Optimal designs for estimating pairs of coefficients in Fourier regression models

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1 Optial designs for estiating pairs of coefficients in Fourier regression odels Holger Dette Ruhr-Universität Bochu Faultät für Matheati 70 Bochu Gerany e-ail: Viatcheslav B. Melas St. Petersburg State University Departent of Matheatics St. Petersburg Russia eail: Petr Shpilev St. Petersburg State University Departent of Matheatics St. Petersburg Russia eail: January 00 Abstract In the coon Fourier regression odel we investigate the optial design proble for estiating pairs of the coefficients where the explanatory variable varies in the interval [ π π]. L-optial designs are considered and for any iportant cases L- optial designs can be found explicitly where the coplexity of the solution depends on the degree of the trigonoetric regression odel and the order of the ters for which the pair of the coefficients has to be estiated. Keywords and Phrases: L-optial designs Fourier regression odels paraeter subsets equivalence theore AMS Subject classification: 6K05

2 Introduction Fourier regression odels of the for (. y β 0 + β j sin(jt + j β j cos(jt + ε t [ π π]. j are widely used in applications to describe periodic phenoena. Typical subject areas include engineering [see e.g. McCool (979] edicine [see e.g. Kitsos Titterington and Torsney (9] agronoy [see e.g. Weber and Liebig (9] or ore generally biology [see the recent collection of research papers edited by Lestrel (997]. Applications of trigonoetric regression odels also appear in two diensional shape analysis [see e.g. Young and Ehrlich (977 and Currie et al. (000 aong any others]. An optial design can iprove the efficiency of the statistical analysis substantially and therefore the proble of designing experients for Fourier regression odels has been discussed by several authors. Optial designs with respect to Kiefer s φ p -optiality criteria have been studied by Karlin and Studden (966 page 37 Hill (97 or Wu (00 aong others [see also Puelshei (993 p. ] while Lau and Studden (95 discuss the proble of constructing robust designs if the degree in odel (. is not exactly nown. Designs for identifying the degree have been deterined by Biederann Dette and Hoffann (007 Dette and Haller (99 and Zen and Tsai (00. More recent wor discussed the proble of constructing optial designs for the estiation of a particular coefficient in the Fourier regression odel (. [see Dette and Melas (003 and Dette Melas and Shpilev (007]. These authors also deonstrated that unifor designs are optial for estiating a subset of the coefficients {β i β i... β ir β ir } where i <... < i r r {... }. The ain purpose of the present paper is to obtain further insight into the construction of optial designs for estiating paraeter subsystes in the Fourier regression odel (.. In particular we are interested in the L-optial design proble for estiating pairs of the coefficients {β i β i } and {β j β j } where i i {0... } j j {... }. The precise estiation of specific pairs of coefficients is of particular interest because in any biological applications such as two diensional shape analysis one or two coefficients have a concrete biological interpretation [see e.g. Young and Ehrlich (977 Currie et al. (000]. In Section we introduce the general notation and state several preliinary results. We forulate and prove a particular version of the equivalence theore for L-optial designs which is an iportant tool for the deterination of the optial designs in Section 3. Here L- optial designs are found explicitly for several cases. Finally several exaples are presented in Section in order to illustrate the theoretical results.

3 L-optial designs Consider the trigonoetric regression odel (. define β (β 0 β...β T as the vector of unnown paraeters and f(t (f 0 (t... f (t T ( sin t cos t... sin(t cos(t T. as the vector of regression functions. Following Kiefer (97 we call any probability easure ξ on the design space [ π π] with finite support an (approxiate design. The support points of the design ξ give the locations where observations are taen while the weights give the corresponding proportions of the total nuber of observations to be taen at these points. If the designs ξ puts asses ξ i at the points t i (i... and n uncorrelated observations can be taen then the the quantities ξ i n are rounded to integers such that i n i n [for a rounding procedure - see e.g. Puelshei and Rieder (99] and the experienter taes n i observations at each t i (i.... In this case the covariance atrix of the least squares estiate for the paraeter β in the trigonoetric regression odel (. is approxiately given by σ n M (ξ where ( π (. M(ξ π f(tf T (tdξ(t R + + denotes the inforation atrix of the design ξ [see Puelshei (993]. Note that for a syetric design ξ after an appropriate perutation P R + + of the order of the regression functions the inforation atrix (. will be bloc diagonal that is ( Mc (ξ 0 (. M(ξ P M(ξP 0 M s (ξ with blocs given by (.3 (. For a given atrix M c (ξ M s (ξ ( π π cos(it cos(jtdξ(t ( π sin(it sin(jtdξ(t π ij0 ij. (.5 L l i li T i0 3

