A one site-mobile-immobile model was selected as the direct (forward) problem to describe bio-macromolecular diffusion inside living cell in cylindrical coordinate system 130:

∂ F ∂ t ( ρ , t ) = D F ∂ 2 F ∂ ρ 2 ( ρ , t ) + D F 1 ρ ∂ F ∂ ρ ( ρ , t ) − K a ∗ F ( ρ , t ) + K d C ( ρ , t ) ∂ C ∂ t ( ρ , t ) = K a * F ( ρ , t ) − K d C ( ρ , t ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaWaaSaaaeaacqGHciITcqWGgbGraeaacqGHciITcqWG0baDaaGaeiikaGccciGae8xWdiNaeiilaWIaemiDaqNaeiykaKIaeyypa0Jaemiraq0aaSbaaSqaaiabdAeagbqabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaOGaemOrayeabaGaeyOaIyRae8xWdi3aaWbaaSqabeaacqaIYaGmaaaaaOGaeiikaGIae8xWdiNaeiilaWIaemiDaqNaeiykaKIaey4kaSIaemiraq0aaSbaaSqaaiabdAeagbqabaGcdaWcaaqaaiabigdaXaqaaiab=f8aYbaadaWcaaqaaiabgkGi2kabdAeagbqaaiabgkGi2kab=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@90DA@

Since a circular bleach spot was used to bombard the cell and to track the fluorescently tagged biomolecules inside bleach spot during the time course of a FRAP experiment, system (1) was written in cylindrical coordinate system.

Furthermore, the binding reaction was modeled by primary rate kinetic or single binding site model 113252630:

F + S ⇄ K d K a C MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGgbGrcqGHRaWkcqqGtbWudaGdnaWcbaGaee4saS0aaSbaaWqaaiabbggaHbqabaaaleaacqqGlbWsdaWgaaadbaGaeeizaqgabeaaaOGaayPKHiaawcziaiabboeadbaa@38DB@

subject to the following initial and Neumann boundary conditions:

F ( ρ , 0 ) = { 0 , F e q , 0 < ρ ≤ w w < ρ ≤ R C ( ρ , 0 ) = { 0 , C e q , 0 < ρ ≤ w w < ρ ≤ R ∂ F ∂ ρ | ρ = 0 , R = 0 ∂ C ∂ ρ | ρ = 0 , R = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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f8aYjabg2da9iabicdaWiabcYcaSiabdkfasbqabaGccqGH9aqpcqaIWaamaeaadaabcaqaamaalaaabaGaeyOaIyRaem4qameabaGaeyOaIyRae8xWdihaaaGaayjcSdWaaSbaaSqaaiab=f8aYjabg2da9iabicdaWiabcYcaSiabdkfasbqabaGccqGH9aqpcqaIWaamaaaaaa@8584@

where F is the concentration of free biomolecule, S is the concentration of vacant binding sites, C is the concentration of the bound complex(C = FS), D_F is the molecular diffusion coefficient (L²T^-1) of free biomolecule, K_a is the free biomolecule-vacant binding site association rate coefficient (T^-1), K_d is dissociation rate coefficient (T^-1), Ka∗ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaaa@3033@ = K_aS is the pseudo-association rate coefficient, ρ is radial coordinate (L) in the cylindrical coordinate system, w is the radius of the bleached area, R is the length of the spatial domain, and F_eq and C_eq are equilibrium concentration of F and C 2530:

F e q = K d K a ∗ + K d , C e q = K a ∗ K a ∗ + K d MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemOray0aaSbaaSqaaiabdwgaLjabdghaXbqabaGccqGH9aqpdaWcaaqaaiabdUealnaaBaaaleaacqWGKbazaeqaaaGcbaGaem4saS0aa0baaSqaaiabdggaHbqaaiabgEHiQaaakiabgUcaRiabdUealnaaBaaaleaacqWGKbazaeqaaaaakiabcYcaSaqaaiabdoeadnaaBaaaleaacqWGLbqzcqWGXbqCaeqaaOGaeyypa0ZaaSaaaeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaGcbaGaem4saS0aa0baaSqaaiabdggaHbqaaiabgEHiQaaakiabgUcaRiabdUealnaaBaaaleaacqWGKbazaeqaaaaaaaaaaa@4C2E@

The initial condition implies that photochemical bleaching inactivates the fluorescence tag on the biomolecules in the bleached area but does not change the concentrations of free and bound biomolecules or vacant binding sites. The boundary conditions imply that the diffusive biomolecule flux is zero at the center of the bleach spot and at the cell or nucleus membrane during time course of a FRAP experiment 30.

The forward problem was solved by the finite element method. The weak formulation of the dependent variables F and C were approximated using piecewise linear approximating functions 131373839:

F ' ( ρ , t ) = ∑ j = 1 N F j ( t ) φ j ( ρ ) C ' ( ρ , t ) = ∑ j = 1 N C j ( t ) φ j ( ρ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6501@

in which N is total number of nodes in the spatial domain, ϕ_j(ρ) are the selected linear basis functions, and F_j(t) and C_j(t) are the associated time-dependent unknown coefficients that represent the solution of Equation (1) at nodes within the domain.

