Spatial Functional Data Modeling of Plant Reflectances
Philip A. White, Henry Frye, Michael F. Christensen, Alan E. Gelfand, John A. Silander Jr
SSubmitted to the Annals of Applied Statistics arXiv: arXiv:0000.0000
SPATIAL FUNCTIONAL DATA MODELING OF PLANTREFLECTANCES
By Philip A. White ∗ , † , Henry Frye ‡ , Michael F. Christensen § ,Alan E. Gelfand § , and John A. Silander, Jr. ‡ Brigham Young University † , University of Connecticut ‡ , and DukeUniversity § Plant reflectance spectra – the profile of light reflected by leavesacross different wavelengths - supply the spectral signature for aspecies at a spatial location to enable estimation of functional andtaxonomic diversity for plants. We consider leaf spectra as “responses”to be explained spatially. These spectra/reflectances are functionsover a wavelength band that respond to the environment.Our motivating dataset leads us to develop rich novel spatial mod-els that can explain spectra for genera within families. Wavelength re-sponses for an individual leaf are viewed as a function of wavelength,leading to functional data modeling. Local environmental featuresbecome covariates. We introduce wavelength - covariate interactionsince the response to environmental regressors may vary with wave-length, so may variance. Formal spatial modeling enables predictionof reflectances for genera at unobserved locations with known envi-ronmental features. We incorporate spatial dependence, wavelengthdependence, and space-wavelength interaction (in the spirit of space-time interaction).Our data are gathered for several families from the Cape FloristicRegion (CFR) in South Africa. We implement out-of-sample valida-tion to select a best model, discovering that the model features listedabove are all informative for the functional data analysis. We thensupply interpretation of the results under the selected model.
1. Introduction.
The reflectance of the surface of a material is thefraction of incident electromagnetic radiation reflected at the surface. It is afunction of the wavelength (or frequency) of the light, its polarization, andthe angle of incidence. The reflectance as a function of wavelength is calleda reflectance spectrum. The literature on reflectances is substantial, with alarge portion focused on the interaction of electromagnetic energy with theatmosphere and terrestrial objects, e.g., reflectances associated with differ- ∗ Corresponding Author
Keywords and phrases: environmental regressors, functional data analysis, heterogene-ity, hierarchical model, interaction, kernel weighting, reflectance, spatial confounding a r X i v : . [ s t a t . A P ] F e b P. WHITE ET AL. ent land cover/vegetation types. Typically, they are gathered by satellites,aircraft, and ground-level sensors. The focus of this manuscript is on plantreflectances, i.e., data gathered for plants at leaf level.Such spectra have become an invaluable tool to capture the diversity inleaf traits that have accumulated over the course of seed plant evolution(Reich et al., 2003; Cornwell et al., 2014) enabling estimation of functionaldiversity (Kokaly et al., 2009; Schneider et al., 2017) and taxonomic diver-sity (Clark, Roberts and Clark, 2005; Cavender-Bares et al., 2016). Theyprovide drivers for ecosystem processes (Schweiger et al., 2018) and guideconservation (Asner et al., 2017).Traits can be detected using reflectance spectra (Kokaly et al., 2009;Serbin et al., 2014) but complication arises because reflectance spectra inte-grate leaf traits in complex ways (Jacquemoud and Baret, 1990; F´eret et al.,2017) and multiple traits can affect the same spectral region (Curran, 1989).Our intent here is not to connect reflectances to traits. Rather, we viewthe reflectances as a “response” to be explained spatially, by genus, withinfamily. They are functions over a wavelength band and can be viewed as an“uber” trait that is expected to respond to environment. We do not seek todisentangle the integration of traits which results in the observed reflectancesat a given spatial location.From a scientific perspective, our contributions include modeling reflectanceat genus level and, viewing the set of wavelength responses for an individualleaf as a function of wavelength, we explain reflectances using functionaldata modeling. We incorporate local spatial covariates/environmental fea-tures as regressors for reflectances, adopting additional model componentsthat have not previously been considered in analyses of plant reflectances.We offer an incisive analysis of a real dataset under our modeling.The methodological novelty of our spatial functional models includes thefollowing. We introduce spatial dependence, as well as wavelength depen-dence, both through random effects. Further, we add space-wavelength in-teraction (in the spirit of space-time interaction) by constructing a space-wavelength random effect through wavelength kernel convolutions of spa-tial Gaussian processes. In general, this random effect has nonseparablecovariance and is wavelength nonstationary. We explicitly model the vari-ance to be heterogeneous across wavelength. Also, expecting that the re-flectance response to environmental regressors may vary with wavelength,we include wavelength - covariate interactions. Furthermore, we can predictreflectance for genera at unobserved locations with known environmentalfeatures. Lastly, we present a novel orthogonalization to remove spatial con-founding between random effects and environmental regressors.
