Adaptive Sampling Algorithms for Minimizing the Variance of Parameter Estimates in the Bidirectional Surface Reflectance Model

Transcription

1 Adaptie Sampling Algorithm for Minimizing the Variance of Parameter Etimate in the Bidirectional Surface Reflectance Model Or Intelligent Intrument on Robotic Helicopter A Technical Report Preented to Dr. Kein Wheeler NASA/Ame Reearch Center Team Member: Gabriel Cadden Marian Hofer Shikha Naik Kim Ninh Pinky Thakkar Faculty Superior: Dr. Steen M. Crunk Center for Applied Mathematic and Computer Science (CAMCOS Department of Mathematic San Joe State Unierity December 9 005

2 Adaptie Sampling Algorithm for Minimizing the Variance of Parameter Etimate in the Bidirectional Surface Reflectance Model Intelligent Intrument on Robotic Helicopter ABSTRACT Traditional intrument collect data which are tored and later analyzed. Often uch collected data are not ueful or een wore meaurement that would hae been ueful were not collected. By the time the data are analyzed it i too late for a reearcher to obtain thoe ueful (and ometime crucial meaurement. We introduce a paradigm hift where the intrument and analyi ytem work in concert. By deeloping oftware engine that analyze the data a the meaurement are made we can identify meaurement that would improe the analyi reult and requet them from the intrument. Thi require that three ytem work ymbiotically: the intrument the inference engine and the inquiry engine. We will apply thi methodology to the problem of mapping the urface of the ground with a pectrometer mounted on an autonomou rotorcraft (robotic helicopter. The inference engine will analyze the data from the pectrometer and the inquiry engine will ue the reult of the analyi to direct the helicopter.

3 Table of Content ABSTRACT... I. INTRODUCTION... 4 II. THEORY... 6 A. Linear Regreion (i.e. Linear Leat-Square Fit... 6 B. Nonlinear Regreion (i.e. Nonlinear Leat-Square Fit... 7 C. Maximum Likelihood Etimator... 8 D. Fiher Information Matrix... E. Bayeian Statitic:... F. Information Theory... 5 G. Entropy... 6 H. Maximum Entropy Sampling... 6 III. IMPLEMENTATION... A. Our Model... B. Linearizing the Model... 3 C. Experiment Setup... 4 IV. RESULTS... 5 A. Etimation of the Parameter... 5 B. Maximization of Entropy... 6 V. CONCLUSIONS... 3 VI. RECOMMENDATIONS AND FUTURE RESEARCH... 3 A. Non-Normal Prior... 3 B. Obtaining Poterior Ditribution from Mey Sytem... 3 C. Future Reearch APPENDIX A: Reference APPENDIX B: Matlab function

4 I. INTRODUCTION In order to etimate the amount of carbon equetered by tree in the United State the amount of tanding wood mut be etimated with quantifiable uncertainty (Wheeler 005. Thee etimate come from either atellite image or near ground meaurement. The amount of error in the etimate from thee two approache are currently unknown. To thi end an autonomou helicopter with differential GPS (Global Poitioning Sytem LIDAR (Light Detection and Ranging tereo imager and pectrometer ha been deeloped a a teting platform for conducting further tudie (Wheeler 005. The controlling oftware ytem for thi robotic helicopter mut be deeloped uch that the ground ampling i optimized. The implet ampling method i to end the helicopter to eery poible location to gather data of interet. Howeer thi method i highly inefficient a it will require a large amount of time and money. The other poible method employ random ampling. The helicopter could take random ample until the etimate are reaonably correct. Although there i a poible chance that the helicopter will take ample from the location that offer the mot amount of information and therefore minimizing the needed ample et there i no guarantee that thi ample et will be taken eery time. Therefore the mot efficient olution i to take only a few ample from key location that offer the mot amount of information. In order to optimize the ampling proce the oftware mut work in tandem with the ampling hardware to control the helicopter poition (Wheeler 005. Once a ample ha been taken the data i fed into the oftware ytem which then calculate the next bet poition to gather further data. Initially the oftware ytem aume an 4

5 empirical model for the ground being examined. With each addition of data from the intrument the oftware ytem employ the Bayeian Probability Theory to update it model and calculate the helicopter next poition. The ytem repeat thi proce until the uncertaintie of it model are within a atifactory range. Thi method allow the ytem to be adaptie during ampling and enure adequate ground coer. The model ued in thi project i baed on the BRDF Bidirectional Reflectance Ditribution Function (Wolfgang Lucht 000. Thi function permit the calculation of the amount of reflection baed on the reflectance of the object and the poition of the un and the iewer. With thi function the ytem can compenate for different poition of the un during ampling. Thi project i baed on two reearch paper Bayeian Adaptie Exploration written by Thoma J. Loredo and Experimental Deign to Maximize Information written by Paola Sebatiani and Henry Wynn (00. Both of thee paper follow the Bayeian deign method. Therefore the framework of the implementation preented in thi paper i a combination of the mathematic proided by thee paper. By uing the Bayeian methodology the ytem examine current aailable data with regard to preiouly collected data in order to predict the future data. Thi prediction lead to the identification of which new oberation will yield the mot information. Applying thee method thi paper outline the theory and implementation of the ampling oftware. Firt we decribe the theorie and etablih the mathematical background which include regreion maximum likelihood Fiher information Bayeian tatitic information theory entropy and maximum entropy ampling. Second we proide a thorough explanation of the implementation proce through the 5