Substitution of expressions (3) into Equation (1) will not satisfy the partial differential equation and hence will produce a residual. The goal of the finite element approximation is to minimize this error. This can be accomplished by introducing the weight function, ϕ_i(ρ), and setting the integral of the weighted residuals to zero. In the Galerkin method, which was used in this study, the weighting functions are chosen to be identical to the basis function 131:

∫ Ω [ ∂ F ' ∂ t ( ρ , t ) − D F ∂ 2 F ' ∂ ρ 2 ( ρ , t ) − D F 1 ρ ∂ F ' ∂ ρ ( ρ , t ) + K a ∗ F ' ( ρ , t ) − K d C ' ( ρ , t ) ] φ i ( ρ ) d Ω = 0 ∫ Ω [ ∂ C ' ∂ t ( ρ , t ) − K a ∗ F ' ( ρ , t ) + K d C ' ( ρ , t ) ] φ i ( ρ ) d Ω = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaWaa8qeaeaadaWadaqaamaalaaabaGaeyOaIyRaemOray0aaWbaaSqabeaacqGGNaWjaaaakeaacqGHciITcqWG0baDaaGaeiikaGccciGae8xWdiNaeiilaWIaemiDaqNaeiykaKIaeyOeI0Iaemiraq0aaSbaaSqaaiabdAeagbqabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaOGaemOrayKaei4jaCcabaGaeyOaIyRae8xWdi3aaWbaaSqabeaacqaIYaGmaaaaaOGaeiikaGIae8xWdiNaeiilaWIaemiDaqNaeiykaKIaeyOeI0Iaemiraq0aaSbaaSqaaiabdAeagbqabaGcdaWcaaqaaiabigdaXaqaaiab=f8aYbaadaWcaaqaaiabgkGi2kabdAeagjabcEcaNaqaaiabgkGi2kab=f8aYbaacqGGOaakcqWFbpGCcqGGSaalcqWG0baDcqGGPaqkcqGHRaWkcqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaOGaemOrayKaei4jaCIaeiikaGIae8xWdiNaeiilaWIaemiDaqNaeiykaKIaeyOeI0Iaem4saS0aaSbaaSqaaiabdsgaKbqabaGccqWGdbWqcqGGNaWjcqGGOaakcqWFbpGCcqGGSaalcqWG0baDcqGGPaqkaiaawUfacaGLDbaacqWFgpGzdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiab=f8aYjabcMcaPiabdsgaKjabgM6axbWcbaGaeyyQdCfabeqdcqGHRiI8aOGaeyypa0JaeGimaadabaWaa8qeaeaadaWadaqaamaalaaabaGaeyOaIyRaem4qamKaei4jaCcabaGaeyOaIyRaemiDaqhaaiabcIcaOiab=f8aYjabcYcaSiabdsha0jabcMcaPiabgkHiTiabdUealnaaDaaaleaacqWGHbqyaeaacqGHxiIkaaGccqWGgbGrcqGGNaWjcqGGOaakcqWFbpGCcqGGSaalcqWG0baDcqGGPaqkcqGHRaWkcqWGlbWsdaWgaaWcbaGaemizaqgabeaakiabdoeadjabcEcaNiabcIcaOiab=f8aYjabcYcaSiabdsha0jabcMcaPaGaay5waiaaw2faaiab=z8aMnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIae8xWdiNaeiykaKIaemizaqMaeyyQdCfaleaacqGHPoWvaeqaniabgUIiYdGccqGH9aqpcqaIWaamaaaaaa@B9D8@

where Ω is the study domain. Applying Green's first identity 3739 to equation (4) yields:

∑ e ∫ Ω e ∂ F ' ∂ t ( ρ , t ) φ i ( ρ ) d Ω + D F ∑ e ∫ Ω e ∂ F ' ∂ ρ ( ρ , t ) ∂ φ i ( ρ ) ∂ ρ d Ω − D F ∑ e ∫ Ω e ρ ∂ F ' ∂ r ( ρ , t ) φ i ( ρ ) d Ω + K a ∗ ∑ e ∫ Ω e F ' ( ρ , t ) φ i ( ρ ) d Ω − K d ∑ e ∫ Ω e C ' ( ρ , t ) φ i ( ρ ) d Ω = − q f n ∑ e ∫ Ω e ∂ C ' ∂ t ( ρ , t ) φ i ( ρ ) d Ω − K a ∗ ∑ e ∫ Ω e F ' ( ρ , t ) φ i ( ρ ) d Ω + K d ∑ e ∫ Ω e C ' ( ρ , t ) φ i ( ρ ) d Ω = − q c n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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f8aYjabcYcaSiabdsha0jabcMcaPiab=z8aMnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIae8xWdiNaeiykaKcaleaacqWGLbqzaeqaniabggHiLdGccqWGKbazcqGHPoWvcqGHsislcqWGlbWsdaWgaaWcbaGaemizaqgabeaakmaaqafabaWaa8qeaeaacqWGdbWqcqGGNaWjcqGGOaakcqWFbpGCcqGGSaalcqWG0baDcqGGPaqkcqWFgpGzdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiab=f8aYjabcMcaPiabdsgaKjabgM6axbWcbaGaeyyQdC1aaSbaaWqaaiabdwgaLbqabaaaleqaniabgUIiYdaaleaacqWGLbqzaeqaniabggHiLdGccqGH9aqpcqGHsislcqWGXbqCdaWgaaWcbaGaemOzayMaemOBa4gabeaaaOqaamaaqafabaWaa8qeaeaadaWcaaqaaiabgkGi2kabdoeadnaaCaaaleqabaGaei4jaCcaaaGcbaGaeyOaIyRaemiDaqhaaiabcIcaOiab=f8aYjabcYcaSiabdsha0jabcMcaPiab=z8aMnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIae8xWdiNaeiykaKIaemizaqMaeyyQdCfaleaacqGHPoWvdaWgaaadbaGaemyzaugabeaaaSqab0Gaey4kIipaaSqaaiabdwgaLbqab0GaeyyeIuoakiabgkHiTiabdUealnaaDaaaleaacqWGHbqyaeaacqGHxiIkaaGcdaaeqbqaamaapebabaGaemOrayKaei4jaCcaleaacqGHPoWvdaWgaaadbaGaemyzaugabeaaaSqab0Gaey4kIipakiabcIcaOiab=f8aYjabcYcaSiabdsha0jabcMcaPiab=z8aMnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIae8xWdiNaeiykaKcaleaacqWGLbqzaeqaniabggHiLdGccqWGKbazcqGHPoWvcqGHRaWkcqWGlbWsdaWgaaWcbaGaemizaqgabeaakmaaqafabaWaa8qeaeaacqWGdbWqcqGGNaWjcqGGOaakcqWFbpGCcqGGSaalcqWG0baDcqGGPaqkcqWFgpGzdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiab=f8aYjabcMcaPiabdsgaKjabgM6axbWcbaGaeyyQdC1aaSbaaWqaaiabdwgaLbqabaaaleqaniabgUIiYdaaleaacqWGLbqzaeqaniabggHiLdGccqGH9aqpcqGHsislcqWGXbqCdaWgaaWcbaGaem4yamMaemOBa4gabeaaaaaaaa@372B@

in which Ω_e is the domain of element and q_fn and q_cn are fluxes of the free and bound bio-macromolecules across the boundary out of the element, respectively.

