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Where F T is the transpose of F .
In order to fully specify the model we need to provide the first state θ 0 , as well as the variancecovariance
matrices for the system noise and measurement noise. These terms are distributed
as:
θ 0 ∼ N (m 0 , C 0 ) (13.3)
v t ∼ N (0, V t ) (13.4)
w t ∼ N (0, W t ) (13.5)
Clearly that is a lot of notation to specify the model. For completeness I will summarise all
of the terms here to help you get to grips with it:
• θ t - The state of the model at time t
• y t - The observation of the model at time t
• G t - The state-transition matrix between current and prior states at time t and t − 1
respectively
• F t - The observation matrix between the current observation and current state at time
t
• w t - The system noise drawn from a multivariate normal distribution
• v t - The measurement noise drawn from a multivariate normal distribution
• m 0 - The mean value of the multivariate normal distribution of the initial state, θ 0
• C 0 - The variance-covariance matrix of the multivariate normal distribution of the initial
state, θ 0
• W t - The variance-covariance matrix for the multivariate normal distribution from which
the system noise is drawn
• V t - The variance-covariance matrix for the multivariate normal distribution from from
which the measurement noise is drawn
Now that we have specified the linear state-space model we need an algorithm to actually
solve it. This is where the Kalman Filter comes in. We can use Bayes’ Rule and conjugate priors,
as discussed in the previous part of the book, to help us derive the algorithm.
13.2 The Kalman Filter
This section follows very closely the notation and analysis carried out in Pole et al[81]. I decided
it was not particularly helpful to invent my own notation for the Kalman Filter as I want you to
be able to relate it to other research papers or texts.