4 with vectors l i R + the class Ξ L is defined as the set of all approxiate designs for which the linear cobinations of the paraeters li T β i 0... are estiable that is l i Range(M(ξ; i Siilarly the sets Ξ s and Ξ c are defined as the sets of all designs for which the atrices M s (ξ and M c (ξ are nonsingular respectively. Finally a designs ξ is called L-optial if ξ arg in ξ Ξ L trlm + (ξ where L is a fixed and nonnegative definite atrix and for a given atrix A the atrix A + is the pseudo-inverse which is characterized by the four conditions (a AA + A A (b A + AA + A + (c (AA + T AA + (d (A + A T A + A. [see Rao (96]. The following result gives a characterization of L-optial designs which is particularly useful for designs with a singular inforation atrix. For a nonsingular inforation atrix this theore was forulated and proved in Eraov and Zhigljavsy (97. The theore is stated for a general regression odel y β T f(t + ε with + regression functions and a general design space χ. Theore. Let L R (+ (+ denote a given and nonnegative definite atrix of the for (.5 and assue that the set of inforation atrices is copact. The design ξ is an eleent of the class Ξ L if and only if The design ξ is L-optial if and only if l T i M (ξm(ξ l T i i (.6 ax t χ ϕ(t ξ trlm + (ξ where ϕ(t ξ f T (tm + (ξlm + (ξf(t Moreover the equality (.7 ϕ(t i ξ trlm + (ξ holds for any t i supp(ξ. Proof. The first part of this theore was proved in Rao (96. For a proof of the second part we use the following lea. Lea. If A is a nonnegative definite atrix then A+A + A + A and there is equality if and only only if A A + A.

5 Proof of Lea.. Let A B T B where B R n n. By definition of the pseudo-inverse atrix we have A + B + B +T so that A + A + A + A B T B + B + B +T B + B +T B T B (B T B + (B B +T (B B +T T (B B +T 0 where the third equality is obtained fro the identity B + B +T B T B B + (BB + T B B + BB + B B + B. Now we return to the proof of Theore.. In the first part we show that a L-optial design satisfies the conditions (.6 and (.7 For any α (0 define ξ α ( αξ + αξ t where ξ t denotes the Dirac easure at the point t and assue that the design ξ is L-optial. In this case the inequality trlm + (ξ α trlm + (ξ is satisfied for all α (0 which iplies trlm + (ξ α α0+ 0. α On the other hand we obtain observing the identity the inequality trlm + (ξ α α LM + (ξ α M(ξ α α 0 L M + (ξ α M(ξ α + LM + (ξ α M(ξ α α α { α0+ tr L M } { + (ξ α α0+ tr LM + (ξ α M(ξ } α α α M + (ξ α α0+ trlm + (ξ trlm + (ξ M(ξ t M + (ξ trlm + (ξ f T (sm + (ξ LM + (ξ f(sξ t (ds trlm + (ξ ϕ(t ξ 0. Therefore we have ϕ(t ξ trlm + (ξ for all t. Moreover the equality ϕ(t ξξ(dt trlm + (ξf(tf T (tm + (ξξ(dt trlm + (ξm(ξm + (ξ trlm + (ξ 5