The time derivatives in equation (5) were approximated using an Euler time-marching algorithm (backward finite difference scheme). Inserting equation (3) into equation (5) and integrating over the elements produces a system of time-dependent ordinary differential equations which can be formulated in matrix form:

[ A ] { F } n + 1 − { F } n Δ t + [ B ] { F } n + 1 + [ E ] { F } n + 1 − [ H ] { C } n + 1 = { Q f } n + 1 [ A ] { C } n + 1 − { C } n Δ t − [ E ] { F } n + 1 + [ H ] { C } n + 1 = { Q c } n + 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@A5FE@

where:

[ A 1 ] = ∑ e ∫ Ω e φ i φ j d Ω [ B ] = ∑ e D f ∫ Ω e ∂ φ i ∂ ρ ∂ φ j ∂ ρ d Ω − ∑ e D f ρ e ∫ Ω e ∂ φ i ∂ ρ φ j d Ω [ E ] = ∑ e K a ∗ ∫ Ω e φ i φ j d Ω [ H ] = ∑ e K d ∫ Ω e φ i φ j d Ω { Q f } = { 0 , 0 , ......... , 0 , q f n } { Q c } = { 0 , 0 , ......... , 0 , q c n } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabcUfaBjabdgeabnaaBaaaleaacqaIXaqmaeqaaOGaeiyxa0Laeyypa0ZaaabuaeaadaWdraqaaGGaciab=z8aMnaaBaaaleaacqWGPbqAaeqaaaqaaiabgM6axnaaBaaameaacqWGLbqzaeqaaaWcbeqdcqGHRiI8aaWcbaGaemyzaugabeqdcqGHris5aOGae8NXdy2aaSbaaSqaaiabdQgaQbqabaGccqWGKbazcqGHPoWvaeaacqGGBbWwcqWGcbGqcqGGDbqxcqGH9aqpdaaeqbqaaiabdseaenaaBaaaleaacqWGMbGzaeqaaOWaa8qeaeaadaWcaaqaaiabgkGi2kab=z8aMnaaBaaaleaacqWGPbqAaeqaaaGcbaGaeyOaIyRae8xWdihaaaWcbaGaeyyQdC1aaSbaaWqaaiabdwgaLbqabaaaleqaniabgUIiYdGcdaWcaaqaaiabgkGi2kab=z8aMnaaBaaaleaacqWGQbGAaeqaaaGcbaGaeyOaIyRae8xWdihaaaWcbaGaemyzaugabeqdcqGHris5aOGaemizaqMaeyyQdCLaeyOeI0YaaabuaeaadaWcaaqaaiabdseaenaaBaaaleaacqWGMbGzaeqaaaGcbaGae8xWdi3aaSbaaSqaaiabdwgaLbqabaaaaOWaa8qeaeaadaWcaaqaaiabgkGi2kab=z8aMnaaBaaaleaacqWGPbqAaeqaaaGcbaGaeyOaIyRae8xWdihaaaWcbaGaeyyQdC1aaSbaaWqaaiabdwgaLbqabaaaleqaniabgUIiYdGccqWFgpGzdaWgaaWcbaGaemOAaOgabeaaaeaacqWGLbqzaeqaniabggHiLdGccqWGKbazcqGHPoWvaeaacqGGBbWwcqWGfbqrcqGGDbqxcqGH9aqpdaaeqbqaaiabdUealnaaDaaaleaacqWGHbqyaeaacqGHxiIkaaGcdaWdraqaaiab=z8aMnaaBaaaleaacqWGPbqAaeqaaaqaaiabgM6axnaaBaaameaacqWGLbqzaeqaaaWcbeqdcqGHRiI8aaWcbaGaemyzaugabeqdcqGHris5aOGae8NXdy2aaSbaaSqaaiabdQgaQbqabaGccqWGKbazcqGHPoWvaeaacqGGBbWwcqWGibascqGGDbqxcqGH9aqpdaaeqbqaaiabdUealnaaBaaaleaacqWGKbazaeqaaOWaa8qeaeaacqWFgpGzdaWgaaWcbaGaemyAaKgabeaaaeaacqGHPoWvdaWgaaadbaGaemyzaugabeaaaSqab0Gaey4kIipaaSqaaiabdwgaLbqab0GaeyyeIuoakiab=z8aMnaaBaaaleaacqWGQbGAaeqaaOGaemizaqMaeyyQdCfabaGaei4EaSNaemyuae1aaSbaaSqaaiabdAgaMbqabaGccqGG9bqFcqGH9aqpcqGG7bWEcqaIWaamcqGGSaalcqaIWaamcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqaIWaamcqGGSaalcqWGXbqCdaWgaaWcbaGaemOzayMaemOBa4gabeaakiabc2ha9bqaaiabcUha7jabdgfarnaaBaaaleaacqWGJbWyaeqaaOGaeiyFa0Naeyypa0Jaei4EaSNaeGimaaJaeiilaWIaeGimaaJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiOla4IaeiOla4IaeiOla4IaeiOla4IaeiOla4IaeiOla4IaeiilaWIaeGimaaJaeiilaWIaemyCae3aaSbaaSqaaiabdogaJjabd6gaUbqabaGccqGG9bqFaaaa@F087@

in which ρ_e is the radial position of the centroid of element e.

