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CPS2164 Jonathan Hosking et al.
component, and the stochastic day-of-week effects. A full description of the
model and its state-space form are given in Hosking et al. (2013). The model
has observation vector length m, state vector length 9B+4, and 4 distinct
model parameters, one for each variance component.
The model for hourly demand (m=24) has observation vector length 24
and state vector length 112. Model parameters were estimated using the L-
BFGS-B method as implemented in R function optim. Parameter estimation on
11 months of data took 274 sec using numerical derivatives. With the new
procedure using analytical derivatives, estimation time is reduced to 92 sec,
faster by a factor of 3.0.
The model can be extended by allowing each diagonal element in the
observation and state noise variance matrices to vary independently. This
extended model has 60 (=m+3B) model parameters in the variance matrices.
Starting with parameter values corresponding to the fitted 4-parameter model,
estimation with numerical derivatives took 3246 sec; with analytical derivatives,
81 sec. Using analytical derivatives here delivers the same solution as numerical
derivatives, but 40 times faster.
The same model can be used for demand measured over 15-minute
intervals, now with m=96. The state vector is the same as for the model for
hourly data: it has 112 elements, and their physical interpretations are the same
as for the states in the hourly model. Like the hourly model, in its basic form it
has 4 distinct model parameters, one for each variance component, in the
variance matrices. Using analytical derivatives speeds up the estimation of
these parameters by a factor of approximately 2.9. When the 15-minute model
is extended by allowing each diagonal element in the observation and state
noise variance matrices to vary independently, it has 132 (=m+3B) model
parameters in the variance matrices. Using analytical derivatives speeds up the
estimation of these parameters by a factor of 84. The computational results
are summmarized in Table 1.

7. Conclusions
We have derived an explicit expression for the derivative of the log-
likelihood function for a state space time series model. It enables numerical
procedures for parameter estimation to compute

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