6 is obviously fulfilled for any design ξ Ξ L. It now follows that stateents (.6 and (.7 are satisfied. In order to prove the converse iplication we assue that the design ξ satisfies (.6 but is not L-optial. We denote an L-optial design by ξ and define ξ α ( αξ + αξ. In the following discussion we use the notation M(ξ α M α M(ξ M M(ξ M and obtain fro Lea. the inequality 0 α( αtrl(m M + + M M + M M + M M + M + α which iplies trl(αm M + α M M + + αm M + α M M + M + α + trl( αm M + M M + + α M M + M M + M + α trl(αm M + α M M + + α M M + M M + M + α + trl(αm M + α M M + αm M + M M + + α M M + M M + M + α trl(αm (( αm + + αm + M + α + trl(αm M + α M M + αm M + + α M M + + M M + M M + M + α trl(αm (( αm + + αm + M + α + trl(( αm (αm + + ( αm + IM + α trl((αm + + ( αm + M α IM + α trl(αm + + ( αm + M + α trl(m + α trl(αm + + ( αm +. In other words we have proved that the functional ξ trlm + (ξ is convex on the set Ξ L and consequently it follows that 0 > trlm + (ξ trlm + (ξ trlm + (ξ α α0+ trlm + (ξ ϕ(t ξ ξ (dt. α On the other hand we have by stateent (.6 the inequality trlm + (ξ ϕ(t ξ which yields trlm + (ξ ϕ(t ξ ξ (dt. i.e. a contradiction to the previous inequality. Theore. is proved. In general an analytical deterination of L-optial designs is very difficult. Theore. can be used to chec the optiality of a given design nuerically. Moreover we will use this characterization in the following section for an explicit construction of L-optial designs for special paraeter subsystes in the Fourier regression odel (.. 6

7 3 Analytical solutions of the L optial design proble In the present section we develop explicit solutions of the L-optial design proble in the Fourier regression odel (. for several paraeter subsystes. We begin with the proble of constructing optial designs for estiating two coefficients corresponding to the sinus- or cosinus functions. More precisely define e j R + as the jth unit vector and consider the atrices L ( e et + e et L ( e et + e et We call an L-optial design with the atrix L ( and L ( L-optial design for estiating the pair of coefficients β β and β β respectively. Theore 3. Consider the trigonoetric regression odel (. with > 3. The design with ( ξ ( tn t n... t t... t n n n n n n i π n t i n + ( (i x x arctan( 5 n is L-optial for estiating the pair of coefficients β β. Moreover trlm + (ξ ( For any α [0 ω n ] the design with ξ ( ( π tn... t 0 t... t n π ω n α ω n... ω ω 0 ω... ω n α (i π n t i i... n n ω 0 5 5ω ω n ω i ω i i... n is L-optial for estiating the pair of coefficients β β. 7

8 Moreover trlm + (ξ ( trlm + (ξ ( The L-optial design ξ (0 for estiating the pair of coefficients β 0β coincides with the design ξ ( defined in part. Proof. We will only prove the first part of the theore for the case where the degree of the Fourier regression odel is even. All other stateents of the theore are treated siilarly. In this case and the design in part of Theore. can be rewritten as ( ξ( t t... t t... t with t i i π + ( (i x x arctan( 5. The proof is based on the characterization of L-optial designs given in Theore.. Recall the definition of the atrix M s M s (ξ R in (. and let s[ij] s[ij] (ξ denote the eleent of the atrix M s in the position (i j. We consider the syste of equations (3. (3. s[j ] 0 j... j s[j] 0 j... (3.3 (3. s[ ] sin ( x s[] sin (x. We will show below that the design ξ( satisfies these equations. Under this assuption we obtain for the function ϕ(t ξ in Theore. the representation ϕ(t ξ f T (tm + (ξ LM + (ξ f(t sin ( t sin ( x + sin (t sin (x. Now a straightforward calculation shows trlm + (ξ 5 + 3

