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When investigating the dynamics of three-dimensional multi-body biomechanical systems it is often difficult to derive spatiotemporally directed predictions regarding experimentally induced effects. A paradigm of ‘non-directed’ hypothesis testing has emerged in the literature as a result. Non-directed analyses typically consist of ad hoc scalar extraction, an approach which substantially simplifies the original, highly multivariate datasets (many time points, many vector components). This paper describes a commensurately multivariate method as an alternative to scalar extraction. The method, called ‘statistical parametric mapping’ (SPM), uses random field theory to objectively identify field regions which co-vary significantly with the experimental design. We compared SPM to scalar extraction by re-analyzing three publicly available datasets: 3D knee kinematics, a ten-muscle force system, and 3D ground reaction forces. Scalar extraction was found to bias the analyses of all three datasets by failing to consider sufficient portions of the dataset, and/or by failing to consider covariance amongst vector components. SPM overcame both problems by conducting hypothesis testing at the (massively multivariate) vector trajectory level, with random field corrections simultaneously accounting for temporal correlation and vector covariance. While SPM has been widely demonstrated to be effective for analyzing 3D scalar fields, the current results are the first to demonstrate its effectiveness for 1D vector field analysis. It was concluded that SPM offers a generalized, statistically comprehensive solution to scalar extraction's over-simplification of vector trajectories, thereby making it useful for objectively guiding analyses of complex biomechanical systems.

Biomechanical processes are often manifested as one-dimensional (1D) trajectories. It has been shown that 1D confidence intervals (CIs) are biased when based on 0D statistical procedures, and the non-parametric 1D bootstrap CI has emerged in the Biomechanics literature as a viable solution. The primary purpose of this paper was to clarify that, for 1D biomechanics datasets, the distinction between 0D and 1D methods is much more important than the distinction between parametric and non-parametric procedures. A secondary purpose was to demonstrate that a parametric equivalent to the 1D bootstrap exists in the form of a random field theory (RFT) correction for multiple comparisons. To emphasize these points we analyzed six datasets consisting of force and kinematic trajectories in one-sample, paired, two-sample and regression designs. Results showed, first, that the 1D bootstrap and other 1D non-parametric CIs were qualitatively identical to RFT CIs, and all were very different from 0D CIs. Second, 1D parametric and 1D non-parametric hypothesis testing results were qualitatively identical for all six datasets. Last, we highlight the limitations of 1D CIs by demonstrating that they are complex, design-dependent, and thus non-generalizable. These results suggest that (i) analyses of 1D data based on 0D models of randomness are generally biased unless one explicitly identifies 0D variables before the experiment, and (ii) parametric and non-parametric 1D hypothesis testing provide an unambiguous framework for analysis when one׳s hypothesis explicitly or implicitly pertains to whole 1D trajectories.

A variety of biomechanical data are sampled from smooth n-dimensional spatiotemporal fields. These data are usually analyzed discretely, by extracting summary metrics from particular points or regions in the continuum. It has been shown that, in certain situations, such schemes can compromise the spatiotemporal integrity of the original fields. An alternative methodology called statistical parametric mapping (SPM), designed specifically for continuous field analysis, constructs statistical images that lie in the original, biomechanically meaningful sampling space. The current paper demonstrates how SPM can be used to analyze both experimental and simulated biomechanical field data of arbitrary spatiotemporal dimensionality. Firstly, 0-, 1-, 2-, and 3-dimensional spatiotemporal datasets derived from a pedobarographic experiment were analyzed using a common linear model to emphasize that SPM procedures are (practically) identical irrespective of the data's physical dimensionality. Secondly two probabilistic finite element simulation studies were conducted, examining heel pad stress and femoral strain fields, respectively, to demonstrate how SPM can be used to probe the significance of field-wide simulation results in the presence of uncontrollable or induced modeling uncertainty. Results were biomechanically intuitive and suggest that SPM may be suitable for a wide variety of mechanical field applications. SPM's main theoretical advantage is that it avoids problems associated with a priori assumptions regarding the spatiotemporal foci of field signals. SPM's main practical advantage is that a unified framework, encapsulated by a single linear equation, affords comprehensive statistical analyses of smooth scalar fields in arbitrarily bounded n-dimensional spaces.

