The bivariate (2D) Common Conditional Exceedance Charge, briefly ACER2D, was utilized to analyse FPSO hawser tensions to review excessive bivariate statistics. One benefit of ACER2D is its capability to account for the clustering impact of response peaks inside excessive worth statistics. The clustering impact is vital for the case studied on this paper since hawser tensions possess distinct narrow-band results, as seen in Fig. 7. Notice that each stochastic response processes (H1 and H2 FPSO stern hawsers on this paper) are synchronous in time; the latter is helpful for learning coupling results and bivariate statistics. A quick introduction of the bivariate ACER2D technique is given beneath^{19,32}. See Refs.^{16,33} for different statistical and reliability approaches.

This paper considers bivariate random course of (Z(t)=(X(t),Y(t))), with two scalar processes (X(t),Y(t)), both simulated or measured synchronously, over a time frame ((0,T)). The information pattern (({X}_{1},{Y}_{1}),dots ,({X}_{N},{Y}_{N})) is assumed to be recorded at *N* time equidistant instants ({t}_{1},dots ,{t}_{N}) given commentary time span ((0,T)).

Subsequent, allow us to estimate the CDF (joint cumulative distribution perform) (Pleft(xi ,eta proper):=mathrm{ Prob }left({widehat{X}}_{N}le xi ,{widehat{Y}}_{N}le eta proper)) of the maxima vector (left({widehat{X}}_{N},{widehat{Y}}_{N}proper)), with ({widehat{X}}_{N}=mathrm{max}left{{X}_{j} ;j=1,dots ,Nright}), and ({widehat{Y}}_{N}=mathrm{max}left{{Y}_{j} ;j=1,dots ,Nright}). On this paper (xi) and (eta) being H_{1} and H_{2} hawser tensions correspondingly. The non-exceedance occasion is recognized: ({mathcal{C}}_{kj}left(xi ,eta proper):={{X}_{j-1}le xi ,{Y}_{j-1}le eta ,dots ,{X}_{j-k+1}le xi ,{Y}_{j-k+1}le eta }) for (1le kle jle N+1). In response to the definition of CDF (P(xi ,eta ))

$$start{array}{ll}& P(xi ,eta )=mathrm{ Prob }({mathcal{C}}_{N+1,N+1}(xi ,eta )) & =mathrm{ Prob }({X}_{N}le xi ,{Y}_{N}le eta | {mathcal{C}}_{NN}(xi ,eta ))cdot mathrm{Prob }({mathcal{C}}_{NN}(xi ,eta )) & =prod_{j=2}^{N}mathrm{ Prob }({X}_{j}le xi ,{Y}_{j}le eta | {mathcal{C}}_{jj}(xi ,eta ))cdot mathrm{Prob }({mathcal{C}}_{22}(xi ,eta )).finish{array}$$

(2)

The CDF (P(xi ,eta )) could also be represented as in Refs.^{7,8,34,35,36}

$$Pleft(xi ,eta proper)approx mathrm{exp}left{-sum_{j=ok}^{N}left({alpha }_{kj}left(xi ;eta proper)+{beta }_{kj}left(eta ;xi proper)-{gamma }_{kj}left(xi ,eta proper)proper)proper},mathrm{ for big }xi mathrm{and }eta ,$$

(3)

for a big conditioning parameter (ok) with ({alpha }_{kj}left(xi ;eta proper) : = textual content{ Prob }({X}_{j}>xi | {mathcal{C}}_{kj}(xi ,eta )), {beta }_{kj}left(eta ;xi proper) : = textual content{Prob }({Y}_{j}>eta left|{mathcal{C}}_{kj}left(xi ,eta proper)proper), {gamma }_{kj}left(xi ,eta proper) : = textual content{Prob }({X}_{j}>xi ,{Y}_{j}>eta | {mathcal{C}}_{kj}(xi ,eta ))). Subsequent, the (ok)-th order bivariate ACER2D features are outlined

$${mathcal{E}}_{ok}left(xi ,eta proper)= frac{1}{N-k+1} sum_{j=ok}^{N}left({alpha }_{kj}left(xi ;eta proper)+{beta }_{kj}left(eta ;xi proper)-{gamma }_{kj}left(xi ,eta proper)proper), ok=1, 2,dots$$

