A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter

M.V. Kulikova, I.V. Semushin

School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

E-mail: mkulikova@cam.wits.ac.za

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Introduction

The method of maximum likelihood … Gradient-search optimization algorithms; Gradient of the negative Log Likelihood Function; Kalman Filter ⟹ is known to be unstable;

[P. Park&T. Kailath, 1995] The extended Square Root Information Filter (eSRIF):

avoids numerical instabilities arising from computational errors; appears to be better suited to parallel implementation and to very large scale integration (VLSI) implementation.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 2/??

Problem Statement

Consider the discrete-time linear dynamic stochastic system

x_{t+1} = F_{t}x_{t} + G_{t}w_{t},  t = 0, 1, …, N,  (1)
z_{t} = H_{t}x_{t} + v_{t},  t = 1, 2, …, N,  (2)

with the system state x_{t} ∈ ℝ^{n}, the state disturbance w_{t} ∈ ℝ^{q}, the observed vector z_{t} ∈ ℝ^{m}, and the measurement error v_{t} ∈ ℝ^{m}, such that the initial state x_{0} and each w_{t}, v_{t} of {w_{t} : t = 0, 1, …}, {v_{t} : t = 1, 2, …} are taken from mutually independent Gaussian distributions with the following expectations:

Consider the discrete-time linear dynamic stochastic system

x_{t+1} = F_{t}x_{t} + G_{t}w_{t},  t = 0, 1, …, N,  (1)
z_{t} = H_{t}x_{t} + v_{t},  t = 1, 2, …, N,  (2)

with the system state x_{t} ∈ ℝ^{n}, the state disturbance w_{t} ∈ ℝ^{q}, the observed vector z_{t} ∈ ℝ^{m}, and the measurement error v_{t} ∈ ℝ^{m}, such that the initial state x_{0} and each w_{t}, v_{t} of {w_{t} : t = 0, 1, …}, {v_{t} : t = 1, 2, …} are taken from mutually independent Gaussian distributions with the following expectations:

Consider the discrete-time linear dynamic stochastic system

x_{t+1} = F_{t}x_{t} + G_{t}w_{t},  t = 0, 1, …, N,  (1)
z_{t} = H_{t}x_{t} + v_{t},  t = 1, 2, …, N,  (2)

with the system state x_{t} ∈ ℝ^{n}, the state disturbance w_{t} ∈ ℝ^{q}, the observed vector z_{t} ∈ ℝ^{m}, and the measurement error v_{t} ∈ ℝ^{m}, such that the initial state x_{0} and each w_{t}, v_{t} of {w_{t} : t = 0, 1, …}, {v_{t} : t = 1, 2, …} are taken from mutually independent Gaussian distributions with the following expectations:

Consider the discrete-time linear dynamic stochastic system

x_{t+1} = F_{t}x_{t} + G_{t}w_{t},  t = 0, 1, …, N,  (1)
z_{t} = H_{t}x_{t} + v_{t},  t = 1, 2, …, N,  (2)

with the system state x_{t} ∈ ℝ^{n}, the state disturbance w_{t} ∈ ℝ^{q}, the observed vector z_{t} ∈ ℝ^{m}, and the measurement error v_{t} ∈ ℝ^{m}, such that the initial state x_{0} and each w_{t}, v_{t} of {w_{t} : t = 0, 1, …}, {v_{t} : t = 1, 2, …} are taken from mutually independent Gaussian distributions with the following expectations:

Consider the discrete-time linear dynamic stochastic system

x_{t+1} = F_{t}x_{t} + G_{t}w_{t},  t = 0, 1, …, N,  (1)
z_{t} = H_{t}x_{t} + v_{t},  t = 1, 2, …, N,  (2)

with the system state x_{t} ∈ ℝ^{n}, the state disturbance w_{t} ∈ ℝ^{q}, the observed vector z_{t} ∈ ℝ^{m}, and the measurement error v_{t} ∈ ℝ^{m}, such that the initial state x_{0} and each w_{t}, v_{t} of {w_{t} : t = 0, 1, …}, {v_{t} : t = 1, 2, …} are taken from mutually independent Gaussian distributions with the following expectations:

Consider the discrete-time linear dynamic stochastic system

x_{t+1} = F_{t}x_{t} + G_{t}w_{t},  t = 0, 1, …, N,  (1)
z_{t} = H_{t}x_{t} + v_{t},  t = 1, 2, …, N,  (2)

with the system state x_{t} ∈ ℝ^{n}, the state disturbance w_{t} ∈ ℝ^{q}, the observed vector z_{t} ∈ ℝ^{m}, and the measurement error v_{t} ∈ ℝ^{m}, such that the initial state x_{0} and each w_{t}, v_{t} of {w_{t} : t = 0, 1, …}, {v_{t} : t = 1, 2, …} are taken from mutually independent Gaussian distributions with the following expectations:

E \{[ \begin{matrix}x_{0} \\ w_{t} \\ v_{t} \end{matrix} ]\} = [ x̄_{0} ; 0 ; 0 ]