6 preentation of ariou formulae application of the method decribed by Sebatiani and Wynn prior and poterior method regreion likelihood and olution. The concluding dicuion include comparion of olution from different method adantage and diadantage of thi particular implementation aenue for further reearch and a dicuion of non-normal parameter and their implication. II. THEORY In thi ection of the paper we dicu the mathematical background of our method from etimating model parameter to etablihing an algorithm for maximum entropy ampling. A. Linear Regreion (i.e. Linear Leat-Square Fit Linear model repreent the relationhip between a continuou dependent ariable and one or more independent ariable (either continuou or categorical in the form: y = X + ε (Montgomery and Peck 99. where: y i an n-by- ector of oberation of the dependent ariable. X i the n-by-p deign matrix determined by the independent ariable. i a p-by- ector of unknown parameter to be etimated. ε i an n-by- ector of random diturbance independent of each other and uually haing a normal ditribution. Example: y = 0 + x + x + 3x3 6

7 y = + x + x + co( 0 3 x linearized model y = 0 + x + x + 3x3 where x = x and x 3 = co( x y = 0 exp( x log( y = log( 0 + x linearized model z = B0 + x where B 0 = log( 0 Soling for the unknown parameter i achieed by oling an oer-determined linear ytem of equation uing leat-quare technique. Leat-quare minimize r = Ax y which i the quared Euclidean norm of the reidual. Matlab ue a numerically table algorithm that utilize the QR factorization of A where Q i orthogonal and R i upper triangular to ole thi problem. B. Nonlinear Regreion (i.e. Nonlinear Leat-Square Fit Like linear model nonlinear model repreent the relationhip between a continuou dependent ariable and one or more independent ariable. Howeer in nonlinear model the unknown parameter do NOT enter the model linearly. Thi model i of the form: y = ƒ(x + ε (Matlab where: y i an n-by- ector of oberation of the dependent ariable. ƒ i any function of X and. X i the n-by-p deign matrix determined by the independent ariable. i a p-by- ector of the unknown parameter. ε i an n-by- ector of random diturbance. 7

8 Example: 3 y = 5 (Hougen-Waton model for reaction kinetic + x x + x 3 x + x 4 3 Nonlinear model are more difficult to fit requiring iteratie method that tart with an initial gue of the unknown parameter ector. Each iteration alter the current gue until the algorithm conerge. In Matlab nonlinear leat-quare fitting i performed uing the Gau-Newton algorithm with Leenberg-Marquardt modification for global conergence. C. Maximum Likelihood Etimator Maximum likelihood etimation (MLE i a tatitical technique that help u make inference about the unknown parameter of a ditribution gien a particular obered et of ample data. Suppoe that we hae collected a ample data et x x x 3 x n from a population with a known probability ditribution function ƒ and an unknown ditributional parameter. We define the likelihood function to be: Likelihood = Probability (that we ample alue x x x n = ƒ(x x x n The likelihood function tell u how likely the obered ample i a a function of the parameter (Wackerly et al. 00. Maximizing the likelihood with repect to gie the parameter alue for which the obered ample i mot likely to hae been generated that i the parameter alue that agree mot cloely with the obered data. The alue ˆMLE that maximize the likelihood function i known a the maximum 8

9 likelihood etimator (MLE of the true parameter. Thi concept can be generalized for ditribution that hae multiple parameter. Example (uing a dicrete ditribution Conider toing a coin 50 time and recording each time the alue of the outcome; H for head and T for tail. Suppoe the final outcome wa 3 head and 8 tail. Let P(H = p P(T = p. Suppoe that thi coin came from a box that contained 3 coin: one ha p(h = 0.35 the econd ha p(h = 0.50 and the third ha p(h = Which coin did we ue in thi experiment? Uing maximum likelihood etimation we can calculate which coin it mot likely wa gien the outcome that we obered. Uing our three poible alue of parameter p the likelihood function baed on our definition earlier take thee three alue: 50 L(3 head p = 0.35 = ( ( x L(3 head p = 0.50 = ( ( L(3 head p = 0.65 = ( ( Note that the likelihood function i maximized when p = Thu our coin i mot likely the one with p(h = 0.65 which i the maximum likelihood etimator of the true parameter p. Example (uing a continuou ditribution In the continuou ditribution cae maximizing the likelihood i equialent to maximizing the logarithm of the likelihood ince log i a continuou trictly-increaing pˆ MLE 9

10 function oer the range of the likelihood. Algebraically thi boil down to taking the firt deriatie of log(likelihood with repect to the parameter etting thi deriatie equal to zero and oling for ˆMLE. If the ditribution ha multiple parameter we mut take the partial deriatie of log(likelihood with repect to each parameter equate them to zero and ole the reulting ytem of equation. Conider the normal ditribution function: ƒ( x µ σ exp ( x µ = πσ σ where µ and σ are the unknown mean and ariance (repectiely of thi ditribution. The correponding likelihood function i: Likelihood = ƒ(x x x n µ σ = ( µ n x exp i = πσ σ i = n πσ exp n i= ( x µ i σ Since maximizing the likelihood i equialent to maximizing it log Maximize the natural log of the equation aboe. ( n n x l = log (Likelihood = i µ log πσ i= σ In order to maximize l we mut take it partial deriatie l l and et µ σ l l = 0 and = 0 and ole thi ytem of two equation with two unknown for µ σ µˆ MLE and σ MLE. The following are the reult produce by thi proce: 0

