shahid lecture-12- mkag1273

MAL1303: STATISTICAL HYDROLOGYStochastic Methods in Hydrology

Dr. Shamsuddin ShahidDepartment of Hydraulics and Hydrology

Faculty of Civil Engineering, Universiti Teknologi Malaysia

Room No.: M46-332; Phone: 07-5531624; Mobile: 0182051586 Email: [email protected]

11/23/2015 Shamsuddin Shahid, FKA, UTM

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Markov Transition Matrix




For four class, there will be four cumulative distribution functions.

Cumulative distribution functions for each class is calculated as,

Fj (x) = P [next day rainfall < x; when rainfall today belongs to class Cj].

For Example,

FR5(x) = P [next day rainfall < x; when rainfall today belongs to class R5].

Cumulative Distribution Functions




Fj (x) = P [next day rainfall < x;

when rainfall today belongs to class Cj].

For Example:FR5(x) = P [next day rainfall < x; when rainfall today belongs to class R5].

P [next day rainfall < 5] = 2P [next day rainfall < 4] = 2P [next day rainfall < 3] = 2P [next day rainfall < 2] = 1P [next day rainfall < 1] = 1

Rainfall10

516

234320

2052304310





FR5(x) = P [next day rainfall < x; when rainfall today belongs to class R5].

P [next day rainfall < 5] = 2P [next day rainfall < 4] = 2P [next day rainfall < 3] = 2P [next day rainfall < 2] = 1P [next day rainfall < 1] = 1


Find the distribution and distribution parameters.

Consider, we found distribution is exponential,FR5(x) = exp (-)

Where,

= 0.105




Calculate Daily Monsoon Rainfall

First, we need to define the initial condition.

Consider, Initial condition

R5 --- R10 --- R20 --- R>20

(1/4) (1/4) (1/4) (1/4)





(1/4) (1/4) (1/4) (1/4)

[0.25 0.25 0.25 0.25] X

0.39 0.21 0.27 0.14




R5 R10 R20 R>20

0.39 0.21 0.27 0.14

FR5(x) = exp (-x)

Where,

= 0.105

Cumulative Distribution,

1 - exp (-x)

Rainfall in Day1 (x) =

0.39 = 1 -0.105exp(-0.105x)






0.39 0.21 0.27 0.14 X

0.41 0.24 0.24 0.11




General equation is,

u(n) = u Pn

Or

u(n) = u(n-1) P


0.39 0.21 0.27 0.14 X




Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determinedboth by the process's predictable actions and by a random element.

Stochastic hydrology is mainly concerned with the assessment ofuncertainty in model predictions

Stochastic Process




Application of Stochastic Process in Hydrology

Stochastic hydrology is an essential base of water resourcessystems analysis, due to the inherent randomness of the input,and consequently of the results.

Stochastic process is applied for forecasting of hydrologicalphenomena such as, flood, droughts, etc.

Stochastic process is applied for forecasting rainfall, riverdischarge, etc.

Stochastic hydrology is very important in decision-makingprocess regarding the planning and management of watersystems.




A stationary time series is one whose statistical properties such asmean, variance, autocorrelation, etc. are all constant over time.

Most statistical forecasting methods are based on the assumptionthat the time series can be rendered approximately stationary throughthe use of mathematical transformations.

A stationarized series is relatively easy to predict: you simply predictthat its statistical properties will be the same in the future as theyhave been in the past.

Stationary Time Series




Linear Stochastic Models

1. Moving Average (MA)2. Auto Regression (AR)3. Auto Regressive Moving Average (ARMA)4. Auto Regressive Integrated Moving Average (ARIMA)

Stochastic Models




Moving Average

The concept underlying moving average is that the k most recenttime periods is a good predictor of the current and next periodvalues.

The process is called moving averages because each average iscalculated by dropping the oldest observation and including thenext observation.




• The moving average removes some of the non-randomness in the data.

• Therefore, the moving average merely smooth the fluctuations in thedata.

