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

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DATA EXTENSION & FORECASTING 3   Docsity.com Data Extension & Forecasting e.g., Stream flow records for reservoir planning Data forecasting 4   1970 2010 2040 Available data Extension of data We are here Inflow, (Forecast) Time, t Time, t+1 Inflow, It (known) [ ]1 1ˆ , ,...........t t tI f I I+ −=1t̂I + Docsity.com Example-1 7   Data Forecast 105 - 115 103 108 120 97 110 121 117 79 110 107.67 107.75 110.2 108 108.28 109.87 110.67 107.5 Docsity.com Method of Moving Averages (MA) • As a new observation becomes available, new average is computed by dropping the oldest observation and including the newest one. • No. of data points used for computing the average remains the same • Uses the latest ‘T’ periods of known data Data Extension & Forecasting 8   T+1 T T T+2 T T+3 Docsity.com Example-2 9   Data MA (3) 105 - 115 - 103 - 108 120 97 110 121 117 79 107.67 108.67 110.33 108.33 109 109.33 116 Docsity.com If the correlogram indicates that the time series is purely random •  Xt , Xt-k are independent •  Distribution of Xt is known •  Generate Xt using data generation technique to follow given distribution with parameters estimated from sample Data Generation – Uncorrelated Data 12   k     ρk   ρk=0, v k ≠ 0   Mainly used for flood peaks, storm intensities, short duration rainfall etc . Not useful for stream flows, seasonal rainfall, and such long time processes.   Docsity.com •  Most hydrologic time series exhibit serial dependence e.g., X(t) correlated with X(t-τ) ρk ≅ (ρ1)k ρk → 0, k → ∞ Data Generation – Serially Correlated Data 13   ρk     k     Exponentially decaying   First order Markov process   Docsity.com First order Markov process: Xt+1 = µx +ρ1 (Xt – µx ) + εt+1 ε ∼ Mean 0 and variance σε2 This model is stationary w.r.t both mean and variance Data Generation – Serially Correlated Data 14   Deterministic component   Random component   Docsity.com If X ∼ N(µx, σx2) then ε ∼ N(0, σε2) If {ut } ∼ N(0, 1) , {utσε} i.e., is N(0, σε2) Data Generation – Serially Correlated Data 17   ( )211t xuσ ρ− ( ) 21 1 1 11t x t x t xX X uµ ρ µ σ ρ+ += + − + − Standard normal deviate   First order stationary Markov model Or Thomas Fiering model (Stationary)   Docsity.com To generate data using First order Markov model, • Known sample estimates of µx, σx, ρ1 • Assume X1 (may be assumed to be µx) • Generate values X2, X3, X4, X5 …… • Generate a large set of values and discard first 50-100 values to ensure that the effect of initial value dies down • Negative value: retain it for generating next value, set it to zero, in applications. Data Generation – Serially Correlated Data 18   ( ) 21 1 1 11t x t x t xX X uµ ρ µ σ ρ+ += + − + − Docsity.com Consider the annual stream flow data (in cumecs) at a river for 29 years Example-3 19   S.No. Data S.No. Data S.No. Data 1 1093.31 11 1042.33 21 1444.97 2 1636.87 12 1492.13 22 1203.08 3 1485.67 13 1205.90 23 910.73 4 1579.51 14 1245.77 24 883.59 5 1443.00 15 1197.81 25 970.98 6 1327.40 16 1754.55 26 1001.92 7 1108.70 17 1108.56 27 1434.91 8 928.10 18 957.64 28 1635.00 9 840.83 19 1425.80 29 1875.78 10 1447.03 20 1128.62 Docsity.com 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 to some extent •  A main application of this 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 22   Docsity.com First order Markov model with non-stationarity, for stream flow generation: ρj is serial correlation between flows of jth month and j+1th month. ti, j+1 ∼ N(0, 1) Data Generation – Serially Correlated Data 23   ( )1 2, 1 1 , 1 1 1ji j j j ij j i j j j j X X t σ µ ρ µ σ ρ σ + + + + += + − + − StaHonary  First  order   Markov  Model  ( ) 2 1 1 1 11j x j x j xX X tµ ρ µ σ ρ+ += + − + − Docsity.com The monthly stream flow (in cumec) for a river is available for 29 years (only 12 years data is given here) Example-4 24   SL. YEAR JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY NO. 1 1979-80 54.60 325.40 509.50 99.40 53.50 25.80 12.50 5.60 3.10 2.20 0.90 0.81 2 1980-81 220.78 629.16 591.32 120.33 43.33 14.83 8.41 4.05 1.73 1.12 0.85 0.96 3 1981-82 131.30 538.89 574.21 151.06 53.03 19.49 8.38 4.51 1.89 1.11 0.74 1.06 4 1982-83 100.19 630.02 702.07 83.29 32.45 16.60 6.80 3.33 2.03 1.23 0.85 0.65 5 1983-84 171.30 444.30 512.30 211.00 62.40 24.00 8.40 4.50 2.30 1.10 0.80 0.60 6 1984-85 147.80 636.20 293.50 127.70 79.70 22.10 10.10 4.60 2.70 1.40 0.70 0.90 7 1985-86 174.50 323.30 393.20 75.40 100.60 21.80 10.90 4.00 1.90 1.40 1.00 0.70 8 1986-87 126.40 288.30 395.30 54.40 29.80 21.40 6.40 2.60 1.70 0.70 0.60 0.50 9 1987-88 60.50 291.00 269.60 95.09 80.84 26.39 10.37 3.68 1.65 0.71 0.62 0.38 10 1988-89 40.95 620.00 427.60 251.80 74.73 17.71 7.05 3.33 1.51 0.87 0.59 0.90 11 1989-90 167.10 398.80 277.80 102.70 61.10 19.54 6.79 3.33 1.52 0.96 0.77 1.93 12 1990-91 150.80 591.50 471.20 197.00 35.67 25.62 10.52 4.02 2.10 1.22 1.32 1.16 Docsity.com Assume X1 = µ1 = 117.49; σ1 = 52.24, ρ1 = 0.348 µ2 = 474.5, σ2 = 150.18, X1,2 = = = 521.67 Example-4 (contd.) 27   ( ) 222 1 1,1 1 1,2 2 1 1 1X tσµ ρ µ σ ρ σ + − + − ( ) 2 150.18474.5 0.348 117.49 117.49 52.24 0.335*150.18 1 0.348 + − + − Docsity.com X1,2 =521.67, µ2 = 474.5; σ2 = 150.18, ρ2 = 0.154 µ3 = 421.39, σ3 = 126.53, X1,3 = = 474.64 Example-4 (contd.) 28   ( ) 2 126.53421.39 0.154 521.67 474.5 150.18 0.377*126.53 1 0.154 + − + − Docsity.com X1,3 =474.64, µ3 = 421.39; σ3 = 126.53, ρ3 = 0.169 µ4 = 145.94, σ4 = 77.65, X1,4 = = 180.45 Example-4 (contd.) 29   ( ) 2 77.65145.94 0.169 474.64 421.39 126.53 0.379*77.65 1 0.169 + − + − Docsity.com