Component and the Least Square Estimation of Mean and Covariance Functions of Biperiodically Correlated Random Signals

Abstract

The component and the least square (LS) estimators of mean and covariance functions of biperiodically correlated random processes (BPCRPs) as the model of the signal with binary stochastic recurrence are analyzed. The formulae for biases and the variances for estimators are obtained and the sufficient condition for the mean square consistency of mean function and Gaussian BPCRP covariance function are given. It is shown that the leakage errors are absent for the LS estimators in contrast to the component ones. The comparison of the bias and variance of the component and the LS estimators is carried out for BPCRP particular case.

Appendices

Appendix A

Proof of Proposition 2.1

For the mathematical expectation of (4), we have

$$ E\hat{m}\left( t \right) = \sum\limits_{{l,n = - N_{1} }}^{{N_{1} }} {e^{{i\lambda _{{ln}} t}} \left[ {\frac{1}{T}\int\limits_{0}^{T} {m\left( s \right)e^{{ - i\lambda _{{ln}} s}} ds} } \right]} . $$

Taking into account representation (8) and integrating, we obtain

$$ \frac{1}{T}\int\limits_{0}^{\theta } {m\left( s \right)ds} = m_{{ln}} + \sum\limits_{\begin{subarray}{l} k,r = - N_{1} \\ k \ne l,r \ne n \end{subarray} }^{{}} {m_{{kr}} \varphi \left( {\lambda _{{k - l,r - n}} T} \right).} $$

Hence, we have (9) and (10).

The variance of estimator (4) is equal to

$$ Var\left[ {\hat{m}\left( t \right)} \right] = \sum\limits_{{l,n = - N_{1} }}^{{N_{1} }} {\sum\limits_{{k,r = - N_{1} }}^{{N_{1} }} {e^{{i\lambda _{{l - r,n - k}} t}} } } \left[ {\frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {b\left( {s_{1} ,s_{2} - s_{1} } \right)e^{{i\left( {\lambda _{{ln}} s_{1} - \lambda _{{kr}} s_{2} } \right)}} ds_{1} ds_{2} } } } \right]. $$

Introduce a new variable $u = s_{2} - s_{1}$, change the order of integration, and take into consideration the equality $b\left( {s, - u} \right) = b\left( {s - u,u} \right)$. Then,

$$ \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {b\left( {s_{1} ,s_{2} - s_{1} } \right)e^{{i\left( {\lambda _{{ln}} s_{1} - \lambda _{{kr}} s_{2} } \right)}} ds_{1} ds_{2} } = \frac{1}{{T^{2} }}} \int\limits_{0}^{T} {\int\limits_{{ - s}}^{{T - s}} {b\left( {s,u} \right)e^{{i\lambda _{{l - k,n - r}} s}} e^{{ - i\lambda _{{kr}} u}} duds} } $$

$$ = \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{{T - u}} {b\left( {s,u} \right)e^{{i\lambda _{{l - k,n - r}} }} \left( {e^{{i\lambda _{{ln}} u}} + e^{{ - i\lambda _{{kr}} u}} } \right)dsdu} } . $$

After substituting into last expressions the representation

$$ b\left( {s,u} \right) = \sum\limits_{{p,q = - N_{2} }}^{{N_{2} }} {B_{{pq}} \left( u \right)e^{{i\lambda _{{pq}} s}} } $$

and temporal integration in the first approximation, we come to (11). It follows from (11) that $Var\left[ {\hat{m}\left( t \right)} \right] \to 0$ as $T \to 0$, i.e., estimator (4) is the mean square consistent.