E \{[ \begin{matrix}(x_{0} − x̄_{0}) \\ w_{t} \\ v_{t} \end{matrix} ] [ \begin{matrix}(x_{0} − x̄_{0}) \\ w_{t} \\ v_{t} \end{matrix} ]^{T} \} = [ \begin{matrix}P_{0} & 0 & 0 \\ 0 & Q_{t} & 0 \\ 0 & 0 & R_{t} \end{matrix} ]

and E {w_{t}w_{t′}^{T} } = 0, E {v_{t}v_{t′}^{T} } = 0 if t ≠ t′. Assume the system is parameterized by a vector θ ∈ ℝ^{p} of unknown system parameters. This means that all the above characteristics, namely F_{t}, G_{t}, H_{t}, P_{0} ≥ 0, Q_{t} ≥ 0 and R_{t} > 0 can depend upon θ (the corresponding notations F_{t}(θ), G_{t}(θ) and so on, are suppressed for the sake of simplicity).

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 4/??

E \{[ \begin{matrix}x_{0} \\ w_{t} \\ v_{t} \end{matrix} ]\} = [ x̄_{0} ; 0 ; 0 ]

E \{[ \begin{matrix}(x_{0} − x̄_{0}) \\ w_{t} \\ v_{t} \end{matrix} ] [ \begin{matrix}(x_{0} − x̄_{0}) \\ w_{t} \\ v_{t} \end{matrix} ]^{T} \} = [ \begin{matrix}P_{0} & 0 & 0 \\ 0 & Q_{t} & 0 \\ 0 & 0 & R_{t} \end{matrix} ]

and E {w_{t}w_{t′}^{T} } = 0, E {v_{t}v_{t′}^{T} } = 0 if t ≠ t′. Assume the system is parameterized by a vector θ ∈ ℝ^{p} of unknown system parameters. This means that all the above characteristics, namely F_{t}, G_{t}, H_{t}, P_{0} ≥ 0, Q_{t} ≥ 0 and R_{t} > 0 can depend upon θ (the corresponding notations F_{t}(θ), G_{t}(θ) and so on, are suppressed for the sake of simplicity).

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 4/??

For example, (1), (2) may describe a discrete autoregressive (AR) process observed in the presence of additive noise

x_{t+1} = [ \begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & 1 \\ θ_{1} & θ_{2} & θ_{3} & … & θ_{p} \end{matrix} ] x_{t} + [ 0 ; 0 ; ⋮ ; 0 ; γ ] w_{t},

z_{t} = [ 0 0 … 1 ] x_{t} + v_{t}.

In this case θ are the unknown AR parameters which need to be estimated.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 5/??

The negative Log Likelihood Function (LLF) for system (1), (2) is given by

L_{θ} (Z_{1}^{N}) = \frac{1}{2} ∑_{t=1}^{N} \{\frac{m}{2} ln(2π) + ln(det(R_{e,t})) + e_{t}^{T} R_{e,t}^{−1}e_{t}\}

with e_{t} =^{def} z_{t} − H_{t}x̂_{t} being the zero-mean innovations whose covariance is determined as R_{e,t} =^{def} E{e_{t}e_{t}^{T} } = H_{t}P_{t}H_{t}^{T} + R_{t} through matrix P_{t}, the error covariance of the time updated estimate x̂_{t} generated by the Kalman Filter.

Let l_{θ}(z_{t}) denote the negative LLF for the t-th measurement z_{t} in system (1), (2), given measurements Z_{1}^{t−1} =^{def} {z_{1}, …, z_{t−1}}, then

l_{θ}(z_{t}) = \frac{1}{2} \{\frac{m}{2} ln(2π) + ln(det(R_{e,t})) + e_{t}^{T} R_{e,t}^{−1}e_{t}\} . (3)

By differentiating (3) we obtain

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] i = 1, … p. (4)

As can be seen the computation of (3) and (4) leads to implementation of a Kalman Filter (and its derivative with respect to each parameter) which is known to be unstable.

Let l_{θ}(z_{t}) denote the negative LLF for the t-th measurement z_{t} in system (1), (2), given measurements Z_{1}^{t−1} =^{def} {z_{1}, …, z_{t−1}}, then

l_{θ}(z_{t}) = \frac{1}{2} \{\frac{m}{2} ln(2π) + ln(det(R_{e,t})) + e_{t}^{T} R_{e,t}^{−1}e_{t}\} . (3)

By differentiating (3) we obtain

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] i = 1, … p. (4)

As can be seen the computation of (3) and (4) leads to implementation of a Kalman Filter (and its derivative with respect to each parameter) which is known to be unstable.

