Derivation of a State-Space Model by Functional Data Analysis

Summary

By approximating a stochastic process by means of spline interpolation of its sample-paths, a time dependent state-space model is introduced. Then we derive the expression of the associated transition matrix that allows to obtain a discrete model useful in applications. In order to essay the behaviour of the proposed models simulations on a narrow-band process are developed. Finally, the paper includes an application with real data obtained from the Stock Market of Madrid.

Appendix

In order to calculate the coefficients α_i that appear in (3) we take into account the expression of the cubic B-splines:

$$\begin{array}{*{20}{c}} {B_i^{(3)}({t_l}) = \left\{ {\begin{array}{*{20}{c}} {0\;\;\;\;,} \\ {{a_{{t_l},l - 1}},} \\ {{a_{{t_l},l - 2}},} \\ {{a_{{t_l},l - 3}},} \end{array}} \right.}&{\begin{array}{*{20}{c}} {\text{if}\;i \ne l - 1,\;l - 2,\;l - 3} \\ {\text{if}\;i = l - 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {\text{if}\;i = l - 2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {\text{if}\;i = l - 3\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}} \end{array}$$

where

$$\begin{array}{*{20}c} {{a_{{t_l},l - 1}}{\rm{ }} = \,\,\,{{{{\left( {{t_l} - {t_{l - 1}}} \right)}^2}} \over {\left( {{t_{l + 2}} - {t_{l - 1}}} \right)\left( {{t_{l + 1}} - {t_{l - 1}}} \right)}}\,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ {{a_{{t_l},l - 2}}{\rm{ }} = \left( {{{{t_l} - {t_{l - 2}}} \over {{t_{l + 1}} - {t_{l - 2}}}}} \right)\left( {{{{t_{l + 1}} - {t_l}} \over {{t_{l + 1}} - {t_{l - 1}}}}} \right) + \left( {{{{t_{l + 2}} - {t_l}} \over {{t_{l + 2}} - {t_{l - 1}}}}} \right)\left( {{{{t_l} - {t_{l - 1}}} \over {{t_{l + 1}} - {t_{l - 1}}}}} \right)} \\ {{a_{{t_l},l - 3}} = {{{{\left( {{t_{l + 1}} - {t_l}} \right)}^2}} \over {\left( {{t_{l + 1}} - {t_{l - 2}}} \right)\left( {{t_{l + 1}} - {t_{l - 1}}} \right)}}.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ \end{array} $$

Then expression (3) is reduced to the linear system:

$${\alpha _{l - 3}}{a_{{t_l},l - 3}} + {\alpha _{l - 2}}{a_{{t_l},l - 2}} + {\alpha _{l - 1}}{a_{{t_l},l - 1}} = {X_\omega }\left( {{t_l}} \right)\quad \forall l = 0,1, \ldots ,n$$

((11))

and in order to guarantee an unique solution for this system it is necessary to impose that all the sample-paths have second order derivative in the extremes of the definition interval, that is, for l = 0 and l = n:

$${\alpha _{l - 3}}{a_{l - 3}} + {\alpha _{l - 2}}{a_{l - 2}} + {\alpha _{l - 1}}{a_{l - 1}} = {X''_\omega }({t_l}),$$

((12))

with:

$$\begin{array}{*{20}c} {{a_{l - 1}} = {t_{l + 1}} - {t_{l - 2}}\quad \quad \quad \quad \quad \,\,\,} \\ {{a_{l - 2}} = - \left( {{t_{l + 2}} + {t_{l + 1}} - {t_{l - 1}} - {t_{l - 2}}} \right)} \\ {{a_{l - 3}} = {t_{l + 2}} - {t_{l - 1}}.\quad \quad \quad \quad \quad \,\,} \\ \end{array} $$

Then, Prenter (1975) ensures that there exists only one cubic spline like (3) whose coefficients verify (11) and (12). We also assume that ${X''_\omega }({t_0}) = {X''_\omega }({t_n}) = 0$ because it is not common to observe such values in each sample and it is not a restrictive hypothesis as prove Aguilera et al. (1996).

Let us denote

$$\Gamma \omega = {\left( {\quad {\alpha _{ - 3}}\quad {\alpha _{ - 2}}\quad {\alpha _{ - 1}}\quad \ldots \quad {\alpha _{n - 2}}\quad {\alpha _{n - 1}}\quad } \right)^T}$$

to the coefficients vector of the cubic B-spline interpolation for the sample ω and by

$${X_\omega } = {\left( {{{X''}_\omega }({t_0})\;\;\;{X_\omega }({t_0})\;\;\;{X_\omega }({t_1})\;\;\;...\;\;\;{X_\omega }({t_n})\;\;\;{{X''}_\omega }({t_n})} \right)^T}$$

to the vector containing the observations of the sample-path ω and its second order derivative in the points 0 and T. Then, the system equations (11) and (12) can be expressed as:

$${\mathbb A}{\Gamma _\omega } = {\chi _\omega },$$

where the matrix ${\mathbb A}$ is given by:

$$\mathbb{A}=\left(\begin{array}{cccccccc}{a_{-3}} & {a_{-2}} & {a_{-1}} & {0} & {\cdots} & {0} & {0} & {0} \\ {a_{t_{0},-3}} & {a_{t_{0},-2}} & {a_{t_{0},-1}} & {0} & {\cdots} & {0} & {0} & {0} \\ {0} & {a_{t_{1},-2}} & {a_{t_{1},-1}} & {a_{t_{1}, 0}} & {\cdots} & {0} & {0} & {0} \\ {0} & {0} & {a_{t_{2},-1}} & {a_{t_{2}, 0}} & {\cdots} & {0} & {0} & {0} \\ {\cdots} & {\cdots} & {\cdots} & {\cdots} & {\ddots} & {\cdots} & {\cdots} & {\cdots} \\ {0} & {0} & {0} & {0} & {\cdots} & {a_{t_{n}, n-3}} & {a_{t_{n}, n-2}} & {a_{t_{n}, n-1}} \\ {0} & {0} & {0} & {0} & {\cdots} & {a_{n-3}} & {a_{n-2}} & {a_{n-1}}\end{array}\right)$$

and when the knots are equally spaced it is reduced to:

$$\mathbb{A}=\left(\begin{array}{cccccccc}{1} & {-2} & {1} & {0} & {\cdots} & {0} & {0} & {0} \\ {\frac{1}{6}} & {\frac{4}{6}} & {\frac{1}{6}} & {0} & {\cdots} & {0} & {0} & {0} \\ {0} & {\frac{1}{6}} & {\frac{4}{6}} & {\frac{1}{6}} & {\cdots} & {0} & {0} & {0} \\ {0} & {0} & {\frac{1}{6}} & {\frac{4}{6}} & {\cdots} & {0} & {0} & {0} \\ {\cdots} & {\cdots} & {\cdots} & {\cdots} & {\ddots} & {\cdots} & {\cdots} & {\cdots} \\ {0} & {0} & {0} & {0} & {\cdots} & {\frac{1}{6}} & {\frac{4}{6}} & {\frac{1}{6}} \\ {0} & {0} & {0} & {0} & {\cdots} & {1} & {-2} & {1}\end{array}\right).$$

The inverse ${\mathbb T} = {{\mathbb A}^{ - 1}}$ is called B-spline interpolation matrix.