PATIAL MODELING OF REFLECTANCE Functional data analysis (FDA) is well established for analyzing datarepresenting curves/surfaces varying over a continuum. The physical con-tinuum over which these functions are defined is often time but here, itis wavelength. Pioneering work for FDA is attributed to Ramsey and Sil-verman (e.g., Ramsay, 2005; Ramsay and Silverman, 2007). The field hasundergone rapid growth, and numerous applications have been found inareas such as imaging (Locantore et al., 1999) (including MRI brain imag-ing (Tian et al., 2010)), finance (Laukaitis, 2008), climatic variation (Besse,Cardot and Stephenson, 2000), spectrometry data (Reiss and Ogden, 2007),and time-course gene expression data (Leng and M¨uller, 2006). For a morecomprehensive overview of applications, see Ullah and Finch (2013).Explicit modeling of functional data is usually carried out by specifyingfunctions in one of two ways: (i) as finite linear combinations of some set ofbasis functions or (ii) as realizations of some stochastic process. A key fea-ture of functional data analysis implementation is some version of dimensionreduction to specify functions. Here, we have random functions over a wave-length span as well as over a spatial region. We combine both approaches,using basis functions over wavelength with process realizations over spaceto build space by wavelength regressions over environment.We work with plant reflectances gathered from the Cape Floristic Region(CFR) in South Africa. We present an extensive cross-validation study formodel selection across a rich collection of models to demonstrate the abilityof our space-wavelength modeling to predict reflectances well for generawithin a family at unobserved locations. We present and discuss our findingsfor three plant families found within the CFR.The format of the paper is as follows. Section 2 describes the collecteddata. Section 3 undertakes a broad exploratory data analysis to motivate thefeatures we incorporate in our modeling. Section 4 explains our modeling,model comparison, and presents a novel orthogonalization for functional re-gression coefficients. Section 5 presents the analysis of the CFR data while abrief Section 6 offers a summary and suggestion for future work. Substantialdetail of our exploratory analysis, as well as model sensitivity analysis, hasbeen placed in the Supplemental Material.
2. The Dataset.
We work with plant reflectances gathered from theCape Floristic Region (CFR) in South Africa, see Figure 1. Reflectances weremeasured with a USB-4000 Spectrometer (manufactured by Ocean Optics)using a leaf clip attachment. Sun leaves from the top of each selected canopywere measured. The spectrometer has a range of 450-950 nanometers (nm)with a total of 500 reflectance measurements. We study plant reflectance
P. WHITE ET AL. viewed as a function of wavelength t , across the window t ∈ [450 ,
3. Exploratory Data Analysis and Modeling.
We explore the char-acteristics of plant reflectances for the three families given above (Aizoaceae,Asteraceae, Restionaceae) in the HTR and Cederberg areas. We retain theentire dataset because it is somewhat small from a spatial perspective. Notethat the domains for the three families do not overlap well (Figure 1) so wewill fit each family separately when implementing our spatial modeling.The number of genera with observed reflectances within each family is:Aizoaceae - 16, Asteraceae - 38, and Restionaceae - 10. The SupplementalMaterial provides: (i) the proportion of sites where each family is present,(ii) the number of sites with one, two, or three families, and (iii) a moredetailed breakdown of family co-occurrence. To summarize, Aizoaceae andRestionaceae rarely co-occur; in fact, Restionaceae is mostly limited to theCederberg region apart from a few HTR observations. Replication at thegenus level is uncommon and even more uncommon at the species level.3.1.
Data Locations.
In Figure 1, we show all locations, where reflectancesare observed with sites coded by region (shape) and family (color). We alsoplot locations coded by the number of families observed at that site. In theHTR and Cederberg regions, only 22 of the 133 sites have more than one
PATIAL MODELING OF REFLECTANCE reflectance spectrum for a given species; only 27 of the 183 total species(across all families) are observed at more than one site. This suggests thatspecies-level modeling is infeasible. The Supplemental Material offers morecommentary on data locations and duplication. −32.50−32.25−32.00−31.75−31.50−31.25 19.2 19.6 20.0 20.4longitude l a t i t ude Family
AIZOACEAEASTERACEAERESTIONACEAE
Region
CederbergHTR −32.50−32.25−32.00−31.75−31.50−31.25 19.2 19.6 20.0 20.4 longitude l a t i t ude number_fams Fig 1 . Locations (Left) colored by family with region-specific shapes and (Right) colorednumber of families observed at the site.
Reflectance spectra.
To visualize the form and variability in reflectancespectra, we plot all of the curves by family in Figure 2 along with plots ofthe genus-specific means. We can see that the family-specific means do notcapture the spread of the variability seen in all the curves while the genus-specific means show nearly the same variability for all of the curves. l og ( R e f l e c t an c e ) m ean_ l og_ r e f l e c t an c e l og ( R e f l e c t an c e ) m ean_ l og_ r e f l e c t an c e l og ( R e f l e c t an c e ) m ean_ l og_ r e f l e c t an c e Fig 2 . AIZOACEAE (Top-Left) All Curves, (Top-Right) Genus-specific means.ASTERACEAE (Middle-Left) All Curves, (Middle-Right) Genus-specific means.RESTIONACEAE (Bottom-Left) All Curves, (Bottom-Right) Genus-specific means.