The inverse problem was formulated as a nonlinear optimization problem in which model parameters p = [D_f, Ka∗ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaaa@3033@, K_d] were optimized by minimizing a penalty function representing the discrepancy between the observed and predicted average fluorescence intensity recovery time series inside the bleach spot 1. If the measurement errors asymptotically follow a multivariate normal distribution with zero mean and covariance matrix,V, the likelihood function can be formulated as 40:

L ( p ) = ( 2 π ) − N / 2 det ⁡ [ V ] − 1 / 2 exp ⁡ [ − 1 2 ( U ∗ − U ( p ) ) T V − 1 ( U ∗ − U ( p ) ) ] MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@66FD@

where L(β) is the likelihood function, N is the number of observations, p is the vector of the parameters being optimized, U* is a vector and/or matrix of observations, and U is a corresponding vector and/or matrix of model predictions as a function of the parameters being optimized. This vector is obtained by solving the forward problem. The maximum likelihood estimator consists of those values of the unknown parameters that maximize the magnitude of the same likelihood function 40. Since logarithm is a monotonically increasing function of its argument (the value of p that maximizes L(p) also maximizes ln L(p)), and because ln L(p) is simpler and much easier to use than L(p) itself, therefore ln L(p) is usually used in optimization:

ln ⁡ L ( p ) = − N 2 ln ⁡ ( 2 π ) − 1 2 det ⁡ [ V ] − 1 2 ( U ∗ − U ( p ) ) T V − 1 ( U ∗ − U ( p ) ) ] MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@64FD@

The maximum of the likelihood function must satisfy the set of equations ∂ ln L(p)/∂p = 0. When the error covariance matrix is known, maximization of equation (8) is equivalent to the minimization of the following weighted least square problem 1:

ϕ(p) = [(U* - U(p))^TV^-1(U* - U(p))]

where ϕ(p) is the objective or penalty function. If there is information about the values and distributions of parameters, it can be incorporated in the objective function as well 1:

φ ( p ) = [ ( U ∗ − U ( p ) ) T V − 1 ( U ∗ − U ( p ) ) ] + [ ( p ∗ − p ^ ) T V p − 1 ( p ∗ − p ^ ) ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6314@

in which p* is parameter vector containing a priori information, p^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGWbaCgaqcaaaa@2E25@ is the corresponding predicted parameter vector, and V_p is the covariance matrix for the parameter vector. This kind of optimization is known as Bayesian estimation. The second term in equation (10), which is sometimes called the plausibility criterion 1 insures that the optimized values of the parameters remain in some feasible region around p*. Matrices V and V_p, which are sometimes called weighting matrices, provide information about the measurement accuracy as well as any possible correlation between measurement errors and between parameters.

A limitation of equation (10) is that the error covariance matrix is generally not known. A common approach to overcoming this problem is to make some a priori assumptions about the structure of the error covariance matrix. In the absence of any additional information regarding the accuracy of input data, the simplest and most recommended way is to assume that observation errors are uncorrelated which implies setting V equal to the identity matrix and V_p to zero. In this case the optimization problem collapses to the well known ordinary least squares formulation 4142:

φ ( p ) = 1 2 r ( p ) T r ( p ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFgpGzcqGGOaakcqWGWbaCcqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabikdaYaaacqWGYbGCcqGGOaakcqWGWbaCcqGGPaqkdaahaaWcbeqaaiabdsfaubaakiabdkhaYjabcIcaOiabdchaWjabcMcaPaaa@3EF7@

where r is the residual (differences between the observed and predicted state variable) column vector.

Minimization of equation (11) was carried out iteratively by first starting with an initial guess of parameter vector, {p^(k)} and updating it at each iteration until the termination criteria were met 13132:

p^(k+1) = p^(k) + α^(k)Δp^(k)

where α^(k) is a scalar step length and Δp^(k) is the direction of the search or step direction 41.

Using QR decomposition 43 the linear least square problem below, which is the Osborne-Moré extended version of the Levenberg-Marquardt algorithm, was solved to obtain the search direction in each iteration:

min ⁡ ‖ ( r ( p k ) 0 ) + ( J ( p ) k ( λ k D k ) 1 2 ) Δ p k ‖ 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacyGGTbqBcqGGPbqAcqGGUbGBdaqbdaqaamaabmaaeaqabeaacqWGYbGCcqGGOaakcqWGWbaCdaahaaWcbeqaaiabdUgaRbaakiabcMcaPaqaaiabicdaWaaacaGLOaGaayzkaaGaey4kaSYaaeWaaqaabeqaaiabdQeakjabcIcaOiabdchaWjabcMcaPmaaCaaaleqabaGaem4AaSgaaaGcbaGaeiikaGccciGae83UdW2aaWbaaSqabeaacqWGRbWAaaGccqWGebardaahaaWcbeqaaiabdUgaRbaakiabcMcaPmaaCaaaleqabaWaaSaaaeaacqaIXaqmaeaacqaIYaGmaaaaaaaakiaawIcacaGLPaaacqGHuoarcqWGWbaCdaahaaWcbeqaaiabdUgaRbaaaOGaayzcSlaawQa7amaaCaaaleqabaGaeGOmaidaaaaa@5429@

where λ is a positive scalar known as Marquardt's parameter or Lagrange multiplier 36, J is the Jacobian or sensitivity matrix, and D is a positive definite scaling matrix that ensures the descent property of the algorithm even if the initial guess is not "smart". For non-zero values of λ, the Hessian approximation is always a positive definite matrix, which ensures the descent property of the algorithm 41.

