Hydro-Mechanical Properties of an Unsaturated Frictional Material

3.2. STEPS OF MODEL BUILDING 73
- A basic assumption to statistical methods for process modeling is that the described
process is a statistical process. Thus the model process should include random variation.
- The most methods of process modeling rely on the availability of observed responses
(here volumetric water content measurements) that are on average directly equal to the
regression function value. This means the random standard error at each combination
of explanatory variable values is zero.
- Due to the presence of random variation it is difficult to determine if the data are
from equal quality. The most process modeling procedures treat all data equally when
estimating the unknown parameters in the model. Therefore it is assumed that the
random errors have a constant standard deviation.
- Process modeling involves the assumption of random variation. But not all intervals of
the regression function include the true process parameters. To check these intervals
the form of the distribution and the probability of the random errors must be known.
It is assumed the random errors follow a normal distribution.
Several plots are used to appreciate wether or not the model fits the experimental data well:
- Plots of residual versus predicted variables and observed versus predicted variables:
The random variation is checked using scatterplot of residuals versus predicted variables
(volumetric water content θ) and observed versus predicted values are used for checking
the sufficiency of the functional form of the proposed model and the assumption of
constant standard deviation of random error. The plots allow the comparison of the
amount of random variation of the entire range of data. If the plot of residuals versus
predicted variables (volumetric water content θ) is randomly distributed the model fits
the experimental data well. The scatter of the residuals should be constant across the
whole range of the predictors. Any systematic structure in the plot is an indiction for
the need to improve the model. A comparison of the amount of random variation is also
possible when plotting observed values versus predicted values. This relation should be
non-random in structure and linearly related.
- Normal probability plot:
Normal probability plots are used to check wether or not the errors are distributed nor-
mally. The normal probability plot gives the sorted values of the residuals versus the
associated theoretical values from the standard normal distribution. Unlike the residual
scatter plots (i.e. residual versus predicted variables and observed versus predicted val-
ues) a random scatter of points does not indicate a normal distribution. Instead if the
plotted points lie close to the straight line the errors are normally distributed. Another
curvature indicates that the errors are not normally distributed. Significant deviations