This article provides two unique methodologies that may be coupled to study the dependability of multidimensional nonlinear dynamic systems. First, the structural reliability approach is well suited for multidimensional environmental and structural reactions and is either measured or numerically simulated over sufficient time, yielding lengthy ergodic time series. Second, a unique approach to predicting extreme values has technical and environmental implications. In the event of measurable environmental loads, it is also feasible to calculate the probability of system failure, as shown in this research. In addition, traditional probability approaches for time series cannot cope effectively with the system's high dimensionality and cross-correlation across dimensions. It is common knowledge that wind speeds represent a complex, nonlinear, multidimensional, and cross-correlated dynamic environmental system that is always difficult to analyze. Additionally, global warming is a significant element influencing ocean waves throughout time. This section aims to demonstrate the efficacy of the previously mentioned technique by applying a novel method to the Norwegian offshore data set for the greatest daily wind cast speeds in the vicinity of the Landvik wind station. This study aims to evaluate the state-of-the-art approach for extracting essential information about the extreme reaction from observed time histories. The approach provided in this research enables the simple and efficient prediction of failure probability for the whole nonlinear multidimensional dynamic system.

Extreme value predictions typically originate from certain functional classes of statistical distributions to fit the data and are subsequently extrapolated. This paper describes an alternative method for extrapolation that is based on the intrinsic properties of the data set itself and that does not pre-assume any extrapolation functional class. The proposed novel extrapolation method can be utilized in engineering design. To illustrate this, this study uses two examples to showcase the advantages of the proposed method. The first example used synthetic data from a non-linear Duffing oscillator to illustrate the new method. The second example was an actual container ship sailing between Europe and America and experiencing large deck panel stresses in severe weather. In this example, actual onboard measured data were used in the present study. This example represents a real and physical case that is challenging to model due to the non-stationary and highly non-linear natures of the wave-ship load responses. This is especially so in the case of extreme responses, where the roles of second and higher-order responses tend to be more prominent and have higher contributions. The prediction accuracy of the proposed method was also validated versus the Naess–Gaidai extrapolation method. Finally, this study discusses new methods for generic smoothing of distribution tail irregularities due to underlying scarcity in the data set.

Cancer is a worldwide illness that causes significant morbidity and death and imposes an immense cost on global public health. Modelling such a phenomenon is complex because of the non-stationarity and complexity of cancer waves. Apply modern novel statistical methods directly to raw clinical data. To estimate extreme cancer death rate likelihood at any period in any location of interest. Traditional statistical methodologies that deal with temporal observations of multi-regional processes cannot adequately deal with substantial regional dimensionality and cross-correlation of various regional variables. Setting: multicenter, population-based, medical survey data-based biostatistical approach. Due to the non-stationarity and complicated nature of cancer, it is challenging to model such a phenomenon. This paper offers a unique bio-system dependability technique suited for multi-regional environmental and health systems. When monitored over a significant period, it yields a reliable long-term projection of the chance of an exceptional cancer mortality rate. Traditional statistical approaches dealing with temporal observations of multi-regional processes cannot effectively deal with large regional dimensionality and cross-correlation between multiple regional data. The provided approach may be employed in numerous public health applications, depending on their clinical survey data.

The paper describes a novel structural reliability method, particularly suitable for multi-dimensional environmental systems, either measured or numerically simulated over a sufficient period, resulting in sufficiently long ergodic time series. This study illustrates the efficiency of the proposed methodology by applying it to predict extreme wind speeds of a group of selected measured sites in Southern Norway in the region near the Landvik wind station. It is well known that wind speeds at different locations are highly non-linear, multi-dimensional and cross-correlated dynamic environmental responses, which can be challenging to analyse accurately. Unlike other environmental reliability methods, the new method does not require restarting the simulation each time the system fails, e.g., in the case of numerical simulation. In the case of measured environmental system response, an accurate prediction of system failure probability is also possible, as illustrated in this study. Moreover, in contrast to classical reliability methods, the proposed method can handle systems with high dimensionality and cross-correlation between the different dimensions.

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