11 = = = n i i MLE x n x ˆµ and = = = = n i i n i MLE i MLE n x x n x ( ˆ ( ˆ µ σ where x i the ample mean. Thu the ector of etimated parameter i = = = n i i MLE MLE MLE n x x x ( ˆ ˆ σ µ r. D. Fiher Information Matrix F = p p p p p l L l l O M M l L l l l L l l Fiher Information Matrix = F = log ƒ(x r r where r i the ector of the unknown parameter of ditribution f (Myung and Naarro 004. The Fiher information matrix help meaure the accuracy of the etimated parameter. The Cramér-Rao inequality tate that the reciprocal of the Fiher information i a lower bound on the ariance etimator of r (Spall 004. F - ˆ ( ˆ ˆ ( ˆ ˆ ( ˆ ˆ ˆ ( ˆ ˆ ( ˆ ˆ ( ˆ ˆ ( ˆ ( p p p p p V Co Co V Co Co Co V L O M M l L L The tandard error of our approximation are the quare root of the diagonal of F -.

12 E. Bayeian Statitic: The mot common way to iew tatitic i in Frequentit term where probability i deried from obered (or defined frequency ditribution or proportion of population. Thu frequentit tatitician do not look at quetion in which thing might be peronally belieed only at what i known. In contrat Bayeian would ue ome degree of belief to weigh the known frequency and later update their belief in the light of new information uch updating i known a Bayeian inference. Wherea both a Bayeian and a Frequentit would iew the probability of pulling an ace out of a deck of card a /3 becaue that i known to be true and known not to change only a Bayeian would aign a probability to omething like a tree making a ound when nobody i around to hear it. Subjectiim i an element that belong only in the Bayeian world. Thi moement of uing ubjectie belief wa firt brought to light by Reerend Thoma Baye. Hi paper An Eay Toward Soling a Problem in the Doctrine of Chance publihed in 764 introduced a theorem which ha become the backbone behind the Bayeian world thi theorem i commonly known a Baye Theorem. In the two random eent cae Baye Theorem tate that the probability of the random eent A occurring gien that we know that a random eent B ha occurred i gien by the relation of the product between the probability of eent A occurring on it own with the probability of eent B occurring gien that we know eent A ha occurred. All of that diided by the probability of B occurring on it own. P ( A B = P( B A P( A P( B Bayeian Statitic i eentially an extenion of Baye Theorem wherein we combine known information often time referred to a a prior ditribution with new

13 information which i alo known a a likelihood to obtain new expectation which we call a poterior ditribution. Mathematically the poterior ditribution i equal to the product of a prior ditribution and a likelihood function diided by a normalizing contant. In thi project Baye Theorem i implemented a follow: ( model obered data P = P(obered data model P(model P(obered data Thu the probability of what the model parameter hould be gien the obered data i equal to the likelihood of the obered data occurring gien our model parameter multiplied by the probability of our model parameter being what they are all diided by the total probability of the data ample being obered regardle of anything ele. Baye Theorem i often iewed a a way of reiing belief in the light of newly obtained information. The application of Baye Theorem are jut about endle. To illutrate the imple ue of Baye Theorem we hae created a mall example. Keep in mind thi example i not intended to ole anything that i not obiou; it i jut here to how the ue of the theorem. Example: Conider a hore race featuring two hore: Stallion. Gumhoe. Preiouly both hore had participated in 0 race where Stallion won 4 of them and Gumhoe won the other 6. Howeer upon further examination of preiou race track condition it turn out that 5 out of the 6 winning of Gumhoe occurred when it had rained the night before the race cauing the track to be oft the next day. Suppoe that you know that it rained lat night. What the chance of Gumhoe winning the race today? 3

14 We want to etimate the probability of Gumhoe winning gien that it ha rained the night before thi i our poterior. It i equal to the probability that it ha rained the night before gien that gumhoe ha won which we aid wa 5 out of 6 race; thi i the likelihood. We then multiply that by the probability of Gumhoe winning the race with no knowledge of anything ele which out of 0 race he ha won 6 of them; thi i alo known a our prior. We then need to diide all that by the probability that it ha rained which we know out of the 0 race it ha rained 7 time o we ealuate thi equation and find that our poterior i 7.4%. Thu the odd are pretty decent in faor of Gumhoe winning thi race; not o obiou when you jut look at their oerall match-up. P (Gumhoe W Rain = P ( Rain Gumhoe W x P(Gumhoe W P(Rain (5/6(6/0 P (Gumhoe W Rain = (7/0 = 7.4% What thi goe to how i that in Bayeian Statitic the more information you know the more accurate your poterior i going to be. Bayeian inference which relie heaily on conditional probabilitie i baically performed by ealuating the probability of a hypotheized model gien obered data. In doing o one create what could be iewed a an artificial learning ytem where belief reiion i ued. Thu a prior ubjectie probability i replaced by a poterior probability that incorporate newly acquired information. Thi ytem ha two tage: firt ome of the probabilitie are directly altered by ome non-inferential proce uch a peronal experience or intuition. The econd tage inole updating the ret of the prior 4