Let l_{θ}(z_{t}) denote the negative LLF for the t-th measurement z_{t} in system (1), (2), given measurements Z_{1}^{t−1} =^{def} {z_{1}, …, z_{t−1}}, then

l_{θ}(z_{t}) = \frac{1}{2} \{\frac{m}{2} ln(2π) + ln(det(R_{e,t})) + e_{t}^{T} R_{e,t}^{−1}e_{t}\} . (3)

By differentiating (3) we obtain

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] i = 1, … p. (4)

As can be seen the computation of (3) and (4) leads to implementation of a Kalman Filter (and its derivative with respect to each parameter) which is known to be unstable.

Let l_{θ}(z_{t}) denote the negative LLF for the t-th measurement z_{t} in system (1), (2), given measurements Z_{1}^{t−1} =^{def} {z_{1}, …, z_{t−1}}, then

l_{θ}(z_{t}) = \frac{1}{2} \{\frac{m}{2} ln(2π) + ln(det(R_{e,t})) + e_{t}^{T} R_{e,t}^{−1}e_{t}\} . (3)

By differentiating (3) we obtain

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] i = 1, … p. (4)

As can be seen the computation of (3) and (4) leads to implementation of a Kalman Filter (and its derivative with respect to each parameter) which is known to be unstable.

The eSRIF

[P. Park&T. Kailath, 1995]: assume that Π_{0} > 0, R_{t} > 0 and F_{t} are invertible. Given P_{0}^{−T/2} = Π_{0}^{−T/2} and P_{0}^{−T/2}x̂_{0} = Π_{0}^{−T/2}x̄_{0}, then

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 & −ē_{t} \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

where O_{t} is any orthogonal transformation such that the matrix on the right-hand side of the formula is block lower triangular.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 8/??

The eSRIF

The eSRIF is a modification of the conventional one:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} \\ 0 & 0 & I_{q} \end{matrix} ] | & \begin{matrix}−R_{t}^{−T/2}z_{t} \\ P_{t}^{−T/2}x̂_{t} \\ 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 \\ ∗ & ∗ & ∗ \end{matrix} ] | & \begin{matrix}−ē_{t} \\ P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ \end{matrix} ] \end{matrix}

where O_{t} is any orthogonal transformation such that the matrix on the right-hand side of the formula is lower triangular. is block lower

triangular. A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 9/??

The eSRIF

The eSRIF is a modification of the conventional one:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} \\ 0 & 0 & I_{q} \end{matrix} ]| & \begin{matrix}−R_{t}^{−T/2}z_{t} \\ P_{t}^{−T/2}x̂_{t} \\ 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 \\ ∗ & ∗ & ∗ \end{matrix} ]| & \begin{matrix}−ē_{t} \\ P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ \end{matrix} ] \end{matrix}

where O_{t} is any orthogonal transformation such that the matrix on the right-hand side of the formula is lower triangular. is block lower

triangular. A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 9/??

The eSRIF

Remark. The predicted estimate now can be found from the entries of the post-array by solving the triangular system

\binom{}{P̃_{t+1}^{−1/2}} (x̂_{t+1}^{−}) = (P̃_{t+1}^{−1/2}x̂_{t+1}^{−}) .  (5)

The LLG in terms of the eSRIF

The Log Likelihood Gradient (LLG):

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] , i = 1, 2, … p,

in terms of the eSRIF is given by

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}^{1/2}))] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, 2, … p,

where R_{e,t}^{1/2} is a square-root factor of the matrix R_{e,t}, i. e. R_{e,t} = R_{e,t}^{T/2}R_{e,t}^{1/2}, and ē_{t} are the normalized innovations, i. e. ē_{t} = R_{e,t}^{−T/2}e_{t}.

The Log Likelihood Gradient (LLG):

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] , i = 1, 2, … p,

in terms of the eSRIF is given by

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}^{1/2}))] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, 2, … p,

where R_{e,t}^{1/2} is a square-root factor of the matrix R_{e,t}, i. e. R_{e,t} = R_{e,t}^{T/2}R_{e,t}^{1/2}, and ē_{t} are the normalized innovations, i. e. ē_{t} = R_{e,t}^{−T/2}e_{t}.

The Log Likelihood Gradient (LLG):

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{1}{2} \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}))] + \frac{1}{2} \frac{∂}{∂θ_{i}} [e_{t}^{T} R_{e,t}^{−1}e_{t}] , i = 1, 2, … p,

in terms of the eSRIF is given by

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = \frac{∂}{∂θ_{i}} [ln(det(R_{e,t}^{1/2}))] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, 2, … p,

where R_{e,t}^{1/2} is a square-root factor of the matrix R_{e,t}, i. e. R_{e,t} = R_{e,t}^{T/2}R_{e,t}^{1/2}, and ē_{t} are the normalized innovations, i. e. ē_{t} = R_{e,t}^{−T/2}e_{t}.

