskurve q(n)
gemessenes Zeitverhältnis ●
●●●
●●
●●
●●
●●
●●●
●●
● ●●
●
●
●
●
0 1000 2000 3000 4000
●
Datensatzlänge n
●
●
●
●
●

p
p

p
p

p

(f,g) = min(f(x),g(x)) dx
(p+1)
(f,g) = |f(x) − g(x)| dx
(p +1)
2= f + g = |f − g| +2 min(f,g) =ABC(f,g)+2PSR(f,g)
f
g

f(x)
f(x)
f(x)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00 0.05 0.10 0.15 0.20 0.25 0.30
−2 0 2 4 6
x
−2 0 2 4 6
x
−4 −2 0 2 4 6 8 10
x
f(x)
f(x)
f(x)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00 0.05 0.10 0.15 0.20 0.25 0.30
−2 0 2 4 6
x
−2 0 2 4 6
x
−4 −2 0 2 4 6 8 10
x

π1 =0.85 π2 =0.15
f(x)
0.0 0.1 0.2 0.3 0.4
y 1 = 0.85 N(2, 1.2)
y 2 = 0.15 N(2, 1)
−2 0 2 4 6
x
f(x)
0.0 0.1 0.2 0.3 0.4
y 1 = 0.85 N(2, 1.2)
y 2 = 0.15 N(3, 0.35)
−2 0 2 4 6
x

fi fj i, j ∈{1, 2} i = j
αij =
fi(x)fj(x)dx
f 2 i (x)dx
αij =
|Σi| 1
2
| 1
2 (Σi +Σj)| 1
2
exp − 1
2 (μi − μj) T (Σi +Σj) −1
(μi − μj)
Σ1 = Σ2
α21 = α12
α21 = α12
α21 = α12 =exp − 1
4 (μ1 − μ2) T Σ −1
1 (μ
1 − μ2)
α12 α21
f1
f2
α12 α21
f1 f2 α12 α21
α = √ α12α21 =
f1(x)f2(x)dx
1
f 2 2
1 (x)dx f 2 1
2
2 (x)dx

f1 f2
α =
|Σ1Σ2| 1
4
| 1
2 (Σ1 +Σ2)| 1
2
exp − 1
2 (μ1 − μ2) T (Σ1 +Σ2) −1
(μ1 − μ2) .
α12 = α21
f1 f2 π1f1 π2f2
α12 α21
f(x) =π1f1(x)+π2f2(x) π1 =0.9 π2 =0.1
f1 f2
μ 1 = (140, 4.2) T , μ 2 = (142, 5.2) T
Σ1 =
10 2
2 0.5
, Σ2 =
2 1
0.5 0.25
α
α ≈ 0.5016 50%
f1 f2
α12
α21
α12 =0.1234 α21 =2.0395.
π1f1 π2f2 12%
α21 ≥ 1

z
0.00 0.01 0.02 0.03 0.04 0.05 0.06
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●●●●●●●●●●● 130 135 140 145 150
x
y
y
0 2 4 6 8
130 135 140 145 150
12%
50%
πifi πjfj i, j ∈{1, 2} i = j
αij = πj
fi(x)fj(x)dx
πi f 2
i (x)dx
α12 α21
αij = πj
πi
|Σi| 1
2
| 1
2 (Σi +Σj)| 1
2
exp − 1
2 (μi − μj) T (Σi +Σj) −1
(μi − μj) .
x