A false positive is the mistake of inferring an effect when none exists, and although α controls the false positive (Type I error) rate in classical hypothesis testing, a given α value is accurate only if the underlying model of randomness appropriately reflects experimentally observed variance. Hypotheses pertaining to one-dimensional (1D) (e.g. time-varying) biomechanical trajectories are most often tested using a traditional zero-dimensional (0D) Gaussian model of randomness, but variance in these datasets is clearly 1D. The purpose of this study was to determine the likelihood that analyzing smooth 1D data with a 0D model of variance will produce false positives. We first used random field theory (RFT) to predict the probability of false positives in 0D analyses. We then validated RFT predictions via numerical simulations of smooth Gaussian 1D trajectories. Results showed that, across a range of public kinematic, force/moment and EMG datasets, the median false positive rate was 0.382 and not the assumed α=0.05, even for a simple two-sample t test involving N=10 trajectories per group. The median false positive rate for experiments involving three-component vector trajectories was p=0.764. This rate increased to p=0.945 for two three-component vector trajectories, and to p=0.999 for six three-component vectors. This implies that experiments involving vector trajectories have a high probability of yielding 0D statistical significance when there is, in fact, no 1D effect. Either (a) explicit a priori identification of 0D variables or (b) adoption of 1D methods can more tightly control α.

Risk assessment of earth dams is concerned not only with the probability of failure but also with the corresponding consequence, which can be more difficult to quantify when the spatial variability of soil properties is involved. This study presents a risk assessment for an earth dam in spatially variable soils using the random adaptive finite element limit analysis. The random field theory, adaptive finite element limit analysis, and Monte Carlo simulation are employed to implement the entire process. Among these methods, the random field theory is first introduced to describe the soil spatial variability. Then the adaptive finite element limit analysis is adopted to obtain the bound solution and consequence. Finally, the failure probability and risk assessment are counted via the Monte Carlo simulation. In contrary to the deterministic analysis that only a factor of safety is given, the stochastic analysis considering the spatial variability can provide statistical characteristics of the stability and assess the risk of the earth dam failure comprehensively, which can be further used for guiding decision-making and mitigation. Besides, the effects of the correlation structure of strength parameters on the stochastic response and risk assessment of the earth dam are investigated through parametric analysis.

The random rock failure process analysis (RRFPA) method was developed in this research to characterize the material spatial variability and uncertainty in rock failure modelling. The random field theory (RFT) was integrated with the traditional rock failure process analysis (RFPA) to model rock heterogeneity. In this approach, the variation of rock properties is represented as a function of relative distance, such that the influence of material intrinsic correlation on its fracturing behaviour can be appropriately captured. To validate the theory, 300 RRFPA simulations were conducted to investigate the failure characteristics of rock samples under compressive loading. The results showed that by incorporating a spectrum of material properties, the numerical outcomes exhibited distinct upper and lower bounds of stress across all testing scenarios, closely aligning with the experimental relationships. The histograms for uniaxial compressive strength and elastic modulus showed that both properties followed normal distributions, with the average values of 10.099 MPa and 1.818 GPa, respectively. The corresponding coefficients of variation were 0.450 and 0.038. The localized failure tended to result in a more rapid release of acoustic emission energy, but generated smaller cumulative energy compared to the overall failure pattern. In general, the maximum relative error of the RRFPA model was only 0.66% for uniaxial compressive strength, elastic modulus, and critical axial strain.