Taking into account that matrix R_{e,t}^{1/2} is upper triangular, we can show

\frac{∂}{∂θ_{i}} [ln (det(R_{e,t}^{1/2}))] = tr [R_{e,t}^{−1/2} \frac{∂(R_{e,t}^{1/2})}{∂θ_{i}} ], i = 1, 2, …, p

where tr [ · ] is a trace of matrix. Finally, we obtain the expression for the LLG in terms of the eSRIF:

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] +ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, 2, …, p. (6)

Taking into account that matrix R_{e,t}^{1/2} is upper triangular, we can show

\frac{∂}{∂θ_{i}} [ln (det(R_{e,t}^{1/2}))] = tr [R_{e,t}^{−1/2} \frac{∂(R_{e,t}^{1/2})}{∂θ_{i}} ], i = 1, 2, …, p

where tr [ · ] is a trace of matrix. Finally, we obtain the expression for the LLG in terms of the eSRIF:

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] +ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, 2, …, p. (6)

Taking into account that matrix R_{e,t}^{1/2} is upper triangular, we can show

\frac{∂}{∂θ_{i}} [ln (det(R_{e,t}^{1/2}))] = tr [R_{e,t}^{−1/2} \frac{∂(R_{e,t}^{1/2})}{∂θ_{i}} ], i = 1, 2, …, p

where tr [ · ] is a trace of matrix. Finally, we obtain the expression for the LLG in terms of the eSRIF:

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] +ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, 2, …, p. (6)

LLG Evaluation

\frac{∂l(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p. (6),

The eSRIF:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}& 0 & 0 \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

\frac{∂l(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p. (6),

The eSRIF:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}& 0 & 0 \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

LLG Evaluation

\frac{∂l(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p. (6),

The eSRIF:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 & −ē_{t} \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 13/??

LLG Evaluation

\frac{∂l(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p. (6),

The eSRIF:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 & −ē_{t} \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 13/??

LLG Evaluation

\frac{∂l(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2} \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T} \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p. (6),

The eSRIF:

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 & −ē_{t} \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 13/??

Lemma 1. Let  QA = L  (7)

where Q is any orthogonal transformation such that the matrix on the right-hand side of formula (7) is lower triangular and A is a nonsingular matrix. If the elements of A are differentiable functions of a parameter θ then the upper triangular matrix U in

Q′_{θ}Q^{T} = Ū^{T} − Ū  (8)

is, in fact, the strictly upper triangular part of the matrix QA′_{θ}L^{−1}:

QA′_{θ}L^{−1} = L̄ + D + Ū  (9)

where L̄, D and Ū are, respectively, strictly lower triangular, diagonal and strictly upper triangular.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 14/??

LLG Evaluation

Lemma 1 easily leads to a method for computing L′_{θ}. By differenting QA = L we obtain

L′_{θ} = QA′_{θ} + Q′_{θ}A  (10)

Multiply by L^{−1}:

L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}AL^{−1} ⇒ L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}Q^{T} (11)

By substituting (8) and (9) into (11), we find that

L′_{θ}L^{−1} = \underbrace{(L + D + U)}_{QA′_{θ}L^{−1}} + \underbrace{(U^{T} − U)}_{Q′_{θ}Q^{T}} ⟹ L′_{θ} = (L + D + U^{T})L^{−1}.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 15/??

LLG Evaluation

Lemma 1 easily leads to a method for computing L′_{θ}. By differenting QA = L we obtain

L′_{θ} = QA′_{θ} + Q′_{θ}A  (10)

Multiply by L^{−1}:

L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}AL^{−1} ⇒ L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}Q^{T} (11)

By substituting (8) and (9) into (11), we find that

L′_{θ}L^{−1} = \underbrace{(L + D + U)}_{QA′_{θ}L^{−1}} + \underbrace{(U^{T} − U)}_{Q′_{θ}Q^{T}} ⟹ L′_{θ} = (L + D + U^{T})L^{−1}.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 15/??

LLG Evaluation

Lemma 1 easily leads to a method for computing L′_{θ}. By differenting QA = L we obtain

L′_{θ} = QA′_{θ} + Q′_{θ}A  (10)

Multiply by L^{−1}:

L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}AL^{−1} ⇒ L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}Q^{T} (11)

By substituting (8) and (9) into (11), we find that

L′_{θ}L^{−1} = \underbrace{(L + D + U)}_{QA′_{θ}L^{−1}} + \underbrace{(U^{T} − U)}_{Q′_{θ}Q^{T}} ⟹ L′_{θ} = (L + D + U^{T})L^{−1}.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 15/??

LLG Evaluation

Lemma 1 easily leads to a method for computing L′_{θ}. By differenting QA = L we obtain

L′_{θ} = QA′_{θ} + Q′_{θ}A  (10)

Multiply by L^{−1}:

L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}AL^{−1} ⇒ L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}Q^{T} (11)

By substituting (8) and (9) into (11), we find that

L′_{θ}L^{−1} = \underbrace{(L + D + U)}_{QA′_{θ}L^{−1}} + \underbrace{(U^{T} − U)}_{Q′_{θ}Q^{T}} ⟹ L′_{θ} = (L + D + U^{T})L^{−1}.

