4.1.2. Signal Reconstruction
The AE has been first employed to reduce the dimensionality of the pseudo-experimental vibrational recordings. Each MTS
consists of two TS reporting the sampled time evolution of the lateral floor displacements for
, with a sampling rate of
. Each
is associated to a loading condition, whose governing parameters
and
are sampled from two uniform pdfs
and
. Two different building configurations, termed A and B and characterized by different vibration frequencies, have been considered to assess the effects of the interaction between the mentioned structural vibration frequencies and the accuracy of the results of the AE-based identification procedure. In
Table 1, the data related to the two configurations are collected; note that the chosen sampling frequency for the pseudo-experimental measurements avoids signal aliasing to occur.
To train the AE, MTS have been generated for both the configurations: of these samples have been used to train the AE and back-propagate the error; the remaining have been employed as validation set. Based on the loss computed on the validation set, an early-stopping strategy may be exploited. The training of the AE has been started from scratch for both configurations A and B, hence without relying on transfer learning.
For testing the trained AE, a further test set of 512 MTS has been generated. The reconstruction capacity of the AE is assessed not only qualitatively but also quantitatively, by computing for each MTS the two error measures reported in
Table 2, and the relevant mean values and scattering around it over the whole training, validation and test sets.
Results in the following refer to AE hyperparameters gathered in
Table 3: training has been repeated for different sets of hyperparameters, and the one adopted led to the minimum loss
and has been finally selected for the analyses. As for the kernels of the convolutional layers, their dimensions have been set to cope with the fundamental period of structural vibrations, so as to exploit the capacity of convolutional layers to detect a correlation within a TS. Alternatives to this trial-and-error procedure are represented by the Bayesian methods [
46] or by multi-objective optimization [
47], which are anyway computationally infeasible in practical situations of interest for our study. The drawbacks of both approaches may be mitigated by their combination, as suggested in [
48].
Regarding the impact of
P on the reconstruction capacity of the AE, it has been assessed by means of its effects on mimicking the structural response in the frequency domain, starting from the reduced representation. As previously discussed, a lower bound on
P can be assumed to be equal to 2, i.e., equal to the number of generative factors in
. Given the ill-posedness of the inverse problem, by adopting
we expect to have beneficial effects on the AE performance; therefore, we have tested the cases
. As shown in
Figure 8,
Figure 9,
Figure 10 and
Figure 11 the reconstruction error gets progressively reduced by increasing
P for both the considered configurations, but unevenly and showing different correlations with the load frequency
. This behavior is due to the stochastic nature of the training algorithm and also to the strong nonlinearity of
. More specifically:
Figure 8 and
Figure 9 report the error, via the standardized
norm, in reconstructing the displacement
of the first floor as a function of
, when the input signals, respectively, belong to the training and validation sets;
Figure 10 and
Figure 11 report instead the error, via the standardized
norm, in reconstructing the displacement
of the second floor, when the input signals belong to the test set only. For comparison with these plots,
Figure 12 and
Figure 13 further provide, for configurations A and B, a sketch of the reconstruction capacity for the training and validation sets via the standardized
norm, and a sketch of the reconstruction capacity for the test set via the standardized
norm, both for
. Similar results have been obtained for the other values of
P, but are not reported here for the sake of brevity. Such results are shown since, if the reconstruction capacity for the training and validation sets were greater than the one related to the test set, overfitting would have probably spoiled the AE performance: the NN would not acquire any generalization capacity, being limited to reproduce the instances seen during the training.
The investigated reconstruction capacity is less affected by the load amplitude
, as shown in
Figure 14, due to the linearity of the structural behavior. Indeed, when the standardized
norm is considered, as it measures the inaccuracy in the peak reconstruction, larger errors are found for values of
smaller than 2000 N. In spite of the data normalization procedure preceding training, the structural displacements under excitations featuring small values of
have small peaks too, and their incorrect reconstruction results less penalized during the training.
The link between the reconstruction error and the load frequency
varies with
P, and depends on the adopted error measure.
Figure 8 and
Figure 9 have shown that the standardized
error is larger when
gets closer to the structural vibration frequencies
and
, that is when the load induces a resonant response of the structure. This outcome is somehow expected, as the relevant beats in the displacement recordings are signal characteristics hard to catch by the AE. The larger error found for
results as a consequence of the
difficulty of reproducing the long-range temporal correlation characterizing the first vibration mode.
Figure 10 and
Figure 11 have shown instead that the standardized
error is still large when
gets close to the second structural vibration frequency, while becomes rather small, roughly by ten times, for
. An analysis of the dynamics of the two configurations suggests the reason behind this result. During training, the loss function allows modifying more largely the weights
, when the AE fails to reconstruct the vibration mode that has a larger impact on the dynamic response of the structure. The excitation frequency
is sampled from
and the mentioned modes can have a different impact for different instances. To compute the impact that the vibration modes
,
, have on the solution, we first solve a non-standard eigenvalue problem in the form [
49]:
the stiffness and mass matrices of the structure being:
and enforce
as normalization rule.
The equations governing the dynamics of the structure read:
where:
and
are the vectors of storey accelerations and displacements, respectively;
is the vector of the external loads. For each load case, by sampling
at the two floors, we obtain
,
and the instance
.
