Approximation of random processes by cubic splines
Approximation of some classes of random processes by cubic splines with given accuracy and reliability is considered. Estimations of deviation of approximating spline from original process are obtained. A few examples of approximation are considered. Application of splines for simulation of processe...
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Інститут математики НАН України
2008
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| Цитувати: | Approximation of random processes by cubic splines / O. Kamenschykova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 53-66. — Бібліогр.: 7 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859469914188808192 |
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| author | Kamenschykova, O. |
| author_facet | Kamenschykova, O. |
| citation_txt | Approximation of random processes by cubic splines / O. Kamenschykova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 53-66. — Бібліогр.: 7 назв.— англ. |
| collection | DSpace DC |
| description | Approximation of some classes of random processes by cubic splines with given accuracy and reliability is considered. Estimations of deviation of approximating spline from original process are obtained. A few examples of approximation are considered. Application of splines for simulation of processes is also studied.
|
| first_indexed | 2025-11-24T07:35:45Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.53-66
OLEXANDRA KAMENSCHYKOVA
APPROXIMATION OF RANDOM PROCESSES BY
CUBIC SPLINES
Approximation of some classes of random processes by cubic splines
with given accuracy and reliability is considered. Estimations of
deviation of approximating spline from original process are obtained.
A few examples of approximation are considered. Application of
splines for simulation of processes is also studied.
1. Introduction
Let {X(t), t ∈ T = [a, b]} be a random L2(Ω)-process.
Denote by Δ := {a = t0 < ... < tN = b} – the partition of the segment
[a, b] into N parts. Assume that the values {yi, i = 0, N} of the process
X(t) in corresponding points {ti, i = 0, N} are known.
The problem of approximation of such process X(t) with given accuracy
and reliability in norms of different spaces (C([a, b]), Lp(T ) etc.) by cubic
splines, constructed on known values of the process in partition points, with
given boundary conditions, is considered.
We recall some basic definitions and facts used in the article.
Let (Ω,B, P ) be a standard probability space.
Definition 1. The process X̃(t) approximates the process X(t) with given
accuracy ε > 0 and reliability 1 − δ, 0 < δ < 1 in space A if the next
inequality is satisfied:
P
{∥∥∥X(t) − X̃(t)
∥∥∥
A
> ε
}
≤ δ.
Definition 2. [1] Function SΔ,y(t) (or SΔ(t)), continuous on [a, b] with its
first and second derivatives, which is a cubic polynomial on every segment
[ti−1, ti], i = 1, N, and which satisfies the conditions SΔ(ti) = yi, i = 0, N, is
called a cubic spline on Δ, which interpolates values yi in the knots of Δ.
Denote by YN(t) := X(t) − SΔ(t), t ∈ T, the deviation random process.
2000 Mathematics Subject Classifications. 65C20.
Key words and phrases. Approximation, cubic splines, deviation process.
53
54 OLEXANDRA KAMENSCHYKOVA
Definition 3. [2] A continuous even convex function u = (u(x), x ∈ R)
is called an Orlicz N-function, if it is monotonically increasing for x > 0,
u(0) = 0 and u(x)
x
→ 0 as x→ 0 and u(x)
x
→ ∞ as x→ ∞.
Assumption Q: [3] The assumption Q holds for an Orlicz N-function ϕ, if
limx→0
ϕ(x)
x2
= c > 0.
Remark 1. The constant c can be equal to ∞.
Definition 4. [3] Let ϕ be an Orlicz N-function satisfying the assumption
Q. A zero mean random variable ξ belongs to the space Subϕ(Ω) (the space
of ϕ-sub-Gaussian random variables), if there exists a constant rξ ≥ 0 such
that the inequality
E exp (λξ) ≤ exp (ϕ(rξλ))
holds ∀λ ∈ R.
Proposition 1. [2]-[4] The space Subϕ(Ω) is a Banach space with respect
to the norm
τϕ(ξ) = inf{a ≥ 0 : E exp (λξ) ≤ exp(ϕ(aλ)), λ ∈ R}.
Definition 5. [2] Let T be a parametric space. A random process {X(t), t ∈
T} belongs to the space Subϕ(Ω) if for all t ∈ T X(t) ∈ Subϕ(Ω) and
supt∈T τϕ(X(t)) <∞.
Definition 6. [5] A family Λ of random variables ξ ∈ Subϕ(Ω) is called
strictly Subϕ(Ω) if there exists a constant CΛ > 0 such that for any finite
set I, ξi ∈ Λ, i ∈ I, and for ∀λi ∈ R the inequality holds:
τϕ
(∑
i∈I
λiξi
)
≤ CΛ
⎛⎝E(∑
i∈I
λiξi
)2
⎞⎠1/2
.