15 belief to make them more in line with the newly acquired information. Thi learning ytem lead to later dicuion which inole prior that are mey. F. Information Theory Claude E. Shannon i widely credited a the founder of Information Theory. He looked at the general problem in which a ource of information i to be encoded ent oer a channel and then in turn i decoded by the recipient. The information reolution i one of the mot important deelopment in the twentieth century. An eential ingredient to thi reolution ha been a branch of mathematic called information theory which i concerned with encoding and tranmitting data from one place to another. In term of what we were looking at Shannon defined elf-information of ome ytem a the negatie logarithm of that ytem which i a meaure of the freedom of choice with which a meage i elected from the et of all poible meage. The alue obtained i often called bit thi alue i a way of meauring the capacity of a communication channel. Thi i baed upon the log that i ued in computation and quite frequently in programming quetion are preented in a Bernoulli fahion thu we ue log bae and furthermore decribe information a bit. Shannon Self-Information which i imply a meaure of the amount of information content i defined by the following equation: Info [ p( a ] = log[ p( a ] Information theory proide a great example of how different branche of mathematic hae been applied to problem in the real world. The application a mentioned before are numerou in field uch a coding theory data compreion theory and error correction theory. A the information reolution progree information 5

16 theorit will continue to come up with code neceary for accurate and rapid communication. G. Entropy Entropy i a concept not only in thermodynamic but alo in Information Theory. Entropy i the meaure of randomne in an eent or the number of way a ytem might be arranged and till gie the ame appearance. Thi definition can be applied from both thermodynamic and Information Theory. According to Shannon we can quantify entropy in a tatitical manner by aying that Entropy i equal to the expected alue of information. Since we know the expected alue of omething i the ummation of that omething multiplied by the function of that omething we can look at entropy in the following manner the entropy of a ytem i equal the negatie ummation of the that ytem multiplied by the function of that ytem: E [ Info[ p( a] ] p( a log[ p( a ] = H. Maximum Entropy Sampling Information about the preciion of an etimate i coneyed by the width of the confidence interal. If the confidence leel i high and the reulting interal width i ufficiently narrow then the alue of the parameter i reaonably precie. A ery wide confidence interal indicate a lot of uncertainty concerning the alue of the parameter that we are etimating. For example let x and be the ample mean and ample tandard deiation computed from a random ample obtained from a normal population with mean µ. Then 6

17 a 00(-α% confidence interal for µ i ( x tα / n x + tα / n and o the n n interal width i ( tα / n. In order to make the interal width half of the old width n we need to hae ( tα / n a the new width. In other word in ordinary tatitic 4n we need to collect four time the data to make the interal width half a wide. We ue the principle of Maximum Entropy for analyzing the aailable information in order to determine a unique poterior probability ditribution. Let Θ be a multiariate dicrete ariable taking alue on the domain Ω. Then we can ue the Maximum Entropy Sampling (MES principle to derie a imple deign criterion. The principle of maximum entropy i a method for analyzing the aailable information in order to determine a unique epitemic probability ditribution. The principle of maximum entropy tate that the entropy concept can alo be ued to infer a model for a probability ditribution. It tate that the leat biaed model that encode the gien information i that which maximize the uncertainty meaure H(p while remaining conitent with thi information. The maximum entropy principle i like other Bayeian method in that it make explicit ue of prior information. Thi i an alternatie to the method of inference of claical tatitic. Theorem : Let Y i ( xi = µ ( xi + ε i where i ε are independent and are identically ditributed (iid N(0 σ and σ i known; uppoeθ ha a two point prior with ma w and w 0 < w < at and repectiely. Suppoe that each x i arie in a compact ubet X 7

18 d of R and that µ ( x i a bounded function oer X. Then an optimal deign can be found upported at a ingle point x* in X with: x* = arg X min{ µ ( x x µ ( } The following example illutrate the ue of thi criterion. Example: Conider the imple regreion problem Y i ( xi = µ ( xi + ε i or Yi = 0 + x i + ε i 5 with probability = w 0 7 = 7 = with probability w 3 In figure ( we how a graph of y = 5 + 7x and y = 7 + 3x We want to find the next bet x alue at which to take a ample in order to make the bet poible deciion about which et of parameter i true. After chooing thi new 8

19 x alue we meaure an oberation (i.e. the y-alue. We wih thi y-alue to gie u the mot information about the correct model. Therefore by the gien deign criterion x* = arg X min{ µ ( x x µ ( } x* X + x we get: = arg min{(5 + 7x (7 3 } x* arg X min{44 96x + 6x } = where 44 96x + 6x 70 i a parabola a illutrated in figure (. Figure ( x 5 We can find the minimum of thi parabola at x = 3. There i no maximum of thi parabola. There we find a bounded pace to find the maximum. For example if we wih to find the maximum in the bounded interal [4] then the parabola ha the maximum at x = a illutrated in figure (3. 9

20 70 Figure ( x 5 In addition in figure (4 the circled area how that the two line from figure ( hae great eparation at x =. Thu we find the maximum information by uing the MES principle. That i by chooing to take a new oberation (y-alue at x = we gain the mot information about which of the two line i correct. 70 Figure ( y x 7+3x x 5 0