f1
α12
α12
F2

I
J G
f1 f2 r f1 s
f2
I(r, s)
J(r, s) =2
I(2r, 0) + I(0, 2s)
G(r, s) =
I(r, s)
I(2r, 0) · I(0, 2s)
I(r, s) = (f1(x)) r (f2(x)) s dx
fi i ∈{1, 2} p θi =(μi, Σi)
I
I(r, s) = exp{−(rs/2)(μ 1 − μ 2) T (sΣ1 + rΣ2) −1 (μ 1 − μ 2)}
|Σ1| (r−1)/2 |Σ2| (s−1)/2 |sΣ1 + rΣ2|(2π) p(r+s−1)/2
J G
J(r, s) =2exp − 1
2 (μ1 − μ2) T
−1
Σ1 Σ2
+ (μ1 − μ
r s
2) /
/ |Σ1| (r−1)/2 |Σ2| (s−1)/2 |sΣ1 + rΣ2|(2π) p(r+s−1)/2 ×
|Σ1| 1/2−r (2π) −p(r−1/2) (2r) −p/2 + |Σ2| 1/2−s (2π) −p(s−1/2)
(2s)
−p/2
,
G(r, s) =
exp − 1
2 (μ1 − μ2) T Σ1
r
Σ2
−1
+ s (μ1 − μ2) |Σ1| −1/4 |Σ2| −1/4 |sΣ1 + rΣ2| 1/2 (4rs) −p/4
.

J G
I(2r, 0) = |2πΣ1| (1−2r)/2 (2r) −p/2
I(0, 2s) = |2πΣ2| (1−2s)/2 (2s) −p/2
J(r, s) ≤ 1 G(r, s) ≤ 1 J(r, s) =G(r, s) =1
f1(x) =f2(x)
r s
J G
I F G
r s < 1
J G
0.5
J G
0.65
T = (140, 4.2)

y
0 2 4 6 8 10
130 135 140 145 150 155 160
x
y
0 2 4 6 8 10
130 135 140 145 150 155 160
J G
J
G
x

0.5
y
0 2 4 6 8 10
120 130 140 150 160
x
y
0 2 4 6 8 10
120 130 140 150 160
J G
μ = (140, 5.2) T
x

0.5
y
0 2 4 6 8 10
120 130 140 150 160
x
y
0 2 4 6 8 10
120 130 140 150 160
J G
μ = (140, 5.2) T
x

J
G

ˆ θ
μ
n

z
ˆθ
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
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●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●
●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● 2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A

z

Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
n
z
•
•

Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A

Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A

p
p
p(p+3)
2 μ
Σ
μ =(μ1,...,μp) T ⎛
⎞
Σ11 Σ12 Σ13 ··· Σ1p
⎜
⎟
⎜
⎜Σ21
Σ22 Σ23 ··· ⎟ Σ2p⎟
⎜
⎟
Σ= ⎜Σ31
Σ32 Σ33 ⎜
··· Σ3p ⎟
⎜
⎟
⎝
⎠
Σp1 Σp2 Σp3 ··· Σpp
(μ1,μ2, Σ11, Σ21, Σ22) T

z

z

n = 4000
n = 4000

μ 1 Σ1 π1 μ 2 Σ2 π2 α12
4.30
0.13
0.03
1.15 0.03 0.05
4.30
0.13
0.03
1.15 0.03 0.05
4.30
0.63
0.20
1.15 0.20 0.21
4.30
0.13
0.03
1.15 0.03 0.05
4.30
0.23
0.03
1.15 0.03 0.05
4.30
0.23
0.03
1.15 0.03 0.05
4.30
0.23
0.03
1.15 0.03 0.05
5.80
0.25
0.1
1.90 0.1 0.15
5.60
0.25
0.1
1.60 0.1 0.15
5.25
0.25
0.1
1.75 0.1 0.35
5.20
0.25
0.1
1.60 0.1 0.15
5.00
0.25
0.1
1.50 0.1 0.25
4.70
0.35
0.1
1.30 0.1 0.25
4.70
0.35
0.1
1.40 0.1 0.1
JRS
GRS

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
π 1 = 0.75
π 2 = 0.25
3 4 5 6 7
π 1 = 0.9
π 2 = 0.1
Laborparameter A
nicht−patholgisch
pathologisch
3 4 5 6 7
π 1 = 0.75
π 2 = 0.25
Laborparameter A
nicht−patholgisch
pathologisch
3 4 5 6 7
Laborparameter A
nicht−patholgisch
pathologisch
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
π 1 = 0.75
π 2 = 0.25
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
3 4 5 6 7
Laborparameter A
π 1 = 0.75
π 2 = 0.25
3 4 5 6 7
π 1 = 0.75
π 2 = 0.25
Laborparameter A
nicht−patholgisch
pathologisch
3 4 5 6 7
π 1 = 0.75
π 2 = 0.25
nicht−patholgisch
pathologisch
Laborparameter A
nicht−patholgisch
pathologisch
3 4 5 6 7
Laborparameter A
nicht−patholgisch
pathologisch