CΛ is called a determinative constant.
Definition 7. [5] ϕ-sub-Gaussian random process {X(t), t ∈ T} is called
strictly Subϕ(Ω) if the family of random variables {X(t), t ∈ T} is strictly
Subϕ(Ω).
Definition 8. [2] Let f = (f(x), x ∈ R) be a real-valued function. The
function f ∗ = (f ∗(x), x ∈ R) defined by the formula f ∗(x) = supy∈R(xy −
f(y)) is called the Young-Fenchel transform of the function f.
APPROXIMATION OF PROCESSES BY CUBIC SPLINES 55
2. Estimations of the deviation process
Denote by Mj := S ′′
Δ(tj), j = 0, N – the so-called ”moments” of spline
SΔ(t); hj = tj−tj−1, j = 1, N, λj =
hj+1
hj+hj+1
, μj = 1−λj , j = 1, N − 1.
As the spline SΔ(t) is a cubic polynomial on every segment of partition
and as SΔ, S
′
Δ, S
′′
Δ are continuous we obtain the system of N − 2 equations
for the moments of spline Mj, j = 0, N ([1]):
μjMj−1 + 2Mj + λjMj+1 = 6 · (yj+1 − yj)/hj+1 − (yj − yj−1)/hj
hj + hj+1
(1)
For unique solution of this system we specify 2 additional – boundary
conditions – in the next form:
2M0 + λ0M1 = d0 μNMN−1 + 2MN = dN , (2)
where λ0, d0, μN , dN are given values.
The equalities (1) and (2) defining the spline can be written in the matrix
form:
A
−→
M =
−→
d , (3)
where
A :=
⎛⎜⎜⎜⎜⎜⎜⎝
2 λ0 0 . . . . . . . . . . . . . . . .
μ1 2 λ1 0 . . . . . . . . . .
0 μ2 2 λ2 . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 0 μN−1 2 λN−1
. . . . . . . . . . . . . . . . . μN 2
⎞⎟⎟⎟⎟⎟⎟⎠ ,
−→
M :=
⎛⎜⎜⎜⎜⎜⎜⎝
M0
...
...
...
...
MN
⎞⎟⎟⎟⎟⎟⎟⎠ ,
−→
d =
⎛⎜⎜⎜⎜⎜⎜⎝
d0
...
...
...
...
dN
⎞⎟⎟⎟⎟⎟⎟⎠ ,
dj , j = 1, N − 1 – right sides of (1).
Denote by |Δ| = maxj hj the diameter of the partition Δ, and β such
finite number that max1≤j≤N
|Δ|
hj
≤ β.
Theorem 1. Let X(t) be L2(Ω)-process which satisfies the inequality
sup
t∈T
E|X(t+ h) −X(t)|2 ≤ σ2(h), (4)
where σ(h), h > 0 is a known function such that σ(h), h > 0 increases
monotonically, h
σ(h)
, h > 0 is non-decreasing and σ(h) ↓ 0, h ↓ 0. Suppose
SΔ(t) is the spline which interpolates this process and satisfies the boundary
conditions (2). Then ∀t ∈ T
(EY 2
N(t))1/2 ≤
(
2
9
√
3
‖ A−1 ‖ [6β2 + c0|d0| + c0|dN |] +
3
2
)
σ(|Δ|), (5)
56 OLEXANDRA KAMENSCHYKOVA
where c0 = (b−a)2
σ(b−a) .
Remark 2. By the norm of matrix A we mean ‖ A ‖= maxi
∑
j |aij |.
Proof. For t ∈ [tj−1, tj ], j = 1, N we receive the equalities ([1]):
SΔ(t) = Mj−1(tj − t)
(tj − t)2 − h2
j
6hj
+Mj(t− tj−1)
(t− tj−1)
2 − h2
j
6hj
+
+
yj−1 + yj
2
+
yj − yj−1
2hj
(2t− (tj + tj−1)). (6)
From (3) we obtain: Mj =
∑N−1
i=1 A
(−1)
ij di + A
(−1)
j0 d0 + A
(−1)
jN dN , where
A
(−1)
ij are the elements of the matrix A−1. Coefficients before Mj−1,Mj∣∣∣∣(tj − t)
(tj − t)2 − h2
j
6hj
∣∣∣∣ ≤ h2
j
9
√
3
,
∣∣∣∣(t− tj−1)
(t− tj−1)
2 − h2
j
6hj
∣∣∣∣ ≤ h2
j
9
√
3
.
Besides,
h2
j (Ed
2
i )
1/2
6
=
h2
j
hi + hi+1
(
E
(
yi+1 − yi
hi+1
− yi − yi−1
hi
)2
)1/2
≤ β2σ(|Δ|),
(
E
(
yj−1 + yj
2
−X(t) +
yj − yj−1
2hj
(2t− (tj + tj−1))
)2
)1/2
≤ 3
2
σ(|Δ|).
