STATISTICAL method is one branch of mathematical

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1 40 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 Optimizig Forest Samplig by usig Lagrage Multipliers Suhud Wahyudi, Farida Agustii Widjajati ad Dea Oktaviati Abstract To obtai iformatio from a populatio, we use a samplig method Oe of samplig techiques that we ca use is double samplig Double samplig is a samplig techique based o the iformatio of first phase which is used as a additioal iformatio obtaiig estimates for the secod phase I this case, we discuss the model of double samplig with regressio estimator The, to obtai the optimal umber of samples for the first ad secod phases, we use Lagrage multipliers The model aalysis result is a formula to calculate the optimal umber of samples for the first phase ad the secod phase Implemetatio of this method is simulated by usig teak stads data from previous studies at Forest Maagemet Uit FMU Madiu which cosists of Sectio Forest Maagemet Uits FSMU Dagaga ad Dugus The calculatio result of data from FSMU Dagaga, we get optimal umber of plots must be observed i image iterpretatio are 49 plots ad field survey are 4 plots Ad with the data from FSMU Dugus, we get optimal umber of plots to be observed i image iterpretatio are 53 plots ad field survey are 0 plots Idex Terms Double samplig, Lagrage multiplier optimizatios, regressio estimator I INTRODUCTION STATISTICAL method is oe brach of mathematical sciece that focuses o the data collectio techique, processig or aalyzig data, ad deductio based o the data I data processig, we aalyze the relatioship betwee two or more variables, ad decide which oe is the most importat To aalyze the data we ca use regressio ad correlatio, i order to determie which variables are itercoected Oe of the discussio i statistical methods is samplig techique I statistical iferece, if we wat to obtai coclusios about the populatio, although without comprehesive observatio, the compositio of idividuals i the populatio, we ca use samplig techique [] We use samplig techique because of time ad cost efficiecy, large eough populatio, the precisio i the executio of observatio, ad the value of beefits I the process, samplig has may techiques that we ca use i various implemetatios of samplig, oe of them is double samplig Double samplig is a samplig techique based o the iformatio of first phase which is used as a additioal iformatio obtaiig estimates for the secod phase Oe implemetatio of double samplig is i forest ivetory [] However, we have to cosider the cost factor of samplig, so that we eed a optimal allocatio betwee the umber Mauscript received March 6, 07; accepted May 4, 07 The authors are with the Departmet of Mathematics, Istitut Tekologi Sepuluh Nopember, Surabaya 60, Idoesia suhud@matematikaitsacid of samples i the first phase ad secod phase To determie the optimal umber of samples, we ca use optimizatio by miimizig the cost fuctio ad defie the variace estimator fuctio as costrait The result of optimizatio is a optimal umber of samples for the first phase ad the secod phase [] We ca use the Lagrage multipliers method to optimizatio Lagrage multiplier method or Lagrage multipliers are itroduced by Joseph Louis de Lagrage Lagrage multiplier method is a method to maximize or miimize a fuctio of several variables by usig λ as its Lagrage multipliers The extesio of the method to a geeral problem of variables with m costraits has bee discussed i [3] Kitikidou explais that samplig optimizatio by usig Lagrage multipliers are computed by miimizig the cost fuctio ad defiig variace estimator fuctio as costraits [3] I this paper, we discuss a samplig optimizatio usig Lagrage multipliers method ad apply it to the forest observatio, especially the teak forest ivetory II DOUBLE SAMPLING, LINEAR REGRESSION MODEL AND REGRESSION ESTIMATOR IN DOUBLE SAMPLING A Two Phase Samplig Double Samplig Double samplig is oe of samplig techiques with two phases I the first phase, we choose uits umber of samples, ad i the secod phase we choose uits that are part of the first phase We use the first phase as a estimator for the secod phase I this case, we use regressio estimator Mea of regressio estimator is [3]: where ŷ = y + bx x y : mea of y from sub sample x : mea of x from sample x : mea of x from sub sample b : estimator of β iace of regressio estimator is [3]: ŷ = Sy r S y : variace of y o subsample r : correlatio coefficiet betwee y ad x : the umber of first sample which is take from N : the umber of subsamples from Optimum allocatio from cost fuctio is [3]: C = C + C