21 III. IMPLEMENTATION A. Our Model Our model i baed on the emi-empirical MISR (multi-angle imaging pectrometer BRDF (bi-directional reflectance ditribution function Modified Rahman model. It ha 3 parameter r 0 k and b and 4 input ariable φ φ where i the zenith angle and φ i the azimuth angle. Each angle i meaured with repect to the point of iew of the oberer or intrument taking the oberation (a indicated by the ubcript or the un (a indicated by the ubcript a can be een in figure 5 below. The model i: [ ] ( (. (..exp } co( {co( co( co( ( 0 k h p b r R φ φ φ φ Ω + = where ( R φ φ i the reflectance ( ( 0 G r h φ φ φ φ + + = co( tan( tan( ( tan ( tan ( G φ φ φ φ + = and co( in( in( co( co( ( p φ φ + = Ω Figure (5

22 After obtaining initial atellite data we will diide our algorithm into two part:. Regreion Analyi: Regreion i performed and the following quantitie are calculated: a. Etimate of the parameter r 0 k and b. b. R - : The coariance matrix of the etimated parameter c. σ: the etimated tandard deiation of the error which are aumed to be normal. Maximization of Entropy: In thi tep we find the bet location (point to which to end the helicopter. Thi i done by uing the technique uggeted by Sebatiani and Wynn: To maximize the amount of information about the poterior parameter we hould maximize the entropy of the ditribution function. Mathematically maximizing the entropy i achieed by maximizing the quantity log[det( X Σ X + R] where X i the deign matrix of new oberation to be taken X i the tranpoe of the X matrix Σ i the ariance matrix of the error term in the regreion and R i the inere of the prior coariance matrix of the parameter (Sebatiani and Wynn 000. In our cae the X matrix i a function of the input data ( φ φ R i taken to be the inere of the poterior ariance matrix of the parameter taken from the regreion in tep and a the error term are aumed to be independent and identically ditributed their coariance matrix Σ i an identity matrix multiplied by the ariance of the error which again i taken to be the etimated alue from tep thu we take Σ = σ I.

23 With thee alue and noting that finding the et of alue that maximize the log of a function i equialent to finding the parameter that maximize the function itelf we are left with maximizing: ( X Σ X + R det = det X X + R = det(r det I σ - + X XR a contant with repect to the parameter to be etimated thi lead u to σ. Since R i maximizing det I + σ X XR -. Hence the determinant aboe i maximized by finding the alue of the input data ( φ for a gien poition of the un at the current time ( φ that maximize thi lat quantity. The robotic helicopter i programmed to go to the coordinate defined at the end of tep in order to collect new oberation. Thee oberation are ued to upplement the original data et and we return to tep one. Thi loop i repeated until the parameter etimate achiee the deired leel of accuracy in their meaurement a meaured by the poterior coariance matrix of the parameter R. B. Linearizing the Model A near linear fit of thi model i accomplihed by taking the natural logarithm of R φ φ which reult in the following equation: ( [ co( co( {co( + co( }] b. ( ln R( φ φ = ln r0 + ( k ln p Ω + ln h( φ φ Note that aide from term ln h( φ φ the function i linear in all three parameter r 0 k and b. Linearization of ln h (which contain a nonlinear i r 0 3

24 accomplihed by uing the alue of r 0 from the preiou iteration where at iteration n in the linear leat-quare fit: h ( n ( n r0 ( φ φ = + n + G ( φ φ where (0 r 0 i et equal to zero. In Matlab function regre and regtat return the etimated linear fit of the parameter along with their confidence interal (i.e. error bar coariance matrix and other tatitic about the fit uch a etimated function alue and mean quared error. The function nlinfit etimate the coefficient of a nonlinear regreion function uing leat quare. In order to get confidence interal the output of nlinfit( are ued with nlparci(. C. Experiment Setup We aume that the un i moing lightly between oberation. For implicity we aume the un moe through π radian in a day o it moe through π radian per minute. Thu we allow for the poition of the un moe 0.0 radian between meaurement (i.e. aume about minute before the helicopter can moe between location to take new meaurement. Auming 0 random oberation location for φv and V (zenith and azimuth of the iewpoint and that φs and S (zenith and azimuth of the un take on the alue: π ( k for k = 0. Thee 0 random oberation can then be examined 4 in and of themele or be taken to be the initial random oberation with which a firt 4

25 regreion analyi can be performed in order to begin the maximum entropy proce outlined aboe. Our imulation produce the following table of alue: Oberation R φ S φ V S V IV. RESULTS A. Etimation of the Parameter We perform the three tatitical method we mentioned in Chapter II uing the model parameter r 0 = 0. k = 0.9 and b = -0. for thi imulation. With each parameter the tandard error for linear regreion are directly calculated from the regreion proce. Howeer with nonlinear regreion and MLE we ue the Fiher Information Matrix a an approximation tool (lower bound. 5

26 Point Standard 95% CI Method Parameter Etimate Error (Confidence Interal CI Width Linear Regreion r Non L. Regreion r MLE r Linear Regreion k Non L. Regreion k MLE k Linear Regreion b Non L. Regreion b MLE b According to the table of reult Nonlinear regreion proide the bet point etimate for parameter r 0 and k while MLE and Linear Regreion proide the bet point etimate for parameter b. Howeer Linear Regreion proide the narrowet 95% CI for all three parameter. Although linear regreion did not proide the bet point etimate for r 0 and k it etimate were ery cloe to thoe of Nonlinear Regreion and MLE. Thu we conclude there i no practical reaon why we hould not chooe Linear Regreion for parameter etimation. Nonlinear regreion and MLE hae the ame tandard error (SE ince both utilize the Fiher information matrix to obtain etimate of the tandard error. In theory thoe etimate of the tandard error are lower bound on the true error of the model. With eery parameter we find: SE(uing Fiher > SE(linear regreion. B. Maximization of Entropy We implemented Bayeian adaptie algorithm with following aumption: 6