P. WHITE ET AL.
To assess within reflectance function variability as well as between-functionvariability, we calculate binned standard deviations for every curve. For thesebinned standard deviations, we estimate a smooth family-specific averagestandard deviation. Additionally, we calculate the family-specific between-curve standard deviation. These are plotted in Figure 3 and show that vari-ability within reflectance spectrum changes with wavelength and, perhaps,with family. In addition, the variability between curves changes as a func-tion of wavelength and differs by family. These findings lead us to imposeheterogeneity in variance across wavelength, adopting wavelength varyingvariance curve models on the log scale.Given these plots, we are led to four modeling needs: (i) to allow forfamily and genus differences, (ii) to model heterogeneity for the reflectancespectrum because within-curve variability changes across wavelength (iii) tocapture between-curve variability through spatial modeling and/or environ-mental variables, and (iv) to adopt heteroscedastic errors since reflectancesat lower wavelengths ( <
500 nm) appear to be more volatile. B i nned S t anda r d D e v i a t i on B e t w een − S pe c t r u m S D Family
AIZOACEAEASTERACEAERESTIONACEAE
Fig 3 . (Left) 25-nm binned standard deviations for each reflectance spectrum withsmoothed family-specific curves. (Right) Family-specific between-spectrum standard devia-tion as a function of wavelength.
Environmental Variables and Reflectance Spectra.
For each family,we calculate the correlations between the environmental variables (see Sup-plemental Material) and the observed log-reflectances, using wavelength bins(See Figure 4), to assess whether this relationship changes with wavelength.We find consequential changes in correlation as a function of wavelength.The strongest correlations are of magnitude 0.3 to 0.4.
4. Spatial Wavelength Modeling.
Functional data modeling for ourspatial reflectance spectra was motivated by the foregoing exploratory anal-
PATIAL MODELING OF REFLECTANCE AIZOACEAEASTERACEAERESTIONACEAE ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] Wavelength Bin F a m il y −0.4−0.20.00.20.4 Elevation
AIZOACEAEASTERACEAERESTIONACEAE ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] Wavelength Bin F a m il y −0.4−0.20.00.20.4 Mean Annual Precipitation
AIZOACEAEASTERACEAERESTIONACEAE ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] Wavelength Bin F a m il y −0.4−0.20.00.20.4 January Average Minimum Temperature
AIZOACEAEASTERACEAERESTIONACEAE ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] ( , ] Wavelength Bin F a m il y −0.4−0.20.00.20.4 Rainfall Concentration
Fig 4 . (Top-Left to Bottom-Right) elevation, mean annual precipitation, average minimumtemperature in January, rainfall concentration. yses. Families are modeled separately, at genus level, treating species withingenus as replicates. We utilize the following environmental predictors: ele-vation, annual precipitation, rainfall concentration, and minimum Januarytemperature. We introduce wavelength dependent variances to account forevident heterogeneity. Model choice focuses on four issues: (i) Do we needwavelength dependent regression coefficients? (ii) Do we need genus specificwavelength random effects? (iii) Do we need genus specific spatial randomeffects? (iv) How do we specify space-wavelength interaction?In Sections 4.1 and 4.2, we elaborate the models, while Section 4.3 takesup model comparison yielding the model for which results are presented.4.1.
Model development.
For a given family, let i denote genera withinthe particular family, let j denote replicates/species within genus. Let s denote spatial location and t denote wavelength. There is severe imbalancein the data. The genera observed vary across locations and the number ofreplicates observed within a genus varies considerably across the locations.Altogether, our most general model for log reflectance takes the form:(1) Y ij ( s , t ) = µ i ( s , t ) + γ i ( t ) + α i ( s ) + η ( s , t ) + (cid:15) ij ( s , t )Specifically, with regard to the site level covariates, X ( s ), we write the mean µ i ( s , t ) = α i + X T ( s ) β ( t ), where, hierarchically, α i | α ∼ N ( α, σ α ). We havefamily level regression coefficients, β ( t ), which vary with wavelength. So, afirst model choice clarification is whether constant coefficients are adequate P. WHITE ET AL. or whether wavelength varying coefficients are needed. Our EDA (Figure4) suggests the latter, and there is also supporting evidence/suggestion inthe literature (Jacquemoud and Ustin, 2019b). We do not consider thesecoefficients at genus level; with the very irregular observation (includingabsence) of genera across locations, we cannot learn about coefficients atgenus scale. However, we can learn about genus specific intercepts, the α i .Further, we introduce genus level spatial ( α i ( s )) and wavelength ( γ i ( t ))random effects but family level space-wavelength interaction effects, η ( s , t ).In the Supplemental Material, we note different spatial patterns for differentwavelength bins, as well as residual dependence by genus and wavelength.Thus, an additive model (removing η ( s , t )) seems inadequate; the η ’s allowthe functional model for the reflectances to vary more adaptively over space.However, η ( s , t ) is not genus specific. While we have enough data to examineadditivity in wavelength and spatial random effects at genus scale, we areunable to find genus level explanation for the interaction. Then, two modelchoice comparisons are whether the γ ’s and whether the α ’s should still begenus specific?As is customary, heterogeneity in the variance arises through the (cid:15) ij ( s , t )terms where we would have var( (cid:15) ij ( s , t )) = σ ( t ). We can accommodate thisusing a log GP for σ ( t ), or perhaps just binned variances over suitablewavelength bins. For simplicity and flexibility, we specify log( σ ( t )) to bepiecewise linear with knots every 20 nm from 440 - 960 nm. For all knotselections, we use boundary knots slightly beyond the wavelength range.4.2. Explicit Specifications.