We get:
h2
j ((EM
2
j−1)
1/2 + (EM2
j )
1/2) ≤ 2 ‖ A−1 ‖ [6β2σ(|Δ|) + |Δ|2(|d0| + |dN |)]. (7)
Thus
(EY 2
N(t))1/2 ≤ (
E(M2
j−1)
1/2 + E(M2
j )
1/2
) h2
j
9
√
3
+
3
2
σ(|Δ|) ≤
≤ 2
9
√
3
‖ A−1 ‖ [6β2σ(|Δ|) + |Δ|2(|d0| + |dN |)] + 3
2
σ(|Δ|).
As |Δ| ≤ b− a and h
σ(h)
decreases on h, then
|Δ|2
σ(|Δ|) ≤ (b− a)
|Δ|
σ(|Δ|) ≤ (b− a)2
σ(b− a)
=: c0,
so
APPROXIMATION OF PROCESSES BY CUBIC SPLINES 57
(EY 2
N(t))1/2 ≤
(
2
9
√
3
‖ A−1 ‖ [6β2 + c0|d0| + c0|dN |] +
3
2
)
σ(|Δ|).
Corollary 1. In the case of uniform partition of T and the next boundary
conditions λ0 = d0 = μN = dN = 0, i.e. M0 = MN = 0, the inequality (5)
becomes
(E((SΔ(t) −X(t))2)1/2 ≤
(
4
3
√
3
+
3
2
)
σ
(
1
N
)
.
Proof. The corollary results from Theorem 1 and the next proposition.
Proposition 2. [1] If for the matrix A |λ0| < 2, |μN | < 2, then ‖ A−1 ‖≤
max[(2 − λ0)
−1, (2 − μN)−1, 1].
Theorem 2. Let X(t) be L2(Ω)-process which satisfies (4), SΔ(t) be the
spline which interpolates this process and satisfies boundary conditions (2),
A, YN(t), c0, β are defined above. Then ∀t, t + h ∈ [a, b], ∀h > 0 the estima-
tion holds true:
(E(YN(t+ h) − YN(t))2)1/2 ≤
≤ σ(h)
(
4 +
4β
3
‖ A−1 ‖ (6β2 + c0(|d0| + |dN |))
)
. (8)
Proof. Consider 2 possible cases.
Case 1. Let t, t+ h ∈ [tj−1, tj ], j = 1, N, then
(E(YN(t+ h) − YN(t))2)1/2 ≤ (E(SΔ(t+ h) − SΔ(t))2)1/2 + σ(h).
From (6) we get
I := (E(SΔ(t+ h) − SΔ(t))2)1/2 = (E(Mj−1I1 +MjI2 − I3)
2)1/2,
|I1| =
∣∣∣∣(tj − t)(h2 − 2h(tj − t)) − h((tj − (t+ h))2 − h2
j)
6hj
∣∣∣∣ ≤ h2
j ·
h
3hj
,
|I2| =
∣∣∣∣(t− tj−1)(h
2 + 2h(t− tj−1)) + h((t+ h− tj−1)
2 − h2
j)
6hj
∣∣∣∣ ≤ h2
j ·
h
3hj
,
(EI2
3 )1/2 = (E((yj − yj−1)
h
hj
)2)1/2 ≤ σ(hj)
h
hj
≤ σ(h)
58 OLEXANDRA KAMENSCHYKOVA
From (7) and the inequalities above we obtain:
I ≤ h
3hj
· 2 ‖ A−1 ‖ [6β2σ(|Δ|) + c0σ(|Δ|)(|d0| + |dN |)] + σ(h) ≤
≤ σ(h)
(
1 +
2β
3
‖ A−1 ‖ (6β2 + c0(|d0| + |dN |))
)
(as |Δ|
β
≤ hj , β ≥ 1, then 1
hj
≤ β
|Δ| ; as h ≤ |Δ| then h
|Δ|σ(|Δ|) ≤ σ(h)).
(E(YN (t + h) − YN (t))2)1/2 ≤ σ(h)
(
2 +
2β
3
‖ A−1 ‖ (6β2 + c0(|d0| + |dN |))
)
.
Case 2. Now suppose t ∈ [ti−1, ti], i = 1, N − 1, t + h ∈ [tj−1, tj ], j =
2, N, i < j, then again (E(YN(t+ h) − YN(t))2)1/2 ≤ I + σ(h), where
I = (E(SΔ(t+ h) − SΔ(t))2)1/2 = (E(SΔ(t+ h) − SΔ(tj−1))
2)1/2 +
+ (E(SΔ(tj−1) − SΔ(i))2)1/2 + (E(SΔ(i) − SΔ(t))2)1/2 ≤
≤ σ(h)
(
3 +
4β
3
‖ A−1 ‖ (6β2 + c0(|d0| + |dN |))
)
.
