Download Locating a Shift in the Mean of a Time Series

Locating a Shift in the Mean of a
Time Series
Melvin J. Hinich
Applied Research Laboratories
University of Texas at Austin
[email protected]
www.la.utexas.edu/~hinich
Localizing a Single Change in the Mean
A statistical uncertainty principle for the
localization of a single change in the
mean
of a bandlimited stationary random process
GOAL
The smallest mean squared error for any
estimate of the time of change
Discrete-Time Sampling
It is common in time series analysis to begin
with a discrete-time sample of the time series
Apply a linear bandlimited filter to the signal
Then decimate the filtered output
to obtain the discrete-time sample
Linear Bandlimited Filter
The filter is linear and causal
The filter impulse response function is
h t 
.
y  t   0 h  s  x  t  s  ds

The filter smoothes the input since the
filter removed frequency components of
the input for f  f o
Mean Shift
If the mean of the signal has an
abrupt shift from  to   
at an unknown time  o
The shift in the mean of the output is
 H t  o 
H  t   o   0t  h  s  ds
o
Integrated Impulse Response
Cumulative Impulse Response
1
0.9
0.8
0.7
Amplitude
0.6
0.5
0.4
0.3
0.2
0.1
0
1
201
401
601
Time
801
Ideal Bandpass Filter
H ( f )  1 for  f o  f  f o
H  f   0 otherwise
Impulse response of the ideal filter - sinc function
sin  1t 
t
1

2 fo
Meanshift
The shift in the mean of x(tn) is

1

n  1 o sin

F n   o  
 v  dv
v
We will now derive the least squares
estimate of the location of the shift for
x  tn    F  n    o   e  tn 
1
Maximum Likelihood Estimate
ˆ
- the least squares estimate of
 x  t1  ,
o
, x  t N 


x  tn    F n   1 o  e  tn 
e  tn 
ˆ
i.i.d. gaussian variates with variance  e2
is the value that maximizes the statistic
N


S      F n   1 x  tn 
n 1
Least Squares Estimate
The least squares estimate of o is the value that
maximizes
S      F  n   1  x  tn 
N
n 1
The standard deviation of the estimate is
approximately
1
ˆ
   

E
2 fo
Asymptotic Standard Deviation
E - the total energy of the white noise
Area under its bandlimited white noise spectrum
1
ˆ
   

E
2 fo
Hinich Test for a Changing Slope Parameter
y  tn   a  t n   b  t n  x  t n   e  t n 
a  tn   a  1tn
b  tn   b  2tn
y  tn   a  1tn  bx  tn   2tn x tn   e tn 
Regress y  tn  on tn , x  tn  , tn x  tn 
Test the significance of ˆ1 and ˆ2