Appendix B

Proof of Proposition 2.2

Rewrite statistics (7) in the form

$$ \hat{B}_{{ln}} \left( u \right) = \frac{1}{T}\int\limits_{0}^{T} {\left[ {\mathop \xi \limits^{ \circ } \left( s \right)\mathop \xi \limits^{ \circ } \left( {s + u} \right) - \mathop {\hat{m}}\limits^{ \circ } \left( s \right)\mathop \xi \limits^{ \circ } \left( {s + u} \right) - \mathop \xi \limits^{ \circ } \left( s \right)\mathop {\hat{m}}\limits^{ \circ } \left( {s + u} \right) + \mathop {\hat{m}}\limits^{ \circ } \left( s \right)\mathop {\hat{m}}\limits^{ \circ } \left( {s + u} \right)} \right]} e^{{ - i\lambda _{{ln}} s}} ds, $$

where $\mathop {\hat{m}}\limits^{ \circ } \left( s \right) = \hat{m}\left( s \right) - m\left( s \right)$. The mathematical expectation of every component is equal to

$$ E\left\{ {\frac{1}{T}\int\limits_{0}^{\theta } {\mathop \xi \limits^{ \circ } \left( s \right)\mathop \xi \limits^{ \circ } \left( {s + u} \right)e^{{ - i\lambda _{{ln}} s}} ds} } \right\} = B_{{ln}} \left( u \right) + \varepsilon \left[ {B_{{ln}} \left( u \right)} \right], $$

$$ E\left\{ {\frac{1}{T}\int\limits_{0}^{T} {\mathop {\hat{m}}\limits^{ \circ } \left( s \right)\mathop \xi \limits^{ \circ } \left( {s + u} \right)e^{{ - i\lambda _{{ln}} s}} ds} } \right\} = \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {b\left( {s_{1} ,s_{2} - s_{1} + u} \right)q_{{ln}} \left( {N_{1} ,s_{1} ,s_{2} } \right)ds_{1} ds_{2} } } , $$

(B.1)

$$ E\left\{ {\frac{1}{T}\int\limits_{0}^{T} {\mathop {\hat{m}}\limits^{ \circ } } \left( {s + u} \right)\mathop \xi \limits^{ \circ } \left( s \right)e^{{ - i\lambda _{{ln}} s}} ds} \right\} = \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {b\left( {s_{1} ,s_{2} - s_{1} } \right)h_{{ln}} \left( {N_{1} ,s_{1} ,s_{2} ,u} \right)ds_{1} ds_{2} } } , $$

(B.2)

$$ E\left\{ {\frac{1}{T}\int\limits_{0}^{T} {\mathop {\hat{m}}\limits^{ \circ } \left( s \right)\mathop {\hat{m}}\limits^{ \circ } \left( {s + u} \right)e^{{ - i\lambda _{{ln}} s}} ds} } \right\} $$

$$ = \frac{1}{T}\sum\limits_{{m,k = - N_{1} }}^{{N_{1} }} {\sum\limits_{{r,s = - N_{1} }}^{{N_{1} }} {\varphi \left( {\lambda _{{m + r - l,k + s - n}} T} \right)} } \int\limits_{0}^{T} {\int\limits_{0}^{T} {b\left( {s_{1} ,s_{2} - s_{1} } \right)e^{{ - i\left( {\lambda _{{mk}} s_{1} + \lambda _{{rs}} \left( {s_{2} + u} \right)} \right)}} ds_{1} ds_{2} } } . $$

(B.3)

Taking into consideration the representation (12) after transformation, we obtain in the first approximation formulas (13) and (14). Since the functions $h\left( {N_{1} ,u_{2} } \right)$ and $\tilde{h}\left( {N_{1} ,u,u_{1} } \right)$ are bounded, then $\varepsilon \left[ {\hat{b}\left( {t,u} \right)} \right] \to 0$ as $T \to \infty$ when conditions (8) are satisfied.

Appendix C

Proof of Proposition 2.3

For Gaussian BPCRP

$$ G\left( {s_{1} ,s_{1} + u,s_{2} ,s_{2} + u} \right) = b\left( {s_{1} ,s_{2} - s_{1} } \right)b\left( {s_{1} + u,s_{2} - s_{1} } \right) $$

$$ + b\left( {s_{1} ,s_{2} - s_{1} + u} \right)b\left( {s_{1} + u,s_{2} - s_{1} - u} \right). $$