(4)

Then, when (N” ok)

$$Pleft(xi ,eta proper)approx textual content{exp}left{ – left(N-k+1right){mathcal{E}}_{ok}left(xi ,eta proper)proper} ;textual content{ for big }xi textual content{ and }eta .$$

(5)

From Eq. (5), it’s seen that the correct estimation of the bivariate CDF (P(xi ,eta )) is predicated on an equally correct estimation of ACER2D features.

Determine 10 illustrates the correlation sample between neighbouring FPSO stern hawser tensions, H1 and H2, in a bivariate plot of the sampled information; see Fig. 3 for H1 and H2 places. It’s clear from Fig. 10 that the H1 and H2 tensions are barely non-linearly correlated.

By contemplating the hawser rigidity PSDs proven in Fig. 6, it may be seen that the strain is characterised by a number of narrow-band elements. The key element has its pure interval of considerably lower than 20 s. Due to this fact, to watch the dependence impact within the hawser rigidity time collection, the ACER_{100} perform ought to be thought of. The latter is as a result of 20 s corresponds to 100 pattern factors at this paper’s 0.2-s discrete time step. The key goal is conditioning above the biggest slim band pure interval.

Determine 11 presents bivariate ACER2D features ({widehat{mathcal{E}}}_{ok}(xi ,eta )) empirically calculated for numerous conditioning values of (ok) on a decimal logarithmic scale. ({widehat{mathcal{E}}}_{ok}(xi ,eta )) with (ok=2) is given by the higher floor, whereas two decrease surfaces correspond to (ok=50), be aware that floor with (ok=100) shouldn’t be plotted as it’s virtually indistinguishable from the floor with (ok=50), as seen in Fig. 12, subsequently convergence has been achieved. In Fig. 12, marginal ACER_{ok} features are plotted on the decimal logarithmic scale for ranges of conditioning (ok=2, 50). It’s seen that ACER_{2} considerably deviates from ACER_{50} which means that there’s a robust clustering impact, which has been captured already on the conditioning degree (ok=50), for the reason that ACER_{100} perform with conditioning degree (ok=100) is indistinguishable from the ACER_{50} perform with the extent of conditioning (ok=50).

The bottom possibilities are proven in Fig. 12 as being equal to the worth ({N}^{-1}) with (N) being the variety of equidistant temporal factors current in studied time collection, Eqs. (3)–(5). Determine 13 presents optimized Uneven logistic (AL) ({mathcal{A}}_{ok}(xi ,eta )) and optimized Gumbel logistic (GL) ({mathcal{G}}_{ok}(xi ,eta )) fashions contour strains, fitted to corresponding empirical bivariate perform ({widehat{mathcal{E}}}_{ok}(xi ,eta )), with (ok=50). Destructive numbers marking contour strains in Fig. 13 being likelihood ranges of (P(xi ,eta )) on the decimal logarithmic scale. Determine 13 signifies that empirical bivariate ACER2D floor ({widehat{mathcal{E}}}_{50}) captures properly a powerful correlation between two response elements. Optimized fashions ({mathcal{G}}_{50}) and ({mathcal{A}}_{50}) displaying easy contours that match empirical ACER2D contours. Determine 13 reveals high-quality settlement between optimized AL and GL surfaces and corresponding bivariate ACER2D surfaces.

As an alternative of the pair of uncoupled univariate design factors sometimes used within the business, bivariate contours present bivariate design factors with the identical return interval. The latter technique may lead to a multi-dimensional design level that’s much less conservative, which might lead to decrease building prices^{33,36}.

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