Sunday, March 16, 2008

Statistical analysis of Hydrologic variables relation to tree ring growth

Author: T. Rios

Statistical methods that may be utilized to determine hydrologic effects on indexed tree growth.

The ability to determine the effects of multiple hydrologic and climatic variables such as temperature, precipitation, stream flow, and groundwater on the growth of the Western Sycamore requires the use of statistical regression methods and may require more complex methods such as multiple linear regression, factor analysis and Principal Component Analysis (PCA).

A note about time scales Since the fundamental dependent variable, tree ring growth, is measured on an annual time scale, any direct correlation to hydrologic variables is limited also to the one-year time scale. Obviously smaller time scales are used and measured to determine variables such as precipitation, groundwater level, temperature, and stream flow. In order to utilize the finer scale of measurements within the hydrologic data sets, the method of aggregating the smaller time scales into seasonal changes, trends or indices may be used. For instance, if ground water levels are measured quarterly of monthly, the creation of a ground water index that aggregates the values into mean groundwater level, maximum recession, or greatest change between wet period and dry period, effectively creates three subsets of groundwater levels that can describe the finer scale of variability observed within the yearly record without information loss to the dependent variable time scale.

Multiple Linear Regression

This type of statistical method combines all the independent variables (precipitation, stream flow, temperature etc.) into one linear polynomial that models the mean response of the dependent variable (in this case tree ring growth). A simplified mathematical representation using three independent variables x, x2, x3 to model the mean response m is as follows:

where b is a fitting parameter to each independent variable. For application to tree ring growth, the mean response of the tree population on a yearly time scale would be modeled by n independent variables such as the hydrologic variables described above. The regression model would utilize a least square fit to the data that minimizes the sum of the squares of the vertical deviations from each data point to the model line resulting in a residual sum of squared error. The basic premise of multiple linear regression is that the independent variables and the dependent variables have similar variance in the distribution of their values and are normally distributed (e.g. not a skewed distribution).

Principal Component Analysis for Hydrologic Variables

The useful qualities of this method of statistical analysis and application to the tree ring analysis is that it can compress many independent variables into a few principal components that capture the maximum variability of the data. As an example, precipitation, stream flow and groundwater may all be correlated to each other but the multiple linear regression of each with tree growth may not be immediately obvious or feasible because of different measured scales and lags in the time series. If however, one variable could be created from the three that captures the maximum variability of the majority of independent values then it could be correlated to the tree ring data directly. In practice, the principal component method creates several principal components of the independent variables (PC1, PC 2, PC3…) that account for total variability of the original data sets. These principal components are then correlated to the dependent variable to determine if any relationship exists. It is important to keep in mind that the existence of any relationship does not imply on its own a mechanism for the relationship between the dependent and independent variables.

About the author: Hydraulic and environmental Fluid Mechanics Engineer

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