Intepetation Of Wind Coponents As Copositional Vaiables Pablo Buenestado-Caballeo, Eusebi Jaauta-Bagulat, Cae Heada-ala Depataent de Mateàtica Aplicada III, Escola Uniesitàia Engineia Tècnica Industial de Bacelona Uniesitat Politècnica de Cataluna (UPC. Bacelona, pain. Pablo.Buenestado@upc.edu Depataent de Mateàtica Aplicada III, Escola Tècnica upeio Engines Cains, Canals i Pots de Bacelona Uniesitat Politècnica de Cataluna (UPC. Bacelona, pain. Eusebi.Jaauta@upc.edu Depataent de Física i Engineia Nuclea, Escola Uniesitàia Engineia Tècnica Industial de Teassa Uniesitat Politècnica de Cataluna (UPC. Teassa, pain. Cae.Heada@upc.edu Abstact The classical statistical stud of the wind speed in the atospheic suface lae is ade geneall fo the analsis of the thee habitual coponents that pefo the wind data, that is, the coponent W-E, the coponent -N and the etical coponent, consideing these coponents independent. When the goal of the stud of these data is the Aeolian eneg, so is when wind is studied fo an enegetic point of iew and the squaes of wind coponents can be consideed as copositional aiables. To do so, each coponent has to be diided b the odule of the coesponding ecto. In this wok the theoetical analsis of the coponents of the wind as copositional data is pesented and also the conclusions that can be obtained fo the point of iew of the pactical applications as well as those that can be deied fo the application of this technique in diffeent conditions of weathe.. Intoduction In the atospheic suface lae the behaiou of wind is geneall studied stating fo data elated to its thee coponents. If we denote b the west-east coponent of wind data, b to the south-noth coponent and to the etical coponent, we can define a ecto with those thee coponents. This is the wind ecto = (,,. In icoeteoolog, these coodinates ae usuall denoted as u,, w. This pape analses the possibilit of studing the wind coponents as copositional aiables, when the squaes of wind coponents ae consideed. This esults when wind is studied fo an enegetic point of iew. Tansfoations of copositional data in this case ae also studied. We ecall that we call an obseation of coponents an eleent of the set { (,,..., ; j 0, j,,..., } = = > = + We call coposition of coponents an eleent of the subset + defined b
{ ;... } = + + + = C + + In the aboe definition C is a constant naed closue constant. The alue of the closue constant depends on the units of data. Fo eaple, C = 00 if data ae 6 pecentages, C = 0 if data ae pat pe illion (pp, C = if data ae pats of the unit, j etc. Because it is eified that j = C, then = ; so, fo the sake of j= j= C siplicit, we can assue that C =.. The wind coponents as copositional data In the stud of the dnaics in the atospheic suface lae it is ipotant to egiste the wind in thee othogonal diections. Consequentl, the wind is epessed though thee coodinates, two in the hoiontal plane and the thid one othogonal to this plane. As a atte of conenience, in the hoiontal plane, the paallel coodinate is defined accoding in a west-east diection to the equato. The pependicula coodinate in a south-noth diection and the coodinate pependicula to both coodinates inceasing fo the floo. These coodinates ae consideed foing a tihedon positiel guided. o, accoding to these definitions, we can conside the wind ecto as = (,, ; we call it sipl, the wind. The Euclidean no o the odule of the wind, is gien b the epession: = = + + Consideing the quotient between the wind and its odule, we obtain the u ecto as: u = = (,, Accoding to its definition, it will be a unit ecto, that is It is also tue that u = u = Haing done the aboe eposition, we can aie to the copositional aiable of the wind pesented in the following section.
