REVISED CRITERIA FOR CHOICE OF SUBSET SIZE IN TREND ANALYSIS AND PREDICTION

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1 Journal of Informaton, Control and Management Systems, Vol. 4, (006), No. 69 REVISED CRITERIA FOR CHOICE OF SUBSET SIZE IN TREND ANALYSIS AND PREDICTION Vanya MARKOVA Insttute of Control and System Research, BAS P.O.Box. 79, 3 Sofa, Bulgara e-mal: markovavanya@yahoo.com Abstract Ths paper dscussed some aspects of crtera for choce of subset sze. The man goal s proposed a revsed crteron when s nvestgated ar qualty based on coeffcent of determnaton and MSE and crteta of addtve regresson. We clam the crtera are sophstcated estmators of relable and large-scale web-based system for analyss and predctng. Keywords: revsed crtera, addtve regresson, ar qualty, subset sze, INTRODUCTION The most crtcal weakness of the weather models came n a lttle-known area called data assmlaton. As meteorologst Per Unden, of the Swedsh Meteorologcal and Hydrologcal Insttute, wrote a month later, The forecast problems [were] most lkely due to data assmlaton dffcultes only. Data assmlaton s the glue that bnds raw data wth the physcs-based equatons that go nto computer weather models. These equatons, lke all dfferental equatons, requre ntal values to be fed nto them. If you tell them the temperature, velocty, and pressure of the ar n every cubc nch of the Earth s atmosphere, the equatons can predct how those states wll evolve. The problem s that nobody knows the correct ntal condtons. The goal of ths paper s to study trend predcton of ar polluton and nvestgate selecton of set sze observaton and propose a revsed estmator as coeffcent for more balanced forecastng va regresson. In prncple, of the uses of regresson, predcton and estmaton of mean responses are the most tolerant toward elmnatng varables. At the same tme, t s relatvely unmportant whether the varables are causally related or the model s realstc. It s tactly assumed that predcton and estmaton are to be wthn the observaton-space of the data and that the system contnues to operate as t dd when the data were collected. Thus, any varables that contan predctve nformaton on the dependent varable, and for whch nformaton can be obtaned at a reasonable cost, are useful varables.

2 70 Revsed Crtera for Choce of Subset Sze n Trend Analyss and Predcton Of course, more fath could be placed n predctons and estmates based on establshed causal relatonshps, because of the protecton such models provde aganst nadvertent extrapolatons and unrecognzed changes n the correlatonal structure of the system. Extrapolaton requres more care n choce of varables. There should be more concern that all relevant varables are retaned so that the behavor of the system s descrbed as fully as possble. Extrapolatons are always dangerous but can become dsastrous f the equaton s not a reasonably correct representaton of the true model. Any extrapolaton carres wth t the assumpton that the correlatonal structure observed n the sample contnues outsde the sample space. Valdaton and contnual updatng are essental for equatons that are ntended to use for extrapolatons (such as forecasts). One should also be conservatve n elmnatng varables when estmaton of parameters s the objectve. Ths s to avod the bas ntroduced when a relevant varable s dropped The rest of ths paper s organzed as follows: an overvew of trend analyss and trend predcton and background about ts statstcal estmaton crterons of set sze of observatons. Fnally, we presented the revsed estmator for forecastng va regresson and smulaton results. TREND ANALYSIS By extractng the access record of a system of nvestgaton object of an system, we could obtan the number of observaton data ( y, y,... y n ) for at any partcular tme ( x, x,... x n ). Ths knd of data can be vewed as tme seres data. Forecastng wth tme seres data has many real-world applcatons n areas lke analyzng bologcal data, dentfyng models of economcs, weather predcton, etc. [4]. In ths paper, we use tme-seres approach to analyze and predct the concentraton of ar pollutants. The objectve of trend analyss s to fnd out a curve that well fts or sutably approxmates the number of requests aganst tme usng statstcal and emprcal methods. In ths paper, trend predcton process attrbutes the presence of new ncomng data together wth analyzed trend from hstorcal data of smlar category. Intutvely, there may be a close relaton between the past vewng trend and the current or future vewng patterns. Therefore, usng past trend could help us to predct the future trend. The trend analyss related wth these facts: Seasonal fluctuatons often appear n tme seres. To elmnate the seasonal varatons from the data and to have better shape for short term analyss, movng averages can be used to deseasonalzed observatons as follows:

3 Journal of Informaton, Control and Management Systems, Vol. 4, (006), No. 7 y j= l+ y =, = l, l +,..., n l where l s the seasonal perod, n s the total number of data. When calculatng these averages the process move along by droppng the most remote observaton n the prevous average and ncludng the next observaton n the new average. Successve movng averages are computed smlarly untl the fnal observaton. Any method of fttng equaton to a set of data could be called regresson [3]. Such equatons are valuable for at least two reasons,.e. analyzng the trend and makng predctons. Of the varous methods of performng regresson, least square fttng s the most wdely used [3]. In ths secton, three frequently used models and the two proposed addtve regresson models utlzng the least square fttng to extrapolate the best ft are studed: Lnear regresson s a basc regresson model where there s only one explanatory varable [3]. The regresson functon s lnear and the two parameters, slope term b and ntercept term a, of that straght lne have to be found such that the sum of squared predcton errors (SSE), where SSE = ( y ), s mnmum. Exponental regresson assumes that the predcted trend s an exponental functon where there s only sngle explanatory varable. It s smlar to the lnear regresson after takng an ant-logarthm on both sdes. Another frequently used model s the power regresson, whch expressed n terms of a power functon. The model consders the stuaton where heavy tal exsts n the trend. In contrast wth exponental functon, whch mples unbounded drop, power functon experences a rapd ntal drop and gradual fnal decrease. As mentoned above, exponental regresson produces a model wth unbounded drop n the whole cycle whle power regresson produces a model wth rapd ntal drop and a gradual decrease n tal. For a data set wth exponental drop ntally and a flat tal at the end, the method of combnng these two regresson models can be consdered. On the other hand, for another data set wth exponental drop ntally and an exponental tal but wth slower rate at the end, the method of combnng two exponental regresson models can be consdered. In ths paper, we wll explore the addtve regresson model of these two knds: The general form of addtve regresson model s ˆ + y = [ f ( x ) f ( x )] r where x and x are the explanatory varables, f and f are ther regresson functons respectvely, and r s the weghtng factor appled to adjust the magntude after the summng of two terms. The regresson model chooses r such that the Sum of Squared Error (SSE) s mnmzed.

4 7 Revsed Crtera for Choce of Subset Sze n Trend Analyss and Predcton Exponental-exponental-sum (EES) addtve regresson s used to determne the predcted trend that composed of two exponental terms Thus, the basc formulaton of the predcton values can be expressed as the followng functonal form: y = [ еˆ ( x) еˆ ( x)] r ˆ + where r s the weghtng factor. In order to prevent the shft of magntude from summng up the two regresson functons, least squares fttng s appled. By adjustng r, least squares fttng select the lne for whch the sum of squared dstances from all observaton ponts to the lne s mnmzed. Then, r can be obtaned as {[ eˆ ( x) + eˆ ( x)] y( x)} x r = {[ eˆ ( x) + eˆ ( x)]} x Fnally, the sum of two exponentally,.e. y ˆ( x) s used to forecast the trend of y (x). The basc formulaton of exponental-power-sum (EPS) addtve regresson s smlar to EES. Nevertheless, EPS assumes the predcted trend s composed of an exponental term and a power functon term x e ˆ ˆ( = a x b ( x) = ae ; p x) b where а, b, a, b are the results obtaned by the exponental regresson and the power regresson. The formula for EPS s denoted as y ˆ = [ еˆ ( x) + pˆ( x)] r where r s the weghtng factor. Therefore, r can be obtaned as x {[ eˆ ( x) + pˆ( x)] y( x)} r = {[ eˆ ( x) + pˆ( x)]} x Crtera for Choce of Subset Sze Many crtera for choce of subset sze have been proposed. These crtera are based on the prncple of parsmony, whch suggests selectng a model wth small resdual sum of squares wth as few parameters as possble [3]. Most of the crtera are monotone functons of the resdual sum of squares for a gven subset sze and, consequently, gve dentcal rankngs of the subset models wthn each subset sze. However, the choce of crtera may lead to dfferent choces of