• The moving average technique is a good forecasting approach to use ifthe data is stationary.

kY....YYYYF kttttt

t1321

1

Where, Ft+1 is the forecast for period t+1, andYt is the actual value of period t

Moving Average




Moving Average

48.059.069.368.067.359.051.041.030.731.030.739.049.061.068.368.065.359.0




Moving Average

48.059.069.3 53.568.0 64.267.3 68.759.0 67.751.0 63.141.0 55.030.7 46.031.0 35.830.7 30.839.0 30.849.0 34.961.0 44.068.3 55.068.0 64.765.3 68.259.0 66.7




Moving Average

kY....YYYYL kttttt'

t1321

Moving Average, Lt

48.059.0 53.569.3 64.268.0 68.767.3 67.759.0 63.151.0 55.041.0 46.030.7 35.831.0 30.830.7 30.839.0 34.949.0 44.061.0 55.068.3 64.768.0 68.265.3 66.759.0 62.1




Double Moving Average

kL....LLLLL

'kt

't

't

't

't"

t1321

48.059.0 53.569.3 64.2 58.868.0 68.7 66.467.3 67.7 68.259.0 63.1 65.451.0 55.0 59.141.0 46.0 50.530.7 35.8 40.931.0 30.8 33.330.7 30.8 30.839.0 34.9 32.849.0 44.0 39.461.0 55.0 49.568.3 64.7 59.868.0 68.2 66.465.3 66.7 67.459.0 62.1 64.4





Difference between Actual value and first moving average is called Lag1.

Second Lag or Lag2 can be calculated as,

/kt

/t LLlag

212

For example, if first moving average is calculate for K=3, then

/t

/t

/t

/t LLLLlag 1

2132




Data Forecast Error MA Lag1 Lag210.012.0 11.014.0 11.0 3.0 13.0 1.0 2.016.0 13.0 3.0 15.0 1.0 2.018.0 15.0 3.0 17.0 1.0 2.020.0 17.0 3.0 19.0 1.0 2.022.0 19.0 3.0 21.0 1.0 2.024.0 21.0 3.0 23.0 1.0 2.026.0 23.0 3.0 25.0 1.0 2.028.0 25.0 3.0 27.0 1.0 2.030.0 27.0 3.0 29.0 1.0 2.032.0 29.0 3.0 31.0 1.0 2.034.0 31.0 3.0 33.0 1.0 2.036.0 33.0 3.0 35.0 1.0 2.038.0 35.0 3.0 37.0 1.0 2.040.0 37.0 3.0 39.0 1.0 2.042.0 39.0 3.0 41.0 1.0 2.044.0 41.0 3.0 43.0 1.0 2.0


For constant trend, the error is contact.

Double moving average is used to remove the constant trend.

Error is the sum of lag1 and lag2.

Therefore,

211 laglagMAFt




Double Moving Average: Forecasting

Double moving average can be used for forecasting using followingformulas:

mbaF ttt 1

Where,

//t

/tt

//t

/t

/tt

LLk

b

and]LL[La

12




Data L'(t) L"(t) Lag2 Trend Forecast Error10.012.0 11.014.0 13.0 12.0 1.0 2.016.0 15.0 14.0 1.0 2.0 16.0 0.018.0 17.0 16.0 1.0 2.0 18.0 0.020.0 19.0 18.0 1.0 2.0 20.0 0.022.0 21.0 20.0 1.0 2.0 22.0 0.024.0 23.0 22.0 1.0 2.0 24.0 0.026.0 25.0 24.0 1.0 2.0 26.0 0.028.0 27.0 26.0 1.0 2.0 28.0 0.030.0 29.0 28.0 1.0 2.0 30.0 0.032.0 31.0 30.0 1.0 2.0 32.0 0.034.0 33.0 32.0 1.0 2.0 34.0 0.036.0 35.0 34.0 1.0 2.0 36.0 0.038.0 37.0 36.0 1.0 2.0 38.0 0.040.0 39.0 38.0 1.0 2.0 40.0 0.042.0 41.0 40.0 1.0 2.0 42.0 0.044.0 43.0 42.0 1.0 2.0 44.0 0.0


