F D D D D F. smoothed value of the data including Y t the most recent data.

Transcription

1 Module 2 Forecasig 1. Wha is forecasig? Forecasig is defied as esimaig he fuure value ha a parameer will ake. Mos scieific forecasig mehods forecas he fuure value usig pas daa. I Operaios Maageme forecasig is used exesively o esimae fuure demad of produc(s) 2. Wha is ime series forecasig? Time series forecasig uses pas daa o esimae he fuure values. Here he performace wih respec o ime is cosidered. Time is he idepede variable. 3. Meio some simple forecasig models for ime series daa? Some simple forecasig models usig ime series daa are simple average, movig average ad simple expoeial smoohig. 4. Wha is movig average? Movig average is a simple ime series forecasig model based o averages of a chose umber of periods. I is used o forecas a cosa model or level daa. 5. Wrie he basic equaio for simple expoeial smoohig? F +1 = αd + (1 α)f. This equaio ca be used whe simple expoeial smoohig is used as a forecasig model. Here F represes he forecas for period, D is he kow demad for period ad α is he smoohig cosa. The basic equaio for simple expoeial smoohig is Y = αy + (1 α)y 1. Here Y is he smoohed value of he daa icludig Y he mos rece daa. 6. Is expoeial smoohig a form of weighed average? How? Simple expoeial smoohig ca be see as a form of weighed movig average. F = αd + α F, we ge Expadig he geeral equaio + 1 ( 1 ) ( ) ( ) 2 = α + α 1 α + α 1 α α( 1 α) 1 + ( 1 α) F D D D D F As is large ad eds o ifiiy, he erm (1- α) eds o zero. The res of he erms are all erms ivolvig D j. I ca be see ha he F +1 value is a weighed average of 2 1 α, α 1 α, α 1 α,... + α 1 α. If 0 α 1, he erms D o D 1 wih weighs ( ) ( ) ( ) each weigh is smaller ha 1 ad is decreasig. The highes weigh is give o he mos rece poi ad he weighs progressively decrease by a facor (1 α) as he daa ges older. As eds o ifiiy, he weighs are α, α( 1 α), α( 1 α) 2,... This is a ifiie geomeric series whose firs erm is α ad he commo erm is (1 α). α The sum of all he erms of he progressio is 1 (1 (1 α )) =. 7. Wha are he implicaios of usig small α? A small value of α implies ha iiial weigh give o he rece daa is small ad he subseque weighs are smaller. This meas ha more erms coribue o he forecas. This also meas ha more weigh is give o he forecas ha o he demad.

2 8. I he equaio Y = a + b + ε, wha does ε represe? Wha ca you say abou he mea ad variace of ε? The symbol ε represes he error erm. I is assumed o be ormally disribued wih mea = 0 ad wih small variace. This meas ha he errors are expeced o cacel ou each oher. The error erm is also expeced o be small. 9. Wha do a ad b represe i he equaio Y = a + b + ε? I he liear equaio Y = a + b + ε, b is he slope ad a is he y iercep he poi i which he lie ouches he y axis. 10. Wrie he equaios for Hol s model? The basic equaio for Hol s model is F +1 = a + b. Here a is called he level which represes he smoohed value up o ad icludig he las daa. The slope of he lie is give by b ad herefore he forecas for he ex period F +1 = a + b. The values of a ad b are updaed usig a = α D + (1 α)(a -1 + b -1 ) ad b = β (a a - 1) + (1 β)b How is he Hol s model differe from he liear regressio model? Hol s model is differe from liear regressio because i compues differe values of he slope ad iercep a differe pois usig simple expoeial smoohig 12. Wrie he equaios for he Wier s model? F +1 = (a + b )C +1 where a ad b are he level ad red as described i he Hol s model. C +1 is he seasoaliy idex for he period ha we are forecasig. The equaios are a +1 = α(d +1 /C +1 ) + (1-α)(a + b ) b +1 = β(a +1 - a ) + (1 - β)b C +p+1 = γ(d +1 /a +1 ) + (1 - γ)c Wha is seasoaliy idex ad how is i calculaed? Seasoaliy idex capures he effec of he seaso o he daa. I ca be defied as S i = D i /Average. For example, if here are 4 seasos, we ca compue he average of he demads of four seasos. The demad i a period divided by he average gives he seasoaliy idex for he period. I is impora o kow he umber of periods ha cosiue a seaso. 14. Meio some measures of goodess of forecass? Some measures of goodess of forecass iclude, mea squared deviaio, mea absolue deviaio, mea perceage deviaio ec. 15. Wha is a causal model? I a causal model, here is a idepede variable or a causal variable ha impacs he depede variable (demad). 16. Wrie equaios for causal model?