A combination of "one-sided" and "two-sided" finite difference methods 303132 was used to calculate the partial derivatives of the state variable with respect to model parameters and to construct the Jacobian matrix. The "One-sided" finite difference method estimates the partial derivatives of the state variable with respect to model parameters by solving the forward problem (Eq. 1) p+1 times (p is number of parameters to be estimated). On the other hand, the "two-sided" finite difference method estimates the partial derivatives of the state variable with respect to model parameters by solving the forward problem (Eq. 1) 2p+1 times. At the early stages of the optimization, where the search is far from the solution, the "one-sided" finite difference scheme, which is computationally cheap but not as accurate as the "two-sided" approach, was used. As the optimization proceeds in the descent direction, the algorithm switches to a more accurate but computationally expensive approach in which the partial derivatives of the state variable with respect to the model parameters are calculated using a two-sided finite difference scheme. The switch was made when ϕ(p) ≤ 1 × 10^-2. A detailed description of the procedure to update the Jacobian matrix is presented in 130.

In order to update λ at each iteration, the optimization starts with an initial parameter vector and a large λ(λ = 1). As long as the objective function decreases in each iteration, the value of λ is reduced. Otherwise, it is increased. The approach avoids calculation of λ and step length in each iteration and is therefore computationally cheap. A detailed description of the code for updating λ is given in 31.

Finally, the following stopping rule was used to end the search 13132:

i f   ( ∇ φ ( p ) | p = p ^ ≤ δ 1 & Δ φ ( p ) φ ( p ) ≤ δ 2 & φ ( p ) ≤ δ 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGPbqAcqWGMbGzcqqGGaaicqGGOaakdaabcaqaaiabgEGirJGaciab=z8aMjabcIcaOiabdchaWjabcMcaPaGaayjcSdWaaSbaaSqaaiabdchaWjabg2da9iqbdchaWzaajaaabeaakiabgsMiJkab=r7aKnaaBaaaleaacqaIXaqmaeqaaOGaeiOjayYaaSaaaeaacqqHuoarcqWFgpGzcqGGOaakcqWGWbaCcqGGPaqkaeaacqWFgpGzcqGGOaakcqWGWbaCcqGGPaqkaaGaeyizImQae8hTdq2aaSbaaSqaaiabikdaYaqabaGccqGGMaGjcqWFgpGzcqGGOaakcqWGWbaCcqGGPaqkcqGHKjYOcqWF0oazdaWgaaWcbaGaeG4mamdabeaakiabcMcaPaaa@5CEC@

    Stop

else

    Continue Optimization Loop

end

where ∇φ(p)|p=p^ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaabcaqaaiabgEGirJGaciab=z8aMjabcIcaOiabdchaWjabcMcaPaGaayjcSdWaaSbaaSqaaiabdchaWjabg2da9iqbdchaWzaajaaabeaaaaa@38B8@ is the gradient of the penalty function at solution, Δφ(p)φ(p) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabfs5aeHGaciab=z8aMjabcIcaOiabdchaWjabcMcaPaqaaiab=z8aMjabcIcaOiabdchaWjabcMcaPaaaaaa@37CD@ is relative changes in the magnitude of the parameters in two consecutive iterations, and δ₁, δ₂, and δ₃ are user defined small values.

The accuracy of the optimization was assessed by goodness-of-fit analysis. The Root Mean Squared Error (RMSE) and Coefficient of Determination (R²) were calculated for every set of optimized parameters 4445:

RMSE = (r^Tr/(N - p))^1/2

R 2 = [ ∑ I ^ i I i − ∑ I ^ i ∑ I ^ i ] 2 [ ∑ I ^ 2 − ( ∑ I ^ i ) 2 ] [ ∑ I 2 − ( ∑ I i ) 2 ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@645D@

where I_i and I^i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGjbqsgaqcamaaBaaaleaacqWGPbqAaeqaaaaa@2F5E@ are the observed and predicted total normalized average fluorescence intensity, F + C, inside bleached area during time course of FRAP experiment, respectively and N is the number of observations on FRAP time series.

Before being incorporated into the framework of the developed inverse modeling strategy, the numerical model was first validated against the exact solution of Sadegh Zadeh and Montas (unpublished) and the semi-analytic solution of 25. The results are depicted in Figures 1a, 1b, and 1c (for the exact solution of Sadegh Zadeh and Montas) and in Figures 1d (for the solution of 25). To validate our model with the solution of 25, the average of the normalized fluorescence intensity within the bleach spot during the time course of the FRAP experiment, I(t), was calculated by 130:

I ( t ) = 2 w 2 ∫ 0 w ρ [ F ( ρ , t ) + C ( ρ , t ) ] d ρ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGjbqscqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiabikdaYaqaaiabdEha3naaCaaaleqabaGaeGOmaidaaaaakmaapedabaacciGae8xWdiNaei4waSLaemOrayKaeiikaGIae8xWdiNaeiilaWIaemiDaqNaeiykaKIaey4kaSIaem4qamKaeiikaGIae8xWdiNaeiilaWIaemiDaqNaeiykaKIaeiyxa0LaemizaqMae8xWdihaleaacqaIWaamaeaacqWG3bWDa0Gaey4kIipaaaa@4FF5@

While Figure 1a shows spatial and temporal distributions of free GFP-GR, F(ρ, D_f, Ka∗ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaaa@3033@, K_d, t), inside and outside bleach spot after photochemical bleaching during time course of a FRAP experiment, Figure 1b presents distributions of bound protein, C(ρ, D_f, Ka∗ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaaa@3033@, K_d, t), at times of 0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, and 2 seconds. Both Figures indicates that there is excellent agreement between the analytic solution and numerical simulator. Spatial and temporal distributions of total fluorescently tagged GFP-GR, [F + C](ρ, D_f, Ka∗ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaaa@3033@, K_d, t), inside and outside the bleached area are given in Figure 1c, which presents excellent agreement between the analytic and numerical solutions. The same time range was used to perform the comparison. As Figure 1d indicates, there is excellent agreement between the two solutions in simulating the average normalized fluorescence intensity within the bleach spot during time course of the FRAP experiment.