5 Journal of Informaton, Control and Management Systems, Vol. 4, (006), No. 73 subset sze, and they may gve dfferent mpressons of the magntude of the dfferences among subset models. The latter may be partcularly relevant when the purpose s to dentfy several competng models for further study. Sx commonly used crtera are dscussed brefly. In addton, the choce of F-toenter and F-to-stay, or the correspondng sgnfcance levels SLE and SLS are revewed. The sx commonly used crtera to be dscussed are coeffcent of determnaton R, resdual mean square estmaton MSE, adjusted coeffcent of determnaton Radj, Mallows Cp statstc, and two nformaton crtera AIC and SBC. Of the choce of subset sze regardng stoppng rules are of mportance. Frst, the use of all ndependent varables s a very poor rule unless n p s very large. Second, most of the stoppng rules do poorly f n p = 0. The Cp statstc does poorly when n p = 0 (but s recommended for n p = 40). Thrd, the lack-of-.t test of the (t p ) varables that have been dropped s generally very poor as a stoppng rule regardless of the sgnfcance level used. Fnally, an unbased verson of the coeffcent of determnaton generally dd poorly unless n p was large. Ths suggests that R, and perhaps Radj and MSE, may not serve as good stoppng rules for subset sze selecton. Mallows Cp statstc and sgnfcance levels appear to be the most favored crtera for subset sze selecton. The Cp statstc was not the optmum choce of Bendel and Aff n the ntermedateszed data sets and t dd poorly for very small samples. Sgnfcance level as a crteron dd slghtly better than Cp n the ntermedate-szed studes. The poor performance of Cp n the small samples should not be taken as an ndctment. Frst, none of the crtera dd well n such studes and, second, no varable electon routne or model buldng exercse should be taken serously when the sample sze es are as small as n-p < 0. Selecton of Subset sze of Observaton In ths paper, we proposed revsed crterons of estmaton well correlaton relaton between forecast and predctors that follow n the tran of ntutve and emprcal expermentaton especally: on the case wth one explanatory varable n regresson model C = R ( MSEnorm ), C [0,], where MSE MSE mn MSE norm = n hole subset sze. And then n the nterval MSEmax MSEmn [ Cmax ε. Cmax ; Cmax + ε. Cmax ], s more nformatve of the forecastng and gves well balance between the varaton on the forecasts and t s predctors and ε s a constant of accuracy.

6 74 Revsed Crtera for Choce of Subset Sze n Trend Analyss and Predcton -the extraordnary varables x, x n addtve regresson model could be obtaned from to dfferent areas of the same set of data. The way of choosng two areas could be crtcal to the ftness n the trend analyss and predcton. Let there are totally n-day observaton data and we propose three methods on choosng the two subset sze observaton as follows: - begnnng and endng p number of the data set; - the begnnng p an remanng (n-p) number of the data set; - the whole data set and endng p numbers of the data set. Subset sze wll be selected from the one whch has the C [ Cmax ε. Cmax ; Cmax + ε. Cmax ]. Important note s that we submt for dscusson a local extreme about C, MSE n the perod of the learnng. TREND PREDICTION After the process of trend analyss s carred out, we are gong to study trend predcton. Bascally, projecton or extrapolaton of the trend value does trend predcton. In the proposed algorthm, predcton on the movng average s based on regresson models obtaned from the trend analyss. In ths secton, four trend predcton approaches are proposed. Fxed Regresson Selectng The predcton result s gven by p ( ) = ( ) m 0d0 where m0 and d0 are the method and ts correspondng parameter respectvely whch mnmze the SSE of hstorcal trend. Contnuous Regresson Updatng The predcton result s gven by ( ) ( ) p = m d where m and d are the method and ts correspondng parameter respectvely whch mnmze the SSE of new ncomng movng averages. Note that CRU analyses the new ncomng data and determnes the Fxed Regresson Selectng (FRS) assumes that the regresson model, whch best ft the hstorcal data, also ft the new ncomng data,.e. both sets of data follow the same trend. FRS uses the same method and correspondng parameters, whch wll be used for the whole process of predcton, to forecast the new trend. Contnuous Regresson Updatng (CRU) concerns the behavor changes n the new ncomng trend. It keeps track on the new ncomng trend n terms of fttng method and parameter. Snce the trend of the new ncomng data maybe changed contnuously, CRU plays an mportant role on automatc compensaton of the dssmlarty between the new trend and

7 Journal of Informaton, Control and Management Systems, Vol. 4, (006), No. 75 method mand control parameter d whch s the most sutable for the current stuaton. hstorcal trend. Moreover, the selected parameters and the method can be changed contnuously upon the changng behavors of the new ncomng data. Hstorcal Updatng The predcton result s gven by ( ) = ( ) α + ( ) p α = n / n 0 m d 0 0, s = n = y( ) / n = m d 0 0 y ( ) where y () be the new ncomng request, y 0( ) be the hstorcal requests, n be the number of days of the new ncomng requests, n 0 be the number of days of the hstorcal requests. Instead of only consderng the regresson model and the correspondng parameters, hstorcal trend can be appled, HU consders the hstorcal trend y ˆ0 ( ) as well. It sums up the newly-ftted trend y ˆ( ) of the new ncomng trend as well as the hstorcal trend usng two weghtng factors α and s. An α s used to make a balance between the amount of hstorcal data and new ncomng data. Contnuous Regresson wth Hstorcal Updatng p 0 ( The predcton result wll become ( ) = ( ) α + ( ) ( α) α = n / n 0 m d, s = n = y( ) / n = m d 0 0 y ( ) where a and s are the same as HU. The method has the advantages of both HU 0 Hstorcal Updatng (HU) assumes that the new ncomng trend s smlar to the hstorcal trend. It takes the hstorcal data nto account and uses those data to predct the new trend. Ths method s especally useful when there are few data for the new trend. HU makes dfferent weghtng to the newly predcted trend as well as hstorcal trend. When the amount of new ncomng data s small, the result wll rely more on the hstorcal data and vce versa. s s another weghtng factor whch s used to adjust the overall magntude of the hstorcal trend to match up wth the new ncomng trend. Contnuous Regresson wth Hstorcal Updatng (CRHU) s a method combnng CRU and HU. It tres to take advantages of both methods. There are two man components n ths approach: hstorcal trend and new ncomng trend. For the hstorcal trend, the control factors, the model and the parameters, are obtaned based on the analyzed result of the hstorcal data; so that the trend produced by these factors best ft the hstorcal data. For the new ncomng trend, the control factors, the