(E(YN (t + h) − YN (t))2)1/2 ≤ σ(h)
(
4 +
4β
3
‖ A−1 ‖ (6β2 + c0(|d0| + |dN |))
)
.
Thus from cases 1,2 we obtain (8).
Corollary 2. In the case of uniform partition of the segment T and bound-
ary conditions λ0 = d0 = μN = dN = 0, i.e. M0 = MN = 0, the inequality
(8) becomes
(E(YN(t+ h) − YN(t))2)1/2 ≤ 12σ(h), t, t+ h ∈ T, h > 0.
Denote mj := S ′
Δ(tj), j = 0, N, then for t ∈ [tj−1, tj], j = 1, N, the next
equality is satisfied ([1]):
SΔ(t) = mj−1
(tj − t)2(t− tj−1)
h2
j
−mj
(t− tj−1)
2(tj − t)
h2
j
+
+ yj−1
(tj − t)2(2(t− tj−1) + hj)
h2
j
+ yj
(t− tj−1)
2(2(tj − t) + hj)
h2
j
,
whence the system of N − 2 equations follows:
APPROXIMATION OF PROCESSES BY CUBIC SPLINES 59
λjmj−1 +2mj +μjmj+1 = 3λj · yj − yj−1
hj
+3μj
yj+1 − yj
hj+1
, j = 1, N − 1. (9)
We set the next boundary conditions:
2m0 + μ0m1 = c0 λNmN−1 + 2mN = cN , (10)
where μ0, c0, λN , cN are the specified values.
The equalities (9) and (10), which define the spline, can be written in
the matrix form:⎛⎜⎜⎜⎜⎜⎜⎝
2 μ0 0 . . . . . . . . . . . . . . . .
λ1 2 μ1 0 . . . . . . . . . .
0 λ2 2 μ2 . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 0 λN−1 2 μN−1
. . . . . . . . . . . . . . . . . λN 2
⎞⎟⎟⎟⎟⎟⎟⎠ ·
⎛⎜⎜⎜⎜⎜⎜⎝
m0
...
...
...
...
mN
⎞⎟⎟⎟⎟⎟⎟⎠ =
⎛⎜⎜⎜⎜⎜⎜⎝
c0
...
...
...
...
cN
⎞⎟⎟⎟⎟⎟⎟⎠ , (11)
where cj , j = 1, N − 1 are the right sides of (9).
Theorem 3. Let X(t) be L2(Ω)-process, SΔ(t) – spline, which interpolates
it and satisfies the boundary conditions:
2m0 = 2y′0, 2mN = 2y′N , (12)
where y′0 = X ′(t0), y′N = X ′(tN ). Assume there ∃X ′(t), t ∈ T, and
(E(X ′(t) −X ′(s))2)1/2 ≤ ω1(h), ∀t, s ∈ T : |t− s| ≤ h, (13)
where ω1(h) is a positive monotonically increasing function, h
ω1(h)
↑, ω1(h) ↓
0, h ↓ 0. Then
(E(YN(t))2)1/2 ≤ 21
4
|Δ| · ω1(|Δ|), ∀t ∈ T,
(E(Y ′
N(t))2)1/2 ≤ 21
2
ω1(|Δ|), ∀t ∈ T.
Proof. Denote the matrix of coefficients in (11) by B under boundary con-
ditions (12) and equations 3m0 = 3y′0 = c0, 3mN = 3y′N = cN instead of the
first and the last equation of the system (11).
From (11) we get: B(−→m− 1
3
−→c ) = (I− 1
3
B)−→c , where −→m = (m0, ..., mN)T ,−→c = (c0, ..., cN)T . But
60 OLEXANDRA KAMENSCHYKOVA
(I − 1
3
B)−→c =
⎛⎜⎜⎜⎜⎜⎜⎝
0
λ1(c1 − c0) − μ1(c2 − c1)
...
...
λN−1(cN−1 − cN−2) − μN−1(cN − cN−1)
0
⎞⎟⎟⎟⎟⎟⎟⎠ ,
whence
(E ‖ (I − 1
3
B)−→c ‖2)1/2 ≤ max
k=1,N−1
(λkω1(3|Δ|) + μkω1(3|Δ|)) ≤ 3ω1(|Δ|),
(E ‖ −→m − 1
3
−→c ‖2)1/2 ≤‖ B−1 ‖ ·(E ‖ (I − 1
3
B)−→c ‖2)1/2 ≤ 3ω1(|Δ|).