The specification for each α i ( s ) is a genus-level mean 0 Gaussian process with mean of 0 and exponential covariancefunction. The GPs are conditionally independent across genera given ashared decay and shared scale parameter. We specify γ i ( t ) using processconvolution of normal random variables (Higdon, 1998, 2002). We adoptprocess convolutions because of their simple connection to GPs; the kernelsof the process convolution connect the low-rank process to the GP covari-ance (Higdon, 1998). We adopt wavelength knots t γ , ..., t γJ γ , spaced every 25nm from 437.5-962.5 nm (22 in total).Specifically, we let γ i ( t ) = (cid:80) J γ j =1 k t γj ( t − t γj ; θ ( γ ) t γj ) γ ∗ i ( t γj ), where γ ∗ i ( t γj ) areindependent, normally distributed, and centered on a common γ ∗ ( t γj ). Weuse Gaussian kernels for k t j ( · ; θ ( γ ) t j ) with bandwidths θ ( γ ) t j (standard devi-ation of the Gaussian pdf) varying over wavelength. We assume that thelog-bandwidths follow a multivariate normal distribution with global log-bandwidth and Cov (cid:104) log (cid:16) θ ( γ ) t j (cid:17) , log (cid:16) θ ( γ ) t j (cid:48) (cid:17)(cid:105) = v γ exp (cid:0) −| t j − t j (cid:48) | /φ γ (cid:1) , yield- PATIAL MODELING OF REFLECTANCE ing a non-stationary process because of the heterogeneous bandwidth. Wefound that this nonstationary specification outperformed a full-rank station-ary GP with squared-exponential covariance (See Supplemental Material).We specify β ( t ) using kernel convolutions, where β ( t ) = BK β ( t ). With p covariates, B supplies a p × q matrix representation of the p regressioncoefficient functions β ( t ). Here, the kernel convolution has knots every 25nm from 437.5 - 962.5 nm. As with γ i ( t ), we use Gaussian kernels to specify K β ( t ); however, unlike γ i ( t ), we assume common bandwidths for all kernels,for all wavelengths, and for each coefficient function.Turning to η ( s , t ), we use wavelength kernel convolutions of spatially-varying variables. That is, we consider low-rank but heterogeneous andnonstationary (in the wavelength domain) specifications. We select a setof wavelength knots t η , ..., t ηJ η , spaced every 25 nm from 437.5-962.5 nm (22,in total). We define the space-wavelength function as(2) η ( s , t ) = K ( t ) T z ( s ) = J η (cid:88) j =1 k t ηj ( t − t ηj ; θ ( η ) ) z t ηj ( s ) , where z t ηj ( s ) are spatially-varying random variables associated with Gaus-sian wavelength kernels k t ηj ( · ; θ t ηj ). Unlike the kernel structure for γ ( t ), weuse a common bandwidth θ ( η ) for all knots. The construction in (2) allowsheterogeneity and nonstationarity in wavelength space, where the hetero-geneity is introduced through z t ηj ( s ) (See White, Keeler and Rupper, 2021,for a similar construction in the context of spatial monotone regression).As an aside, we remark on choosing the form η ( s , t ) = K T ( t ) z ( s ) vs. η ( s , t ) = K T ( s ) z ( t ). With n sites, the former introduces J η n random effects,the latter 500 n random effects. With J η relatively small, the former is pre-ferred computationally. More importantly, it yields much better fits to thedata (see the Supplemental Material).While we may want dependence between components in z ( s ) at s , thatdependence should have nothing to do with the t ηj ’s. We are capturing as-sociation with regard to the distances between wavelength knots throughthe K ’s and our objective for the z ’s is to obtain perhaps nonseparable andnonstationary covariance structure for η ( s , t ). So, we write z ( s ) = A w ( s )where A is J η × r and the components of w ( s ) are independent mean 0 GP’swith variance 1 and correlation functions, ρ r ( s − s (cid:48) ).When r = J η , we have the familiar linear model of coregionalization(Wackernagel, 1998). We consider using A = I and A p × r , for various r , aswell as a separable specification for z ( s ), where, with V a positive definitematrix, Cov ( z ( s ) , z ( s (cid:48) )) = exp ( − φ z (cid:107) s − s (cid:48) (cid:107) ) V . With A p × r , we constrain P. WHITE ET AL. the decay parameters of the ( w ( s ) , ..., w r ( s )) T to be increasing (see Whiteand Gelfand, 2020), so that the latent GPs have different spatial decays ( φ z ).The resulting processes for z ( s ) are very flexible. We compare the variouschoices through out-of-sample prediction in Section 4.3.Under the general form η ( s , t ) = K T ( t )( s ) A w ( s ), cov( η ( s , t ) , η ( s (cid:48) , t (cid:48) )) = K T ( t ) A Σ w ( s ) , w ( s (cid:48) ) A T K ( t (cid:48) ) . If A = I , we have Σ w ( s ) , w ( s (cid:48) ) = D ( s − s (cid:48) ), a J η × J η diagonal matrix with entry d jj = ρ j ( s − s (cid:48) ). Thus, cov( η ( s , t ) , η ( s (cid:48) , t (cid:48) )) = K T ( t ) D ( s − s (cid:48) ) K ( t (cid:48) ) = (cid:80) j k t ηj ( t − t j ) k t ηj ( t (cid:48) − t ηj ) ρ j ( s − s (cid:48) ). The covariance isalways nonseparable and, if A is unconstrained it is nonstationary.As an illustration, if we take A to be J η ×
2, we have Σ w ( s ) , w ( s (cid:48) ) = (cid:18) ρ ( s − s (cid:48) ) 00 ρ ( s − s (cid:48) ) (cid:19) . Now, with a and a the two columns of A ,cov( η ( s , t ) , η ( s (cid:48) , t (cid:48) )) = ( K T ( t ) a )( K T ( t (cid:48) ) a ) ρ ( s − s (cid:48) ) + ( K T ( t ) a )( K T ( t (cid:48) ) a ) ρ ( s − s (cid:48) ). We achieve both dimension reduction and space-wavelength inter-action. Further, we have nonseparability and nonstationarity (in the wave-lengths) if there are different bandwidths for the different t j . If we set r = 1,we have separability but still nonstationarity in the wavelengths.4.3. Model Comparison.