α12
π1 π2
Σ1 Σ2
JRS GRS

p
p
p
p
HZ
HZ
HZ
p
p
p

T =(4.6, 0.73)
T =(4.65, 0.735)

z
z
z

z
p

z

z

z

p
p
z
HZ
p
p
p
p p
p
HZ
z HZ
p
HZ

p
z
z
z
p
HZ
z
z
z
30% p
z < 2% HZ
> 11%
0.03
0.25 z

z
p
HZ

z
p
HZ

z
p
HZ

z
p
HZ

95%
z

99.1%

99.1%

99.1%

99.1%

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

z
z
3.5% 24.8%
25%

z
z
z
z
z
z
z
z
80%
z

50%
θ
Σ21
θ =(μ1,μ2, Σ11, Σ21, Σ22) T .
π1
π1
π1

p
p
Σ21
HZ
p
p

n = 4000
μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

p
p
z
p

p
p
z
p

100
p
n1 = 100
1000
0.1
100 100 p
z

z
z
z

z
z
0.0400
0.1938

z
z

p
p
p

z

z

z

z

−2.4745
z −0.1938
0.03
z
z
0.01
p
p
p

−2.4745
z −0.1938
0.03
z
z
0.01
p
p
p

−2.4745
z −0.1938
0.03
z
z
0.01
p
p
p

z

z

z

z

99.5%
0.5%
p
p
1.8 2.4% 3.0%
1.8 6.3% 1.8 5.6%
p 100
p
0 2.1%
p

4%
p 10%

p
0.5%
0.1
30%
99.5%
z
80%

p
0.5%
0.1
30%
99.5%
z
80%

μ1 μ2 Σ11 Σ21 Σ22

0.0400
μ1 μ2 Σ11 Σ21 Σ22

π1 =0.9
μ1 μ2 Σ11 Σ21 Σ22

(μ1,μ2, Σ11, Σ21, Σ22)
0.02
p
0.08
Σ21 =0.03
p
0.16
Σ21 =0.03
Σ22
Σ22 =0.05
0.08
p

ˆ θ θ

p
p
p
p

Σ22
p 0.8
μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

z
10−3
unteres RL von A
unteres RL von B
−0.016 −0.012 −0.008
−0.008 −0.004
●
●
●
●●
●
●
●
●
●
●
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● ●
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●
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● ●●●●●●●
●
●●
●● ● ●●●●●●●●●●●●●●● ● ●●●● ●●●
●
●
0 20 40 60 80 100
Anzahl Iterationen
●
●
●
● ●
● ●
●●●
●
●
●
●
●
●
●
●
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●● ●●●
●● ●●● ●● ●●●●●●
●
0 20 40 60 80 100
Anzahl Iterationen
oberes RL von A
oberes RL von B
0.052 0.056 0.060
0.012 0.016
●
●
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●
●
● ●● ●
● ● ●●●
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●● ●●●● ●●●●● ● 0 20 40 60 80 100
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●
●
●
Anzahl Iterationen
●● ●●● ●
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0 20 40 60 80 100
Anzahl Iterationen
z

unteres RL von A
unteres RL von B
−0.018 −0.014 −0.010
−0.008 −0.004
●
●
●
●
●
● ●
●
●
●●● ● ●
●●
●
●●●●
● ● ●
●
● ●●●●●●●●
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●
●
●
0 20 40 60 80 100
●
●
Anzahl Iterationen
●
●
●
●
●
●
●
●● ●●
●
●
●●
● ●
●
● ●●●● ● ●●● ● ● ●●●●●●● ●● ●●●● ●●●●●●●●●● ●●●
● ● ●●●●
●● ●●●●● ● ●●●●
0 20 40 60 80 100
Anzahl Iterationen
oberes RL von A
oberes RL von B
0.054 0.058 0.062
0.012 0.016 0.020
●
●
● ●
●
●
●
●
●
● ●
●
●●●●●
●
●
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●
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●
0 20 40 60 80 100
●
●
●
●
●
●
●
Anzahl Iterationen
● ●
● ●
●
●
●
●
●●●
●
●
●●
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● ●●
●
● ●●●●●●●
●
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●● ●●●●●●● ●● ●●● ●●●● 0 20 40 60 80 100
Anzahl Iterationen
z