9 and it is easy to prove that the condition (.6 is satisfied calculating the solution of the equation ϕ(tξ 0. Moreover a siilar calculation shows that the equalities t trlm + (ξ ϕ(t i ξ sin ( x + 5 sin (x + 3 are satisfied and it reains to show the identities (3. - (3.. For a proof of these identities we note that s[j ] sin(jt sin( t ω sin(jt sin( t and that sin( t ( sin( x. Consequently we obtain s[j ] sin( x ( sin(jt. Next we prove the identity ( sin(jt 0 if j. For this purpose we use the definition of t and obtain ( sin(jt sin(jx + sin ( πj ( + ( sin ( πj + ( ( ( sin ( ( πj [ sin(jx (πj cos(jx cos sin(jx (πj cos(jx + cos sin(jx + ( (j++ ( sin(jx 0 ( cos( πj + ( j ( cos(jx + cos ( πj ]. sin(jx Now applying standard trigonoetric forulae it is easy to calculate the su in the last expression if j i.e. 0 ( cos( πj cos( jπ 0 ( cos( πj cos(jπ cos( jπ ((cos(jπ + cos(3jπ (cos(3jπ + cos(5jπ ( (cos( + ( j+. ( 3jπ ( jπ + cos( 9

10 Therefore we have proved that ( sin(jt sin(jx ( + ( +j + ( j+ 0 which shows that the first equation in (3. is satisfied. The equality s[j] 0 (j follows by siilar arguents that is s[j] sin(jt sin(t sin(x ( sin(jt ( sin(jx cos( πj + ( j 0 sin(jx ( ( j + ( j 0. Finally we obtain by a direct calculation that s[ ] sin ( x ( sin ( x s[] sin (x ( sin (x which copletes the proof of Theore 3.. Rear 3. Note that Theore 3. is only correct for trigonoetric regression odels of degree > 3. For exaple if the L-optial design ξ(3 is given by ( π + x x x π x ξ (3 where x arctan( 5 and trlms (ξ(3 5 design ξ(0 is given by ( π π π 0 ξ (0 5 5 α For any α [ 5 5 π 5 α and trlmc (ξ( If 3 the L-optial design ξ (3 is given by ( π + x x x π x ξ (3 0 ] the L-optial

11 where x arctan( 5 and trlm s + (ξ 3+ 5 while the L-optial design ξ (0 is given by ( ξ (0 π π + x x 0 x π x π z α z z z z z α for any α [ 0 z] where z x and trlm c (ξ( The optiality of the designs can be easily checed by an application of Theore.. Rear 3. Note that by Theore 3. the su of variances of the estiates for the corresponding coefficients in a Fourier regression odel of degree > 3 is given by for the L-optial design while the D-optial design yields for this su. Thus the su of variances obtained by the L-optial design is approxiately 65% saller than the su obtained by the D-optial design. There are two other cases where L-optial designs for the trigonoetric regression odel (. can be constructed explicitly. They are stated in the following two theores. For the sae of brevity only a proof of Theore 3.3 is given here. Theore 3. Consider the trigonoetric regression odel (. with 3. The design ( ξ t t... t t... t ω ω... ω ω... ω with t π x t π t 3 π + x t i t i 3 + π i 5... ω z ω z ω 3 z ω i ω i 3 i 5... is L-optial for estiating one of the pairs of the coefficients {β β } {β β 6 } {β β 6 }. Here only the values x and z depend on the particular pair under consideration and are defined as the solution of the syste (3.5 trlm c (ξ x trlm c (ξ z 0 0. Siilarly for any α [0 ω ] the design ( π ξ t... t 0 t... t π ω α ω... ω ω 0 ω... ω α

12 3 { } { 6 } { 6 } x π/ π/0 z / 0. (3 5/ trlm s /3.70 ( 5 + 3/ Table : The solutions of x and z of the the syste (3.5. The L-optial design for estiating the specific pair of coefficients in the Fourier regression odel (. is specified in the first part of Theore {0 } {0 } {0 6} { } { 6} { 6} x π/ π/ z 0.59 / / trlmc ( 5 + 3/ Table : The solutions of x and z of the the syste (3.6. The L-optial design for estiating the specific pair of coefficients in the Fourier regression odel (. is specified in the second part of Theore 3.. with t 0 0 t x t π x t i t i 3 + π i 3... ω 0 z ω z ω z ω i ω i 3 i 3... is L-optial for estiating one of the pairs {β 0 β } {β 0 β } {β 0 β 6 } {β β } {β β 6 } {β β 6 }. Here only the values x and z depend on the particular pair under consideration and are defined as the solution of the syste (3.6 trlm s (ξ x trlm s (ξ z 0 0. Note that Theore 3. shows that all designs for estiating one of the considered pairs have the sae structure. Only the values x and z depend on the particular pair under consideration. Soe nuerical values for the paraeters x and z in Theore 3.. the Tables and.