27 . We hae no initial data. The un moe lightly between oberation (0.0 radian between meaurement 3. The helicopter moe between location to take the new meaurement with the time interal of two minute After obtaining 0 random (a decribed in the ection aboe oberation location for φv and V (zenith and azimuth of the iewpoint and φs and S (zenith and azimuth of the un alue changing lightly between oberation we obtain the following table of alue of parameter etimate and their tandard error. Parameter Etimate Standard error chooing random oberation r k b Alternatiely uing our Bayeian adaptie algorithm we can chooe the initial ten unique oberation location with the ame poition of un (moing lightly between oberation. We take the identity matrix a an uninformatie prior matrix R - = σ I and aume that error ariance σ i about 0-6 baed on preiou knowledge of the meaurement intrument. The flowing table gie the location for the iewpoint uing location decider algorithm. 7

28 Oberation φ The robotic helicopter i ent to thee location and collect reflectance data. Through the iteratie regreion proce decribed earlier the following parameter etimate are obtained. Parameter Etimate Standard error chooing random oberation r K B We can compare the tandard error for the etimation of the parameter by chooing random location at which to take oberation or by chooing location according to the Bayeian adaptie algorithm to determine where to collect the reflectance data. Our reult how that the tandard error uing Bayeian adaptie algorithm are much maller than the tandard error for the etimate when the location 8

29 of the oberation were random o the parameter etimate are more accurate in uing the Bayeian adaptie algorithm if a fixed number of oberation are to be taken. Alternatiely aume that we hae ome initial data from a atellite. If we hae ten initial data point then we hae two option.. We can take more oberation at random or. We can pick only a few well choen location uing the Bayeian adaptie algorithm at which to take more oberation that would gie u approximately the ame parameter etimate. We take the next oberation baed on Bayeian adaptie algorithm that ue the prior ditribution for the ariance of the parameter matrix R - and the etimated ariance of the parameter from the regreion on the firt ten data point. It alo calculate the ariance of error uing the aboe input argument. Let u firt take 30 more oberation at random. So we hae 40 data point including the 0 initial data point. The table below gie the numeric alue of the parameter etimate and the tandard error. Parameter Etimate Standard error chooing random oberation r k b Compared to our preiou experiment with 0 random location the tandard error for 30 random location are about half a large which i expected. Recall that we had noted earlier that in random ampling approximately four time a much data i 9

30 required in order to obtain etimate that are twice a accurate when uing random ampling. To inetigate the Bayeian adaptie algorithm we pick 0 key location at which to take oberation. We proceed a follow:. Start with the firt ten initial data point. Ue the regreion output a the prior ditribution R - and uing the MSE for 3. Pick 0 new location uing the location decider algorithm (till auming the un i moing lightly between oberation After deciding upon the location we collect reflectance data. We get the following parameter etimate with the collected data. σ Parameter Etimate Standard error chooing random oberation r k b The table aboe how that 0 carefully choen location at which to collect the reflectance oberation proide approximately the ame parameter etimate and tandard error but the Bayeian adaptie algorithm i a lot more efficient to chooe location a only 0 additional location were required rather than 30 additional random location. 30

31 V. CONCLUSIONS One of the main adantage of thi particular implementation i it higher efficiency when compared to random ampling. The reduction rate of the tandard error or the error bar are much fater than thoe offered by mere random ampling. With thee algorithm future poition that will yield the mot amount of information are found much more quickly. Thi efficiency amount to aing of time and money during actual data collection and analyi. The linear regreion method utilized in thi implementation offered a big adantage. Linear regreion yield ytem of linear equation. Thee ytem are in turn much impler to ole. Howeer thi implementation i not a true linear regreion. It i in fact a near linear regreion. The initial guee are needed in order to complete the calculation and then iteratie procedure are employed. VI. RECOMMENDATIONS AND FUTURE RESEARCH A. Non-Normal Prior A big part of the future work that hould be done i to inetigate the iue of working with data parameter that are not normal. In our model we make the aumption that the parameter r 0 k and b are to be normally ditributed. Thi aumption hould not be interpreted a far fetched becaue of the central limit theorem which tate that with the collecting of more information on a ytem that the etimate of the parameter of the ytem will become normally ditributed hence the name normal. In making aumption though quetion often arie a to what to do if your aumption are incorrect. 3

32 In term of what we were doing an intereting quetion arie a far a what to do if the ditribution i not normal or if the ditribution i completely unknown. A mentioned in the preiou ection on Bayeian Statitic we want to ue our learning proce on our ytem. In that ection it wa dicued a to how to ue thi ytem when our prior are known but now they are not known and that i where the problem lie. We need a way to garner the bet poible etimate of thoe prior. To ole problem like thi we can ue ampling that i ampling in term of Bayeian Etimation Method. Uing Bayeian Etimation Method baically run a recurie Bayeian ytem. It i a method for etimating unknown probability ditribution oer time uing new information. B. Obtaining Poterior Ditribution from Mey Sytem A problem that one might run into i obtaining a poterior ditribution from a ytem in which the ditribution of it parameter i unknown becaue high-dimenional ytem are often difficult to normalize. They can reult in integral that cannot be computed directly. In thi cae one would want to ue Monte Carlo Marko Chain method to ealuate the poterior. Thee are the ampling Bayeian Etimation Method that were dicued earlier. There are eeral option in doing o; the mot prealent would be the Metropoli-Hating Algorithm which generate a random walk uing a denity proportion and a rejection method for next moe. In pecial cae of thi Gibb Sampling i uually preferred. It i preferable becaue unlike Metropoli-Hating Algorithm Gibb doe not hae the random walk propertie. Gibb Sampling relie on conditional probabilitie. Before getting into thee one hould firt hae an undertanding of Marko Chain and the Marko proce. 3