Then ∀j = 1, N − 1 ∃ξj ∈ [tj−1, tj] :
cj
3
= X ′(ξj) and c0
3
= X ′(a),
cN
3
= X ′(b), so denoting
−→
X ′ = (X ′(t0), ..., X ′(tN ))T = (y′0, ..., y
′
N)T we ob-
tain:
(E ‖ −→m−−→
X ′ ‖2)1/2 ≤ (E ‖ −→m−1
3
−→c ‖2)1/2+(E ‖ 1
3
−→c −−→
X ′ ‖2)1/2 ≤ 4ω1(|Δ|).
Let’s use the next representation and inequalities: for t ∈ [tj−1, tj], j =
1, N
S ′
Δ(t) − yj − yj−1
hj
=
[
3
h2
j
(
t− tj−1 + tj
2
)2
− 1
4
]
[(mj−1 − y′j−1) +
+ (mj − y′j)
(
y′j−1 + y′j − 2 · yj − yj−1
hj
)
] +
1
hj
(t− tj−1 + tj
2
) ×
× [(mj − y′j) − (mj−1 − y′j−1) + (y′j − y′j−1)], (14)
∣∣∣∣∣ 3
h2
j
(
t− tj + tj−1
2
)2
− 1
4
∣∣∣∣∣ ≤ 1
2
,
1
hj
∣∣∣∣t− tj + tj−1
2
∣∣∣∣ ≤ 1
2
,
(
E
(
mj−1 − y′j−1
)2
)1/2 ≤ 4ω1(|Δ|),
(
E
(
y′j−1 −
yj − yj−1
hj
)2
)1/2
≤ ω1(|Δ|),
APPROXIMATION OF PROCESSES BY CUBIC SPLINES 61
(
E
(
mj − y′j
)2
)1/2
≤ 4ω1(|Δ|),
(
E
(
y′j −
yj − yj−1
hj
)2
)1/2
≤ ω1(|Δ|),
then from (14) for t ∈ [tj−1, tj ], j = 1, N :
(
E
(
S ′
Δ(t) − yj − yj−1
hj
)2
)1/2
≤ 1
2
(4 + 4 + 1 + 1)ω1(|Δ|) +
+
1
2
(4 + 4 + 1)ω1(|Δ|) =
19
2
ω1(|Δ|),
whence ∀t ∈ T (t∗ is the closest of ends tj−1, tj to t):
(E(X ′(t) − S ′
Δ(t))2)1/2 ≤ ω1(|Δ|) +
19
2
ω1(|Δ|) =
21
2
ω1(|Δ|),
(E(X(t) − SΔ(t))2)1/2 = (E(
∫ t
t∗
(X ′(u) − SΔ(u))du)2)1/2 ≤ 21
4
|Δ|ω1(|Δ|).
Theorem 4. Let X(t) be a L2(Ω)- process, SΔ(t) – spline, which interpo-
lates it under the boundary conditions:
(i) 2M0 +M1 = 6
h1
(
y1−y0
h1
− y′0
)
, 2MN +MN−1 = 6
hN
(
y′N − yN−yN−1
hN
)
or
(ii) 2M0 = 2y′′0 , 2MN = 2y′′N ,
where y′0 = X ′(t0), y′N = X ′(tN), y′′0 = X ′′(t0), y′′N = X ′′(tN ). Assume there
∃X ′′(t), t ∈ T, and
(E(X ′′(t) −X ′′(s))2)1/2 ≤ ω2(h), ∀t, s ∈ T : |t− s| ≤ h, (15)
where ω2(h) is a positive monotonically increasing function, h
ω2(h)
is non-
decreasing, ω2(h) ↓ 0, h ↓ 0. Then
(E(YN(t))2)1/2 ≤ 5
2
|Δ|2 · ω2(|Δ|), ∀t ∈ T,
(E(Y ′
N(t))2)1/2 ≤ 5|Δ|ω2(|Δ|), ∀t ∈ T,
(E(Y ′′
N(t))2)1/2 ≤ 5ω2(|Δ|), ∀t ∈ T.
Proof. From (3) under (i) boundary conditions we obtain:
A(
−→
M − 1
3
−→
d ) = (I − 1
3
A)
−→
d ,
62 OLEXANDRA KAMENSCHYKOVA
(I − 1
3
A)
−→
d =
1
3
⎛⎜⎜⎜⎜⎝
d0 − d1
μ1(d1 − d0) − λ1(d2 − d1)
μ2(d2 − d1) − λ2(d3 − d2)
...
dN − dN−1
⎞⎟⎟⎟⎟⎠ . (16)
Note that for (ii) the arguments will be the same as for (i) except that the
first and the last elements of the vector (16) will be equal to zero, but all
the inequalities will remain true.