8 76 Revsed Crtera for Choce of Subset Sze n Trend Analyss and Predcton and CRU methods snce they complment each other. model and the parameters are obtaned based on the factors, whch best ft, the new ncomng data snce they are obtaned by analyzng the new ncomng data. Numercal results show that CRHU s the best n most stuatons n terms of SSE. A case study: mpact of length of learnng perod on accuracy of predcton As a prmer we are gong to study the nfluence of the length of tme s seres about accuracy of forecasts. When tme s seres trend have been modeled so that results for forecastng can be acheved, often the ssue about the learnng perod occurs. There are two opposte hypotheses: -the predcton s more accurate when the learnng perod s longer and contra hypotheses s - the precson of predcton could be aggravated due to mpact of hstorcal data. Such hstorcal data often are based on some factors that acted n remoter past and later dsappear; hence they don t have to be taken n consderaton. Generally these opposte hypotheses can nfluence the accuracy of the forecast. Our task s to prove or deny the statement that: the longer the learnng perod s the more accurate the predcton s. To resolve the task presented above means that the optmum relatonshp between perod of learnng and perod of predcton has to be determned. The measures of accuracy predcton are the coeffcent of determnaton rout mean of squares MSE. The model of lnear regresson s used. value. R and y = α + βx + ε ; y s the ozone concentraton and s calculated as movng average max daly x s the max daly temperature, chosen as a predctor. α, β are coeffcents of regresson The subscrpt ndcates the partcular observatonal unt, =,,, n. The random error ε NID(0, σ ); ε ; assumptons are frequently stated as

9 Journal of Informaton, Control and Management Systems, Vol. 4, (006), No. 77 where NID stands for normally and ndependently dstrbuted.[4] We use data from staton Ndda, Germany from January.003 to 30 November 004 as a perod of learnng and the perod from December 004 to 3 December 004 as a perod of forecastng. The perods: k prop k prop s the coeffcent of proporton between learnng and forecastng = learnng perod /forecastng perod and k prop vares from to 7 mounts. EXPERIMENTS AND RESULTS Results: k prop, R and MSE for lnear regresson wth predctor: MSE learnng perod Fgure MSE lnear regresson predctor - temperature R Fgure learn n g perod R lnear regresson predctor - temperature We observe strong relaton between length of learnng perod and accuracy of prognoss.

10 78 Revsed Crtera for Choce of Subset Sze n Trend Analyss and Predcton For k > 0 we can state that precson of predcton s sgnfcantly lower than prop for k > 0. prop For values of k < 4 we can see ncrease of prop R and MSE, so we couldn t approve the hgh accuracy of prognoss. For values of k < 4 and k > 0 we can observe mnmum of MSE at k=9 and two local maxmums of prop prop R at k=4 and k=9..5 coeff_c perod of learnng Fgure 3 C revsed coeffcent Fnally we proposed a revsed crterons of estmaton well correlaton relaton between forecast and predctors, that suggest ntutonal and emprcally, especally C norm = R ( MSE ), C [0,] [ Cmax ε. Cmax ; Cmax + ε. Cmax ] that confrms the nformatve nterval founded above that gve well balance between the varaton on the forecasts and t s predctors. REFERENCES [] HUBBARD MC, COBOURN WG (998) Development of a regresson model to forecast ground-level ozone concentraton n Lousvlle, KY, Atmospherc Envronment 3(4), [] MARSILI-LIBELLI ST. Smplfed knetcs of troposphere ozone, Ecologcal modelng, vol. 84, 996,pp [3] RAWLINGS, JOHN O., SASTRY G. PENTULA, DAVID A. DICKEY, Appled regresson analyss: a research tool. nd ed., Sprnger, 998 [4] RAO C.R., H.TOUTENBERG. Lnear Models. Least Squares and Alternatves, (Sprnger,999)(ISBN )(439s).