LLG Evaluation

Lemma 1 easily leads to a method for computing L′_{θ}. By differenting QA = L we obtain

L′_{θ} = QA′_{θ} + Q′_{θ}A  (10)

Multiply by L^{−1}:

L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}AL^{−1} ⇒ L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}Q^{T} (11)

By substituting (8) and (9) into (11), we find that

L′_{θ}L^{−1} = \underbrace{(L + D + U)}_{QA′_{θ}L^{−1}} + \underbrace{(U^{T} − U)}_{Q′_{θ}Q^{T}} ⟹ L′_{θ} = (L + D + U^{T})L^{−1}.

LLG Evaluation

Lemma 1 easily leads to a method for computing L′_{θ}. By differenting QA = L we obtain

L′_{θ} = QA′_{θ} + Q′_{θ}A  (10)

Multiply by L^{−1}:

L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}AL^{−1} ⇒ L′_{θ}L^{−1} = QA′_{θ}L^{−1} + Q′_{θ}Q^{T} (11)

By substituting (8) and (9) into (11), we find that

L′_{θ}L^{−1} = \underbrace{(L + D + U)}_{QA′_{θ}L^{−1}} + \underbrace{(U^{T} − U)}_{Q′_{θ}Q^{T}} ⟹ L′_{θ} = (L + D + U^{T})L^{−1}.

Algorithm LLG-eSRIF

I. For each θ_{i}, i = 1, 2, …, p, apply the eSRIF

\begin{matrix}O_{t} [ \begin{matrix}R_{t}^{−T/2} & −R_{t}^{−T/2}H_{t}F_{t}^{−1} & R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2} & −R_{t}^{−T/2}z_{t} \\ 0 & P_{t}^{−T/2}F_{t}^{−1} & −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2} & P_{t}^{−T/2}x̂_{t} \\ 0 & 0 & I_{q} & 0 \end{matrix} ] \\ = [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 & −ē_{t} \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 & P_{t+1}^{−T/2}x̂_{t+1} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] \end{matrix}

where O_{t} is any orthogonal transformation such that the matrix on the right-hand side of the formula is block lower triangular.

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 16/??

Algorithm LLG-eSRIF

II. For each θ_{i}, i = 1, 2, …, p, calculate

\begin{matrix}O_{t} [ \begin{matrix}\frac{∂}{∂θ_{i}} (R_{t}^{−T/2}) & \frac{∂}{∂θ_{i}} (S_{t}^{(1)}) & \frac{∂}{∂θ_{i}} (S_{t}^{(2)}) & \frac{∂}{∂θ_{i}} (S_{t}^{(3)}) \\ 0 & \frac{∂}{∂θ_{i}} (S_{t}^{(4)}) & \frac{∂}{∂θ_{i}} (S_{t}^{(5)}) & \frac{∂}{∂θ_{i}} (S_{t}^{(6)}) \\ 0 & 0 & 0 & 0 \end{matrix} ] \\ = [ \begin{matrix}X_{i} & Y_{i} & M_{i} & L_{i} \\ N_{i} & V_{i} & W_{i} & K_{i} \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ] , \end{matrix}

O_{t} is the same orthogonal transformation as in the eSRIF and

\begin{matrix}S_{t}^{(1)} = −R_{t}^{−T/2}H_{t}F_{t}^{−1}, & S_{t}^{(2)} = R_{t}^{−T/2}H_{t}F_{t}^{−1}G_{t}Q_{t}^{T/2}, & S_{t}^{(3)} = −R_{t}^{−T/2}z_{t}, \\ S_{t}^{(4)} = P_{t}^{−T/2}F_{t}^{−1}, & S_{t}^{(5)} = −P_{t}^{−T/2}F_{t}^{−1}G_{t}Q_{t}^{T/2}, & S_{t}^{(6)} = P_{t}^{−T/2}x̂_{t}. \end{matrix}

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 17/??

Algorithm LLG-eSRIF

III. For each θ_{i}, i = 1, 2, …, p, compute

J_{i} = [ \begin{matrix}X_{i} & Y_{i} & M_{i} \\ N_{i} & V_{i} & W_{i} \end{matrix} ] [ \begin{matrix}R_{e,t}^{−T/2} & 0 & 0 \\ −P_{t+1}^{−T/2}K_{p,t} & P_{t+1}^{−T/2} & 0 \\ ∗ & ∗ & ∗ \end{matrix} ]^{−1} .

Algorithm LLG-eSRIF

IV. For each θ_{i}, i = 1, 2, …, p, split the matrices

J_{i} = \}m+n \overbrace{\underbrace{[ L_{i} + D_{i} + U_{i}}_{m+n} ∗ ∗ ∗ ]}^{m+n+q}

where L_{i}, D_{i} and U_{i} are the strictly lower triangular, diagonal and strictly upper triangular parts of J_{i}, respectively.