Due to the linear behavior of the structure, through modal superposition Equation (
6) is decoupled as follows:
with:
Since
,
and we obtain:
If the structure is initially at rest and if the entries of the load vector
are defined according to Equation (
4), the time history of
is given by:
where:
actually depends on the structural dynamics (through
) and on the spatial distribution of loads.
At a specific time instant
, the modal response becomes
, whose expected value
can be computed as:
where we have accounted for that
and
, respectively, vary in the ranges
and
. Computing the integrals, we obtain:
where:
The term within curly brackets in Equation (
13) provides the dependence of
on
and
.
At the same time instant, the expected value of the storey displacement
,
, thought of as sampled from the corresponding pdf
, is obtained by exploiting Equation (
8) and the linearity of the expectation rule [
50]:
The contribution to of each mode depends linearly on and , and therefore on and . For the case at hand, the ratio between and is equal to for configuration A, and to for configuration B. Accordingly, the error provided by the AE in reconstructing the contribution of the first vibration mode is, on average, roughly ten times larger than the error linked to the second vibration mode. The loss function leads to the setting of the NN weights in the same way. Due to this rationale, the AE is driven to learn better the first vibration mode.
In such a discussion, we have disregarded the temporal dependence of
; this has an impact on the AE capacity of accounting for each mode of vibration. In the comment to
Figure 8 and
Figure 9, we have already addressed that the tendency to learn better the first mode of vibration is counterbalanced by the long range temporal correlation featured by the first mode. The adopted error measures have been introduced with the purpose of investigating these issues, and seem to adequately address them.
Figure 15 shows a comparison of the reconstructed
and the input
signals taken from the validation set, either for
or
of configuration B and for
, to further get insights into what the two error norms provide. In spite of the rather large reconstruction error measured by the standardized
norm and shown by
Figure 9e for the first resonant frequency,
and of
in
Figure 15a are almost perfectly superposed. This comparison confirms that both error measures bring meaningful information, with the standardized
norm measuring inaccuracies in the reproduction of the frequency content of the input signal, while the standardized
norm highlighting the inability to catch peaks in the same input signal.
To better assess the impact of
P on the reconstruction capacity of the AE, box plots depicting the mean and the scattering around it for the two adopted error norms are reported in
Figure 16 and
Figure 17 for configurations A and B, respectively. In the charts, errors are given for both the training and test sets, to evaluate the generalization capacity of the AE. As a general rule, the values of the load configuration-dependent reconstruction error evaluated for the test set is more scattered than the one evaluated for the training set, while the relevant median values are quite similar; the said difference is larger if measured through the standardized
norm.
According to
Figure 16, the optimal number of latent variables for configuration A results to be
when looking at the standardized
norm, and
when looking at the standardized
norm if outliers are also allowed for. By increasing
P and, therefore, the redundancy in the latent representation, an improvement of the AE reconstruction capacity is not achieved. As shown in
Figure 17, also for configuration B an increase of the value
P does not lead to a monotonic reduction of the reconstruction error. Even if the best AE accuracy has been obtained for
, good performances have been attained with
too, with a slight deterioration for
.
Moving deeper into the assessment of the AE performances, a comparison is reported in
Figure 18 between the reconstruction errors for both configurations A and B. For the standardized
norm, the variation relevant to the error values for configuration B is slightly smaller than that relevant to configuration A. A similar trend can be recognized also for the standardized
norm. This outcome can be linked to the smaller gap between the resonance frequencies
and
featured by configuration A; it is worth mentioning that a similar difficulty was already observed with methods like indipendent component analysis or second order blind identification, when the identification of closely spaced modes is involved [
51].
4.1.3. False Nearest Neighbour Heuristics
If the FNN heuristics is included in the AE loss function formulation, a regularization term
is added to
in accordance with Equation (
3). As an outcome,
Figure 19 reports the variance
, with
, of each latent variable for the training set of configuration A, at varying value of the regularization parameter
in the added term
. Except the case
, for which the regularization term appears to be too small,
allows to automatically turn off some of the latent variables. Increasing values of
are not associated to a clear trend in the number of deactivated latent variables. Indeed, the way in which the plain AE loss function
and the regularization term
affect the solution is made practically unpredictable by the strong nonlinear behavior of
.
Even the reconstruction capacity of the AE does not show a clear trend at varying
, as highlighted in
Figure 20 in terms of the results obtained for the training and test sets relevant to configuration A. Though a non-monotonic variation of the AE performance is obtained, the FNN heuristics can be exploited to set the value of
for which the mean value of the reconstruction error and the variation around it, are minimized. For configuration A, the minimum is obtained for
, irrespective of the handled standardized norm and the dataset. For cases featuring such values of
, the number of active latent variables is
, the same found as optimal using the standardized
error norm. Even though they are not shown here for the sake of brevity, results relevant to configuration B are characterized by a slightly more regular effect of
on the number of active variables in
, with a sub-optimal solution attained with the same range of values for the regularization term, endowed with
. In this case, the FNN heuristics has been able to attain a sub-optimal solution, but not the optimal one, featuring instead
.
The regularization of the loss function based on the FNN heuristics thus allows to basically achieve the same optimal AE settings already found. The strength of the regularization term seems to have a non negligible impact on the results. For this reason, the use of leads to marginal advantages for the proposed cases study, given that the tuning of P is substituted by the tuning of . On the other hand, this approach can be considered useful if there is no a clear understanding on the number of generative factors.