On-line estimation of a smooth regression function Liptser, R. - - PDF document

On-line estimation of a smooth regression function

Liptser, R. jointly with L. Goldentyer Tel Aviv University

• Dept. of Electrical Engineering-Systems

December 19, 2002

SETTING We consider a tracking problem for smooth function f = f(t), 0 ≤ t ≤ T, under observation Xin = f(tin) + σξi, tin = i n n is large ;

• (ξi) is i.i.d., Eξi = 0, Eξ2

i = 1;

• σ2 is a positive constant.

Without additional assumptions on f, it is dif- ficult to create an estimator even n is large.

1

Main assumption f is differentiable k-times and the oldest deriva- tive is Lipschitz continuous. Filtering approach: Bar-Shalom and Li Simulate f(k)(t) with a help WHITE NOISE: d f(k)(t) dt = “white noise”. it sounds as nonsense but works pretty good. Nonparametric Statistic Approach. f ∈ Σ(k, α, L) The Stone-Ibragimov-Khasminskii class contain- ing k-times differentiable function with

• f(k)(t′′) − f(k)(t′)
• ≤ L|t′′ − t′|α,

0 < α ≤ 1.

2

Task: to combine both approaches Since a quality of estimating depends on n, any estimate of f is marked by n, that is

• f(j)

n

(t) are estimates of f(j)(t), j = 0, 1, . . . , k respectively. It is known from Ibragimov and Khasminskii that for a wide class of loss sup

f∈Σ(k,α,L)

EL

• n

k+α−j 2(k+α)+1

• f(j)

n

− f(j)Lp

• < C.

and n

k+α−j 2(k+α)+1, j = 0, 1 . . . , k

is the best rate, uniformly in the class, of esti- mating risk convergence to zero in n → ∞.

3

In particular, the risks E f(j)

n

(t) − f(j)(t)

2, j = 0, 1 . . . , k

have the same rates in n: sup

f∈Σ(k,α,L)

lim

n E

• n

k+α−j 2(k+α)+1|

• f(j)

n

(t)−f(j)(t)|

2

< C. These rates cannot be exceeded uniformly on any nonempty open set from (0, T). Jointly with Khasminskii, we realize on-line fil- ter guaranteeing the optimal rates in n.

4

Here tin and

• f(j)

n

(tin) identify ti and

• f(j)

n

(ti). For j = 0, 1 . . . , k − 1,

• f(j)(ti) =

f(j)(ti−1)+ + 1 n

• f(j+1)(ti−1)

+ qj n

(2(k+α)−j) 2(k+α)+1

• Xi −

f(0)(ti−1)

• and for j = k
• f(k)(ti) =

f(k)(ti−1)+ qk n

(2(k+α)−k) 2(k+α)+1

• Xi−

f(0)(ti−1)

• .

The vector q with entries q0, . . . , qk has to be chosen such that all roots of characteristic poly- nomial pk(u, q) = uk+1+q0uk+q1uk−1+. . .+qk−1u+qk are different and have negative real parts.

5

Two problems

• 1. Choice of an appropriate initial conditions:
• f(0)(0),

f(1)(0), . . . , f(k)(0) to minimize a boundary layer. 2. Choice of the vector q such that the as- sumption about roots of the polynomial pk(u, q) remains valid and C(q) ≥ sup

f∈Σ(k,α,L)

E

• n

k+α−j 2(k+α)+1|

• f(j)

n

(t)−f(j)(t)|

2

is smallest as possible. To manage these problems we need to restrict

• urselves by

α = 1.

6

Boundary layer The left side boundary layer c(q)n−

1 2β+1 log n

where the optimal rates in n might be lost is inevitable. This boundary layer is due to on- line limitations of the above tracking system. One can readily suggest an off-line modifica- tion with the same recursion in the backward time subject to some boundary conditions in- dependent of observation Xi’s. This modifica- tion obeys the right side boundary layer. So, a combination of the forward and back- ward time tracking algorithms allows support the optimal rate in n for [0, T].