The developed methodology was then used to estimate diffusion coefficient and binding rate parameters of GFP-tagged glucocorticoid receptor (GFP-GR). The results are given in Table 1 and Figure 2 (the experimental FRAP time series data are from McNally, personal communication). The Root Mean Squared Error (RMSE) and Coefficient of Determination (R²) were calculated, using equations (14) and (15), for every set of optimized parameters and presented in the last two columns of Table 1. The values for diffusion coefficient, binding rate parameters, and corresponding indices estimated by 25 are given as the first run in Table 1 and Figure 2 for sake of comparison. Table 1 and Figure 2 indicate that many combinations of the three parameters can essentially produce the same error level (RMSE) and yields equally excellent fits. The values obtained by 25 represent only one of the possible solutions. In other words, the inverse problem is not well-posed and has no unique solution. Therefore, one may conclude that the Fluorescence Recovery after Photobleaching technique, though useful in studying the dynamics of biological systems, provide insufficient information to uniquely estimate free diffusion coefficient and binding rate parameters of biomolecule simultaneously.

To illustrate the non-uniqueness of the inversion, we plotted the optimized parameter values in three-dimensional parameter hyper-space (Fig. 3). The plot visibly indicates that the different combinations of model parameters can lead to same penalty (objective) function or error levels. The plot also suggests a potential linear form of correlation between bio-macromolecular diffusion coefficient and pseudo-association rate parameter. The first set of optimized parameters, obtained by 25, shows that 86% of the GFP-GR is bound to DNA, nuclear matrix or unknown binding sites and only 14% is free. Our analysis, however, suggests that using FRAP, one cannot conclude how much of the bio-macromolecule under study is free and how much is bound. As Table 1 shows, the concentration of free GFP-GR ranges from zero to 100 per cent. The same is true for the concentration of the bound complex. For example, referring to the results obtained in run 11, one may conclude that 100 per cent of the GFP-GR is free, while the results of run 6 show that all of it is bound. Note that both parameter sets produce excellent fits with the same RMSE and coefficient of determination (see Figure 2).

Analysis of Table 1 indicates that the average binding time per vacant site, calculated by t_b = 1/K_d 25, ranges between 0.3 and 500 mili-seconds. Similarly, the average time for diffusion of GFP-GR from one binding site to the next, obtained by t_d = 1/Ka∗ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaqhaaWcbaGaemyyaegabaGaey4fIOcaaaaa@3033@46, varies over several orders of magnitude. The broad range of average binding time and average diffusion time for GFP-GR indicates that it is doubtful that one can infer an average time for macro-molecule diffusion and binding inside living cells from FRAP results alone.

This analysis explains the conflicting parameter values that may have been reported in literature for biomolecules using the FRAP protocol. Based on our findings, using experimental FRAP time series and coupling it with curve-fitting procedures can lead to misleading conclusions regarding binding reaction, slow or rapid mobility of biomolecules, and concentrations of free macromolecule, vacant binding sites and bound complex inside living cells and tissues.

In this study, the choice of a numerical approach rather than an analytic solution and a finite element approach rather than the finite difference scheme was made so that the parameter estimation could be readily extended to arbitrary initial and boundary conditions, complex domain geometry, and especially so that it could be extended to a ternary system of coupled nonlinear partial differential equations governing transport of free bio-macromolecule, bound complex, and vacant binding site where all three transport entities are moving species and it is impossible to obtain an exact solution for the systems of equations.

So far most of the FRAP studies have assumed an infinite domain to specify boundary conditions and to solve the system of partial differential equations governing biomolecule transport in-vivo [13252930 among many others]. This assumption is unrealistic in the context of living cells. In this study, we address and improve this shortcoming by specifying a finite domain and by formulating Neumann boundary conditions on the cell membrane.

The inverse methodology used in this study is based upon the following assumptions: 1) residuals have a mean of zero, 2) residuals have constant variance, 3) residuals are uncorrelated, and 4) residuals are normally distributed. When these assumptions are met, the parameter optimization estimates poses optimal statistical properties 404142. When these conditions are not met the parameter optimization method may no longer produce optimal parameter estimates. Residuals, or errors in parameter optimization, are defined as the difference between the observed and simulated state variable(s). An analysis of the residuals is a useful and key technique to study possible trends, oscillations, and correlation of errors. It is also important in validating the assumptions on which the inverse modeling strategy rests.

To analyze the residuals, they were plotted against average normalized fluorescence intensity, I¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGjbqsgaqeaaaa@2DDF@(t), within the bleach spot during the time course of the FRAP experiment. Since the residuals are time and/or space series, their possible correlation was thoroughly analyzed. Different statistical measures such as error frequency analysis, normal probability plot, and hypotheses tests were explored to make decision about the residuals. The Student's t-test was used to test if the residuals have a mean of zero. Bartlett's test 47 was applied to determine if the residuals have constant variance. To test the normality of the residuals the Chi-square and Kolmogorov-Smirnov one sample tests were employed. Finally, the t-statistic 48 was used to test if the residuals are correlated.

The basic assumption in the hypothesis test on the residuals' mean is that the data come from a normally distributed population with unknown variance. In this study, the following null and alternative hypotheses were formulated:

H 0 : μ = μ 0 H A : μ ≠ μ 0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdcciGae8hVd0Maeyypa0Jae8hVd02aaSbaaSqaaiabicdaWaqabaaakeaacqWGibasdaWgaaWcbaGaemyqaeeabeaakiabcQda6iab=X7aTjabgcMi5kab=X7aTnaaBaaaleaacqaIWaamaeqaaaaaaa@3F1E@

To perform the test the following critical t-statistic (t) was used:

t = x ¯ − μ 0 s / N MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGH9aqpdaWcaaqaaiqbdIha4zaaraGaeyOeI0ccciGae8hVd02aaSbaaSqaaiabicdaWaqabaaakeaacqWGZbWCcqGGVaWldaGcaaqaaiabd6eaobWcbeaaaaaaaa@3828@

in which x¯ MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaqeaaaa@2E3E@, s, and N are the mean, standard deviation, and size of the sample (errors), respectively. μ₀ is the mean of the population which is zero.

