Statistical analysis of Hydrologic variables relation to tree
ring growth
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