z
10−3
μ1
Σ11
0.018 0.020 0.022
0.011 0.013
● ●
●
●
●
●
●
●
●
●
●
●
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●
●●
0 20 40 60 80 100
● ●
Anzahl Iterationen
●
●●
● ● ● ●
●●
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0 20 40 60 80 100
Anzahl Iterationen
Σ22
0.0020 0.0026
μ2
Σ12
0.001 0.004 0.007
0.0038 0.0044 0.0050
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0 20 40 60 80 100
Anzahl Iterationen
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0 20 40 60 80 100
Anzahl Iterationen
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0 20 40 60 80 100
Anzahl Iterationen
z

μ1
Σ11
0.018 0.021 0.024
0.0115 0.0130 0.0145
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Anzahl Iterationen
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● ●●●●● ●●●●●●●●●●●●●●● ●●●●●●● ●●● ●●● 0 20 40 60 80 100
Anzahl Iterationen
Σ22
0.0022 0.0028 0.0034
● ●●
●
μ2
Σ12
0.002 0.006
0.0046 0.0052
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0 20 40 60 80 100
Anzahl Iterationen
0 20 40 60 80 100
Anzahl Iterationen
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Anzahl Iterationen
z

p
p
p
20%
15%

100
150
10−3

p
p
p
p
p
1 : 500
p
p
3.5

50%

z

z

n = 4000

≥
≥
≥
≥
≥
≥
≥
p
p
p
p
HZ
HZ
HZ
p

T =(4.6, 0.73)
T =(4.65, 0.735)

p
p n1 = 100
p
z
z
z
p
HZ

p
5.6
5.7 5.2 5.1
p

z
≥
≥
≥

p
p=0.39
π1 ≈ 1.48
π1 > 1
π1 =1
50
n = 2702
n = 4000
18 − 49 n = 1957

Density
Density
0.0 0.5 1.0 1.5
2 3 4 5 6 7 8
Kalium
0.0 0.2 0.4 0.6 0.8 1.0
2 3 4 5 6 7 8
Kalium
π1 ≈ 1.48
Density
0.0 0.2 0.4 0.6 0.8 1.0
2 3 4 5 6 7 8
Kalium
π1 =1

HZ
Density
Density
0.0 0.5 1.0 1.5
2 3 4 5 6 7 8
Kalium
0.0 0.2 0.4 0.6 0.8 1.0
2 3 4 5 6 7 8
Kalium
π1 ≈ 0.96
p
p

0.1
p
p
n = 2476
n = 4000

z
≥

z
≥ 18
≥
0.1
≥ 50
n>4000
11000

z
≥ 18
≥
0.1
≥ 50
n>4000
11000

z
18 − 49 ≥ 50
p
p
1.76 2.00
0.71 0.77

z
≥
≥
≥
> 1

Density
Density
0.0 0.5 1.0 1.5 2.0 2.5
0 1 2 3 4 5 6
Phosphat
0.0 0.5 1.0 1.5 2.0 2.5
0 1 2 3 4
Phosphat
π1 ≈ 1.48
Density
0.0 0.5 1.0 1.5 2.0
0 1 2 3 4
Phosphat
π1 =1

1
Density
Density
0.0 0.5 1.0 1.5 2.0
0 1 2 3 4 5 6
Phosphat
0.0 0.5 1.0 1.5 2.0
0 1 2 3 4
Phosphat
π1 ≈ 0.96

p
p
HZ
z
z
≥
p

1
p
18 − 49
50
18 − 49 ≥ 50
p
p HZ

z
≥ 18
≥
HZ
18−49 p
p

1 p
> 1
z
z

z
1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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z
z

z
z
z
z
z 5.4% 7.7%
80%
z
z
80%

1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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z
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z