Introduce a new variable $u_{1} = s_{2} - s_{1}$. The function $G\left( {s,s + u,s + u_{1} ,s + u + u_{1} } \right)$ is the biperiodical function of the variable $s$ and it can be represented by the Fourier series

$$ G\left( {s,s + u,s + u_{1} ,s + u_{1} + u} \right) = \sum\limits_{{k,r = - 2N_{2} }}^{{2N_{2} }} {\tilde{B}_{{kr}} \left( {u_{1} ,u} \right)e^{{i\lambda _{{kr}} s}} } . $$

(C.1)

Proceeding from (12), we get

$$ \begin{gathered} \tilde{B}_{{kr}} \left( {u_{1} ,u} \right) = \left\{ \begin{gathered} \sum\limits_{{p_{1} = k - N_{2} }}^{{N_{2} }} {\sum\limits_{{q_{1} = r - N_{2} }}^{{N_{2} }} {B_{{p_{1} q_{1} }}^{{\left( {k,r} \right)}} \left( {u_{1} ,u} \right)} } ,\;k \le 0,\;r \le 0, \hfill \\ \sum\limits_{{p_{1} = k - N_{2} }}^{{N_{2} }} {\sum\limits_{{q_{1} = - N_{2} }}^{{N_{2} - r}} {B_{{p_{1} q_{1} }}^{{\left( {k,r} \right)}} \left( {u_{1} ,u} \right)} } ,\;k \le 0,\;r > 0, \hfill \\ \sum\limits_{{p_{1} = - N_{2} }}^{{N_{2} - k}} {\sum\limits_{{q_{1} = r - N_{2} }}^{{N_{2} - r}} {B_{{p_{1} q_{1} }}^{{\left( {k,r} \right)}} \left( {u_{1} ,u} \right),\;k > 0,\;r \le 0,} } \hfill \\ \sum\limits_{{p_{1} = - N_{2} }}^{{N_{2} - k}} {\sum\limits_{{q = - N_{2} }}^{{N_{2} - r}} {B_{{p_{1} q_{1} }}^{{\left( {k,r} \right)}} \left( {u_{1} ,u} \right)} ,\;k > 0,\;r > 0,} \hfill \\ \end{gathered} \right.. \hfill \\ \hfill \\ \end{gathered} $$

(C.2)

where

$$ B_{{pq}}^{{\left( {k,r} \right)}} \left( {u_{1} ,u} \right) = \left[ {B_{{k + p,r + q}} \left( {u_{1} } \right)\overline{B} _{{pq}} \left( {u_{1} } \right) + B_{{k + p,r + q}} \left( {u_{1} + u} \right)\overline{B} _{{pq}} \left( {u_{1} - u} \right)} \right]e^{{ - i\lambda _{{pq}} u}} . $$

(C.3)

After reducing the double integral

$$ \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {G\left( {s_{1} ,s_{1} + u,s_{2} ,s_{2} + u} \right)} e^{{i\left( {\lambda _{{pq}} s_{2} - \lambda _{{ln}} s_{1} } \right)}} ds_{1} ds_{2} } $$

in iterated integral and integrating respect to $s$ in the first approximation, we obtain formula (15). If conditions (3) are satisfied, then $\tilde{B}_{{pq}} \left( {u_{1} ,u} \right) \to 0$ as $\left| {u_{1} } \right| \to \infty$. Thus, $Var\left[ {\hat{b}\left( {t,u} \right)} \right] \to 0$ as $\theta \to \infty$, i.e., estimator (5) is the mean square consistent.

Appendix D

Proof of Proposition 3.1

Taking into account the property

$$ \sum\limits_{{j = 1}}^{{2L_{1} + 1}} {m_{{jr}} M_{{jk}} } = \left\{ {\begin{array}{*{20}l} {\left| {\mathbf{M}} \right|,} \hfill & {r = k,} \hfill \\ {0,} \hfill & {r \ne k,} \hfill \\ \end{array} } \right. $$

we conclude the mathematical expectation of the elements of matrix ${\hat{\mathbf{m}}}$ $E\hat{m}_{j} = m_{j}$, $j \in \left[ {0,2L_{1} } \right]$, i.e., the estimators of the Fourier coefficients for the mean function are unbiased. Then, estimator (19) is also unbiased.