//t

/tt

//t

/t

/tt

LLk

b

and]LL[La

12

ttt baF 1




Data L'(t) L"(t) Lag2 Trend Forecast10111316182122 15.925 18.027 20.328 22.430 24.431 26.335 28.3 22.2 6.1 2.036 30.3 24.3 6.0 2.0 36.438 32.1 26.3 5.8 1.9 38.339 33.9 28.2 5.6 1.9 39.943 36.0 30.2 5.8 1.9 41.344 38.0 32.1 5.9 2.0 43.847 40.3 34.1 6.2 2.1 45.848 42.1 36.1 6.0 2.0 48.550 44.1 38.1 6.1 2.0 50.251 46.0 40.1 5.9 2.0 52.254 48.1 42.1 6.0 2.0 53.9





Autocorrelation

Autocorrelation is the correlation of a series with itself. This isunlike cross-correlation, which is the correlation of two differentseries.

Autocorrelation is useful for finding repeating patterns in a timeseries, such as determining the presence of a periodic signal orcycle.




Autocorrelation

t = 1




Autocorrelation

t = 3




Autocorrelation

t = 1; r = 0.9

t = 3; r = 0.5

t = 5; r = 0.0




Autocorrelation

t = 0 or t=20; r = 1.0

t = 15; r = 0.0

t = 10; r = -1.0




Autocorrelation




Autocorrelation

Test for significance of autocorrelation coefficient:

Where,

t is the lagr is the autocorrelation coefficient at that lag, and n is the number of observation




Autocorrelation

Hypothesis Testing:

H0: r is attributable to randomness. No cycle present in the time series.HA: A cycle present in the time series.

If the calculated value of Z > 1.96Null hypothesis rejected




Overall Significance: Ljung-Box Statistics

Null hypothesis: At least one correlation is non-zero.

Test for significance of autocorrelation coefficient:

Where,

h is the number of autocorrelation coefficients being tested.r is the autocorrelation coefficient at that lag, and n is the number of observation

If, Qh > 2 (0.05, h), Null hypothesis is rejected.

h

kkh rkn)n(nQ

1

212




10.011.510.016.511.012.514.014.516.014.521.015.515.016.517.020.518.025.518.017.520.020.524.0

Auto Regression (AR)




10.011.510.016.511.012.514.014.516.014.521.015.515.016.517.020.518.025.518.017.520.020.524.0

Auto Regression (AR)1011

915

910111112101610

91010131017

98

101013

Trend = 0.5

xdt = x – (rank x Trend)

= 10 – (0 x 0.5) = 10=11.5 - (1 x 0.5) = 11




1011

915

910111112101610

91010131017

98

101013





1011

915

910111112101610

91010131017

98

101013

lag-1 -0.34061lag-2 -0.01525lag-3 -0.14931lag-4 -0.15717lag-5 0.0482lag-6 -0.30402lag-7 0.940332lag-8 -0.28836lag-9 -0.10714





h

kkh rkn)n(nQ

1

212


h = 9.r is the autocorrelation coefficient at that lagn = 23

Null hypothesis: At least one correlation is non-zero.

Qh = 42.592 (0.05, h) = 16.92

Qh > 2 , Reject H0

At least one correlation is non-zero.

lag-1 -0.34061lag-2 -0.01525lag-3 -0.14931lag-4 -0.15717lag-5 0.0482lag-6 -0.30402lag-7 0.940332lag-8 -0.28836lag-9 -0.10714





