  Understanding the energy consumption patterns of different types of consumers
is essential in any planning of energy distribution. However, obtaining
consumption information for single individuals is often either not possible or
too expensive. Therefore, we consider data from aggregations of energy use,
that is, from sums of individuals' energy use, where each individual falls into
one of C consumer classes. Unfortunately, the exact number of individuals of
each class may be unknown: consumers do not always report the appropriate
class, due to various factors including differential energy rates for different
consumer classes. We develop a methodology to estimate the expected energy use
of each class as a function of time and the true number of consumers in each
class. We also provide some measure of uncertainty of the resulting estimates.
To accomplish this, we assume that the expected consumption is a function of
time that can be well approximated by a linear combination of B-splines.
Individual consumer perturbations from this baseline are modeled as B-splines
with random coefficients. We treat the reported numbers of consumers in each
category as random variables with distribution depending on the true number of
consumers in each class and on the probabilities of a consumer in one class
reporting as another class. We obtain maximum likelihood estimates of all
parameters via a maximization algorithm. We introduce a special numerical trick
for calculating the maximum likelihood estimates of the true number of
consumers in each class. We apply our method to a data set and study our method
via simulation.
