A comprehensive study of the delay vector variance method for quantification of nonlinearity in dynamical systems

Although vibration monitoring is a popular method to monitor and assess dynamic structures, quantification of linearity or nonlinearity of the dynamic responses remains a challenging problem. We investigate the delay vector variance (DVV) method in this regard in a comprehensive manner to establish the degree to which a change in signal nonlinearity can be related to system nonlinearity and how a change in system parameters affects the nonlinearity in the dynamic response of the system. A wide range of theoretical situations are considered in this regard using a single degree of freedom (SDOF) system to obtain numerical benchmarks. A number of experiments are then carried out using a physical SDOF model in the laboratory. Finally, a composite wind turbine blade is tested for different excitations and the dynamic responses are measured at a number of points to extend the investigation to continuum structures. The dynamic responses were measured using accelerometers, strain gauges and a Laser Doppler vibrometer. This comprehensive study creates a numerical and experimental benchmark for structurally dynamical systems where output-only information is typically available, especially in the context of DVV. The study also allows for comparative analysis between different systems driven by the similar input.


A1.1 Delay Vector Variance
The DVV method is a method for detecting the nonlinearity of the time series, which examines the predictability of time series in phase space at different scales, using the method of time delay embedding for representing a time series. The detailed description and testing of the method is presented in Gautama et al. [1][2][3] and Mandic et al. [4]. It has become fundamental tool for nonlinear time series analysis in many different research fields, such as geophysics and physiology, and can be used with any nonlinear statistic that characterises a time series with single number [2,[5][6][7]. Nonlinearity is often assessed as the absence of linearity and in statistical context, a null hypothesis is asserted that the time series is linear, and it is rejected if the time series does not conform to the properties associated with a linear signal. If the metric of the original time series is significantly different from that of surrogates, the null hypothesis is rejected and the original time series is hypothesized to be nonlinear [3]. For every original time series, we generate Ns = 25 surrogates for the nonlinearity tests [11,12]. The test statistics for the original, , and for the surrogates, , = 1, … , are computed and the series of , , is sorted in increasing order, after which the position index / rank r of is determined. Gautama  where N denotes the length of time series and τ denotes time lag (delay).

2) Compute pairwise Euclidian distances between DVs
3) Compute the mean ) * and standard deviation + * over all pairwise Euclidian distances between DVs, a pragmatic approach to determine the scaling region Since the surrogate time series have signal distribution identical to that of the original, the distributions of pairwise distances, and thus, the mean and standard deviation, will be similar. This distribution is approximately Gaussian for high embedding dimensions.
4) The sets 2 * are generated by grouping those DVs that are within a certain Euclidean distance to so that i.e. sets that consist of all DVs that lie closer to than the certain distance 2 * calculated as Considering a variance measurement valid, too few points for computing a sample variance yields unreliable estimates of the true variance. Jianjun et al. [13] suggest that the set of Therefore the variance of the corresponding target of those DVs will be almost equal to that of the original time series [2]. As a result of the standardisation of the distance axes the resulting DVV plots are straightforward to interpret.
6) The resulting DVV plots are plotted with the standardised distance 2 * on horizontal axis and normalised variance + * > on vertical axis. At the extreme right, DVV plots smoothly converge to unity, because for maximum spans, all DVs belong to the same set, and the variance of the targets is equal to the variance of the time series. If this is not the case, the span parameter * should be increased [3]. If the surrogate time series yield DVV plots similar to that of original time series, it indicates that time series is likely to be linear and vice versa. The example of a DVV plot is illustrated in Figure A1.1.  where + , * > 2 * is the target variance at the span 2 * for the i th surrogate, and the average is taken over all spans 2 * that are valid in all surrogate and original DVV plots.

A1.2 Discussion on Parameters
For a correct choice of embedding parameters, which might not be unique, the target variance, + * > , gives information regarding one of the fundamental properties of a signal, i.e. its predictability. Two extreme cases correspond to a white noise (entirely unpredictable) and a deterministic signal (entirely predictable). It is important to determine the embedding dimension and time lag correctly, since in combination with the structured signal, similar delay vectors in terms of their Euclidian distance have similar targets [14]. The embedding dimension m determines how many previous time samples are used for examining the local predictability. It is important to choose m sufficiently large, such that the m-dimensional phase space enables for a proper representation of the dynamic system. We used and compared three different methods when adopting the embedding dimension and time lag.

A1.2.1 Method 1 -Differential Entropy Method
Method 1 determines the optimal embedding parameters of the signal using a differential entropy method proposed by Gautama et al. [14]. The optimal m, and time lag, τ, are simultaneously determined based on estimates of the differential entropy ratio of the phase space representation of a sampled time signal and an ensemble of its surrogates. The entropy ratio method first uses the Kozachenko-Leonenko (K-L) estimate of the differential entropy [15] as where N is the number of samples in the data set, dj is Euclidean distance of j th delay vector to its nearest neighbour, and CE (≈ 0.5772) is Euler constant. To determine the optimal embedding parameters the ratio between K-L estimates for the time delay Nsub is the number of delay vectors, which is kept constant for all values of m and τ under consideration. If the temporal span of ! ⋅^ is too small, the signal variation within the delay vector is mostly governed by noise and either m or τ should be increased. The set of optimal parameters, ! gh8 ,^g h8 , yields a phase space representation which best reflects the dynamics of the underlying signal production system and it is expected that this representation has a minimal differential entropy.
The minimum of the plot of the entropy ratio yields the optimal set of embedding parameters. In order to determine the optimum embedding parameters in all simulations Ns = 5 surrogates were generated using iAAFT method and the entropy ratios were evaluated for m = 2, 3, …, 10 and τ = 1, 2, …, 10 [14]. Increasing the number of surrogates does not affect the results. The proposed method is illustrated in   The ER criterion requires a time series to display clear structure in phase space; i.e. for signals with no clear structure, the method will not generate clear minimum, and a different approach needs to be adopted [14]. In practice, it is common to have fixed time lag (sampling rate) and to adjust the embedding dimension (length of filter) accordingly [7,16].

A1.2.2 Method 2 -Minimal Target Variance Method
Method 2 determines the optimal embedding dimension by running a number of DVV analyses for different values of m, and choosing the one for which the minimal target variance, + F * > , is the lowest, i.e which yields the best predictability. In this work we performed this analysis for embedding dimensions ranging from 2 to 25 based on Gautama et al. [1]. The time lag, τ, for convenience, is set to unity in all simulations.
This choice of τ is conservative in the context of nonlinearity detection. Assuming the embedding dimension is sufficiently high, a linear time series can be accurately represented using τ = 1, while this is not the case for a nonlinear signal, for which time lag plays an important role in its characterisation. Hence, if the null hypothesis of linearity is rejected, one can assume that the time series is nonlinear. Since the linear part was accurately described for time lag equal to unity, the rejection can be attributed to the nonlinear part of the signal. On the other hand, if the null hypothesis is found to hold, the signal is genuinely linear or the phase space is poorly reconstructed using τ = 1, i.e. the signal is actually nonlinear. The example of the method described is shown in Figure A1

A1.2.3 Method 3 -Manual Setting of Parameters
In Method 3, within the context of nonlinearity detection, m is not considered critical and the optimal embedding dimension of the original time series can be set manually. Gautama et al. [2] report this as a desirable property for a robust analysis method relative insensitivity of the DVV method to the parameter choice. The embedding dimension was set to 3, after observation of DVV plots of available experiments, and the time lag is set to unity for convenience. This convenience does not influence the generality of the results.