3 Problems The equaio for a causal model is Y = a + bx where X is he idepede (causal) variable ad Y is he depede variable. This is a liear model. Oher models exis. 1. Give he daa 92, 93, 92, 91, 93, 94, 92 fid he forecas for he eighh period usig simple average, weighed average (weigh of 1 for he firs four periods ad 2 for he remaiig hree), 3 period movig average? Simple average = ( )/7 = Weighed movig average = [ ( ) + 2( )]/10 = 92.6 Three period movig average = ( )/3 = Give he daa 92, 93, 92, 91, 93, 94, 92 fid he forecas for he eighh period usig simple expoeial smoohig? Use α = 0.3 ad iiial forecas usig simple average? Simple average = 92.28; F 1 = 92.28, α = 0.3 F +1 = αd + (1- α)f ; F 2 = , F 3 = 92.44, F 4 = 92.31, F 5 = 91.91, F 6 = 92.24, F 7 = 92.77, F 8 = Give he daa 63, 64, 66, 67, 67, 69, 71, 72 fid he forecas for he eighh period usig simple average, ad 3 period movig average? Is i a good forecas? Why or why o? Simple average = , hree period movig average forecas = ( )/3 = Boh are o good forecass because he daa shows icreasig red while he forecasig models used are for cosa (level) daa ad idicae a ceral value (average). 4. Give he daa 63, 64, 66, 67, 67, 69, 71, 72 fid he forecas for he ih period usig simple expoeial smoohig? Use α = 0.3 ad iiial forecas usig simple average. Is i a good forecas? Why or why o? F 1 = average = , F +1 = αd + (1- α)f ; F 2 = 66.06, F 3 = 65.44, F 4 = 65.61, F 5 = 66.03, F 6 = 66.32, F 7 = 67.12, F 8 = 68.27, F 9 = The forecas is o good because he daa shows icreasig red while simple expoeial smoohig is o be used for cosa (level) daa ad idicaes a ceral value (average). 5. Derive he expressio for a ad b i he equaio Y = a + b? We wish o derive y a b ε = + + ad fid a ad b such ha ( ) 2 = 1 y a b is miimized. Parially differeiaig he residue wih respec o a ad b ad seig he firs derivaive o zero, we ge he equaios y a b = + ad y a b 2 = +. Here a ad b are ukows ad he oher erms ca be compued. Solvig hese equaios we ge he values of a ad b. 6. Give he daa 63, 64, 66, 67, 67, 69, 71, 72 fid he forecas for he ih period usig liear regressio?

