A multidimensional statistical model for wood data analysis, with density estimated from CT scanning data as an example

Abstract

The trunk of a tree can be seen as a spatiotemporal sampling domain from the statistical perspective, where space is represented by direction horizontally and height vertically, and time through annual growth rings. In this framework, wood properties such as density can be the object of data collection for given estimation and testing purposes. We present a multidimensional statistical model, the tensor normal distribution, in which the variation (variance) of and dependency (covariance) between wood property measurements made for different years at various locations in a tree trunk can be inferred. Its application requires a smaller number of replicates (trees) than the traditional vector normal distribution because variances and covariances for directions and growth rings, for example, must be the same at all heights, up to a multiplicative constant. This assumption on the variance-covariance structure is called "separability", and we explain how to test it. An illustration with wood density estimates obtained from computed tomography scanning data for 11 white spruce (Picea glauca (Moench) Voss) trees is presented. This example is completed by assessing differences in mean wood density according to location in the trunk, with analysis-of-variance F-tests adjusted for the estimated variances and covariances obtained by fitting the model.Résumé : Le tronc d'un arbre peut être vu comme un domaine d'échantillonnage spatio-temporel sous l'angle statistique. L'espace y est représenté par la direction à l'horizontale et la hauteur à la verticale, et le temps, via les cernes de croissance annuels. Dans ce cadre de travail, des propriétés du bois telles que la densité peuvent faire l'objet de collectes de données à des fins d'estimation et de test. Nous présentons un modèle statistique multi-dimensionnel, le tenseur aléatoire normal, dans lequel la variation (variance) et la dépendance (covariance) pour des mesures de propriété du bois en différentes années et à différents endroits dans un tronc d'arbre peuvent être inférées. Son application requiert un plus petit nombre de réplicats (arbres) que la distribution du vecteur aléatoire normal traditionnel, parce que les variances et les covariances pour les directions et les cernes de croissance, par exemple, doivent être les mêmes à chaque hauteur, à une constante multiplicative près. Cette condition sur la structure de la matrice de variance-covariance est appelée « séparabilité », et nous expliquons comment la tester. Une illustration avec des mesures de la densité du bois obtenues par tomodensitométrie assistée par ordinateur pour 11 épinettes blanches (Picea glauca (Moench) Voss) est présentée. Cet exemple est complété par l'évaluation des différences dans la densité moyenne du bois selon la localisation dans l'arbre, à l'aide de tests F d'analyse de variance modifiés en utilisant les variances et covariances estimées obtenues en ajustant le modèle.

Estimation and testing for separable variance–covariance structures

“…In closing, most of the estimation and testing procedures discussed here are likelihoodbased and require normal distributions, which are rarely tested in practice. When tested, sample sizes must be larger: n > pq in 2D and n > pqr in 3D, so that multivariate normality test and LRT for double separability can be applied (Manceur, Beaulieu, Han, & Dutilleul, 2012). Finally, multidisciplinary team work is to be privileged for future top research on matrix or tensor, separable variance-covariance structures.…”

How do trees grow? Response from the graphical and quantitative analyses of computed tomography scanning data collected on stem sections

“…Further quantitative and statistical analyses of numerical wood CT scanning data (i.e. CT numbers) have been limited so far, except in [26] but this recent study, which also allowed some tree-ring profiling, was for bark-to-bark sticks instead of wood discs as stem sections.…”

“…With a CT scanner like ours (i.e. a Toshiba XVision), spatial resolution can be decreased to 50 mm by CT scanning the sample in several parts first and then merging the CT scan datasets; this has been done with wood sticks [26] and could be done with wood discs.…”

“…The high-resolution X-ray CT scanner used in this study is the same as the one used in [24][25][26], where technological details about some of our previous plant science applications (i.e. canopies and wood sticks) can be found.…”

“…Using the Helical Scan option with a 0.3-mm thickness, CT images were first constructed every 0.3 mm in the tree stem direction (z-axis), each CT image (x-y plane) consisting of a 512 Â 512 matrix of CT numbers. Then, CT numbers were converted to wood density estimates using the calibration equation, density = 0.993 Â CT number + 1015, for not oven-dried wood in [12] (see also [26]); our wood samples were not oven-dried and the estimated 95% confidence interval for moisture content (4.6 AE 1.8%) included the lower bound of 6% in [12]. To capture the variability in 3D and represent it graphically, one map of means and one map of corresponding standard deviations (in the x-y plane) were computed from the wood density estimates obtained from 25 CT images (along the zaxis) for each of the two wood discs.…”