1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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32.2% 27.4%

HZ
p
n = 1957 n = 2702
n = 4000

2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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HZ p
n = 2476
n = 4000
> 98% 8%

HZ
HZ
p
p

2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
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p
HZ
HZ
> 98%
13.1% 15.0%
z

p
p
•
•

p
p

z
z
z

R

n = 4000
4.30
0.13 0.03
μ = , Σ=
, π1 =0.75
1.15
0.03 0.05
shape1 =7.8, scale1=1.0,
shape2 =4.5, scale2=1.0,
copula =0.9, π2 =0.25

p
z
z
z

99%
3.4%
p
HZ

Laborparameter B
Laborparameter B
0 1 2 3 4
1 2 3 4 5 6 7
Laborparameter A
0 1 2 3 4
1 2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
0 1 2 3 4
1 2 3 4 5 6 7
Laborparameter A
0 1 2 3 4
1 2 3 4 5 6 7
Laborparameter A
z

μ1 μ2 Σ11 Σ21 Σ22
z
z
z

z
p

μ1 μ2 Σ11 Σ21 Σ22
Σ12
Σ12 =0.03
Σ22

p
p
p

πi
ε

ε
TA TB
T =(TA,TB)
T
εA εB
εA εB
p

R tmvtnorm mle.tmvnorm()
100
n = 4000
mle.tmvnorm()

R tmvtnorm
tmvtnorm

R tmvtnorm
tmvtnorm

z
z

20%
p
•
•
> 1 p
3.5

p
HZ
•
•
HZ
p 3.5
HZ
p
•
p
16
•
p
HZ
p

p
p
z
HZ

p
p

x1
x2
1
F (x; θ) =
2πσ1σ2 1 − ρ2 −∞ −∞
×
1
exp −
2(1 − ρ2 t1 − μ1
)
=
x1
x2
−∞ −∞
1
2π det(Σ)
σ1
2
− 2ρ t1 − μ1
σ1
· t2 − μ2
σ2
· exp − 1
2 (t − μ)T Σ −1
(t − μ)
2
t2 − μ2
+
dt2 dt1
dt2 dt1
μ1 μ2 σ1 σ2
ρ
x1
x2
∂
1
F (x; θ) =
∂μ1
2πσ1σ2 1 − ρ2 −∞ −∞
×
1
exp −
2(1 − ρ2 t1 − μ1
) σ1
− 1
1 − ρ2
t1 − μ1
− ρ t2 − μ2
σ1σ2
=
x1
∂
F (x; θ) =
∂μ2
−∞ −∞
x1
x2
x2
−∞ −∞
σ 2 1
f(t) − 1
1 − ρ2 2
t1 − μ1
σ 2 1
− 2ρ t1 − μ1
σ1
dt2 dt1
− ρ t2
− μ2
σ1σ2
f(t) − 1
1 − ρ2
t2 − μ2
σ2 − ρ
2
t1
− μ1
σ1σ2
· t2 − μ2
σ2
dt2 dt1
dt2 dt1
σ2
2
t2 − μ2
+
×
σ2