7

Suitable choice of q Vector q should satisfy multiple requirements regarding

• C(q) the upper bound for the normalized risk;
• c(q) the parameter of the boundary layer;
• roots of polynomial pk(u, q).

These requirements might contradict each other.

8

Example 1, Σ(0, 1, L) The worst f(t) = f(0) ± Lt. Applying the Arzela-Ascoli theorem we find that C(q) = σq 2 + L2σ2 q2 and q◦ := argmin

g>0

C(q) = (2L)2/3σ1/3. Hence, a reasonable estimator is:

• f(ti) =

f(ti−1) +

2L

nσ

2/3

(Xi − f(ti−1)).

9

General case, Σ(k > 0, 1, L) With the worst f(t) such that f(k)(t) = f(k)(0) ± Lt, applying the Arzela-Ascoli theorem we find C(q) = trace

• P(q) + M(g)M∗(q)
• where

M(q) = L(a − qA)−1b and P(q) solves the Lyapunov equation (a − qA)P(q) + P(q)(a − qA)∗ + σ2qq∗ = 0.

10

Here, a =

       

1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 . . .

       

(k+1)×(k+1)

, A =

• 1

. . .

• 1×(k+1) ,

b =

    

. . . 1

    

(k+1)×1

.

11

Conditional minimization A direct minimization of C(q) is useless. A computer implementation is heavy enough. Even if q◦ = argmin

g

C(q) is found, the main requirement, expressed in term of eigenvalues the polynomial pk(u, q), might be not satisfied (numerical computa- tions show that). So, some kind of a conditional minimization procedure in vector q is desirable. The main tool for such minimization is adaptation to Kalman filter design.

12

Kalman filter design In a frame of Bar Shalom idea, set f(k)(ti) = f(k)(ti−1) + n−

k+2 2(k+1)+1 γ ηi

Xi = f(0)(ti−1) + σξi where (ηi) is a white noise, independent of (ξi), with Eη1 = 0, Eη1 = 1; γ is free parameter. For any γ = 0, the Kalman filter possesses an asymptotic form in n → ∞ and, being applied to the original function f(t), guaranties the op- timal rate in n → ∞ for the estimation risk. In

• ther words, that Kalman filter coincides with
• ur proposed filter.

The remarkable fact is that q = q(γ) and for any positive γ roots of polynomial pk(u, q(γ)) are different and have negative real parts.

13

Thus, q(γ) = Q(γ)A∗ σ2 with Q(γ) being solution of the algebraic Ric- cati equation aQ(γ) + Q(γ)a∗ + γ2bb∗ − Q(γ)A∗AQ(γ) σ2 = 0.

• beying the unique positive-definite solution

since block-matrices G1 =

    

A Aa . . . Aak

    

and G2 =

• bb∗

abb∗ . . . akbb∗ are of full ranks (so called, observability and controllability conditions).

14

C(q(γ))-minimization We reduce the minimization of C(q) with re- spect to vector q to minimization of C(q(γ) with respect to a positive parameter γ.

10

−2

10 10

2

10

4

10 10

2

10

4

10

6

10

8

σ = 0.25 γ C (q (γ) ) 10

−2

10 10

2

10

4

10 10

2

10

4

10

6

10

8

σ = 1 γ C (q (γ) ) 10

−2

10 10

2

10

4

10 10

2

10

4

10

6

10

8

σ = 4 γ C (q (γ) ) L=1; σ =0.25; γ=0.74082; C=5.278; L=10; σ =0.25; γ=4.4817; C=102.1574; L=100; σ =0.25; γ=24.5325; C=2695.7356; L=1; σ =1; γ=1; C=18.8975; L=10; σ =1; γ=6.0496; C=260.7145; L=100; σ =1; γ=33.1155; C=5839.3352; L=1; σ =4; γ=1.3499; C=92.0461; L=10; σ =4; γ=8.1662; C=787.7197; L=100; σ =4; γ=49.4024; C=13850.9423; L=1 L=10 L=100

Here, C

• q(γ)
• in logarithmic scale for k = 2

and various L and σ.