Algorithm LLG-eSRIF

V. For each θ_{i}, i = 1, 2, …, p, compute the following quantities:

\begin{matrix}[ \begin{matrix}\frac{∂R_{e,t}^{−T/2}}{∂θ_{i}} & 0 \\ −\frac{∂(P̃_{t+1}^{−T/2}K_{p,t})}{∂θ_{i}} & \frac{∂P̃_{t+1}^{−T/2}}{∂θ_{i}} \end{matrix} ] = [L_{i} + D_{i} + U_{i}^{T} ] \\ × [ \begin{matrix}R_{e,t}^{−T/2} & 0 \\ −P̃_{t+1}^{−T/2}K_{p,t} & P̃_{t+1}^{−T/2} \end{matrix} ] , \end{matrix}  (12)

Algorithm LLG-eSRIF

\begin{matrix}\frac{∂ē_{t}}{∂θ_{i}} = [\frac{∂R_{e,t}^{−T/2}}{∂θ_{i}} − X_{i}] R_{e,t}^{T/2}ē_{t} + Y_{i}F_{t}x̂_{t} − L_{i}, & (13) \\ \begin{matrix}\frac{∂S_{t+1}^{(6)}}{∂θ_{i}} = [\frac{∂(P̃_{t+1}^{−T/2}K_{p,t})}{∂θ_{i}} + N_{i}] R_{e,t}^{T/2}ē_{t} \\ + [\frac{∂P̃_{t+1}^{−T/2}}{∂θ_{i}} − V_{i}] F_{t}x̂_{t} + K_{i}. \end{matrix} & (14) \end{matrix}

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 21/??

\begin{matrix}\frac{∂ē_{t}}{∂θ_{i}} = [\frac{∂R_{e,t}^{−T/2}}{∂θ_{i}} − X_{i}] R_{e,t}^{T/2}ē_{t} + Y_{i}F_{t}x̂_{t} − L_{i}, & (13) \\ \begin{matrix}\frac{∂S_{t+1}^{(6)}}{∂θ_{i}} = [\frac{∂(P̃_{t+1}^{−T/2}K_{p,t})}{∂θ_{i}} + N_{i}] R_{e,t}^{T/2}ē_{t} \\ + [\frac{∂P̃_{t+1}^{−T/2}}{∂θ_{i}} − V_{i}] F_{t}x̂_{t} + K_{i}. \end{matrix} & (14) \end{matrix}

A Log Likelihood Gradient Evaluation by Using the Extended Square-Root Information Filter – p. 21/??

Algorithm LLG-eSRIF

VI. Finally, we have all values to compute the LLG according to (6):

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2}  \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T}  \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p.
The LLG-eSRIF:  The LLG-eSRIF:
Stage  Stages

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2}  \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T}  \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p.
The LLG-eSRIF:  The LLG-eSRIF:
Stage I  Stages II — IV

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2}  \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T}  \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p.
The LLG-eSRIF:  The LLG-eSRIF:
Stage I  Stages II — IV

\frac{∂l_{θ}(z_{t})}{∂θ_{i}} = − tr [ R_{e,t}^{1/2}  \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} ] + ē_{t}^{T}  \frac{∂ē_{t}}{∂θ_{i}} , i = 1, …, p.
The LLG-eSRIF:  The LLG-eSRIF:
Stage I  Stages II — IV

The LLG-eSRIF consists of two parts:

The source filtering algorithm, i.e. eSRIF. The "differentiated" part: Stages II — IV.

Thus the Algorithm LLG-eSRIF is ideal for simultaneous state estimation parameter identification.

The LLG-eSRIF consists of two parts:

The source filtering algorithm, i.e. eSRIF. The "differentiated" part: Stages II — IV.

Thus the Algorithm LLG-eSRIF is ideal for simultaneous state estimation parameter identification.

The LLG-eSRIF consists of two parts:

The source filtering algorithm, i.e. eSRIF. The "differentiated" part: Stages II — IV.

Thus the Algorithm LLG-eSRIF is ideal for simultaneous state estimation parameter identification.

The LLG-eSRIF consists of two parts:

The source filtering algorithm, i.e. eSRIF. The "differentiated" part: Stages II — IV.

Thus the Algorithm LLG-eSRIF is ideal for simultaneous state estimation parameter identification.

The LLG-eSRIF consists of two parts:

The source filtering algorithm, i.e. eSRIF. The "differentiated" part: Stages II — IV.

Thus the Algorithm LLG-eSRIF is ideal for simultaneous state estimation parameter identification.

The LLG-eSRIF consists of two parts:

The source filtering algorithm, i.e. eSRIF. The "differentiated" part: Stages II — IV.

Thus the Algorithm LLG-eSRIF is ideal for simultaneous state estimation parameter identification.

Algorithm LLG-eSRIF

Remark. Since, the matrices in LLG (6) are triangular, only the diagonal elements of R_{e,t}^{1/2} and \frac{∂(R_{e,t}^{−1/2})}{∂θ_{i}} need to be computed. Hence, the Algorithm LLG-eSRIF allows the m × m-matrix inversion of R_{e,t} to be avoided in the evaluation of LLG.