x1
x2
∂
1
F (x; θ) = −
∂σ1
2πσ
−∞ −∞
2 1σ2
1 − ρ2 ×
1
exp −
2(1 − ρ2 t1 − μ1
)
∂
F (x; θ) =
∂σ2
∂
F (x; θ) =
∂ρ
σ1
2
− 2ρ t1 − μ1
σ1
· t2 − μ2
σ2
2
t2 − μ2
+
1
+
2πσ1σ2 1 − ρ2 ×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
t2
2
− μ2 t2 − μ2
+
×
σ2 σ2
1
−
2(1 − ρ2
−2
)
(t1 − μ1) 2
σ3 +2ρ
1
(t1 − μ1)(t2 − μ2)
σ2 1σ2
dt2 dt1
x1
x2
1
= f(t)
1 − ρ2
(t1 − μ1) 2
− ρ (t1 − μ1)(t2 − μ2)
σ2 1σ2
− 1
dt2 dt1
σ1
−∞ −∞
x1
x2
−∞ −∞
x1
x2
−∞ −∞
σ 3 1
1
f(t)
1 − ρ2
(t2 − μ2) 2
σ3 − ρ
2
(t1 − μ1)(t2 − μ2)
σ1σ2
−
2
1
σ2
1
2πσ1σ2(1 − ρ 2 ) 3
2
1
exp −
2(1 − ρ2 t1 − μ1
)
−∞ −∞
σ1
− 1
(−2ρ) ×
2
2
− 2ρ t1 − μ1
σ1
· t2 − μ2
σ2
σ2
dt2 dt1
2
t2 − μ2
+
1
+
2πσ1σ2 1 − ρ2 ×
1
exp −
2(1 − ρ2 t1 2 − μ1
− 2ρ
) σ1
t1 − μ1
·
σ1
t2
2
− μ2 t2 − μ2
+
×
σ2 σ2
1
2(1 − ρ2 ) (−1)(−2ρ)
t1 2 − μ1
− 2ρ
σ1
(t1
2
− μ1)(t2 − μ2) t2 − μ2
σ1σ2
σ2
+ − 1
· −2
σ2
t1
− μ1 t2 − μ2
dt2 dt1
σ1 σ2
=
x1
x2
ρ 1
f(t) +
1 − ρ2 1 − ρ2 2 t1 − μ1
− ρ
− 2ρ
σ1
t1 − μ1 t2 − μ2
σ1 σ2
2
t2 − μ2
+
+
σ2
t1 − μ1
σ1
t2 − μ2
σ2
dt2 dt1
σ2

∂l(θ)
∂μ1
∂l(θ)
∂μ2
∂l(θ)
∂σ1
∂l(θ)
∂σ2
= 1
1 − ρ 2
κ∈K(p)
= 1
1 − ρ 2
κ∈K(p)
= 1
1 − ρ 2
−
n
i=1
x (i)
1
− μ1
σ 2 1
(−1) g(κ)
κ1
n
i=1
x (i)
2
κ2
−∞ −∞
− μ2
σ 2 2
(−1) g(κ)
κ1
n
i=1
κ∈K(p)
κ1 κ2
−∞ −∞
= 1
1 − ρ 2
−
(x (i)
1
κ2
−∞ −∞
− μ1) 2
σ 3 1
n
(−1) g(κ) F (κ; θ)
f(t)
n
i=1
κ∈K(p)
κ1 κ2
−∞ −∞
− μ2
− ρ x(i) 2
σ1σ2
−
f(t) − 1
1 − ρ2 − μ1
− ρ x(i) 1
σ1σ2
−
f(t) − 1
1 − ρ2 − ρ (x(i) 1
κ∈K(p)
κ∈K(p)
t1 − μ1
σ 2 1
κ∈K(p)
t2 − μ2
− μ1)(x (i)
2
σ 2 1 σ2
(−1) g(κ) ×
σ 2 2
n
(−1) g(κ) F (κ; θ) ×
− ρ t2
− μ2
σ1σ2
n
(−1) g(κ) F (κ; θ) ×
− μ2)
− ρ t1
− μ1
σ1σ2
− n
σ1
1
1 − ρ2
(t1 − μ1) 2
σ3 − ρ
1
(t1 − μ1)(t2 − μ2)
σ2 1σ2
− 1
σ1
(x (i)
2
− μ2) 2
σ 3 2
n
(−1) g(κ) F (κ; θ)
f(t)
− ρ (x(i) 1
κ∈K(p)
− μ1)(x (i)
2
σ1σ 2 2
(−1) g(κ) ×
− μ2)
− n
σ2
1
1 − ρ2
(t2 − μ2) 2
σ3 − ρ
2
(t1 − μ1)(t2 − μ2)
σ1σ2
−
2
1
σ2
dt2 dt1
dt2 dt1
dt2 dt1
dt2 dt1