13 Theore 3.3 Consider the trigonoetric regression odel (. with. The design ( ξ t t... t t... t ω ω... ω ω... ω with t x t x t 3 π x ω z ω z ω 3 z ω z t π x t i t i + π i ω i ω i i is L-optial for estiating one of the pairs of coefficients {β β } {β β 6 } {β β } {β β 6 } {β β } {β 6 β }. Here only the values x x and z depend on the particular pair under consideration and are defined as the solution of the syste (3.7 trlm s (ξ x 0 trlm s (ξ z 0 trlm s (ξ x 0 Siilarly if n 5 then for any α [0 ω n ] the design ( π ξ tn... t 0 t... t n π ω n α ω n... ω ω 0 ω... ω n α with t 0 0 t x t x t 3 π x t π x t i t i 5 + π i n ω 0 z z ω ω z ω ω 3 z ω i ω i 5 i n is optial for estiating any of the pairs of the coefficients {β 0 β } {β 0 β 6 } {β 0 β } {β β } {β β 6 } {β β } {β β 6 } {β β } {β 6 β }. Here only the values x x z and z depend on the particular pair under consideration and are defined as the solution of the syste (3. trlm c (ξ x 0 trlm c (ξ z 0 trlm c (ξ x 0 trlm c (ξ z 0. The nuerical values of the quantities x x z and z are listed in the Table 3 and. 3

14 { } { 6 } { } x π/ x π/ π/.9 z (6 6/ trlms (7 + 6/ { 6 } { } {6 } x arctan( 5/ 0.53 x.3 (π arctan( 5/.566 z 0.6 / 0.7 trlms.96 ( 5 + 3/ ( 5 + 3/ Table 3: The solutions x x and z of the the syste (3.7. The L-optial design for estiating the specific pair of coefficients in the Fourier regression odel (. is specified in the first part of Theore 3.3. {0} {06} {0} {} {6} x π/ π/3 π/ π/ 0.93 x π/ π/3 π/ π/ 0.93 z /6 / 0.07 z /6 / 0.07 trlmc ( 5 + 3/ + 9/.70 {} {6} {} {6} x π/ 0. x π/ π/ π/ π/ z z trlmc ( 5 + 3/ ( 5 + 3/ Table : The solutions of x x z and z of the the syste (3.. The L-optial design for estiating the specific pair of coefficients in the Fourier regression odel (. is specified in the second part of Theore 3.3.

15 Proof of Theore 3.3. We will only prove the first part of Theore 3.3 the second part is treated siilarly. We begin with the case i.e. for which it is easy to chec by direct calculations that the design ξ defined in the first part of Theore 3.3 is L-optial. The corresponding nuerical values of the quantities x x and z can be found as the solution of the syste of equations defined by (3.7. Now let and. We consider the syste of equations s[j] 0 j... j j 3 (3.9 s[j] 0 j... j s[] ( z sin (x + (/ z sin (x s[3] (z sin(x sin(3x + (/ z sin(x sin(3x s[] (z sin(x sin(x + (/ z sin(x sin(x s[] ( z sin (x + (/ z sin (x where s[ij] s[ij] (ξ is the eleent of the atrix M s (ξ R in the i-th row and j-th colun. We will prove below that these equalities are satisfied for the design ξ. In this case it follows that the quantities trlms (ξ and ϕ(t ξ f T (tm + (ξ LM + (ξ f(t in Theore. are independent of the value (note that the atrix L is a given diagonal atrix where the non vanishing entries depend on the particular pair of paraeters under consideration. Consequently it is sufficient to prove Theore 3.3 in the case which has been done in the previous paragraph In order to prove that the equalities (3.9 are satisfied we note that for i... i 3 we have s[i] + j sin(t j sin(it j ω j (sin(t j 3 sin(it j 3 ω j 3 + sin(t j sin(it j ω j j (sin(t j sin(it j ω j + sin(t j sin(it j ω j j sin(x z ( j (sin(it j 3 + sin(it j j + sin(x ( z ( j (sin(it j + sin(it j. j 5