We carry out model comparison for Asteraceae,the most abundant family, using 10-fold cross-validation (described below).In the Supplemental Material, we present cross-validation results examiningvarious specifications of the spatial process in η ( s , t ). When comparing mod-els with different specifications of η ( s , t ), all models include spatially-varyinggenus-specific intercepts α i + α i ( s ), a global (not genus-specific) wavelengthrandom effect γ ( t ), and functional regression coefficients β ( t ). For η ( s , t ),we compare separable, independent, and latent factor models. We find thatthe latent factor specification of η ( s , t ) with r = 10 has the best out-of-sample predictive performance and use this for η ( s , t ) in the remainder ofthe manuscript. For this specification of η ( s , t ), we focus our model compar-ison on eight special cases of (1) arising by (i) including or excluding α i ( s ),(ii) using γ i ( t ) or only γ ( t ), and (iii) having functional coefficients β ( t ) orscalar coefficients β .We hold out reflectances imagining the setting where researchers visiteda site but failed to measure reflectances for some genus at that site. So,at random, we leave out spectra that have (i) at least one other observedreflectance spectrum at the same site and (ii) at least one other observedspectrum of the same genus located elsewhere. For Asteraceae, this yields117 candidates out of the 185 in total. Holding out a subset, we fit themodel using Markov chain Monte Carlo, and, with each posterior sample,we predict the hold-out reflectance spectra. We compare models by aver- PATIAL MODELING OF REFLECTANCE aging across the wavelengths to obtain the predicted mean squared error(MSE), mean absolute error (MAE), and the mean continuous ranked prob-ability score (MRCPS), see Gneiting and Raftery (2007). The results aresummarized in Tables 1 and in the Supplement. α i / α i ( s ) γ ( t )/ γ i ( t ) β / β ( t ) MSE MAE MCRPS Relative MCRPS α i γ ( t ) β α i γ ( t ) β ( t ) 0.141 0.293 0.234 1.196 α i γ i ( t ) β α i γ i ( t ) β ( t ) 0.169 0.318 0.262 1.340 α i ( s ) γ ( t ) β α i ( s ) γ ( t ) β ( t ) 0.097 0.237 0.196 1.000 α i ( s ) γ i ( t ) β α i ( s ) γ i ( t ) β ( t ) 0.290 0.420 0.380 1.940 Table 1
Out-of-sample predictive performance model comparison. Models vary by including orexcluding genus-specific terms, as well as comparing scalar and functional coefficients.All models use r = 10 spatial factors to construct η ( s , t ) . Following the results in Table 1 and the Supplemental Material, we adopta model with (1) a global wavelength random effect, (2) a spatially-varyinggenus-specific intercept, (3) functional regression coefficients, and (4) a space-wavelength random effect specified through the wavelength kernel convolu-tion of a multivariate spatial process with 10 latent spatial GPs havingdifferent decay parameters. We use this model to analyze the CFR data.For the sensitivity of model fit to change in other specifications (e.g., knotspacing and GP/process convolution), we use average deviance, the devianceinformation criterion, and estimated model complexity (Spiegelhalter et al.,2002), as supplied in the Supplemental Material. To summarize, we employa heterogeneous process convolution specification of γ ( t ) because it gave abetter fit than a full-rank homogeneous GP with squared-exponential covari-ance and a process convolution with a common bandwidth for all wavelengthknots. We also specify β ( t ) using kernel convolutions where β ( t ) = BK β ( t ),where we space knots every 25 nm from 437.5 - 962.5 nm. We also find thatthe Gaussian kernel, which corresponds to the Gaussian covariance func-tion, was preferred to using double-exponential kernels for γ ( t ), β ( t ), and η ( s , t ). For γ ( t ), the model fit was improved when bandwidths θ ( γ ) t j variedover wavelength; however, a common bandwidth for the kernels was preferedfor η ( s , t ). The knot spacing, discussed in Section 4.2, was also determinedthrough sensitivity analysis.4.4. Confounding and Orthogonalization.