Confidence interval of correlogram,

Z(/2)/n

Z at p = 0.05 = 1.96n = 23

Z(/2)/n = 0.408

Lag = 7





Yt = 0.778Yt-7 + 2.337




1011

915

910111112101610

91010131017

98

101013

10.1215.56

9.348.56

10.1210.1212.4510.2114.45

9.609.00

10.2110.2112.0210.2813.58

9.819.34

10.2810.2811.69

Yt = 0.778Yt-7 + 2.337





4859.0036569.32472

6867.3120758.9817450.9747140.9963530.67528

3130.6879339.0182649.0252961.0036568.32472

6865.3120758.9817448.97471

--





Auto Regression (AR)48

59.0036569.32472

6867.3120758.9817450.9747140.9963530.67528

3130.6879339.0182649.0252961.0036568.32472

6865.3120758.9817448.97471

--11/23/2015 Shamsuddin Shahid, FKA, UTM



Confidence interval of correlogram,

Z(/2)/n

Z at p = 0.05 = 1.96n = 73

Z(/2)/n = 0.2294

Lag = 1, 2, 3, 5, 6, 7, 8, 9





Y(t) Y(t-1) Y(t-2) Y(t-3) Y(t-5) Y(t-6) Y(t-7) Y(t-8) Y(t-9)31.00 30.68 41.00 50.97 67.31 68.00 69.32 59.00 48.0030.69 31.00 30.68 41.00 58.98 67.31 68.00 69.32 59.0039.02 30.69 31.00 30.68 50.97 58.98 67.31 68.00 69.3249.03 39.02 30.69 31.00 41.00 50.97 58.98 67.31 68.0061.00 49.03 39.02 30.69 30.68 41.00 50.97 58.98 67.3168.32 61.00 49.03 39.02 31.00 30.68 41.00 50.97 58.9868.00 68.32 61.00 49.03 30.69 31.00 30.68 41.00 50.9765.31 68.00 68.32 61.00 39.02 30.69 31.00 30.68 41.0058.98 65.31 68.00 68.32 49.03 39.02 30.69 31.00 30.6848.97 58.98 65.31 68.00 61.00 49.03 39.02 30.69 31.0038.00 48.97 58.98 65.31 68.32 61.00 49.03 39.02 30.6932.68 38.00 48.97 58.98 68.00 68.32 61.00 49.03 39.0232.00 32.68 38.00 48.97 65.31 68.00 68.32 61.00 49.0333.69 32.00 32.68 38.00 58.98 65.31 68.00 68.32 61.0041.02 33.69 32.00 32.68 48.97 58.98 65.31 68.00 68.3251.03 41.02 33.69 32.00 38.00 48.97 58.98 65.31 68.0058.00 51.03 41.02 33.69 32.68 38.00 48.97 58.98 65.3169.32 58.00 51.03 41.02 32.00 32.68 38.00 48.97 58.9870.00 69.32 58.00 51.03 33.69 32.00 32.68 38.00 48.9765.31 70.00 69.32 58.00 41.02 33.69 32.00 32.68 38.00

- - - - - - - - -- - - - - - - - -

4859.0036569.32472

6867.3120758.9817450.9747140.9963530.67528

3130.6879339.0182649.0252961.0036568.32472

6865.3120758.9817448.97471

--






98877665543322110

tttttttt

t

YbYbYbYbYbYbYbYbbY

4859.0036569.32472

6867.3120758.9817450.9747140.9963530.67528

3130.6879339.0182649.0252961.0036568.32472

6865.3120758.9817448.97471

--11/23/2015 Shamsuddin Shahid, FKA, UTM



Autocorrelation

Limitations of Autocorrelation:

1. The observations must be regularly spaced through time.2. Any linear trend in the data should be removed in advance. Linear

trends will cause a gradual decline in peaks on theautocorrelogram with increasing lag.

3. In order for there to be sufficient comparisons in the calculation ofthe coefficient, the rules of thumb are: (a) there should be at least50 observations in the time series; and (b) the lag should notexceed n/4

4. Significantly high r values at small lags may not reflect cyclicity butjust smoothness in the data.

5. Although significantly negative Z values are possible, these arenot important as they correspond to negative autocorrelation,themselves due to peak-trough correspondences in the data;these will inevitably occur in association with high positive(peak-peak; trough-trough) autocorrelations and offer no additionalinformation.




Autoregressive Moving-Average (ARMA) models form a class of linear time series models.

ARMA is a combination of AR and MA

Autoregressive Moving-Average (ARMA) =Auto-Regression (AR) + Moving Average (MA)

Auto Regressive Moving Average (ARMA)




eYb.......YbYbYbbY LktLtLtLtt 83322110

LktkLtLtLtt eb.......ebebebbY 3322110


Auto-Regression (AR)

The error term is calculated from Moving average.