Proceeding from (19) for the variance estimator, we obtain

$$Var\left[ {\hat{m}\left( t \right)} \right] = \frac{1}{{\left| {\mathbf{M}} \right|^{2} }}\sum\limits_{{l,n = 0}}^{{2L}} {R_{{\tilde{m}_{l} \tilde{m}_{n} }} f_{l} \left( t \right)f_{n} \left( t \right)} ,$$

where

$$ R_{{\tilde{m}_{l} \tilde{m}_{n} }} = \frac{1}{{\theta ^{2} }}\int\limits_{0}^{\theta } {\int\limits_{0}^{\theta } {b\left( {s_{1} ,s_{2} - s_{1} } \right)} } \left\{ {\begin{array}{*{20}l} {\cos \omega _{l} s_{1} } \hfill & {\cos \omega _{n} s_{2} } \hfill \\ {\sin \omega _{l} s_{1} } \hfill & {\sin \omega _{n} s_{2} } \hfill \\ \end{array} } \right\}ds_{1} ds_{2} . $$

(D.1)

After transformation of the double integral (C.1), we obtain expression (20). Since the inner integral is bounded, then $R_{{\tilde{m}_{l} \tilde{m}_{n} }} \to 0$ as $T \to \infty$, i.e., the least square estimator of BPCRP’s mean function is the mean square consistent.

Appendix E

Proof of Proposition 3.2

Taking into account the series

$$ b\left( {t,u} \right) = B_{0} \left( u \right) + \sum\limits_{{k = 1}}^{{L_{c} }} {\left[ {B_{k}^{c} \left( u \right)\cos \omega _{k} t + B_{k}^{s} \left( u \right)\sin \omega _{k} t} \right]} $$

and the property

$$ \sum\limits_{{r = 1}}^{{2L_{2} + 1}} {d_{{rk}} D_{{rj}} } = \left\{ {\begin{array}{*{20}l} {\left| {\mathbf{D}} \right|,} \hfill & {k = j,} \hfill \\ {0,} \hfill & {k \ne j,} \hfill \\ \end{array} } \right. $$

we have

$$ \frac{{\left[ {D_{{1k}} } \right]^{T} }}{{\left| {\mathbf{D}} \right|}}\left[ {\frac{1}{\theta }\int\limits_{0}^{\theta } {b\left( {t,u} \right)dt} } \right] = B_{0} \left( u \right), $$

$$ \frac{{\left[ {D_{{p + 1,k}} } \right]^{T} }}{{\left| {\mathbf{D}} \right|}}\left[ {\frac{1}{\theta }\int\limits_{0}^{\theta } {b\left( {t,u} \right)\cos \omega _{r} tdt} } \right] = B_{p}^{c} \left( u \right), $$

$$ \frac{{\left[ {D_{{p + L_{2} + 1,k}} } \right]^{T} }}{{\left| {\mathbf{D}} \right|}}\left[ {\frac{1}{\theta }\int\limits_{0}^{\theta } {b\left( {t,u} \right)\sin \omega _{r} tdt} } \right] = B_{p}^{s} \left( u \right),\;p \in \left[ {1,L_{2} } \right]. $$

Thus, leakage error is absent.

It follows from (B.1) and (B.2) that the rest of the components of biases (24) and (25) for estimators of the covariance components are defined by the integrals of the forms

$$ I\left( T \right) = \frac{1}{{T^{2} }}\int\limits_{0}^{T} {\int\limits_{0}^{T} {b\left( {s_{2} ,s_{2} - s_{1} } \right)} } \left\{ {\begin{array}{*{20}l} {\cos \omega _{p} s_{1} } \hfill & {\cos \omega _{q} s_{2} } \hfill \\ {\sin \omega _{p} s_{1} } \hfill & {\sin \omega _{q} s_{2} } \hfill \\ \end{array} } \right\}ds_{1} ds_{2} . $$

If conditions (3) are satisfied, these integrals vanish as $T \to \infty$. So, the LSM-estimators of the covariance components are asymptotically unbiased.