An estimation technique for Time Indexed Gaussian Mixture
Models
Abstract
Assume that there is data x
1
, x
2
, .., x
n
coming from densities N(µ
1
, σ
1
) with probability
α and N(µ
2
, σ
2
) with probability 1 − α and our goal is to estimate the parameters θ =
[µ
1
, σ
1
, µ
2
, σ
2
, α]. The estimation of this parametric mixture model is a well known problem
in statistics and can be solved using the well known Expecation-Maximization procedure for
mixture models.
We now further generalize this problem to the case in which the observations x
1
, x
2
, .., x
n
are
now coming from time-indexed densities N(µ(t)
1
, σ(t)
1
) with probability α and N(µ(t)
2
, σ(t)
2
)
with probability 1−α. Again, we want to estimate the parameters θ = [µ(t)
1
, σ(t)
1
, µ(t)
2
, σ(t)
2
, α].
In this paper, we develop a Time Indexed Expectation Maximization (TSEM) procedure that
can be used to estimate a general class of time indexed mixture models.
Finally, we exend the generalized problem to the the case in which the data x
1
, x
2
, ....x
n
is
not directly observable, but rather a function of noisy observations of the data y
1
, y
2
, ...y
n
is
observed. For this final solution, we develop a novel Particle Smoother Expectation Maximation
(PSEM) procedure that can be used to estimate this case; in particular, we highlight models
that incorporate unknown mixture densities in state equations and nonlinear observations. We
use this procedure to solve the time-indexed Gaussian mixture problem and successfully apply
this algorithm to the estimation of electricity price spikes.
1 Introduction
Electricity price movements is one example of a highly non-linear time series that typically features
both jumps and spikes. Since the widespread deregulations of the electricity markets in the early
90s, the market activity has increased exponentially. This trend has made the evaluations of supply
contracts a topic of serious concern. Electricity markets are known to have much more erratic
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