Notice that
dj
6
=
(yj+1−yj)/hj+1−(yj−yj−1)/hj
hj+hj+1
= y[tj−1, tj , tj+1] – the dif-
ference quotient. So
dj
6
= 1
2
X ′′(ξj) for some point ξj ∈ (tj−1, tj+1). From
Taylor’s formulae with the remainder term in the Lagrange form we get that
for some point ξ0, t0 < ξ0 < t1, the equality is satisfied: d0
6
=
(y1−y0)/h1−y′0
h1
=
1
2
X ′′(ξ0). Similar statement holds true for dN . Thus as
1
3
(E(μk(dk − dk−1) − λk(dk+1 − dk))
2)1/2 ≤ 3ω2(|Δ|), k = 1, N − 1,
1
3
(E(d0 − d1)
2)1/2 ≤ 3ω2(|Δ|), 1
3
(E(dN − dN−1)
2)1/2 ≤ 3ω2(|Δ|),
we receive the estimation:
(E ‖ (I − 1
3
A)d ‖2)1/2 ≤ 3ω2(|Δ|), whence
(E ‖ −→
M − 1
3
−→
d ‖2)1/2 ≤ ‖A−1‖(E(‖ (I − 1
3
A)
−→
d ‖)2)1/2 ≤ 3ω2(|Δ|).
But from the equalities above for di, i = 1, N − 1, d0, dN we get:
(E ‖ −→
X ′′ − 1
3
−→
d ‖2)1/2 ≤ ω2(|Δ|),
so
(E ‖ −→
M −−→
X ′′ ‖2)1/2 ≤ (E ‖ −→
M − 1
3
−→
d ‖2)1/2 + (E ‖ 1
3
−→
d −−→
X ′′ ‖2)1/2 ≤
≤ 4ω2(|Δ|), −→
X ′′ = (X ′′(t0), ..., X ′′(tN))T .
SΔ(t) is a piecewise-linear function, then ∀t ∈ [tj−1, tj ], j = 1, N
(E|X ′′(t) − S ′′
Δ(t)|2)1/2 ≤ (E|X ′′(t) −X ′′(tj)|2)1/2 +
+ (E|X ′′(tj) − SΔ(t)|2)1/2 ≤ 5ω2(|Δ|).
Since SΔ(tj) = X(tj), j = 0, N, then from Rolle’s theorem for any interval
(tj−1, tj) ∃ξj : X ′(ξj) = S ′
Δ(ξj), so ∀t ∈ [tj−1, tj], j = 1, N
APPROXIMATION OF PROCESSES BY CUBIC SPLINES 63
(E(|X ′(t) − S ′
Δ(t)|)2)1/2 = (E(|
∫ t
ξj
(X ′′(u) − SΔ(u))du|)2)1/2 ≤
≤ 5|t− ξj| · ω2(|Δ|) ≤ 5|Δ|ω2(|Δ|),
(E(|X(t) − SΔ(t)|)2)1/2 = (E(|
∫ t
t∗
(X ′′(u) − SΔ(u))du|)2)1/2 ≤
≤ |t− t∗| · 5|Δ|ω2(|Δ|) ≤ 5
2
|Δ|2ω2(|Δ|)
(t∗ is the closest of ends tj−1, tj to t).
3. Examples of approximation (space Lp(T ))
Theorem 5. i) Let {X(t), t ∈ T = [a, b]} be a SSubϕ(Ω)-process with deter-
minative constant CΛ and which satisfies (4), SΔ(t) is the spline interpolat-
ing this process under boundary conditions 2M0 + λ0M1 = d0, μNMN−1 +
2MN = dN , where λ0, d0, μN , dN are given values. Then ∀ε > 0 :
p(−1)
(
ε
c3CΛσ(|Δ|)
)
1
c3CΛσ(|Δ|)
ε
(b− a)1/p
≥ p,
where p(−1)(t), t > 0 is the inverse function for p(t), p(t) is the density of
ϕ(x), ϕ(x) =
∫ x
0
p(t)dt, c3 = 2
9
√
3
‖ A−1 ‖ [6β2 + c0|d0| + c0|dN |] + 3
2
, A, β, c0
are defined above, the next inequality holds:
P
{
‖X(t) − SΔ(t)‖Lp
> ε
}
≤ 2 exp
{
−ϕ∗
(
ε
c3CΛσ(|Δ|)(v − u)1/p
)}
,
where ϕ∗ is the Young-Fenchel transform of the function ϕ, p ≥ 1.
