CASE STUDIES -
ARMA MODELS
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Sakleshpur Annual Rainfall Data - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics
The main points which I found very interesting are: Sakleshpur Annual Rainfall Data, Correlogram, Power Spectrum, Partial Auto Correlation, Parameters for Selected Model, Significance of Residual Mean, Significance of Periodicities, Whittle’s White Noise Test
Typology: Study notes
2012/2013
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CASE STUDIES -
ARMA MODELS
3
Case study – 5
4
Sakleshpur Annual Rainfall Data (1901-2002)
Time (years)
Rainfall in mm
Case study – 5 (Contd.)
6
PAC function Power spectrum
x
5
2 2
k k
N
I k α β
k
k
N
1
2
cos 2
n
k t k
t
x f t
N
α π
=
=
1
2
sin 2
n
k t k
t
x f t
N
β π
=
=
∑
P
p
- φ
p
= ρ
p
Auto Correlations
Partial Auto Correlation
Auto Correlation
function
Case study – 5 (Contd.)
7
Case study – 5 (Contd.)
9
- Significance of residual mean
Model η(e) t
(N )
ARMA(5,0) 0.000005 1.
Significance of periodicities:
Case study – 5 (Contd.)
10
Periodicity η F
(2, N-2 )
st
nd
rd
th
th
th
Case study – 5 (Contd.)
12
- AR(1) 9. Model Likelihood - AR(2) 8. - AR(3) 8. - AR(4) 9. - AR(5) 9.
- AR(6) 9.
- ARMA(1,1) 8.
- ARMA(1,2) 8.
- ARMA(2,1) 7.
- ARMA(2,2) 5.
- ARMA(3,1) 6.
- ARMA(3,2) 6. - AR(1) 1. Model MSE - AR(2) 1. - AR(3) 1. - AR(4) 1. - AR(5) 1. - AR(6) 1.
- ARMA(1,1) 1.
- ARMA(1,2) 1.
- ARMA(2,1) 1.
- ARMA(2,2) 1.
- ARMA(3,1) 1.
- ARMA(3,2) 27.
Case study – 5 (Contd.)
13
- ARMA(1, 2) is selected with least MSE value for
one step forecasting
- The parameters for the selected model are as
follows
1
1
2
Constant = -0.
Significance of periodicities:
Case study – 5 (Contd.)
15
Periodicity η F
(2, N-2 )
st
nd
rd
th
th
th
Case study – 5 (Contd.)
16
- Whittle s white noise test:
Model η F
(n1, N–n1)
ARMA(1, 2) 0.3605 1.
Summary of Case studies
Case study-1:Time series plot
18
Daily rainfall data of Bangalore city Monthly rainfall data of Bangalore city
0 5 10 15 20 25 30 35
500
600
700
800
900
1000
1100
1200
1300
1400
Yearly rainfall data of Bangalore city
Summary of Case studies
Case study-1: Correlogram
19
0 100 200 300 400 500 600
-0.
-0.
0
Lag
Sample Autocorrelation
Sample Autocorrelation Function
Daily rainfall data of Bangalore city Monthly rainfall data of Bangalore city
Yearly rainfall data of Bangalore city
Time (months)
Flow in
cumec
Summary of Case studies
Time series plot
21
- Monthly stream flow data of a river
Time
Flow in
cumec
- Monthly stream flow data for Cauvery
- Sakleshpur Annual Rainfall Data
Summary of Case studies
Correlogram
22
- Monthly stream flow data for Cauvery 4. Monthly stream flow data of a river
- Sakleshpur Annual Rainfall Data