33 A Marko Chain i eentially jut any kind of equence of random ariable; with the range of the probability of any random ariable occurring i the tate pace. Marko Chain are often time built in matrix form becaue computation i eaier and more undertandable thi way. In matrix term our tate pace become a tate matrix where each probability i the probability of moing from one tate to the next. In Bayeian term the original tate matrix can be iewed a the prior ditribution. In a Marko proce we ue Bayeian propertie to change our probabilitie when we are gien new information about the probabilitie. A wa mentioned in the Bayeian ection thi new information i alo know a the likelihood which in the Marko proce i referred to a the tranitional probabilitie. Thee tranitional probabilitie like the tate probabilitie can be formed into a tranition matrix. One would then take the original tate matrix and the tranition matrix and ue matrix multiplication to garner a new matrix. Thi new matrix in Bayeian term would be your poterior in the Marko proce i your new tate matrix. Marko Chain theory tate that by recuriely doing thi proce i.e. taking the new tate matrix and applying the tranitional matrix to it we will eentually reach a conergence alo known a a teady-tate matrix. Our Monte Carlo Marko Chain olution to the problem earlier tated eentially doe the following: you firt build a ditribution from a Marko Chain and then chooe tranition probabilitie to ue that ditribution to create a new ditribution of interet. We then hae a ytem where each random ariable i choen baed on the preiou ariable in the chain and all that i needed i to run the ytem until conergence alo known a walking the Marko chain. In uing thi type of method tough integration i not required and neither i normalization. That how MCMC method work but more 33

34 pecifically how doe Metropoli-Hating Algorithm work? We tart by taking an initial alue from our ytem. It can be anything a long a it lie inide the ytem. Then we take thi initial alue and ample a candidate point from a jumping ditribution. The candidate point i equal to the probability of obtaining the new point gien we hae obered the initial point. Once we hae the point we now need to know whether or not to accept or reject it o we deelop an acceptance ratio. The acceptance ratio i equal to the probability of the candidate point diided by the initial point. If our acceptance ratio i le than we accept the candidate point with a probability of the acceptance ratio. If the acceptance ratio i greater than then we accept the candidate point and et that point a the new point to ue intead of the initial point. We then continue on by repeating thi proce which doe the following: When we accept a candidate point with a probability of the acceptance ratio obtained thi probability become the new probability of the moe in the Marko Chain and by repeating thi uing thee tranition probabilitie we will reach the conergence a earlier dicued. A pecial cae of Metropoli-Hating Algorithm i the Gibb ampler wherein the random alue i alway accepted during our rejection proce. What remain i how to form a Marko Chain when the alue conerge to the target ditribution. One of the main point of the Gibb ampler i that it only conider uniariate conditional ditribution. Gibb ue thee becaue they are far eaier to compute than complex joint ditribution and often time hae impler form and thu it i eaier to conider a equence of conditional ditribution than it i to obtain the marginal by integration of the joint ditribution. The ampler tart out with an initial alue ay y 0 for y and obtain x 0 by generating a random ariable from the conditional ditribution of x gien that y i 34

35 equal to y 0. The ampler then ue that alue obtained to generate a new alue of y. Thi alue come from the conditional ditribution of y gien x i equal to x 0. The ampler proceed like thi eemingly playing tenni where it olley back and forth from one conditional ditribution to the other. Repeating thi proce will eentually like in the other MCMC method bring u to a teady-tate. C. Future Reearch A future reearch project could perhap examine the ituation where rather than ending the helicopter to the next bet location we hould end it to the next bet efficient location in term of time power etc. In addition to the application dicued in thi paper thi reearch ha potential application in other field uch a military medical and robotic for object recognition and tracking. APPENDIX A: Reference Baye' theorem. (005. Wikipedia The Free Encyclopedia. Retrieed December from BRDF Explained. (000 Augut. Retrieed December from Deore J.L. (999. Probability and Statitic for Engineering and the Science (5 th ed.. Brook/Cole Publihing Company. Information theory. (005. Wikipedia The Free Encyclopedia Retrieed December from Loredo T.J. Bayeian Adaptie Exploration Unpublihed technical report. Marko Chain Monte Carlo. (005. Wikipedia The Free Encyclopedia. Retrieed December from 35