∂l(θ)
∂ρ
2ρn 1
= +
1 − ρ2 1 − ρ2 +
−
ρ
(1 − ρ 2 ) 2
κ∈K(p)
n
i=1
n
i=1
x (i)
1
(x (i)
1
2 − μ1
σ1
n
(−1) g(κ) F (κ; θ)
+ 1
1 − ρ2 t1 − μ1
− ρ
+ t1 − μ1
σ1
t2 − μ2
σ2
σ1
− μ1)(x (i)
2
κ∈K(p)
2
dt2 dt1
σ1σ2
+
x (i)
2
− μ2)
2 − μ2
σ2
(−1) g(κ)
κ1
− 2ρ t1 − μ1
σ1
κ2
−∞ −∞
t2 − μ2
σ2
− 2ρ (x(i) 1
ρ
f(t)
1 − ρ2 − μ1)(x (i)
2
σ1σ2
2
t2 − μ2
+
σ2
− μ2)

F : S → Rn×n S := {X|X ∈ Rn×n ,X } F (X) =
X−1
dF = −X −1 (dX)X −1
F : S → R S := {X|X ∈ Rn×n , det(X) > 0} F (X) =
ln(det(X))
dF =tr(X −1 dX)
F : S → R S := {X|X ∈ Rn×n , det(X) > 0,X , dX = 0}
F (X) = ln(det(X))
d 2 F = −tr(X −1 (dX)X −1 (dX)) < 0
F : R p → R p×p F (μ) = n
i=1 (xi − μ)(xi − μ) T xi ∈ R p
i =1,...,n
dF = −(dμ)
n
(xi − μ) −
i=1
n
(xi − μ)(dμ) T
i=1

B Σ
b11
B =
b21
⎛√
b12
Σ11
= ⎝
Σ21
b22
0
Σ22 − Σ2 ⎞
⎠ ,
21
Σ11
Σ11
Σ=
Σ21
Σ12
Σ22
√ Σ11
B Σ
vec dB(Σ; U) = A(Σ) · vec U
⎛
1
⎜ 2
⎜
= ⎜
⎝
√ 0 0 0
Σ11
− Σ21
2 √ Σ11 3 √ 1 0 0
Σ11
0 0 0 0
Σ2 21 − Σ21
1 0
⎛
⎜
dB(Σ; U) = ⎝
=
⎛
⎜
⎝
Σ 2 11
Σ22− Σ2 21
Σ 11
1
2 √ Σ11 u11
− Σ21 √
Σ11 u21
2 √ Σ11 3 u11 + 1
σ22− Σ2 21
Σ 11
1
2 √ Σ11 u11
− Σ21
2 √ Σ11 3 u11 + 1
√ Σ11 u21
Σ2 21
Σ2
11
Σ22− Σ2 Σ21
u11 −
21
Σ22−
Σ11 Σ2 21
Σ11 0
u21 +
Σ22− Σ2 21
Σ 11
0
1
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ · ⎜
⎟ ⎜
⎟ ⎝
⎠
Σ22− Σ2 21
Σ 11
Σ2 21
Σ2
11
Σ22− Σ2 Σ21
u11 −
21
Σ22−
Σ11 Σ2 21
Σ11 u22
u21 +
u11
u21
u12
u22
⎞
⎟
⎠
⎞
⎟
⎠
1
Σ22− Σ2 21
Σ 11
u22
⎞
⎟
⎠

z

z

z

z

z

z

z

z

z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

z
z
z
p
z

z
z
z

z
z
z

z
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
z
z

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
1 2 3 4 5 6 7 8
Laborparameter A
2 3 4 5 6 7
Laborparameter A
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
Laborparameter B
Laborparameter B
Laborparameter B
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A
2 3 4 5 6 7
Laborparameter A

z
z

z

z

z

z

z

z

z

z

z

z

z

z

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

μ1 μ2 Σ11 Σ21 Σ22

π1
π1
π1

1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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z

1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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z

1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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1 2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium (in mmol/l)
Phosphat (in mmol/l)
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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2 3 4 5 6 7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Kalium
Phosphat
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