Modelling Elvis Impersonators

1
Modelling Elvis Impersonators Fresh evidence
that pop stars are more popular dead than alive.
The University of Missouris Jean Gaddy Wilson
told a recent press conference in Dallas that, in
1977 when Elvis Died there where 48 professional
Elvis impersonators. Today there are 7328. If
that growth is projected, by the year 2012 one
person in four on the face of the globe will be
an Elvis Impersonator Royal Statistical Society
News, 1996 Assume Elviss first hit was in 1955
and that the first impersonator started in that
year. Assume that there is an exponential growth
in elvis impersonators i.e. that the model is of
the form EI exp(b1year b0)
2
yLog(EI)b1year b0 We can form a matrix of
independent variables We can also form a
vector of dependent output variables The Least
squares fit to this is The prediction for 2012
is then 168110. If Jean Gaddy Wilson is right,
either there will be a dramatic drop in world
population or growth of Elvis Impersonators is
more dramatic than an exponential model will
allow.
3
(No Transcript)
4
Time series models (ARMAX) General form of the
discrete time model used for system
identification is the ARMAX model.
Autoregressive, Moving Average, Exogeneous
inputs. Autoregressive refers to the fact that
the output is a linear combination of previous
values of the output. Moving Average refers to
the noise model. Exogeneous implies that there is
an input to the system along with knowledge of
its previous values. Thus the model is
5
The picture is
Use a delay block
6





-


-
7
Variants are Autoregressive Moving Average (ARMA)
- No access to knowledge of the
input Autoregressive exogeneous (ARX) - Assume
that only disturbance is white noise Finite
Impulse response (FIR) - Output is a linear
combination of only past input values. The output
will drop to zero in finite time if the input
becomes zero. Note on z transform We can use the
z transform on the ARMAX model and its variants
to specify the z domain transfer function as
8
L.S. parameter calculation of ARMAX models We
can put the general ARMAX model into a vector
form for instant i as For all data values
, taken over a range of data i1, n, form a
vector y , and a matrix F values thus all the
data can be collected together to form the
following
9
As youve guessed it, at time n the least squares
solution to this is
But now add a new input value u and a new output
value y and we need to recalculate the entire
thing.
Recursive identification methods Would like a way
of efficiently recalculating the model each time
we have new data. Ideal form would be
Thus if the model is correct at time n-1 and the
new data at time n is indicative of the model
then the correction factor would be zero.
10

• Advantages of recursive model estimation
• Gives an estimate of the model (all be it poor)
  from the first time step
• Can be computationally more efficient and less
  memory intensive, especially if we can avoid
  doing large matrix inverse calculations
• Can be made to adapt to a changing system, eg
  online system identification allows telephone
  systems to do echo cancellation on long distance
  lines.
• Can be used for fault detection, model estimates
  start to differ radically from a norm
• Forms the core of adaptive control strategies and
  adaptive signal processing
• Ideal for real-time implementations

11
Example Estimation of a constant (scalar)
model
y
i
i
This is the mean level of the signal, derived by
LS method.
12
If we introduce a subscript n to represent the
fact that n data points are used in deriving the
mean, such that
The above equation is the least squares in
recursive form.
13
General form of Recursive algorithms where
is a vector of model parameters
estimate is the difference
between the measured output and the estimated
output at time n is the scaling -
sometimes known as the Kalman Gain