unr * stat 758 * fall 2006. white noise periodogram of white noise
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UNR * STAT 758 * Fall 2006
UNR * STAT 758 * Fall 2006
0 200 400 600 800 1000
-3-1
01
23
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
0.0 0.1 0.2 0.3 0.4 0.5
02
46
810
Frequency, w
Spe
ctru
m,
f(w
)White noise
Periodogram of white noise
0 200 400 600 800 1000
-3-1
01
23
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
White noise
Periodogram of white noise
0.0 0.1 0.2 0.3 0.4 0.5
02
46
810
Frequency, w
Spe
ctru
m,
f(w
)
Smoothed periodogram (spans=3)
0 200 400 600 800 1000
-3-1
01
23
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
White noise
0.0 0.1 0.2 0.3 0.4 0.5
02
46
810
Frequency, w
Spe
ctru
m,
f(w
)
Smoothed periodogram (spans=11)
0.0 0.1 0.2 0.3 0.4 0.5
02
46
810
Frequency, w
Spe
ctru
m,
f(w
)
0 200 400 600 800 1000
-3-1
01
23
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
White noise
Smoothed periodogram (spans=101)
0.0 0.1 0.2 0.3 0.4 0.5
0
e+00
4
e+05
8
e+05
Frequency, w
Spe
ctru
m,
f(w
)
Time, t
Tim
e se
ries,
X(t
)
0 2000 4000 6000 8000 10000
-100
-60
-20
0
UNR * STAT 758 * Fall 2006
Brownian walk
Smoothed periodogram (spans=3)
1 e-04 5 e-04 5 e-03 5 e-02 5 e-01
1
e-02
1
e+02
1
e+06
Frequency, w
Spe
ctru
m,
f(w
)
Time, t
Tim
e se
ries,
X(t
)
0 2000 4000 6000 8000 10000
-100
-60
-20
0
UNR * STAT 758 * Fall 2006
Brownian walk
Smoothed periodogram (spans=3)
0.0 0.1 0.2 0.3 0.4 0.5
01
23
4
Frequency, w
Spe
ctru
m,
f(w
)
0 50 100 150 200
-4-2
02
4
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
MA(1), positive coefficient
Smoothed periodogram (spans=501 = 0.05 length(X))
0.0 0.1 0.2 0.3 0.4 0.5
01
23
4
Frequency, w
Spe
ctru
m,
f(w
)
0 50 100 150 200
-4-2
02
4
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
MA(1), negative coefficient
Smoothed periodogram (spans=501 = 0.05 length(X))
0.0 0.1 0.2 0.3 0.4 0.5
020
6010
0
Frequency, w
Spe
ctru
m,
f(w
)
0 50 100 150 200
-10
-50
5
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
MA(10) = smoothing
Smoothed periodogram (spans=501 = 0.05 length(X))
0.0 0.1 0.2 0.3 0.4 0.5
020
6010
0
Frequency, w
Spe
ctru
m,
f(w
)
0 50 100 150 200
-50
5
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
MA(10) with coeffs (-1,1, -1,1, -1,1, -1,1, -1,1)
Smoothed periodogram (spans=501 = 0.05 length(X))
0 10 20 30 40
0.0
0.4
0.8
Lag
AC
F
Series x
0 50 100 150 200
-20
010
20
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
ACF of the test signal
Test signal
0.0 0.1 0.2 0.3 0.4 0.5
050
010
0015
00
Frequency, w
Spe
ctru
m,
f(w
)
0 50 100 150 200
-20
010
20
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
Spectrum of the test signal
Test signal
0 50 100 150 200
-20
010
20
Time, t
Tim
e se
ries,
X(t
)
UNR * STAT 758 * Fall 2006
Test signal
sin(5 )t tX t Z
Normal with variance 100
UNR * STAT 758 * Fall 2006
0 50 100 150
-12
-8-4
0
t
x
UNR * STAT 758 * Fall 2006
Brownian walk
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Frequency, w
Filt
er,
h(w
)
Low-pass filter
0 50 100 150
-12
-8-4
0
t
x
0 50 100 150
-12
-8-4
0
t
x
UNR * STAT 758 * Fall 2006
Brownian walk
Low-pass filtering
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Frequency, w
Filt
er,
h(w
)
0 50 100 150
-12
-8-4
0
t
x
UNR * STAT 758 * Fall 2006
Brownian walk
Low-pass filter
0 50 100 150
-12
-8-4
0
t
x
0 50 100 150
-12
-8-4
0
t
x
UNR * STAT 758 * Fall 2006
Brownian walk
Low-pass filtering