4 We compue ΣY = 539, Σ = 36, Σ 2 = 204 ad ΣY = The equaios are 539 = 8a + 36b ad 2479 = 36a + 204b. Solvig, we ge b = ad a = F 9 = a + 8b = Give he daa 63, 64, 66, 67, 67, 69, 71, 72 fid he forecas for he ih period usig Hol s model? Use α = β = 0.2. The equaios for Hol s model have bee give earlier. Usig hese equaios, we ge F 2 = 64.29, F 3 = 65.51, F 4 = 66.91, F 5 = 68.23, F 6 = 69.23, F 7 = 70.42, F 8 = ad F 9 = Daa for four quarers for hree years is 81, 62, 76, 55, 85, 65, 79, 60, 90, 69, 84, 64. Fid he forecas for he ex four periods usig a simple seasoaliy model compuig seasoaliy idices? We assume ha here are 4 seasos ad daa for hree years (say). The oal demads for 3 years are 274, 289 ad 307. The forecased oal demad is 323 ad per seaso i is The seasoaliy idices (average) are 1.18, 0.9, 1.09 ad The forecased values are 95.3, 72.68, 88, Daa for four quarers for hree years is 81, 62, 76, 55, 85, 65, 79, 60, 90, 69, 84, 64. Fid he forecas for he ex four periods usig Wier s model. α = β = 0.2, γ = 0.3. The equaios for Wiers model are give: F +1 = (a + b )C +1 where a ad b are he level ad red as described i he Hol s model. C +1 is he seasoaliy idex for he period ha we are forecasig. The equaios are a +1 = α(d +1 /C +1 ) + (1-α)(a + b ); b +1 = β(a +1 - a ) + (1 - β)b ; C +p+1 = γ(d +1 /a +1 ) + (1 - γ)c +1. The compuaios give C 13 = 0.3, C 14 = 0.225, C 15 = 0.275, c 16 = 0.2, a 12 = 313.9, b 12 = 4.22, F 13 = Usig F 13 = D 13, we ge F 14 =71, F 15 = 89.5, F 16 = Sudes believe ha he salary hey ca expec durig a placeme process is relaed o heir academic performace. The CGPA (idicaor of performace) ad he salary obaied by six sudes are (7, 6), (6.8, 5.8), (7.5, 6.5), (8, 7), (8.2, 7.5) ad (8.6, 8). Fid he salary ha a sude wih CGPA 8.7 ca expec? We build a causal model of he form Y = a + bx where Y is he salary ad X is he CGPA. The equaios are ΣY = a + bσx; ΣXY = aσx + bσx 2. Compuig, we ge ΣX = 46.1, ΣY = 40.8, ΣXY = ad ΣX 2 = Solvig, we ge a = 7.11 ad b = For X = 8.7, Y = Give he daa 92, 93, 92, 91, 93, 94, 92 fid he forecas for he eighh period usig simple average. Compue he mea absolue deviaio? Di Fi F 8 = Mea absolue deviaio = MAD =. Compuig, we ge MAD = ( )/7 = Give he daa 83, 87, 90, 92, 96, 99, fid he forecas for he eighh period usig liear regressio? We compue ΣY = 547, Σ = 21, Σ 2 = 91 ad ΣY = The equaios are 547 = 6a + 21b ad 1969 = 21a + 91b. Solvig, we ge b = 3.11 ad a = F 8 = a + 7b = 102

5 13. Give he daa 83, 87, 90, 92, 96, 99 fid he forecas for he eighh period usig Hol s model? Use α = β = 0.2. The equaios for Hol s model are give earlier. Performig he compuaios, we ge F 7 = a 6 + b 6 = = Give he daa 92, 93, 92, 91, 93, 94, 92 fid he forecas for he eighh period usig simple expoeial smoohig? Use α = 0.2 α = 0.7 ad iiial forecas usig simple average? Explai he effec of α o he coribuio of he daa for he various periods A α = 0.2, F 7 = Sice we are buildig a cosa model, F 8 = A α = 0.7, F 7 = Sice we are buildig a cosa model, F 8 = Lower α gives more weigh o forecas while high α gives more weigh o demad. 15. Cosider he daa 92, 93, 92, 91, 93, 94, 92. Fid he mea absolue deviaio for he forecas usig simple expoeial smoohig α = 0.2. The forecass are F 1 = 92.43, F 2 = 92.34, F 3 = 92.47, F 4 = 92.38, F 5 = 92.43, F 6 = 92.43, F 7 = MAD = ( )/7 =