36 Maximum Entropy (005. Wikipedia The Free Encyclopedia. Retrieed December from Montgomery D.C. and Peck E.A. (99. Intorduction TO Linear Regreion Analyi ( nd ed.. New York: Wiley Intercience. Myung J.I and Naarro D. J. (004. Information Matrix Ohio State Unierity webite. Retrieed Noember Sebatiani P. and Wynn H. P. (000. Maximum entropy ampling and optimal Bayeian experimental deign J.R. Statit. Soc. B. 6 Part pp Sebatiani P. and Wynn H. P. (00. Experimental Deign to Maximize Information Technical report Department of Mathematic and Statitic Unierity of Maachuett at Amhert. Spall J.C. (004. The Fiher Information Matrix: Performance Meaure and Monte Carlo-Baed Computation. Retrieed Noember from National Intitute of Standard and Technology etric/permis_003/proceeding/spall.pdf Wackeryly D.D. Mendenhall W. III and Scheaffer R.L. (00. Mathematical Statitic with Application (6 th ed.. Pacific Groe: Duxbury Pre. Weaer W. and Shannon C. E. The Mathematical Theory of Communication Urbana IL: Unierity of Illinoi Pre 949 Wheeler K. R. (005. Parameterization of the Bidirectional Surface Reflectance NASA Ame Reearch Center MS 59-. Wikle C. K. (003. Hierarchical Bayeian model for predicting the pread of ecological procee. Ecology

37 APPENDIX B: Matlab function Matlab function are proided on the attached CD. An oeriew of the function their parameter and their return alue are proided below. [beta co me] = linrahman(ra Input : A = [phi phi theta theta] (angle MUST be in thi particular order R i the BRDF dependent ector Output: Thi function calculate a "linearized" leat quare fit of the Modified Rahman model parameter r 0 k and b. It output the ector beta = [r0_hat k_hat b_hat] (in that order the coariance matrix and me (mean quared error. Included i a code for printing out 95% interal and the tandard error. To actiate the code for 95% CI and SE' imply remoe comment from the code at the end of thi program. beta = nlinrahman(rabeta0 Input: A = [phi phi theta theta] (angle MUST be in thi particular order R i the BRDF dependent ector beta0 = [r0 k b] (in thi order i the initial gue of parameter Output: Thi function calculate nonlinear leat quare fit of the Modified Rahman model parameter r0 k and b. It output the ector beta = [r0_hat k_hat b_hat] after calling modrahman. Included i a code for printing out 95% interal for the etimated beta. To actiate the code for 95% CI imply remoe the comment from the code at the end of thi program. 37

38 R_hat = modrahman(beta0a Input: beta0 = [r0 k b] (in thi order i the initial gue of parameter A = [phi phi theta theta] (angle MUST be in thi particular order Thi functin accept a matrix A = [phi phi theta theta] and an initial gue x3 ector beta0 = [r0 k b] Angle in matrix A MUST be in that particular order and parameter in ector beta0 MUST be in that particular order. Output: It output the correponding BRDF alue of the Modified Rahman model of reflectance (R_hat Thi function i intended to be ued with nlinrahman (it will be called by nlinrahman F = loglik(rax0 Input: R i the BRDF dependent ector A = [phi phi theta theta](angle MUST be in thi order x0 i the ector of initial guee of r0 k b (MUST be in thi order Output: Thi function calculate -log(likelihood of the error~n(0^ gien the ector of the parameter x0=[r0 k0 b ] (in that order where i the tandard deiation of the error of the model Error~N(0^ loglik i intended to be ued w/fminearch to obtain MLE' of r0 k & b uing beta = fminearch(@(x loglik(raxx0 [beta ] = maxlik(rabeta0t Input: R i the BRDF dependent ector A = [phi phi theta theta] (angle MUST be in thi order beta0 i the ector of initial guee = [r0 k b](parameter MUST be in thi order 38

39 t i an initial gue of the tandard deiation of the error. Ouput: Thi function calculate the maximum likelihood etimator of r0 k & b It alo calculate the MLE of the tandard deiation of the error by calling the function loglik then finding it' maximum uing fminearch. The error i aumed to be Normal(0^ [newxmat] = xmatfunction(newtheta newtheta newphi newphi Input: newtheta and newphi are new poition of the Sun and newphi and newphi are the new poition of the iewpoint. Ouput: Thi function calculate the new X matrix where newxmat=[ X X ]. Here X andx are the ector of oberation. Thi function i intended to be ued in maxfunc (. maxdet = maxfunc (newxmat priorcomat arerr Input: newxmat contain three column haing the following format: newxmat=[ X X ] PriorCoMat i the prior coariance matrix arerr i the ariance of tandard error Output: The function calculate the max {det[ I + (/ σ X X R - ]} and tore it in the ariable maxdet. Thi function i called in minfunction( that calculate the minimum of the determinant correponding the new iew alue minalue = minfunction(newviewvalthetasphispriorcomat arerr Input: newviewval contain the poition for the iew in the ame ector. 39

40 ThetaS and PhiS are the poition of the Sun. priorcomat i prior coariance matrix arerr i the ariance of tandard error. Ouput: Thi function calculate the minimum i.e. - max {det[ I + (/ σ X X R - ]} correponding the new iew alue. Thi function i being called in ReadInputMat( [newphiv newthetav] = newpoition(newsuntheta newsunphi priorcomat arerr Input: newsuntheta and newsunphi are the new poition of the Sun. priorcomat i the prior coariance matrix. arerr i ariance of tandard error Output: Thi function calculate the new alue of theta and phi for the iewpoint i.e. new location for the iew at which to take oberation. Thi function ue a in built matlab function fmincon( to find a minimum of a contrained nonlinear multiariable function. fmincon( need an initial gue for the location of the iew and after a maximum number of iteration return the new location for iewpoint. 40