Ill-Conditioned Example Problems

Problem 1 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 0 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1. For the θ = 1, this example illustrates the initialization problems [Kaminski P.G., Bryson A.E., Schmidt S.F., 1971] that result when H_{1}Π_{0}H_{1}^{T} + R_{1} is rounded to H_{1}Π_{0}H_{1}^{T}.

Problem 1 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 0 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1. For the θ = 1, this example illustrates the initialization problems [Kaminski P.G., Bryson A.E., Schmidt S.F., 1971] that result when H_{1}Π_{0}H_{1}^{T} + R_{1} is rounded to H_{1}Π_{0}H_{1}^{T}.

Problem 1 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 0 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1. For the θ = 1, this example illustrates the initialization problems [Kaminski P.G., Bryson A.E., Schmidt S.F., 1971] that result when H_{1}Π_{0}H_{1}^{T} + R_{1} is rounded to H_{1}Π_{0}H_{1}^{T}.

Problem 1 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 0 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1. For the θ = 1, this example illustrates the initialization problems [Kaminski P.G., Bryson A.E., Schmidt S.F., 1971] that result when H_{1}Π_{0}H_{1}^{T} + R_{1} is rounded to H_{1}Π_{0}H_{1}^{T}.

Comparison (Problem 1)

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  (P_{1})′_{θ}|_{θ=1} = [ \begin{matrix}\frac{e^{2}}{1 + e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]

’diff’ IF  (P_{1}^{−1})′_{θ}|_{θ=1} = − [ \begin{matrix}\frac{1 + e^{2}}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} − [ \begin{matrix}\frac{1}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]

LLG- eSRIF  (P_{1}^{−1/2})′_{θ}|_{θ=1} = −\frac{1}{2} [ \begin{matrix}\frac{\sqrt{1 + e^{2}}}{e} & 0 \\ 0 & 1 \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\frac{1}{e} & 0 \\ 0 & 1 \end{matrix} ]

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  (P_{1})′_{θ}|_{θ=1} = [ \begin{matrix}\frac{e^{2}}{1 + e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]

’diff’ IF  (P_{1}^{−1})′_{θ}|_{θ=1} = − [ \begin{matrix}\frac{1 + e^{2}}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} − [ \begin{matrix}\frac{1}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]

LLG- eSRIF  (P_{1}^{−1/2})′_{θ}|_{θ=1} = −\frac{1}{2} [ \begin{matrix}\frac{\sqrt{1 + e^{2}}}{e} & 0 \\ 0 & 1 \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\frac{1}{e} & 0 \\ 0 & 1 \end{matrix} ]

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  (P_{1})′_{θ}|_{θ=1} = [ \begin{matrix}\frac{e^{2}}{1 + e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]

’diff’ IF  (P_{1}^{−1})′_{θ}|_{θ=1} = − [ \begin{matrix}\frac{1 + e^{2}}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} − [ \begin{matrix}\frac{1}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]

LLG- eSRIF  (P_{1}^{−1/2})′_{θ}|_{θ=1} = −\frac{1}{2} [ \begin{matrix}\frac{\sqrt{1 + e^{2}}}{e} & 0 \\ 0 & 1 \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\frac{1}{e} & 0 \\ 0 & 1 \end{matrix} ]

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  (P_{1})′_{θ}|_{θ=1} = [ \begin{matrix}\frac{e^{2}}{1 + e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]

’diff’ IF  (P_{1}^{−1})′_{θ}|_{θ=1} = − [ \begin{matrix}\frac{1 + e^{2}}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]  =^{r} − [ \begin{matrix}\frac{1}{e^{2}} & 0 \\ 0 & 1 \end{matrix} ]

LLG- eSRIF  (P_{1}^{−1/2})′_{θ}|_{θ=1} = −\frac{1}{2} [ \begin{matrix}\frac{\sqrt{1 + e^{2}}}{e} & 0 \\ 0 & 1 \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\frac{1}{e} & 0 \\ 0 & 1 \end{matrix} ]

Problem 2 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 1 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1.

Problem 2 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 1 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1.

Problem 2 Given:

\begin{matrix}P_{0} = [ \begin{matrix}θ & 0 \\ 0 & θ \end{matrix} ] , H = [ 1, 1 ] , R = e^{2}θ, F = I_{2}, Q = 0, \\ G = [ 0, 0 ]^{T} , 0< e « 1 \end{matrix}

where θ is an unknown parameter, I_{2} is an identity 2 × 2 matrix; to simulate roundoff we assume e + 1 ≠ 1 but e^{2} + 1 =^{r} 1. Calculate: (P_{1})′_{θ} at the point θ = 1.