16 For the first su in the last line we obtain + ( j (sin(it j 3 + sin(it j j ( ( j sin ( i (x + (s π + sin ( i (π x + (s π j ( sin ( iπ ( sin ( iπ ( cos ( ix iπ iπ cos ( ix iπ ( cos ( ix + iπ cos ( ix iπ 0. + iπ Siilarly it follows (substituting x for x that the second su also vanishes which iplies s[i] 0 for i... i 3. For the eleent s[i] we find for i... i 3 s[i] + (sin(t j 3 sin(it j 3 ω j 3 + sin(t j sin(it j ω j j (sin(t j sin(it j ω j + sin(t j sin(it j ω j j sin(x z (sin(it j 3 sin(it j j + sin(x ( z A straightforward calculation now yields (sin(it j 3 sin(it j j ( sin(it j sin(it j. j ( sin ( i (x + (s π sin ( i (π x + (s π j ( ( cos ix iπ iπ cos ( ix iπ + iπ + cos ( ix iπ cos ( ix + iπ sin ( 0 iπ and the sae arguents show that the second su also vanishes which iplies s[i] 0. To conclude the proof it reains to calculate the quantities s[ij] i j... for 6

17 which we obtain s[ij] + (sin(it r 3 sin(jt r 3 ω r 3 + sin(it r sin(jt r ω r r (sin(it r sin(jt r ω r + sin(it r sin(jt r ω r r z (sin(i(x + (r π sin(j(x + (r π r + sin(i( x + rπ sin(j( x + rπ ( + z sin(ix sin(jx ( ( (i+j(r + ( (i+jr r ( sin(ix sin(jx z ( + z sin(ix sin(jx ( ( (i+j(r + ( (i+jr. r Fro this representation it is obvious that the quantities s[ij] i j... have the values specified by the syste of equations and the theore has been proved. Rear 3.3 Note that for any with / < and for any β [0 ] the design ( π π + ξ(0 π... π + π π β... β is L-optial for estiating the pair of coefficients {β 0 β }. Moreover in this case it follows that trlm c (ξ (0 [see Dette and Melas (003 Lea.3] Exaples In this section we present several exaples which illustrate the theoretical results obtained in Section 3. 7

18 . L-optial design for estiating the coefficients of sin(t and sin(t in the Fourier regression odel of degree. In this exaple we present the L-optial design for estiating the coefficients β 3 β 7 (i.e. the coefficients at sin(t and sin(t in the trigonoetric regression odel of degree. The L-optial design is directly obtained fro Theore 3. and given by ( π x ξ(37 π x π + x x x π x π + x π x where x arctan ( The corresponding support points of the design ξ (37 are depicted in Figure Figure : Support points of the L-optial design for estiating the coefficients β 3 β 7 in the Fourier regression odel of degree. The atrices M s (ξ (37 M s (ξ(37 and Ms (ξ(37 for this design are given by M s (ξ( respectively. The atrix corresponding to the cosine ters is of no interest but is readily seen that (using an appropriate perutation of the regression functions ( M M + (ξ(37 + c (ξ( Ms (ξ(37.

