Basic HTML Version
300
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 3
Behavior prediction models and control charts for safety control of concrete dams
not be possible to completely eliminate variability, the control chart
helps to reduce it as much as possible.
The control charts very well describe what the statistical process control
is all about. They are usually used for online monitoring and they make
it possible to ascertain whether recent predictions are under control.
When values exceed the limits set in the chart or when the analyzed
variable is undergoing a behavior that may be considered nonrandom,
it is necessary to detect and eliminate the cause of variability.
When a process involves more than one control variable, it is prefer-
able to use control charts for model residuals. The control chart for the
regression residuals allows for the analysis of the influence of more
than one control variable on the dependent variable. Normalized re-
siduals were used hereby since, according to [16], the standardized
residuals are escalated so that their standard deviation is approxi-
mately equal to 1. Therefore, the major residuals will be more obvious
as from the inspection of residual charts. The standardized residuals
were obtained in the Student form, according to equation [13].
(13)
r
i
=
y
i
- y
^
i
σ
^
2
1 - h
ii
, 0
h
ii
1
where
s
2
is the mean square for the errors, calculated as follows:
(14)
σ
^
2
=
SQE
n - p
where:
p is the number of independent variables;
h
ii
is the i-th element of the diagonal of the matrix H and the variance
of the adjusted value
y
^
i
.Thus, it can be interpreted as the distance
of points X
i
in relation to the average of all points of X in the data set.
The matrix H is usually called the hat matrix because it transforms
the values observed by the auscultation instruments into a vector of
estimated values
y
^
i
. The matrix H is calculated as follows:
(15)
H = X(X
T
X)
-1
X
T
The control limits are selected on the basis of the value of the t
Figure 2 – Contribution of the temperature to displacement of the dam crest
Figure 3 – Contribution of concrete creep to displacement of the dam crest
^