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Constructing and Validating the statistical regression model
By: Ousmane NDIAYE & Dr Simon MASON PRESAO-10 : Seasonal Outlook Forum in West Africa Niamey, Niger, 21st May to 1st June 2007
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How does it work ? Very simple : time Period 1 Period 2 Y = a*X + b
Predictand (rainfall) Y Y50 Y51 Y52 Y99 Y98 Y97 Y96 Yk Yk+1 Yk-1 Y53 X X50 X51 X52 X99 X98 X97 X96 Xk Xk+1 Xk-1 X53 Predictor (SST) Very simple : Period 1 Period 2 Y = a*X + b (a,b) ? a*X + b Ŷ Measured Forecasted
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Divide the record into 2 periods
1) Period to construct the model (50 to K) Period to test the model (k+1 to 99) Y50 Y51 Y52 Y99 Y98 Y97 Y96 Yk Yk+1 Yk-1 Y53 X50 X51 X52 X99 X98 X97 X96 Xk Xk+1 Xk-1 X53 2) Y(50:k) = a*X(50:k) + b a*Xk+1 + b Ŷk+1 Forecasted time series a*Xk+2 + b Ŷk+2 (a , b) a*X98 + b Ŷ98 a*X99 + b Ŷ99
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Retrospective forecast
1) Period to built the model : (50 to G) Year to test the model : G+1 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 Y(50:G) = a*X(50:G) + b (a , b) a*XG+1 + b ŶG+1 2) Period to built the model : (50 to G+1) G+2 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 ŶG+2 Y(50:G+1) = a’*X(50:G+1) + b’ (a’ , b’) a’*XG+2 + b’ Forecasted time series N) Period to built the model : (50 to 98) 99 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 Y(50:G) = α*X(50:98) + β (α , β ) α*X99 + β Ŷ99
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Cross Validating window
Cross-Validation Cross Validating window 3 Test buffer Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 Step 1 {52-98}=[50-99]\[ ] Y(52:98) = a1*X(52:98) + b1 (a1 , b1) a1*X50 + b1 Ŷ50 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 Step 2 {53-99}=[50-99]\[ ] Y(53:99) = a*X(53:99) + b (a , b) a*X51 + b Ŷ51 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 Step 3 {50;54-99}=[50-99]\[ ] X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 Y(50;54:99) = a*X(50;54:99) + b (a , b) a*X52 + b Ŷ52
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Cross-Validation (continuing)
Step G+1 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 {50:G-2;G+2-99}=[50-99]- [G-1;G;G+1] Y(50:G-2;G+2:99) = aG*X(50:G-2;G+2:99) + bG (aG , bG) aG*XG + bG ŶG Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 Step for 98 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 {50:96}=[50-99]-[ ] Y(50:96) = a98*X(50:96) + b98 (a98 , b98) a98*X98 + b98 Ŷ98 Y50 Y51 Y52 Y99 Y98 Y97 YG YG+1 YG-1 YG+2 Step for 99 X50 X51 X52 X99 X98 X97 XG XG+1 XG-1 XG+2 {51-97}=[50-99]-[98-99;50] Y(51:97) = a99*X(51:97) + b99 (a , b) a99*X99 + b99 Ŷ99
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