The flexibility of the residualspecification in our best performing model results in annihilation of the P. WHITE ET AL. significance of the spatial regressors. This is a well-documented problem inthe literature (see, e.g., Hodges and Reich, 2010; Khan and Calder, 2020).A solution in the literature is orthogonalization; that is, projection of therandom effects (the spatial residuals) onto the orthogonal complement of themanifold spanned by the spatial covariates. This yields revised regressioncoefficients with direct interpretation in the presence of the random effects.The coefficients are more aligned with those that arise from model fittingignoring spatial random effects.We propose a similar orthogonalization approach here but our setting ismore demanding because we have both space and wavelengths in our resid-uals. We have to introduce orthogonalization with regard to the manifoldspanned by the spatial covariates as well as with regard to the manifoldspanned through the use of kernel functions with knots. We present the de-tails below for the simpler case where we have no replicates at locations.However, in our application, we have replicates associated with the spatiallocations and also with different genera. So, formally, the orthogonalizationrequires us to introduce a location by genus matrix in order to align the num-ber of observed sites with the number of observed reflectances. We presentthe more detailed argument in the Supplemental Material.With n sites and 500 wavelengths, we can express (1) in matrix form as(3) Y = α + XBK Tβ + η ∗ + (cid:15) where Y is the n ×
500 matrix of log-reflectance spectra data by sites, is a n ×
500 matrix of ones, α is the global mean, X is the n × p spatialdesign matrix (with p covariates), B is p × J β with J β knots, K β is the500 × J β kernel design matrix with J β knots. η ∗ is also n ×
500 summing thecorresponding matrix forms for the mean-zero random effects ( γ ( t ), α i ( s ), α i − α , and η ( s , t )). Then, using standard results, we can vectorize (3) to(4) vec ( Y ) = α + ( X ⊗ K β ) vec ( B ) + vec ( η ∗ ) + vec ( (cid:15) )where vec ( Y ) is an n × X ⊗ K β an n × pJ matrix.Now, define the joint projection matrix,(5) P XK β ≡ ( X ⊗ K β )(( X ⊗ K β ) T ( X ⊗ K β )) − ( X ⊗ K β ) T = ( X ( X T X ) − X T ) ⊗ ( K β ( K Tβ K β ) − K Tβ ) = P X ⊗ P K β , and write vec ( η ∗ ) = P XK β vec ( η ∗ ) + ( I − P XK β ) vec ( η ∗ ). Then, we can write vec ( Y ) = ( X ⊗ K β ) vec ( B ∗ ) + ( I − P XK β ) vec ( η ∗ ) + vec ( (cid:15) ), where the updated unconfounded coefficients are (in vec and block form)(6) vec ( B ∗ ) = vec ( B ) + (( X ⊗ K β ) T ( X ⊗ K β )) − ( X ⊗ K β ) T vec ( η ∗ ) ,B ∗ = B + ( X T X ) − X T η ∗ K β ( K Tβ K β ) − . PATIAL MODELING OF REFLECTANCE Here, vec ( B ∗ ) and B ∗ provide the vector and matrix of regression coeffi-cients, respectively, under the orthogonalization. The model is fitted using(1). Then, with the posterior samples of the β ’s, γ ’s, α ’s, and η ’s along withthe X ( s ) and K β ( t ), the unconfounded B ∗ ’s can be obtained using (6).
5. Analysis of the CFR Reflectance Spectra Data.
We focus dis-cussion on a comparison between families but give specific attention to theresults on Asteraceae, the most abundant family. We compare and discussresults from the orthogonalized coefficients using the approach in Section 4.4.In addition, we summarize covariate importance on log-reflectance. Againusing the orthogonalized random effects and unconfounded regression func-tions, we discuss the proportion of variance explained by each model term.The confounding between random effects (genus, wavelength, and spatial)and covariates pushes β ( t ) to zero, obliterating any significant inference withregard to the effect of environmental variables on log-reflectance. For eachMCMC posterior sample, we calculate the proportion of the variance in eachrandom effect ( α i ( s ), γ ( t ), η ( s , t )) explained by X and K β . We orthogonalizeour random effects with respect to X and K β as described in Section 4.4 toremove the diminishing of the effect of the regressors.For the Asteraceae family, we explore the proportion of the variance ex-plained by each of the mean-zero model terms. For every posterior sample,we calculate the empirical variance of all nonorthogonalized and orthogonal-ized terms (See Figure 5 to the 95% credible regions): (cid:15) ij ( s , t ), x ( s ) T β ( t ), α i ( s ) + α i − α , γ ( t ), and η ( s , t ). We take α i ( s ) + α i to capture both genus-specific terms and subtract α to make α i ( s ) + α i − α a mean-zero randomeffect. For orthogonalized terms, γ ( t ) explains slightly under 25% of thevariability of the data, while both η ( s , t ) and x ( s ) T β ( t ) explain over 30%of the total variance. Without orthogonalization of the random effects, theenvironmental regression explains almost no variance. The genus-specificspatially-varying intercept ( α i ( s ) + α i ) − α explains over 10% of the totalvariance while (cid:15) ij ( s , t ) accounts for about 5% of variance in the data.In Figure 5, we plot the proportion of between-spectrum variability ex-plained by all orthogonalized mean-zero terms as a function of wavelength(posterior mean and 95% credible interval). Even though γ ( t ) is commonto all spectra, after orthogonalization, it is no longer a constant term forall spectra. For wavelengths less than 700 nm, we find that unconfoundedenvironmental regression and space-wavelength random effects are most im-portant in explaining between-spectrum variance. For higher wavelengths( >
750 nm), where there is little variation in the wavelength functions;the orthogonalized global wavelength random effects γ ( t ) and the uncon- P. WHITE ET AL. founded environmental regression explain the most between-spectrum vari-ance. The spatially-varying genus-specific offset, ( α i ( s ) + α i ) − α explainsbetween 10-20% of between-spectrum variance for most wavelengths butappears particularly influential for wavelengths between (675-725 nm). The (cid:15) ij ( s , t ) account for the 0 to 10% of unexplained between-spectrum variancein log-reflectance, depending on wavelength. llllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllll llllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllll lllllllllllllll llllllll llllllll e ij ( s ,t ) x ( s ) b ( t ) a i ( s ) + a i − ag ( t ) h ( s ,t ) model term P r opo r t i on V a r i an c e E x p l a i ned Orthogonalizedconfoundedorthogonalized P r opo r t i on V a r i an c e E x p l a i ned model term e ij (s,t)x(s) b (t) a i (s) + a i − ag (t) h (s,t) Fig 5 . (Left) Proportion of variance explained by each model term and (Right) Proportionof between-site variance explained by model terms.