ii) Let {X(t), t ∈ T = [a, b]} be a separable SSubϕ(Ω)-process with deter-
minative constant CΛ, satisfying (4) and assume
∫ ν
0
ψ(ln(σ(−1)(u)))du <∞
for all z and for sufficiently small ν > 0 where ψ(u) = u/ϕ(−1)(u). For
∀t, s ∈ T put EX(t)X(s) = R(t, s) and suppose ∂2R(t,s)
∂t∂s
exists and satisfies
sup
|t−s|≤h
(
∂2R(u, v)
∂u∂v
∣∣∣∣
u=v=t
+
∂2R(u, v)
∂u∂v
∣∣∣∣
u=v=s
− 2∂2R(u, v)
∂u∂v
∣∣∣∣
u=t,v=s
)
≤ ω2
1(h),
where ω1(h) is a positive monotonically increasing function, h
ω1(h)
is non-
decreasing, ω1(h) ↓ 0, h ↓ 0. Then for the spline with boundary conditions
2m0 = 2y′0, 2mN = 2y′N , where y′0 = X(1)(t0), y
′
N = X(1)(tN), ∀ε > 0 :
64 OLEXANDRA KAMENSCHYKOVA
p(−1)
(
ε
21
4
CΛ|Δ|ω1(|Δ|)
)
1
21
4
CΛ|Δ|ω1(|Δ|)
ε
(b− a)1/p
≥ p,
the inequality holds true:
P
{
‖X(t) − SΔ(t)‖Lp
> ε
}
≤ 2 exp
{
−ϕ∗
(
ε
21
4
CΛ|Δ|ω1(|Δ|)(b− a)1/p
)}
.
iii) Let {X(t), t ∈ T = [a, b]} be a separable SSubϕ(Ω)-process with de-
terminative constant CΛ, satisfying (4) and assume
∫ ν
0
ψ(ln(σ(−1)(u)))du <
∞ for all z and for sufficiently small ν > 0 where ψ(u) = u/ϕ(−1)(u). For
∀t, s ∈ T put EX(t)X(s) = R(t, s) and suppose ∂4R(t,s)
∂t2∂s2
exists and satisfies
sup
|t−s|≤h
(
∂4R(u, v)
∂u2∂v2
∣∣∣∣
u=v=t
+
∂4R(u, v)
∂u2∂v2
∣∣∣∣
u=v=s
− 2∂4R(u, v)
∂u2∂v2
∣∣∣∣
u=t,v=s
)
≤ ω2
2(h),
where ω2(h) is a positive monotonically increasing function, h
ω2(h)
is non-
decreasing, ω2(h) ↓ 0, h ↓ 0. Then for the spline with boundary conditions
2M0+M1 = 6
h1
(
y1−y0
h1
− y′0
)
, 2MN+MN−1 = 6
hN
(
y′N − yN−yN−1
hN
)
or 2M0 =
2y′′0 , 2MN = 2y′′N , where y′′0 = X(2)(t0), y
′′
N = X(2)(tN), ∀ε > 0 :
p(−1)
(
ε
5
2
CΛ|Δ|2 · ω2(|Δ|)
)
1
5
2
CΛ|Δ|2 · ω2(|Δ|)
ε
(b− a)1/p
≥ p,
the inequality holds:
P
{
‖X(t) − SΔ(t)‖Lp
> ε
}
≤ 2 exp
{
−ϕ∗
(
ε
5
2
CΛ|Δ|2ω2(|Δ|)(b− a)1/p
)}
.
Proof. The theorem comes from theorems 1,3,4, theorem 3.1 from [6] (in-
equality for norm in Lp(T ) of Subϕ(Ω) process) and theorems 3.7-3.9 from
[7] (conditions for the existence of continuous partial derivatives of a random
field of the space SSubϕ(Ω)).
Example 1. Assume σ(h) = chα, 0 ≤ α ≤ 1, c > 0, p = 2, ϕ(x) = x2/2, T =
[0, 1], c = α = 1, the partition Δ is uniform, CΛ = 1.
Applying theorem 5, let’s obtain the inequalities for the order of partition
N , which should be satisfied in order that the corresponding spline with
given boundary conditions approximates the original process with given
accuracy ε and reliability 1 − δ in the norm of L2.
a) Let the boundary conditions for spline be S ′′
Δ(0) = S ′′
Δ(0) = 0. Then
for ε = 0.03, δ = 0.01 we get N ≥ 244, for ε = 0.1, δ = 0.1 ⇒ N ≥ 55.
APPROXIMATION OF PROCESSES BY CUBIC SPLINES 65
b) assume that the conditions of theorem 5 ii) are satisfied, ω1(h) = h.
Then for ε = 0.03, δ = 0.01 for the corresponding spline we derive N ≥ 24;
for ε = 0.1, δ = 0.1 we have: N ≥ 12.
c) assume that the conditions of theorem 5 iii) are satisfied, ω2(h) = h.