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  \frac{1}{2 + e^{2}} [ \begin{matrix}1 + e^{2} & −1 \\ −1 & 1 + e^{2} \end{matrix} ]  =^{r} \frac{1}{2}[ \begin{matrix}1 & −1 \\ −1 & 1 \end{matrix} ]

’diff’ IF  −\frac{1}{e^{2}} [ \begin{matrix}1 + e^{2} & 1 \\ 1 & 1 + e^{2} \end{matrix} ]  =^{r} −\frac{1}{e^{2}} [ \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ]

LLG- eSRIF  −\frac{1}{2} [ \begin{matrix}\sqrt{\frac{2 + e^{2}}{1 + e^{2}}} & \frac{1}{e\sqrt{1 + e^{2}}} \\ 0 & \frac{\sqrt{1 + e^{2}}}{e} \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\sqrt{2} & \frac{1}{e} \\ 0 & \frac{1}{e} \end{matrix} ]

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  \frac{1}{2 + e^{2}} [ \begin{matrix}1 + e^{2} & −1 \\ −1 & 1 + e^{2} \end{matrix} ]  =^{r} \frac{1}{2}[ \begin{matrix}1 & −1 \\ −1 & 1 \end{matrix} ]

’diff’ IF  −\frac{1}{e^{2}} [ \begin{matrix}1 + e^{2} & 1 \\ 1 & 1 + e^{2} \end{matrix} ]  =^{r} −\frac{1}{e^{2}} [ \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ]

LLG- eSRIF  −\frac{1}{2} [ \begin{matrix}\sqrt{\frac{2 + e^{2}}{1 + e^{2}}} & \frac{1}{e\sqrt{1 + e^{2}}} \\ 0 & \frac{\sqrt{1 + e^{2}}}{e} \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\sqrt{2} & \frac{1}{e} \\ 0 & \frac{1}{e} \end{matrix} ]

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  \frac{1}{2 + e^{2}} [ \begin{matrix}1 + e^{2} & −1 \\ −1 & 1 + e^{2} \end{matrix} ]  =^{r} \frac{1}{2}[ \begin{matrix}1 & −1 \\ −1 & 1 \end{matrix} ]

’diff’ IF  −\frac{1}{e^{2}} [ \begin{matrix}1 + e^{2} & 1 \\ 1 & 1 + e^{2} \end{matrix} ]  =^{r} −\frac{1}{e^{2}} [ \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ]

LLG- eSRIF  −\frac{1}{2} [ \begin{matrix}\sqrt{\frac{2 + e^{2}}{1 + e^{2}}} & \frac{1}{e\sqrt{1 + e^{2}}} \\ 0 & \frac{\sqrt{1 + e^{2}}}{e} \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\sqrt{2} & \frac{1}{e} \\ 0 & \frac{1}{e} \end{matrix} ]

Filter  Exact Answer  Rounded Answer
{’diff’ \atop KF}  \frac{1}{2 + e^{2}} [ \begin{matrix}1 + e^{2} & −1 \\ −1 & 1 + e^{2} \end{matrix} ]  =^{r} \frac{1}{2}[ \begin{matrix}1 & −1 \\ −1 & 1 \end{matrix} ]

’diff’ IF  −\frac{1}{e^{2}} [ \begin{matrix}1 + e^{2} & 1 \\ 1 & 1 + e^{2} \end{matrix} ]  =^{r} −\frac{1}{e^{2}} [ \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ]

LLG- eSRIF  −\frac{1}{2} [ \begin{matrix}\sqrt{\frac{2 + e^{2}}{1 + e^{2}}} & \frac{1}{e\sqrt{1 + e^{2}}} \\ 0 & \frac{\sqrt{1 + e^{2}}}{e} \end{matrix} ] =^{r} −\frac{1}{2} [ \begin{matrix}\sqrt{2} & \frac{1}{e} \\ 0 & \frac{1}{e} \end{matrix} ]

Numerical Results

Example. Let the test system (1), (2) be defined as follows:

x_{t+1} = [ \begin{matrix}1 & Δt \\ 0 & e^{−Δt/τ} \end{matrix} ] x_{t} + [ \begin{matrix}0 \\ 1 \end{matrix} ] w_{t},

z_{t} = [ 1 0 ] x_{t} + v_{t}

where τ is a unknown parameter which needs to be estimated. For the test problem, τ^{∗}=10 was chosen as the true value of parameter τ.

Numerical Results

Example. Let the test system (1), (2) be defined as follows:

x_{t+1} = [ \begin{matrix}1 & Δt \\ 0 & e^{−Δt/τ} \end{matrix} ] x_{t} + [ \begin{matrix}0 \\ 1 \end{matrix} ] w_{t},

z_{t} = [ 1 0 ] x_{t} + v_{t}

where τ is a unknown parameter which needs to be estimated. For the test problem, τ^{∗}=10 was chosen as the true value of parameter τ.

Conclusion

In this paper, the new algorithm for evaluating the Log Likelihood Gradient (score) of linear discrete-time dynamic systems has been developed. The necessary theory has been given and substantiated by the computational experiments. Two ill-conditioned example problems have been constructed to show the superior perfomance of the Algorithm LLG-eSRIF over the conventional approach.