Locating a Shift in the Mean of a Time Series
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Transcript 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 t n
1
Maximum Likelihood Estimate
ˆ
- the least squares estimate of
x t1 ,
o
, x t N
x tn F n 1o 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
n1
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
ˆ
1
E
2 fo
Hinich Test for a Changing Slope Parameter
y tn a tn b tn x tn e tn
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