2 WAHYUDI et al: OPTIMIZING FOREST SAMPLING BY USING LAGRANGE MULTIPLIERS 4 C : the total cost of samplig C : the cost of first phase samplig C : the cost of secod phase samplig Let x,x,,x are radom samples from populatio with mea µ da stadard deviatio σ If we use samplig with replacemet ad ulimited populatio the we get []: µ x = µ σ x = σ µ x : mea of mea samplig distributio σx : variace of mea samplig distributio I double samplig, we assume y have ormal distributio, cofidece iterval for mea is [3]: where ŷ the we get ŷ = ε Z α/ B Liear Regressio Model ŷ Z α/ σ < y < ŷ + Z α/ σ = σ, error of estimatio is [3]: ε = Z α/ σ Liear regressio model of populatio is [4]: Y = α + βx + e j where α ad β are costat populatio parameters, ad β is the regressio coefficiet Regressio coefficiet β is [5]: β = N i= x i Xy i Y N i= x i X = σ xy σ x iace of populatio i regressio is [5]: σ = σ y β σ x Correlatio coefficiet of populatio i regressio is [5]: ρ = N i= x i Xy i Y Ni= x i X Ni= y i Y = σ xy σ x σ y Relatio of correlatio coefficiet with regressio coefficiet is [5]: ρ = β σ x σ y From equatio, we ca write: From equatio, we get: σ = σ y ρ σ y = σ ρ Regressio model i sample is [4]: y = a + bx k + e k For k =,,3,, with a ad b are estimators for α ad β, ad e k is error of estimator for k-th observatio Regressio coefficiet b is [5]: b = i= x i xy i y i= x i x = S xy S x Correlatio coefficiet of sample i regressio is: r = i= x i xy i y i= x i x i= y i y = S xy S x S y 3 Sy = i= y i i= y i C Regressio Estimator i Double Samplig Aother model of Y is [5]: Y = Y + β If we assume y is a estimator of equatio Y, the we get : y = Y + β + e where e is the error, so E e = 0, the we get: We also get: y = Y + β + e 4 E y = E Y + β + e = Y The above equatio of E y shows that y is a ubiased estimator for Y If we assume i= e ix i x = e w, we get the value of b which i= x i x is a estimator of β as follows: b = β + e w Because of E e = 0, so E e w = 0 The E b = β ad E E b = β E e w = E i= e i x i x i= x i x = D Expectatio ad iace i Multivariate Distributio Geeral variace distributio is defied as [5]: y = Ey E y = E y E y E y = y + E y Expectatio ad variace i multivariate distributio is [5]: E = E E m = E E m m + E m E m + E E m E

3 4 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 E Fisher s Distributio Fisher s variable Fm,m is distributed as [5]: Fm,m = Fm,m where m ad m is degree of freedomi Fisher s distributio If we assume m = ad m =, we get: xi X F, = i= xi X Expectatio of distributio is as follows [5]: where m 3 EF m,m = m m F Optimizatio by usig Lagrage Multipliers Method Optimizatio techique of multivariables with equality costrait have the followig geeral form [6]: miimize f X subject to g j X = 0, for j =,,,m where X = {x,x,,x } T where m If m >, the it caot be solved The first step of this method is the costructio of Lagrage fuctio that is defied as [6]: LX,λ = f X + m j= λ j g j X 5 Theorem [6]: Necessary coditio for a fuctio f X with costrait g j X = 0, where j =,,,m such that it has relative miimum at poit x is first partial derivative of Lagrage fuctio defied as L = L{x,x,,x,λ,λ,,λ } has value zero Theorem [6]: A sufficiet coditio for f X to have relative miimum or maximum at the quadratic, Q, defied by: Q = i= j= L dx i dx j x i x j evaluated at x = x must be positive defiite or egative defiite for all values of dx for which the costraits are satisfied Necessary coditio Q = i= j= L x i x j dx i dx j to be positive or egative defiite for all admissible variatios dx is that each root of the polyomial p i, defied by the followig determiat equatio, be positive or egative L p L L 3 L g g g m L L p L 3 L g g g L L L 3 L m p g m g m g m g g g 3 g = 0 g g g 3 g g m g m g m3 g m where L i j = Lx,λ x i x j ad g i j = g ix x j Observe that equatio 6 is a polyomial of order m, i p III RESULTS AND DISCUSSIONS A