Hydro-Mechanical Properties of an Unsaturated Frictional Material

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CHAPTER 3. INTRODUCTION TO PROCESS MODELING - A STATISTICAL
APPROACH
- To satisfy the assumption, that the variances of the residuals are homogeneous.
- To linearize the experimental results as much as possible.
First it is tried to transform the response variable, that is the volumetric-water content θ
to get homogeneous variances. Often the square root ( √ θ, √ ψ), the power of 2 (ψ 2 , θ 2 ),
or logarithm (ln(θ), ln(ψ)) and even double logarithm (ln ln(θ), ln ln(ψ)) work well for such
kind of purpose. After examining these plots the data giving the best linear form is chosen
for model building. Based on the appropriate transformation found finally the corresponding
non-linear model is proposed.
3.2.4 Model Fit
In the present study the Levenberg-Marquardt algorithm, which is an improvement of the
Gauss-Newton method was used for solving non-linear least-squares regression problems. The
Levenberg-Marquardt algorithm provides a numerical solution to the problem by minimizing
a generally non-linear function over a range of parameters of the function. The parameters
β of a model curve are optimized so that the sum of the squares of the deviations becomes
minimum value:
g(β) = � (θi − f(β, ψi)) 2
(3.3)
where: ψi and θi are sets of measured suction and volumetric water content and β represents
the parameters of the model f(x, β) + ε.
3.2.5 Model Assumption and Model Calibration
Model validation is possibly the most important step in model building sequence. The val-
idation of the fit from a selected model does not only include the quoting of the coefficient
of regression determination R 2 . A high value of R 2 does not guarantee that the model suffi-
ciently fits the experimental data. Also the residual analysis is available, that is the primary
tool for model validation. Whereas the coefficient of regression determination R 2 for model
validation is focused on a particular aspect of the relation between the model fit and the
experimental data and the information is compressed in a single value, the residual analysis
illustrates a broad range of complex aspects between the model fit and the experimental data.
The residual is the difference between the responses observed or measured (experimental re-
sults of volumetric water content) and the corresponding responses computed using the model
(calculated results of volumetric water content):
ei = θi − f( −→ ψ i, −→ β ) (3.4)
where: ei is the residual, θi represents the i − th response in the experimental data set and
ψi the corresponding variable in the i − th observation in the data set. If the plots of the
residuals are conform to the assumptions described below the model fitting succeeded: