Data Forecasting - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

Download Data Forecasting - Stochastic Hydrology - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!

Data Generation: Consider AR(1) model, X t = φ 1

X

t-

• e t e.g., φ 1 = 0.5 : AR(1) model is X t

= 0.5X

t-

• e t 3

ARIMA Models

zero mean; uncorrelated

X

1 = 3.0 (assumed) X 2

X

3

And so on… 4

ARIMA Models

Say X 1

X

2

X

3

X

4

= 3.767 and so on... 6

ARIMA Models

Data Forecasting: Consider AR(1) model, X t = φ 1

X

t-

• e t Expected value is considered. 7

ARIMA Models

[ ] 1 [ 1 ] [ ]

1 1

t t t t t

E X E X E e

X X

− −

Expected value of e t is zero

Say X 1

X

2

Error e 2

9

ARIMA Models

2

X = × + ×

Initial error assumed to be zero 3

X = × + ×

Actual value to be used

X

3

Error e 3

and so on. 10

ARIMA Models

4 (^ )

X = × + × −

Case study – 1

12 Rainfall data for Bangalore city is considered.

• Time series plot, auto correlation function, partial auto correlation function and power spectrum area plotted for daily, monthly and yearly data.

Case study – 1 (Contd.)

13 Daily data – Time series plot Time Rainfall in mm

Case study – 1 (Contd.)

15 Daily data – Partial auto correlation function

Case study – 1 (Contd.)

16 Daily data – Power spectrum Wk I(k)

Case study – 1 (Contd.)

18 Monthly data – Correlogram

Case study – 1 (Contd.)

19 Monthly data – Partial auto correlation function

0 5 10 15 20 25 30 35 500 600 700 800 900 1000 1100 1200 1300 1400

Case study – 1 (Contd.)

21 Yearly data – Time series plot Time Rainfall in mm

Case study – 1 (Contd.)

22 Yearly data – Correlogram