For -t_α/2 <t <t_α/2 the null hypothesis (mean is zero) cannot be rejected at the significance level of α. The rejection regions t < -t_α/2 or t_α/2 <t indicate that the null hypothesis can be rejected at the level of significance α.

The mean and standard deviation of the residuals were -0.0029 and 0.0234 with sample size n = 43. The t-statistic was calculated as:

t = − 0.0029 − 0 0.0234 / 43 = − 0.8127 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGH9aqpdaWcaaqaaiabgkHiTiabicdaWiabc6caUiabicdaWiabicdaWiabikdaYiabiMda5iabgkHiTiabicdaWaqaaiabicdaWiabc6caUiabicdaWiabikdaYiabiodaZiabisda0iabc+caVmaakaaabaGaeGinaqJaeG4mamdaleqaaaaakiabg2da9iabgkHiTiabicdaWiabc6caUiabiIda4iabigdaXiabikdaYiabiEda3aaa@47CD@

For 42 degrees of freedom, the tabled t-values for different levels of significance are given in Table 2. The calculated t-statistic was then compared with the tabled t-values at different levels of significance and the results summarized in Table 2. As the Table indicates the null hypothesis (mean of the residuals is zero) can not be rejected even at 20 per cent level of significance. The possibility of committing error type one is extremely slim.

The Bartlett's test statistic was used to verify for equality of variances across sub-groups of a sample against the alternative that variances are not constant. Equal variance across samples is called homogeneity of variances and is usually used in several statistical tests such as analysis of variance and nonlinear optimization which assumes that the errors have constant variance 47:

T = ( N − k ) ln ⁡ s p 2 − ∑ i = 1 k ( N i − 1 ) ln ⁡ s i 2 1 + 1 3 ( k − 1 ) [ ∑ i = 1 k 1 N i − 1 − 1 N − k ] MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6C58@

where si2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@3095@ is the variance of the subgroup, N_i is the sample size of the subgroup, k is the number of subgroups, and sp2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemiCaahabaGaeGOmaidaaaaa@30A4@ is the pooled variance. This variance is a weighted average of the variances:

s p 2 = ∑ i = 1 k ( N i − 1 ) ln ⁡ s i 2 / ( N − k ) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaqhaaWcbaGaemiCaahabaGaeGOmaidaaOGaeyypa0ZaaabCaeaacqGGOaakcqWGobGtdaWgaaWcbaGaemyAaKgabeaakiabgkHiTiabigdaXiabcMcaPiGbcYgaSjabc6gaUbWcbaGaemyAaKMaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdGccqWGZbWCdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaOGaei4la8IaeiikaGIaemOta4KaeyOeI0Iaem4AaSMaeiykaKcaaa@4BB7@

The rejection region is those values of T > χ(α,k−1)2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaqhaaWcbaGaeiikaGIae8xSdeMaeiilaWIaem4AaSMaeyOeI0IaeGymaeJaeiykaKcabaGaeGOmaidaaaaa@36F2@ in which χ(α,k−1)2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaqhaaWcbaGaeiikaGIae8xSdeMaeiilaWIaem4AaSMaeyOeI0IaeGymaeJaeiykaKcabaGaeGOmaidaaaaa@36F2@ is the upper critical value of the chi-square distribution with k - 1 degree of freedom at the level of significance α.

The following null and alternative hypotheses were formulated:

H 0 : σ 1 2 = σ 2 2 H A : σ 1 2 ≠ σ 2 2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdcciGae83Wdm3aa0baaSqaaiabigdaXaqaaiabikdaYaaakiabg2da9iab=n8aZnaaDaaaleaacqaIYaGmaeaacqaIYaGmaaaakeaacqWGibasdaWgaaWcbaGaemyqaeeabeaakiabcQda6iab=n8aZnaaDaaaleaacqaIXaqmaeaacqaIYaGmaaGccqGHGjsUcqWFdpWCdaqhaaWcbaGaeGOmaidabaGaeGOmaidaaaaaaa@4572@

To verify if the residuals have constant variance they were divided into different sections. One of the possible solutions in Table 1 was chosen and the residual plot versus laser beam recovery (Fig. 3) was divided into three regions. The variance in each region was calculated and compared with each other using the Bartlett test. The residuals were divided into three groups as:

S12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaqhaaWcbaGaeGymaedabaGaeGOmaidaaaaa@2FEA@ = S²(r(1:5)) = 9.2651 × 10^-4

S22 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaqhaaWcbaGaeGOmaidabaGaeGOmaidaaaaa@2FEC@ = S²(r(6:24)) = 9.3655 × 10^-4

S32 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaqhaaWcbaGaeG4mamdabaGaeGOmaidaaaaa@2FEE@ = S²(r(25:33)) = 9.3126 × 10^-5

The pooled weighted variance was found to be Sp2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaqhaaWcbaGaemiCaahabaGaeGOmaidaaaaa@3063@ = 7.1030 × 10^-4. The Bartlett's statistic was calculated as T = 9.5454 which is less than the upper critical value of the χ² for two degrees of freedom (k = 3) at one per cent level of significance (χ(0.01,2)2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaqhaaWcbaGaeiikaGIaeGimaaJaeiOla4IaeGimaaJaeGymaeJaeiilaWIaeGOmaiJaeiykaKcabaGaeGOmaidaaaaa@36BD@ = 10.60). It is, however, more than the tabled value for five per cent level of significance. At one percent level of significance the null hypothesis (the residuals have constant variance) cannot be rejected. Based on this test and analysis of residual plot versus laser beam recovery, it is concluded that the residuals have equal variance.