. The wind kinetic coefficient We define the wind kinetic coefficient, that we denote b E, as the ecto: E = ( E, E, E =,, This ecto is coposed b thee coponents; each one of the has no diension (in phsic sense. The ecto E is elated to the popotion of kinetic eneg associated to the wind in each one of its diections. Also, fo thei definition, the ecto E is such that its coponents ae copositional aiables, because the su of these is. In fact: + + + + = + + = = = E E E In the figue we pesent an eaple of tena diaga of the kinetic coefficient of the wind. 0 0. 0.8 E Ais 0.4 0.6 E Ais 0.6 0.4 0.8 0. 0 0 0. 0.4 0.6 0.8 E Ais Figue : Eaple of tena diaga of the kinetic coefficient of the wind B this wa we hae been able to associate a copositional coponent to the wind ecto. It is sought to stud its applicational possibilities to the dnaic chaacteistics of the atospheic suface lae. The atheatical and statistical analsis of the copositional aiables is usuall ade b a peious tansfoation applied to data. In the following section we pesent soe eaples of these.
4. Tansfoations of the wind kinetic coefficient The tansfoations oe coonl applied to copositional data alues ae: the additie logatio tansfoation (ALR, the cented logatio tansfoation (CLR, both defined b Aitchison (986, and the ost ecent the isoetic logatio tansfoation (ILR defined b Egocue et al. (00. In this section we pesent how these tansfoations ae applied to the kinetic coefficient. 4. Additie logatio tansfoation ALR The ALR tansfoation is defined as: ALR: ( n n,,..., ln,... In the case of the kinetic coefficient the application is: ALR: E ALR E = E E = A A ( ( ln, E E Then, iplifing ALR E ( ( ( E E = = ( ln ln E E ALR( E = ln ln = B logaith ules ALR( E = (ln ln,(ln ln ( A A coponents of the tansfoation ALR of the kinetic coefficient, is epessed as follows: In consequence the (, 4
A = (ln ln A = (ln ln Finall, the ati epession of this sste is: 4. Cented logatio tansfoation CLR The CLR tansfoation is defined as: CLR: ( ln A 0 = ln A 0 ln,..., ln,... g ( g ( g ( In this epession, g( is the geoetic ean of the coponents. In ou case: CLR: E E E E CLR E = = C C C ge ( ge ( g( E ( ( ln,, In this epession ge ( is the geoetic ean of the wind kinetic coefficient: g( E / = EEE = = Then, ( ( CLR( E = ln ge ( ge ( ge ( iplifing CLR( E = ln 5
4 4 4 CLR( E = ln Finall, we hae: 4 C = ln ln ln 4 C = ln + ln ln 4 C = ln ln + ln The ati epession of this linea sste is: C 4/ / / ln C / 4/ / = ln C / / 4/ ln 4. Isoetic logatio tansfoation ILR The ILR tansfoation is defined as: ILR: ( In this case, we hae: H n g (,..., n,..., ln, ln,..., ln n + ILR: H E ILR E = E EE = I I ( ( ln, ln, E E ( ( ( ILR( E ln, ln = 6
ILR( E = ln, ln ILR( E = (ln ln, (ln + ln ln Consequentl, the ILR coponents ae: I = (ln ln I = (ln + ln ln The ati epession of this linea sste is: I ln / / 0 = ln I / / / ln 5. Conclusions. Wind coponents can be consideed as copositional aiables, consideing the squaes of wind ecto coponents.. Fo this consideation, we can define a wind kinetic coefficient though which is possible to analse the atospheic dnaics accoding to the ethodolog based on copositional data.. Usual copositional data tansfoations applied to wind kinetic coefficient can be intepeted fo enegetic point of iew in the wind studies. 6. Refeences ( Aitchison, J. (986. The statistical Analsis of Copositional Data. Chapan and Hall, 46 pp. ( Egocue, J.J., Pawlowsk-Glahn, V., Mateu-Figueas, G. and Baceló-Vidal, C. (00. Isoetic logatio tansfoations fo copositional data analsis. Matheatical Geolog, ol. 5, 79-00. ( Jaauta-Bagulat, E., P. Buenestado-Caballeo and C. Heada-ala (005. Tansfoation and back tansfoation of copositional data: a siple ethodolog using linea sstes and algebaic ethods. ubited fo publication. 7