Mahalanobis Distance and Longitudinal Growth

Mahalanobis distance is unperturbed by inclusion of highly correlated additional variable, unlike Euclidean distance (Dasgupta 2008). With an application of law of iterated logarithm, an almost sure result is proved in this direction for convergence of sample $$D^2$$ statistic to $$\Delta ^2,$$ the population Mahalanobis distance squared under mild assumptions, improving earlier results. A rate of convergence $$O_e( n^{-1}(\log \log n)^2(1 -\rho ^2)^{-1})$$ a.s., depending on sample size n and magnitude of correlation $$|\rho |$$ , is proved. The results are of relevance in analysis of longitudinal growth data where some of the variables may be highly correlated. Growth experiments on coconut trees are continued in Sunderban to examine its adaptability in saline soil. Moderate salinity of soil is known to be conducive for coconut tree growth. To examine adaptability of coconut trees in saline soil of Sunderban, longitudinal growth of 42 plants at two time points is observed on six growth characteristics per plant and relevant $$D^2$$ statistic is computed to assess tree growth with gradual proximity of plantations towards a saline water river Bidyadhari. Principal components of $$p=6$$ variables at two time points on the year 2015 and 2017 of $$n=42$$ trees are compared to examine the growth status. The angle $$\theta $$ between two growth vectors x and y defined by the relation $$\cos \theta = \frac{(x.y)}{||x||.||y||}\in [-1,1]$$ is examined to check direction of growth variability. The values of $$\cos \theta $$ quantify directional variation present in principal components. These are used to study the growth patterns of coconut trees on a number of aspects, e.g. growth in size, shape. As the variables are scaled in computation of angle, relative variability of the characteristics is examined. When stability of growth is attained, variation over time of each coordinate variable would be negligible after scaling, and the angle between the two growth vectors corresponding to two time points would be small. We compute $$n=42$$ angles corresponding to 42 trees, between two growth vectors of $$p=6$$ principal components at two time points, and examine whether growth has reached stability. Angle based on principal components may detect minute growth variations, as principal components maximise variance maintaining orthogonality of components to each other. We examine growth in different orthogonal directions, as the first principal component usually represents the size; the remaining components represent shape and other characteristics. The estimates of angles are independent for 42 plants, and large values of the angles indicate variability of growth direction over time. The angles computed from the growth vectors of original variables are also considered. The angles made by growth vectors of original variables are usually of less variation than those based on principal components. With increase in number of growth variables, variation in growth is seen to be more prominent. In Dasgupta (2017), moderate salinity of soil is found to be conducive for growth of coconut trees. In the present study, in conformity with previous findings, we observe that for some plants the angle between growth vectors increases, when proximity of plants to the river of saline water increases to a moderate level. This is so reflected in variation of angles with respect to the increase in plant’s serial numbers from 1 to 30, in 42 plants. Higher serial numbers represent gradual proximity of plants towards saline water river in the group of 42 plants. Clustering of points with large values of angle is observed in the scatter diagrams near serial number 30, where salinity is moderate. The presence of ‘whiskers’ in a specific region is indicated. The outliers seem to be present near the plant no. 30 with moderate salinity of soil. The plant with largest identification serial number 42 in the riverbank is closest to the river with salinity of water at about 33 g/l, before onset of monsoon.