The following null and alternative hypotheses were used to test possible correlation among the residuals:

H 0 : ρ = 0 H A : ρ ≠ 0 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdcciGae8xWdiNaeyypa0JaeGimaadabaGaemisaG0aaSbaaSqaaiabdgeabbqabaGccqGG6aGocqWFbpGCcqGHGjsUcqaIWaamaaaa@3B6E@

where ρ is the correlation coefficient in the population. For n > 2 these hypotheses can be tested using the following t-statistic 48:

t = r s 1 − r s 2 N − 2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGaemiDaqNaeyypa0ZaaSaaaeaacqWGYbGCdaWgaaWcbaGaem4CamhabeaaaOqaamaakaaabaWaaSaaaeaacqaIXaqmcqGHsislcqWGYbGCdaqhaaWcbaGaem4CamhabaGaeGOmaidaaaGcbaGaemOta4KaeyOeI0IaeGOmaidaaaWcbeaaaaaaaa@4B96@

in which r_s is the correlation coefficient in the sample.

The null hypothesis (correlation coefficient is zero) is rejected when the absolute value of the t-statistic is greater than the critical t-value (t < -t_α/2 or t_α/2 <t) at the level of significance α.

The residuals were first divided into two sub-groups:

r₁ = r(1 : end - 1)

r₂ = r(2 : end)

The correlation coefficient (r_s = 0.1931) was then calculated and was used to obtain the critical t-statistics (with sample size n = 42):

t = 0.1931 1 − 0.0373 42 − 2 = 1.2447 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDcqGH9aqpdaWcaaqaaiabicdaWiabc6caUiabigdaXiabiMda5iabiodaZiabigdaXaqaamaakaaabaWaaSaaaeaacqaIXaqmcqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIZaWmcqaI3aWncqaIZaWmaeaacqaI0aancqaIYaGmcqGHsislcqaIYaGmaaaaleqaaaaakiabg2da9iabigdaXiabc6caUiabikdaYiabisda0iabisda0iabiEda3aaa@470A@

These critical t-statistics were then compared with the tabled t-values at different levels of significances and the results presented in Table 3. As the Table indicates the null hypothesis (residuals are uncorrelated) can not be rejected even at 20 per cent level of significance. The possibility of having error type one is almost zero and, therefore, we didn't perform autocorrelation/serial correlation analysis.

One of the assumptions of the least squares theory, which was applied in this study, is the normality of the residuals. In other words, it is assumed that the errors are normally distributed. To analyze the normality of the errors, two qualitative and two quantitative methods were used: 1) Error frequency analysis and normal probability plots, and 2) Hypothesis tests on the normality of the residuals using the chi-square goodness of fit test, which is based on the differences between the observed (o_i) and expected (e_i) error frequencies will be used 49:

χ 2 = ∑ i = 1 k ( o i − e i ) 2 e i MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaahaaWcbeqaaiabikdaYaaakiabg2da9maaqahabaWaaSaaaeaacqGGOaakcqWGVbWBdaWgaaWcbaGaemyAaKgabeaakiabgkHiTiabdwgaLnaaBaaaleaacqWGPbqAaeqaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaakeaacqWGLbqzdaWgaaWcbaGaemyAaKgabeaaaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdaaaa@440F@

where k is the number of intervals or cells.

Error frequency analysis was first performed by constructing residuals' histogram. The histogram is presented in Figure 4. The Figure visibly shows that the errors are normally distributed. This was confirmed by the analysis of the normal probability plot (Fig. 5) and the chi-square hypothesis test on the normality of the random variable.

Residual frequency analysis and normal probability plots, though useful in figuring out the underlying probability distribution function, are only qualitative means to study possible normal distribution of random variable. To verify normality of the errors, the chi-square test was used. To perform the test, first the residuals were grouped into different cells (histograms). The number of residuals in every cell were counted which is e_i. Then using the upper limit of the cells, the mean, standard deviation, and the cumulative normal distribution, the expected frequencies were calculated. The cells were merged when the observed error frequencies were less than 5. Then the χ² index was calculated and compared with χ(1−α,v)2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaqhaaWcbaGaeiikaGIaeGymaeJaeyOeI0Iae8xSdeMaeiilaWIaemODayNaeiykaKcabaGaeGOmaidaaaaa@3708@50. This information was used in the hypothesis test. The null and alternative hypotheses were formulated as:

H 0 : r = N ( μ , σ ) H A : r ≠ N ( μ , σ ) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOJaemOCaiNaeyypa0JaemOta4KaeiikaGccciGae8hVd0MaeiilaWIae83WdmNaeiykaKcabaGaemisaG0aaSbaaSqaaiabdgeabbqabaGccqGG6aGocqWGYbGCcqGHGjsUcqWGobGtcqGGOaakcqWF8oqBcqGGSaalcqWFdpWCcqGGPaqkaaaa@4742@

or;

H₀ : r = N(0.0012,0.0224)

H_A : r ≠ N(0.0012,0.0224)

where μ and σ are the mean and the standard deviation of the residuals in population.

Since the calculated χ² (3.6190) is less than the tabled value (χ(0.80,2)2 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFhpWydaqhaaWcbaGaeiikaGIaeGimaaJaeiOla4IaeGioaGJaeGimaaJaeiilaWIaeGOmaiJaeiykaKcabaGaeGOmaidaaaaa@36CC@ = 4.61), the null hypothesis (the residuals are normally distributed) cannot be rejected even at 20 per cent level of significance implying that the residuals are normally distributed.

The chi-square test is a powerful test when the sample size is large. However, combining cells when the expected error frequencies are less than five loses information and hence decreases the power of the test. Furthermore, for very small samples this test is not applicable 49. To overcome these limitations, the Kolmogorov-Smirnov one sample test is usually used since it treats each observation separately and does not lose information through merging of categories. This test is more powerful than the chi-square test when sample size is not large.

The Kolmogorov-Smirnov one sample test was used to verify if the residuals are normally distributed. Results (not shown here) indicate that the null hypothesis (the errors are normally distributed) cannot be rejected even at 20 per cent level of significance implying that the residuals are strongly normally distributed.

In conclusion, detailed residual analysis indicates that: 1) residuals have zero mean, 2) residuals have constant variance, 3) residuals are normally distributed, and 4) residuals are uncorrelated. Therefore, the necessary and sufficient criteria for least square parameter optimization, which was used in this study, were met.