We get: for ε = 0.03, δ = 0.01 for the corresponding spline the order of the
partition should meet N ≥ 7, for ε = 0.1, δ = 0.1 ⇒ N ≥ 4.
4. Application of splines for simulation of gaussian processes
Consider the problem of simulation of centered gaussian process {ξ(t),
t ∈ T = [0, 1]}, with known covariance function B(t, s) = Eξ(t)ξ(s), t, s ∈
T, in Lp(T ) with given accuracy ε > 0 and reliability 1 − δ, 0 < δ < 1 by
spline with corresponding boundary conditions (2).
Then the algorithm of such simulation is following:
1) for given B(t, s) choose such function σ(h) = chα, c > 0, 0 < α ≤ 1,
that sups,t∈T :|t−s|≤h(B(t, t) − 2B(t, s) +B(s, s)) ≤ σ2(h).
2) for given ε, δ and obtained σ(h) find the order of spline N :
P
(‖ ξ(t) − SΔ(t) ‖Lp(T )> ε
) ≤ δ.
(using theorem 5 i)). Calculate the points of the partition tk = k
N
, k = 0, N
and nonnegatively definite matrix R = (Rij)
N
i,j=0, Rij = B(ti, tj).
3) get eigenvalues b20, ..., b
2
N and eigenvectors −→x0, ...,
−→xN of R.
Then as it is known the matrix A =
⎛⎝−→x0
...−→xN
⎞⎠ reduces R to diagonal form
i.e. D = ARAT , where D =
⎛⎜⎜⎝
b20 0 . . . . .
0 b22 0 ...
. . . . . . . . . . . . .
. . . . . 0 b2N
⎞⎟⎟⎠
4) simulate vector
−→
ζ = (ζ0, ..., ζN , ), ζi ∼ N(0, 1) are independent, cal-
culate −→η = (η0, ..., ηN), where ηi = biζi.
5) find vector
−→̃
ξ = A−→η , construct corresponding spline with boundary
conditions (2) – desired model of the process ξ(t).
References
1. Ahlberg, J., Nilson, E., Walsh, J., The theory of splines and their applica-
tions, M., (1972).
2. Buldygin, V. V., Kozachenko, Y. V., Metric characteristics of random vari-
ables and random processes, American Mathematical Society, Providence,
R I, (2000).
3. Guiliano Antonini, R., Kozachenko, Yu., Nikitina, T., Spaces of ϕ-Sub-
gaussian Random Variables, Memorie di Matematica e Applicazioni, 121,
Vol. 27, fasc. 1, (2003), 95–124.
66 OLEXANDRA KAMENSCHYKOVA
4. Kozachenko, Yu. V., Ostrovskyi, E. I. Banach spaces of random variables
of Sub-gaussian type, Theor. Probability and Math. Statist. 32, (1985),
42–53.
5. Kozachenko, Yu. V., Kovalchuk Y. O. Boundary problems with random ini-
tial conditions and functional series from Subϕ(Ω). I, Ukr. math. journal.
50, (1998), No 4, 504–515.
6. Giuliano Antonini, R., Kozachenko, Yu. V., Sorokulov, V. V., On accu-
racy and reliability of simulation of some random processes from the space
Subϕ(Ω), Theory of Stochastic Processes, vol. 9(25), No 3-4, 2003, 50–57.
7. Kozachenko, Yu. V., Slivka, G. I., Justification of the Fourier method
for hyperbolic equations with random initial conditions, Theor. Prob-
ability and Math. Statist. 69, (2003), 67–83.
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail: kamalev@gmail.com
|
| id | nasplib_isofts_kiev_ua-123456789-4568 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-24T07:35:45Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kamenschykova, O. 2009-12-07T15:34:10Z 2009-12-07T15:34:10Z 2008 Approximation of random processes by cubic splines / O. Kamenschykova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 53-66. — Бібліогр.: 7 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4568 Approximation of some classes of random processes by cubic splines with given accuracy and reliability is considered. Estimations of deviation of approximating spline from original process are obtained. A few examples of approximation are considered. Application of splines for simulation of processes is also studied. en Інститут математики НАН України Approximation of random processes by cubic splines Article published earlier |
| spellingShingle | Approximation of random processes by cubic splines Kamenschykova, O. |
| title | Approximation of random processes by cubic splines |
| title_full | Approximation of random processes by cubic splines |
| title_fullStr | Approximation of random processes by cubic splines |
| title_full_unstemmed | Approximation of random processes by cubic splines |
| title_short | Approximation of random processes by cubic splines |
| title_sort | approximation of random processes by cubic splines |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4568 |
| work_keys_str_mv | AT kamenschykovao approximationofrandomprocessesbycubicsplines |