1011

915

910111112101610

91010131017

98

101013

10.1215.56

9.348.56

10.1210.1212.4510.2114.45

9.609.00

10.2110.2112.0210.2813.58

9.819.34

10.2810.2811.69

Yt = 0.778Yt-7 + 2.337

Auto Regressive Moving Average (ARMA




Data L'(t) L"(t) Lag21011

915

9101111 10.7512 1110 10.87516 11.75 1 0.76610 11.125 0.125 -0.152

9 11.125 0.25 -0.30510 11.125 -0.625 -0.15210 11 -0.125 -0.15213 11.25 0.125 0.30710 11 -0.125 -0.15217 11.875 0.875 0.919

9 11 -0.25 -0.3058 10.75 -0.25 -0.458

10 10.875 -1 -0.15210 10.875 -0.125 -0.15213 11.25 0.5 0.307






Data Lag16 0.76610 -0.1529 -0.305

10 -0.15210 -0.15213 0.30710 -0.15217 0.9199 -0.3058 -0.458

10 -0.15210 -0.15213 0.307




1011

915

910111112101610

91010131017

98

101013

10.1215.56

9.348.56

10.1210.1212.4510.2114.45

9.609.00

10.2110.2112.0210.2813.58

9.819.34

10.2810.2811.69


10.2514.86

9.598.93

10.2510.2512.2310.3313.92

9.829.30

10.3310.3311.8710.3913.18

9.999.59

10.3910.3911.58




ARMA10.2514.86

9.598.93

10.2510.2512.2310.3313.92

9.829.30

10.3310.3311.8710.3913.18

9.999.59

10.3910.3911.58

1011

915

910111112101610

91010131017

98

101013

AR10.1215.56

9.348.56

10.1210.1212.4510.2114.45

9.609.00

10.2110.2112.0210.2813.58

9.819.34

10.2810.2811.69





Auto Regressive Moving Average (ARMA)1011

915

910111112101610

91010131017

98

101013

ARMA10.2514.86

9.598.93

10.2510.2512.2310.3313.92

9.829.30

10.3310.3311.8710.3913.18

9.999.59

10.3910.3911.58




21.7526.8622.0921.9323.7524.2526.7325.3329.4225.8225.8027.3327.8329.8728.8932.1829.4929.5930.8931.3933.08

10.011.510.016.511.012.514.014.516.014.521.015.515.016.517.020.518.025.518.017.520.020.524.0





Non-stationary Time Series

The models are applicable to stationary time series only.

If the parameters like autocorrelation varies with time, thesemodels can not be used




Auto Regressive Integrated Moving Average (ARIMA)

Most naturally-occurring time series in hydrology are not at all stationary (at least when plotted in their original units). Instead they exhibit various kinds of trends, cycles, and seasonal patterns.




• The best strategy may not be to try to directly predict the level of the seriesat each period.

• Instead, it may be better to try to predict the change that occurs from oneperiod to the next (i.e., the quantity Y(t)-Y(t-1)).

• In other words, it may be helpful to look at the first difference of the series,to see if a predictable pattern can be discerned there.

• For practical purposes, it is just as good to predict the next change as topredict the next level of the series, since the predicted change can always beadded to the current level to yield a predicted level

Differencing




The seasonal difference of a time series is the series of changes from oneseason to the next. For monthly data, in which there are 4 seasons, theseasonal difference of Y at period t is Y(t)-Y(t-4).

The first difference of the seasonal difference of a monthly time series Y atperiod t is equal to (Y(t) - Y(t-4)) - (Y(t-1) - Y(t-5). Equivalently, it is equal to(Y(t) - Y(t-1)) - (Y(t-4) - Y(t-5)).

Seasonal Differencing




Several approaches are there to identify, measure and remote the trendand seasonal components of the time series data.

One of the easiest and most common method is differencing.

The first difference,

Y’t = Yt – Yt-1

is one way to ca capture and remove the effect of the trend.

Seasonal Differencing




ARIMA models are, in theory, the most general class of models forforecasting a time series which can be stationarized bytransformations such as differencing and logging.

A ARIMA model is classified as an ARIMA(p,d,q) model, where:

p is the number of autoregressive terms, d is the number of nonseasonal differences, and q is the number of lagged forecast errors in the prediction equation.

ARIMA(1,1,1)ARIMA(1,0,1)ARIMA(2,1,2)






Box-Jenkins methodology.

1. Model Selection2. Parameter Estimations3. Model Checking

Many cases it is a iterative processes.