Mea of Regressio Estimator i Double Samplig Liear regressio estimator ca be defied as: Y = α + βx Y = α + βx 7 Equatio 7 is liear regressio populatio average If we estimate liear regressio from its sample, where a is a estimator for α, ad b is a estimator for β, so we obtai: y = a + bx 8 y = a + bx 9 Equatio 9 is a liear regressio average equatio i sample From equatio 9, we obtai: a = y bx 0 By substitutig equatio 0 to equatio 8, we obtai estimator regressio equatio as follows: y = y + bx bx = y + bx x If the value of x is ukow, the to compute its estimator, we ca use x = i= x i we obtai: ŷ = y + bx x Equatio is mea estimator equatio of liear regressio i double samplig B iace of Regressio Estimator i Double Samplig After we obtai equatio, substitutig y from equatio 4 ad value of b from equatio, so we obtai aother model of mea estimator equatio of liear regressio i double samplig It is give by: ŷ = Y + β + e + β + e w x x = Y + β x X + e w x X e w + e To obtai the variace of regressio estimator, we use the trivariate distributio theory Mea of trivariate distributio is as follows: E Ŷ = E E 3 ŷ We assume: E 3 ŷ = Y + β x X So we obtai E 3 ŷ, which is a ubiased estimator The we determie E E 3 ŷ as follows: E E 3 ŷ = E Y + β x X = Y + β x X For the ext step, we assume x is ot costat, we obtai: E E 3 ŷ = Y + β x X = Y

4 WAHYUDI et al: OPTIMIZING FOREST SAMPLING BY USING LAGRANGE MULTIPLIERS 43 Equatio is a ubiased estimator of the trivariate distributio The, we get variace of trivariate distributio by the formula : Ŷ = E 3 ŷ E E 3 ŷ + E 3 ŷ + 3 The we determie the secod part of equatio 3, so we obtai: E 3 ŷ = Y + β x X = 0 So that equatio 4 becomes: = E e + x X E + x X E E ee w + x X E ee w x X E 5 The, we solve each part of equatio 5, the first part of equatio 5 is: E e = e E e = σ e = σ Next, we solve the secod part of equatio 5 as follows: x X E = σ E σ F, = 3 The, we solve the third part of equatio 5 as follows: x X E = σ E σ F, = 3 The fourth part of equatio 5 is as follows: E ee w = E 0 = 0 Next, the fifth part of equatio 5 is give by: x X E ee w = x X 0 = 0 From the sixth part of equatio 5, we obtai: x X E = X X E X X E =0 The result from all previous steps is: E 3 ŷ = σ + σ 3 + σ 3 6 If the umber of sample is too big, the 3 ad equatio 6 becomes: = σ + + From aalysis result, equatio 3 becomes: Ŷ = σ β σ x 7 Third part of equatio 3 is: E E 3 ŷ = Y + β x X If ad, the 0 ad 0 I this case, equatio 7 becomes: = β x x = β σ x Ŷ = σ y ρ 8 Next, we determie: Equatio 8 is a variace equatio of regressio estimator E 3 ŷ i populatio If we estimate equatio 8 i sample, the we = E 3 Y + β x X + e w x X ca write: e w + e = E e + x X ŷ = Sy 4 r 9 Equatio 9 is variace equatio of regressio estimator i sample C Samplig Optimizatio by usig Lagrage Multipliers Method First step of optimizatio is determie the objective fuctio ad costrait The objective fuctio is observatio cost fuctio, that ca be writte as: f = C + C The costrait is the variace of regressio estimator i sample as follows: g = Sy r ε Zα/ = 0 The we costruct Lagrage fuctio as i equatio 5: L = C + C + λ Sy r ε Zα/ = C + C + λsy λs yr + λs yr λ ε Optimal coditio for equatio 0 is: 0 L = C λsyr = 0 L = C λsy r = 0 L λ = S y S yr + S yr ε Zα/ = 0 3

5 44 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 From equatio, we obtai: From equatio, we obtai: λ = C S yr 4 λ = C S y r From equatios 4 ad 5, we obtai: 5 = C r r C 6 = From equatio 3, we get: r C C r Sy S yr + S yr = ε 7 8 The, substitutig equatio 7 to equatio 8, we obtai: S y ε r = S yr + S y C C r r C r C C r + r C r r = 9 ε By solvig the above problem usig Lagrage multipliers method, we get equatio 9 which is a formula for calculatig the umber of optimal plots o first phase We ca write it as follows: C Syr + Sy C r r opt = ε 30 Next, by substitutig equatio 6 to equatio 8, we obtai: S y r + S y C ε C r r = 3 Equatio 3 is the formula for calculatig the umber of optimal plots o secod phase We ca write it as follows: S y r C + Sy C r r opt = ε 3 D Implemetatio of Samplig Optimizatio by usig Lagrage Multipliers Method The method is implemeted by usig simulatio of image iterpretatio data ad field survey data Image iterpretatio data is the data forest picture