After updating β ( t ) in the presence of orthogonalization, we present ourinference on covariates for all families. In Figure 6, we show the posteriormean, 95% credible interval for each element of β ( t ). All coefficient functionsare significantly non-zero for most wavelengths (around 99% for all wave-lengths). Because covariates are centered and scaled, (i) we can interpreteffects as the expected change in log-reflectance for a one standard devi-ation change in the covariate, holding the other covariates constant, and,more importantly, (ii) we can compare the scales of the coefficient functionsamong covariates.The four covariates have positive effects for some wavelengths, negativeeffects for others, with a transition around 700 nm, a threshold/boundarybetween visible (450-700nm) and nearinfrared regions (NIR, 700-1400nm) ofthe spectrum. The visible region is most strongly affected by differences inplant pigment composition/concentration while the NIR is most affected bystructural properties related to the cell wall, to air interface within the leaf(Jacquemoud and Ustin, 2019b). Traits can exhibit uniform effects acrossmultiple parts of the spectrum (e.g., often in water content) or can causeincreased reflectance in parts of the spectrum and decreased reflectance inothers (Feng et al., 2008; Jacquemoud and Ustin, 2019a). Different sets oftraits acting in concert in response to environment likely drive the positiveand negative shifts across the 700 nm threshold in Figure 6.For Asteraceae, we estimate that higher elevations are associated with PATIAL MODELING OF REFLECTANCE lower reflectance levels at wavelengths less than 700 nm but higher re-flectance at wavelengths above 700 nm. The relationships of precipitationand temperature with reflectance are similar. On the other hand, rainfallconcentration is positively correlated with reflectance at low wavelengths andbecomes negatively correlated with reflectance as wavelength increases. Wenote that rainfall concentration, the environmental feature that reflectanceresponds differently to, is largely longitudinally driven in comparison to theother features. Specifically, the extreme western and to some extent the ex-treme eastern sample sites have significantly higher rainfall concentrationsthan more central locations. Because there is between-covariate correlation,the coefficient functions must be interpreted as partial slopes, i.e., holdingall other covariates constant.To compare covariate importance, we calculate the mean integrated ab-solute coefficient over the wavelength domain, | β j | = (cid:82) | β j ( t ) | dt ≈ (cid:80) i =1 | β j ( t i ) | , for each covariate. This metric weights the contribution ofthe coefficient equally regardless of sign or wavelength. We calculate | β j | forevery posterior sample and plot these in Figure 7. In terms of | β j | , elevationand temperature are more influential on reflectance than precipitation andrainfall concentration.5.1. Comparison across families.
We compare the regression coefficientfunctions for the three families in this study (posterior mean and 95% cred-ible interval): Aizoaceae, Asteraceae, and Restionaceae (See Figure 6). Theregression coefficient functions are clearly distinct across the families. How-ever, between-covariate correlation or different spatial sites covered by eachfamily may account for some of these differences.The estimated effects of elevation, annual precipitation, and tempera-ture are opposite in direction for all wavelengths between Asteraceae andAizoaceae. For these covariates, we see positive effects on Aizoaceae log-Reflectance for wavelengths <
700 nm and negative effects for wavelengths >
700 nm, with opposite patterns for Asteraceae. For Asteraceae, the esti-mated effects of rainfall concentration are positive for lower wavelengths andnegative for higher wavelengths, while they are nearly zero for Aizoaceae.Restionaceae has very small estimated temperature effects. For elevationand rainfall concentration, Restionaceae shows significant effects on log-reflectance for wavelengths <
700 nm, but essentially no effect for higherwavelengths. The estimated effect of precipitation for Restionaceae is simi-lar to Aizoaceae in pattern but is smaller in magnitude.In Figure 6, we also plot the variance function for (cid:15) ij ( s , t ) for each fam-ily (posterior mean and 95% credible interval). Asteraceae has the highest P. WHITE ET AL. estimated variance for most low wavelengths (450 - 700 nm), a trend thatmatches the between spectrum variance patterns in Figure 3. Restionaceaehas the lowest estimated variance for (450 - 700 nm). All families have verylow estimated variance for most high wavelengths (700 - 950 nm). llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
500 600 700 800 900 − . − . . . wavelength (nm) E ff e c t o f E l e v a t i on AsteraceaeAizoaceaeRestionaceae llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
500 600 700 800 900 − . − . . . . wavelength (nm) E ff e c t o f A nnua l P r e c i p i t a t i on AsteraceaeAizoaceaeRestionaceae llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
500 600 700 800 900 − . − . . . . wavelength (nm) E ff e c t o f R a i n f a ll C on c en t r a t i on AsteraceaeAizoaceaeRestionaceae llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
500 600 700 800 900 − . − . − . − . . . . . wavelength (nm) E ff e c t o f A v g M i n J an T e m p AsteraceaeAizoaceaeRestionaceae llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
500 600 700 800 900 . . . . wavelength (nm) s ( t ) AsteraceaeAizoaceaeRestionaceae