15

Explicit minimization procedure Entries of Q(γ) obey the following presentation Qij(γ, σ) = Uijσ2

γ

σ

i+j+1

k+1 , i, j = 0, 1, . . . , k,

where Uij are entries of the matrix U also being solution of the algebraic Riccati equation free

• f σ and γ:

aU + Ua∗ + bb∗ − UA∗AU = 0. We have q0(γ) = U00

γ

σ

1/k+1

q1(γ) = U01

γ

σ

2/k+1

................................. qk(γ) = U0k

γ

σ

• .

16

For k = 0, . . . , 4 k U00 U01 U02 U03 U04 1 NA NA NA NA 1 √ 2 1 NA NA NA 2 2 2 1 NA NA 3

• 4 +

√ 8 2 + √ 2

• 4 +

√ 8 1 NA 4 1 + √ 5 3 + √ 5 3 + √ 5 1 + √ 5 1

17

Roots of pk(u, q) k = 0 : −

γ

σ

• k = 1 :

−

γ

σ

1/2 1

√ 2 ± i 1 √ 2

• k = 2 :

−

γ

σ

1/3

1; 1 2 ± i √ 3 2

• k = 3 :

−

γ

σ

1/4

0.924 ± i0.383; 0.383 ± i0.924

• k = 4 :

−

γ

σ

1/5

1; 0.809 ± i0.588; 0.309 ± i0.951

• .

18

Example 2 k = 2, L = 100, σ = 0.25.

• f(0)(ti) =

f(0)(ti−1) + 1 n

• f(1)(ti−1)

+ 9.225 n6/7

• Xi −

f(0)(ti−1)

• f(1)(ti) =

f(1)(ti−1) + 1 n

• f(2)(ti−1)

+ 42.550 n5/7

• Xi −

f(0)(ti−1)

• f(2)(ti) =

f(2)(ti−1) + 98.132 n4/7

• Xi −

f(0)(ti−1)

• .

19

Forward and backward time tracking with n = 2000

0.2 0.4 0.6 0.8 1 −8 −6 −4 −2 2 Time Value Tracking 3−rd order Lipshitz continuous function Observations Signal Forward time tracking 0.2 0.4 0.6 0.8 1 −8 −6 −4 −2 2 Time Value Tracking the function Signal Forward time tracking Combined time tracking 0.2 0.4 0.6 0.8 1 −20 −15 −10 −5 5 10 Time Value Tracking the 1−st derrivative Signal Forward time tracking Combined time tracking 0.2 0.4 0.6 0.8 1 −40 −30 −20 −10 10 20 30 Time Value Tracking the 2−nd derivative Signal Forward time tracking Combined time tracking

20

Comparison with spline n = 2000

0.2 0.4 0.6 0.8 1 −8 −6 −4 −2 2 Time Value Tracking 3−rd order Lipshitz continuous function Observations Signal Spline estimation 0.2 0.4 0.6 0.8 1 −8 −6 −4 −2 2 Time Value Tracking the function Signal Forward time tracking Spline estimation 0.2 0.4 0.6 0.8 1 −20 −15 −10 −5 5 10 Time Value Tracking the 1−st derrivative Signal Forward time tracking Spline estimation 0.2 0.4 0.6 0.8 1 −100 100 200 300 400 Time Value Tracking the 2−nd derivative Signal Forward time tracking Spline estimation

21

Application to Finance

200 400 600 800 1000 1200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Volatility Original data Reconstructed data

Volatility is assumed to be Lipschitz continu-

• us functions.

22

200 400 600 800 1000 1200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Volatility Original data Reconstructed data

Tracking with adaptation.

23