obtaied from observatios with remote sesig The result of remote sesig is calculated by software util we get the diameter, desity, ad umber of trees per plot, the we ca also calculate tree volume per plot The, we check the result of image iterpretatios i the field To determie the potetial of a forest, it is impossible to observe all objects i forest Thus, we eed to take some samples I previous research of Fathia Amalia R D, she takes 76 plot samples for first phase samplig which is i image iterpretatio ad 38 plot samples for secod phase without kowig whether the umber of samples is optimum or ot I this paper, we eed to calculate the optimal umber of samples i image iterpretatio ad i the field Samples were observed i the form of plots where the plot cosists of several trees Data of previous observatio result ca be used for calculatig the optimal umber of samples which must be observed o image iterpretatio ad field We use data from FMU Perum Perhutai Madiu II, East Java, which icludes data from FSMU Dagaga ad Dugus For calculatig the umber of optimal samples, we use the followig parameters: TABLE I SUM AND AVERAGE OF TREE VOLUME Locatio Parameter m 3 FSMU Dagaga FSMU Dugus /0 ha Sum of V image samples V f ield Sum of V image samples Sum of V f ield Sum of Vimage samples Sum of Vf ield Average of V f ield Average V image Average V image samples Observatio cost cosists of two types: image observatio cost ad field observatio cost Image observatio cost is the total of cost which is used to buy image, image processig cost, ad image map pritig cost Field observatio cost is icluded i trasportatio cost, employee salary ad etc So that, we obtai the cost per hectare: TABLE II OBSERVATION COST Locatio FSMU Dagaga FSMU Dugus Cost Rp/ha Image Iterpretatio Field The to determie the optimal umber of samples i the first phase opt ad secod phase opt, we calculate S y value first by usig formula i equatio 3, ad also calculate r by usig formula i equatio 9 Next, we calculate opt by usig formula i equatio 30 ad calculate opt by usig formula i equatio 3 We calculate all step by usig Matlab, for locatio FSMU Dagaga we obtai optimal umber of samples that must be

6 WAHYUDI et al: OPTIMIZING FOREST SAMPLING BY USING LAGRANGE MULTIPLIERS 45 observed for first phase opt is 49 plots image iterpretatio ad umber of samples that must be observed for the secod phase opt is 4 plots field survey With the same way, for FSMU Dugus the umber of optimal samples that must be observed is 53 plots image iterpretatio ad 0 plots field survey IV CONCLUSIONS Based o aalysis result ad discussio, we obtai the followig coclusios: Result from formula aalysis i samplig ad optimizatio by usig Lagrage multipliers method, we obtai the umber of optimal samples i the formula for first phase ad secod phase is: C Syr + Sy C r r opt = ε S y r C + Sy C r r opt = ε S y : variacey from the secod phase sample r : correlatio coefficiet C : cost of first phase samplig C : cost of secod phase samplig ε : error i estimatio Z α/ : value of radom variables i stadard ormal distributio From the calculatio results, the umber of optimal samples i image iterpretatio ad field survey with FSMU Dagaga data, we obtai the umber of optimal plots that must be observed i image iterpretatio is 49 plots ad i field survey is 4 plots The other side, with FSMU Dugus data we obtai the umber of optimal plots that must be observed i image iterpretatio is 53 plots ad i field survey is 0 plots So that, if the umber of samples is suitable with that calculatio result, the we obtai a optimal samplig REFERENCES [] R Walpole, Pegatar statistika, edisi ke-3 Itroductio to statistics PT Gramedia Pustaka Utama, 990 [] P Malamassam, Modul Mata Kuliah Ivetarisasi Huta Makassar: Hasauddi Uiversity, 009 [3] K Kitikidou, Optimizig forest samplig by usig Lagrage multipliers, America Joural of Operatios Research, vol, o, pp 94 99, 0 [4] S Makridakis, S Wheelwright, ad V McGee, Metode da Aplikasi Peramala Jilid Ir Utug Sus Ardiyato, M Sc & Ir Abdul Basith, M Sc Terjemaha Jakarta: Erlagga, 999 [5] P de Vries, Samplig Theory for Forest Ivetory Wageige: Wageige Agricultural Uiversity, 986 [6] D Lukato, Pegatar Optimasi No Liier Yogyakarta: Gajah Mada Uiversity, 000