Fig 6 . Between family comparison of (Top-Left to Middle-Right) environmental regressioncoefficient functions and (Bottom) wavelength-varying variance σ ( t ) . The differing responses in visible and near-infrared reflectance to environ-ment between Aizoaceae and Asteraceae 6 likely indicate that genera withinthe two families employ different adaptive strategies in response to their localenvironments across the landscape. The Aizoaceae family consists of smallsucculent stemmed and leafed plants while the Asteraceae family largelyconsists of non-succulent leafed herbs and shrubs. Both plant families adaptvia other traits tied to aridity tolerance (e.g., water storage for periods ofdrought) and avoidance (e.g., leaf hairs, wax, and anthocyanin pigmentationthat block UV radiation). The adaptive traits in the respective ”evolutionarytoolboxes” of Aizoaceae and Asteraceae are constrained by their phyloge-netic ancestry, resulting in differing strategic responses to environment intheir traits and thus, reflectances. In contrast, the Restionaceae consist of
PATIAL MODELING OF REFLECTANCE grass-like plants with tough fibrous photosynthetic stems that vary less thanthe other two families in adaptation to drought.We show the posterior distribution (boxplot) for | β j | across all covariatesand families (See Figure 7). Since | β j | represents the relative importanceof covariates for log-reflectance, we see that the covariates are more impor-tant in describing log-reflectances for Asteraceae than Aizoaceae and moreimportant for Aizoaceae than for Restionaceae, with the exception of rain-fall concentration. Perhaps the relative importance | β j | may be higher forAizoaceae and Asteraceae because these have more expansive spatial distri-butions and thus experience higher variability in environmental variables. lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll E l e v P r e c i p R F L C on c T e m p covariate R e l a t i v e I m po r t an c e family AizoaceaeAsteraceaeRestionaceae
Fig 7 . Coefficient importance | β | for all families. Despite the differences in spatial ranges, the families differ in terms ofwhich environmental variables have the highest relative importance to theirreflectance signals. The most important variable for Asteraceae was eleva-tion. Likely, elevation is a proxy for several environmental factors; prominentamong them is the biome shift from the higher elevation Fynbos biomewithin the Cederberg mountains to the lower elevated Succulent Karoobiome. These biomes differ widely in their environments with the Fynbosbiome having nutrient-poor soils and a regular fire cycle while the Succu-lent Karoo is largely arid with low levels of rainfall. Asteraceae was theonly family of the three to fully span both biomes in large numbers andthese biomes feature a wide difference in environments. The most importantvariable for Aizoaceae was the minimum average temperate in January (thepeak austral summer month), a strong indicator of the maximum tempera- P. WHITE ET AL. ture a plant can tolerate. This suggests that the major driver of Aizoaceaereflectances are underlying adaptations related to heat tolerance/avoidance.While more limited in its spatial extent, the Restionaceae reflectance spec-tra responded most to rainfall concentration. Under the notion that higherconcentrations of rainfall in fewer months out of the year would lead to moredramatic periods without water, much of the differences in Restionanceaereflectance may be in response to underlying traits managing water duringtimes of drought.
6. Summary and Future Work.
We have offered plant reflectancemodeling to capture variation over space between reflectance across gen-era within a family. We incorporate wavelength heterogeneity, spatial de-pendence, and also wavelength - covariate interaction as well as space -wavelength interaction. We have fitted these models to reflectances fromthe Cape Floristic Region in South Africa, demonstrating successful modelperformance and revealing a range of novel inference as well as successfulspatial prediction.This work has several future applications and opportunities for furtherdevelopment. Our current data only included the visible and near-infraredreflectance spectra of leaves. These data could be expanded to include thereflectance of plant canopies across a broader spectral range to make predic-tions relevant to the reflectance spectra collected by broader band sensorsaboard aerial and satellite remote sensing platforms. Our spatially explicitpredictions of plant reflectance would be highly relevant for spectral unmix-ing analyses which seek to predict the abundances of spectral end members,i.e., individual species, in a canopy of vegetation. Future modeling effortsinclude exploring reflectance signatures following evolutionary history, ex-plicitly taking into account phylogeny among different groups of plants.Our space-wavelength model could also be adapted for space-time appli-cations. For suitable spatiotemporal settings, it may be useful to constructspatial kernel convolutions of wavelength/temporal GPs. Also, our approachto spatial orthogonalization for functional regression coefficients could beapplied to dynamic regression in spatiotemporal settings.
Acknowledgement.
We thank Matthew Aiello-Lammens, Douglas Euston-Brown, Hayley Kilroy Mollmann, Cory Merow, Jasper Slingsby, Helga vander Merwe, and Adam Wilson for their contributions in the data collec-tion and curation. Special thanks to Cape Nature and the Northern CapeDepartment of Environment and Nature Conservation for permission to col-lection leaf spectra and traits. Data collection efforts were made possibleby funding from National Science Foundation grant DEB-1046328 to J.A.
PATIAL MODELING OF REFLECTANCE Silander. Additional support was provided by NASA FINESST grant award19-EARTH20-0266 to H.A. Frye and J.A. Silander.SUPPLEMENTARY MATERIALExtended data analysis, residual analysis, orthogonalization, and results.(LINK ADDED LATER).
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