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I am trying to implement the log-likelihood expression Eq(7) from the paper, [Parameter Estimation for Linear Dynamical Systems (1996)][1]Parameter Estimation for Linear Dynamical Systems (1996).

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ and $k$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: According to theory, the log-likelihood function is maximized at the true parameter values. I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you. [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.5997

I am trying to implement the log-likelihood expression Eq(7) from the paper, [Parameter Estimation for Linear Dynamical Systems (1996)][1].

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ and $k$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: According to theory, the log-likelihood function is maximized at the true parameter values. I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you. [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.5997

I am trying to implement the log-likelihood expression Eq(7) from the paper, Parameter Estimation for Linear Dynamical Systems (1996).

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ and $k$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: According to theory, the log-likelihood function is maximized at the true parameter values. I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you.

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edited Mar 15, 2015 at 4:48

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I am trying to implement the log-likelihood expression Eq(7) from the paper, [Parameter Estimation for Linear Dynamical Systems (1996)][1].

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ and $k$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: According to theory, the log-likelihood function is maximized at the true parameter values. I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you. [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.5997

I am trying to implement the log-likelihood expression Eq(7) from the paper, [Parameter Estimation for Linear Dynamical Systems (1996)][1].

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you. [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.5997

I am trying to implement the log-likelihood expression Eq(7) from the paper, [Parameter Estimation for Linear Dynamical Systems (1996)][1].

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ and $k$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: According to theory, the log-likelihood function is maximized at the true parameter values. I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you. [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.5997

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asked Mar 14, 2015 at 3:52

SKM

787
1
7
26

Conceptual question on log-likelihood value

I am trying to implement the log-likelihood expression Eq(7) from the paper, [Parameter Estimation for Linear Dynamical Systems (1996)][1].

Re-writing,

For the model,

$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$

$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$

$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$

The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$

where $\pi_1, V$ is the mean and variance of the initial condition of $h$.

Q1: In the paper, I could not see what $p$ is. Can somebody show how this likelihood expression is coming from the joint pdf?

Q2: I implemented this expression using MATLAB and getting a 2by2 matrix

-13.4165 Inf; 

Inf -13.4165.

Will the log-likelihood be negative and the off-diagonals all infinity? What Is the implication of this?

Shall appreciate an intuitive answer for a beginner level. Thank you. [1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.5997

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