19 A straightforward calculation shows that the atrix M + (ξ(37 LM + (ξ(37 is given by ( M + (ξ(37lm + (ξ( Ms (ξ(37 L sms (ξ(37 where L s is the corresponding bloc of the atrix L. Therefore the function ϕ(t ξ (37 is given explicitly by ϕ(t ξ(37 f T (tm + (ξ(37lm + (ξ(37f(t ( + 5 sin (t + (3 + 5 sin (t. 5 0 The equalities (.7 are checed by a direct calculation and the function ϕ(t ξ(37 is depicted in Figure. Figure : The function ϕ(t ξ(37 defined in Exaple... L-optial design for estiating the coefficients of cos(t and cos(3t in the Fourier regression odel of degree. We consider again the trigonoetric regression odel of degree and use Theore 3.3 to deterine the L-optial design for estiating the pair of coefficients β and β 6 which correspond to the ters cos(t and cos(3t. The L-optial design for estiating these coefficients is given by ( ξ(6 π.3 π π π 0.75 α α and the support points of the optial design are depicted in Figure 3. We finally note that a straightforward calculation yields for the function ϕ(t ξ (6 in Theore. ϕ(t ξ (6 9

20 f T (tm + (ξ (6 LM + (ξ (6 f(t cos(t cos(t cos(6t cos(t. This function is depicted in Figure. π π.3 π π.0 Figure 3: The support points of the L-optial design for estiating pair of coefficients β β 6 in the Fourier regression odel of degree. Figure : The function ϕ(t ξ(6 defined in Exaple.. Acnowledgeents. The support of the Deutsche Forschungsgeeinschaft (SFB 75 Koplexitätsredution in ultivariaten Datenstruturen is gratefully acnowledged. The wor of the authors was also supported in part by an NIH grant award IR0GM0776:0A and by the BMBF project SKAVOE. 0

21 References [] Biederann S. Dette H. and Hoffann P. (007. Constrained optial discriinating designs for Fourier regression odels. Annals of Statistical Matheatics in press. [] Currie A.J. Ganeshananda S. Noition D.A. Garric D. Shelbourne C.J.A. Oragguzie N. (000. Quantiative evalution of apple (Malus x doestica Borh. fruit shape by prinicple coponent analysis of Fourier descriptors. Euphytica 9-7. [3] Dette H. Haller G. (99. Optial Designs for the Identification of the Order of a Fourier Regression. Annals of Statistics [] Dette H. and Melas V.B. (003. Optial designs for estiating individual coefficients in Fourier regression odels. The Annals of Statistics [5] Dette H. Melas V.B. and Shpilev P.V. (007. Optial designs for estiating the coefficients of the lower frequencies in trigonoetric regression odels. Annals of the Institute of Statistical Matheatics [6] Eraov M. Zhigljavsy A. (97. Matheatical Theory of Optial Experient. Naua Moscow. (in Russian. [7] Hill P.D.H. (97. A note on the equivalence of D-optial design easures for three rival linear odels. Bioetria [] Karlin S. Studden W.J. (966. Tchebycheff systes: with applications in analysis and statistics. Interscience New Yor. [9] Kiefer J.C.(97. General equivalence theory for optiu designs (approxiate theory. The Annals of Statistics [0] Kitsos C.P. Titterington D.M. Torsney B. (9. An optial design proble in rhythoetry. Bioetrics [] Lestrel P.E. (997. Fourier descriptors and their applications in biology. Cabridge: Cabridge University Press. [] Lau T.S. Studden W.J. (95. Optial designs for trigonoetric and polynoial regression. Ann. Statist [3] McCool J.I. (979. Systeatic and rando errors in least squares estiation for circular contours. Precision Engineering 5-0.

22 [] Puelshei F. Rieder S. (99. Efficient rounding of approxiate designs. Bioetria [5] Puelshei F. (993. Optial Design of Experients. Wiley New Yor. [6] Rao C.R. (96. Linear statistical inference and its applications. Wiley New Yor. [7] Weber W.E. Liebig H.P. (9. Fitting response functions to observed data. Electronische Datenverarbeitung in Medizin und Biologie -9. [] Wu H. (00. Optial designs for first-order trigonoetric regression on a partial circle. Statistica Sinica [9] Young J.C. Ehrlich R. (977. Fourier bioetrics: haronic aplitudes as ultivariate shape descriptors. Systeatic Zoology [0] Zen M.-M. and Tsai M.-H. (00. Criterion-robust optial designs for odel discriination and paraeter estiation in Fourier regression odels. Journal of Statistical Planninng and Inference 75-7.

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