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Stationary Markov Model - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

Avinashilingam University Mathematical Statistics

The main points which I found very interesting are: Stationary Markov Model, Thomas Fiering Model, Data Generation, Serially Correlated Data, Standard Normal Deviate, Variance and Lag-One, Serial Correlations, Stream Flow Generation, Standard Deviation, Logarithms

Typology: Study notes

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Uploaded on 04/20/2013

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Data Generation – Serially Correlated

Data

Standard normal deviate%

First order stationary Markov model

Thomas Fiering model (Stationary)%

( )

11 11

jx jxjx

XXt

µρµσ ρ

=+ −+−

• Stationary w.r.t mean, variance and lag-one

correlation

• Known sample estimates of µx, σx, ρ1

• Assume X1 (= µx)

• Generate values X2, X3, X4, X5 ……

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Data Generation – Serially Correlated

Data

Standard normal deviate

First order stationary Markov model Or Thomas Fiering model (Stationary) ( )

j x j x j x

X μ ρ X μ t σ ρ

= + − + −

Stationary w.r.t mean, variance and lag-one correlation
Known sample estimates of μ x , σ x , ρ 1
Assume X 1 (= μ x )
Generate values X 2 , X 3, X 4, X 5 ……

First order Markov model with non-stationarity:

First order stationary Markov model assumes that the process is stationary in mean, variance and lag- one auto correlation.
The model is generalized to account for non- stationarity (mainly due to seasonality/periodicity) in hydrologic data.
A main application of this generalised model is in generating the monthly stream flows with pronounced seasonality.
Periodicity may affect not only the mean, but all the moments of data including the serial correlations.

Data Generation – Serially Correlated

Data

The monthly stream flow (in cumec) for a river is available for 29 years (12 years data is given here) Example- 6

SL.

YEAR JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY

NO.

Time series of monthly stream flow for 29 years 7 0 100 200 300 400 500 600 700 800 900 0 50 100 150 200 250 300 350 400

Assume X 1 = μ 1 = 117.49; σ 1 = 52.24, ρ 1 = 0. μ 2 = 474.5, σ 2 = 150.18, X 1, = = = 521. 9 ( )

X t 1 σ μ ρ μ σ ρ σ

− + − ( )

474.5 0.348 117.49 117.

0.335*150.18 1 0.

X 1, =521.67, μ 2 = 474.5; σ 2 = 150.18, ρ 2 = 0. μ 3 = 421.39, σ 3 = 126.53, ρ 3 = 0. X 1, = = 474. 10 ( )

421.39 0.154 521.67 474.

0.377 *126.53 1 0.

0 50 100 150 200 250 300 350 400 450 500 JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY Mean

Observed

Generated

-‐0. -‐0. -‐0. 0

JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY Lag-1 correlation

Observed

Generated

Where Y

i,j+

= ln (X

i, j+

refer to the mean, standard deviation

and lag one correlation of logarithms of original data

Data Generation – Serially Correlated

Data

1 1 1

j j j j j j j

y

i j y y ij y i j y y

y

Y Y t

y (^) j y (^) j yj

 - Lag- S.No. Month Mean Stdev.

1 JUN 117.49 52.24 0. correlation
2 JUL 474.50 150.18 0.
3 AUG 421.39 126.53 0.
4 SEP 145.94 77.65 0.
5 OCT 66.61 30.67 0.
6 NOV 22.99 13.26 0.
7 DEC 10.30 9.82 0.
8 JAN 5.55 9.16 -0.
9 FEB 1.91 0.74 0.
10 MAR 1.09 0.54 0.
11 APR 0.76 0.51 0.
12 MAY 0.80 0.60 -0. - Lag- S.No. Month Mean Stdev.
- 1 JUN 125.69 59.30 0. correlation
- 2 JUL 469.36 142.10 -0.
- 3 AUG 365.98 130.60 0.
- 4 SEP 140.40 78.60 0.
- 5 OCT 65.28 33.89 0.
- 6 NOV 22.33 12.40 0.
- 7 DEC 10.68 8.30 0.
- 8 JAN 7.69 7.13 -0.
- 9 FEB 1.95 0.59 0.
10 MAR 0.95 0.52 0.
11 APR 0.60 0.47 0.
12 MAY 0.68 0.50 0. - Lag- S.No. Month Mean Stdev.
- 1 JUN 4.64 0.54 0. correlation
- 2 JUL 6.11 0.33 0.
- 3 AUG 6.00 0.31 0.
- 4 SEP 4.86 0.49 0.
- 5 OCT 4.10 0.44 -0.
- 6 NOV 2.71 2.02 0.
- 7 DEC 1.83 1.91 0.
- 8 JAN 1.13 1.77 0.
- 9 FEB 0.12 2.15 0.
10 MAR -0.66 2.42 0.
11 APR -1.05 2.32 0.
12 MAY -1.08 2.35 -0. - Lag- S.No. Month Mean Stdev.
- 1 JUN 4.74 0.64 0. correlation
- 2 JUL 6.09 0.31 -0.
- 3 AUG 5.86 0.33 0.
- 4 SEP 4.82 0.50 0.
- 5 OCT 4.08 0.48 0.
- 6 NOV 2.64 1.45 0.
- 7 DEC 1.76 1.40 0.
- 8 JAN 1.25 1.31 0.
- 9 FEB 0.33 1.93 0.
10 MAR -1.10 2.41 0.
11 APR -1.56 2.42 0.
12 MAY -1.60 2.42 0.

3 JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY Standard Deviation

Observed

Generated

-‐0. -‐0. 0

JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY Lag-1 Correlation

Stationary Markov Model - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

Related documents

Partial preview of the text

Download Stationary Markov Model - Stochastic Hydrology - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!

Data Generation – Serially Correlated

Data

Standard normal deviate

j x j x j x

Data Generation – Serially Correlated

Data

SL.

YEAR JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY

NO.

Observed

Generated

Observed

Generated

Where Y

= ln (X

refer to the mean, standard deviation

and lag one correlation of logarithms of original data

Data Generation – Serially Correlated

Data

y

i j y y ij y i j y y

y

Y Y t

Observed

Generated

Observed

Generated