One-dimensional (1D) kinematic, force, and EMG trajectories are often analyzed using zero-dimensional (0D) metrics like local extrema. Recently whole-trajectory 1D methods have emerged in the literature as alternatives. Since 0D and 1D methods can yield qualitatively different results, the two approaches may appear to be theoretically distinct. The purposes of this paper were (a) to clarify that 0D and 1D approaches are actually just special cases of a more general region-of-interest (ROI) analysis framework, and (b) to demonstrate how ROIs can augment statistical power. We first simulated millions of smooth, random 1D datasets to validate theoretical predictions of the 0D, 1D and ROI approaches and to emphasize how ROIs provide a continuous bridge between 0D and 1D results. We then analyzed a variety of public datasets to demonstrate potential effects of ROIs on biomechanical conclusions. Results showed, first, that a priori ROI particulars can qualitatively affect the biomechanical conclusions that emerge from analyses and, second, that ROIs derived from exploratory/pilot analyses can detect smaller biomechanical effects than are detectable using full 1D methods. We recommend regarding ROIs, like data filtering particulars and Type I error rate, as parameters which can affect hypothesis testing results, and thus as sensitivity analysis tools to ensure arbitrary decisions do not influence scientific interpretations. Last, we describe open-source Python and MATLAB implementations of 1D ROI analysis for arbitrary experimental designs ranging from one-sample t tests to MANOVA.

In this article, a microscopic constitutive model is established that includes friction, plastic, and spring elements and has clear physical meaning. The friction unit reflects the mutual friction between crack surfaces, the plastic unit reflects the development of concrete plasticity, and the fracture of the spring unit reflects the formation and expansion of interior cracks in concrete. In addition, the integration of the random field theory into this model uncovers the physical underpinnings of the relationship between concrete’s nonlinearity and randomness. The multi-scale modeling of the concrete static damage constitutive model is then realized once the parameters of the random field are discovered using the macro test results. In order to apply the model’s applicability in finite element programs, a subroutine was ultimately constructed. The experimental data and the anticipated values from the numerical simulation are in good agreement, supporting the model’s realism.

Nonuniform (non-constant) temporal smoothness can arise in biomechanical processes like impacts, and heterogeneous smoothness (unequal smoothness across observations) can arise in mechanically diverse comparisons such as padded vs. unpadded impacts, where padded dynamics are generally smoother than unpadded dynamics. It has been reported that statistical parametric mapping’s (SPM’s) probability values can be invalid for such cases. The purpose of this paper was to clarify the scope of validity for SPM analysis of nonuniformly and heterogeneously smooth one-dimensional (1D) data. We simulated a variety of nonuniformly and heterogeneously smooth Gaussian 1D data over a range of smoothness values, and computed Type I error rates across 10,000 simulation iterations for each smoothness type. Results showed that, in all cases, SPM accurately controlled error at the prescribed α=0.05. Moreover, the distribution of false positives was uniform across time, implying that all regions are equally likely to produce false positives, irrespective of local roughness. We nevertheless show that cluster-level inferences (i.e., p values specific to local regions of significance), while never exceeding alpha (by definition), may be over-or-underestimated by approximately 0.01 for the currently simulated scenarios. We conclude that SPM’s null hypothesis rejection decisions are valid for both nonuniform and heterogeneous 1D data, but that clusters’ p values may be marginally too small/large in rough/smooth regions, respectively. Since cluster-level p values never exceed α, these p value errors are negligible for hypothesis testing purposes. Nevertheless, inter-cluster p value comparisons should be avoided. Implications for statistical power and general results interpretation are discussed.

Eye movement data analyses are commonly based on the probability of occurrence of saccades and fixations (and their characteristics) in given regions of interest (ROIs). In this article, we introduce an alternative method for computing statistical fixation maps of eye movements— i Map—based on an approach inspired by methods used in functional magnetic resonance imaging. Importantly, i Map does not require the a priori segmentation of the experimental images into ROIs. With i Map, fixation data are first smoothed by convolving Gaussian kernels to generate three-dimensional fixation maps. This procedure embodies eyetracker accuracy, but the Gaussian kernel can also be flexibly set to represent acuity or attentional constraints. In addition, the smoothed fixation data generated by i Map conform to the assumptions of the robust statistical random field theory (RFT) approach, which is applied thereafter to assess significant fixation spots and differences across the three-dimensional fixation maps. The RFT corrects for the multiple statistical comparisons generated by the numerous pixels constituting the digital images. To illustrate the processing steps of i Map, we provide sample analyses of real eye movement data from face, visual scene, and memory processing. The i Map MATLAB toolbox is editable and freely available for download online ( www.unifr.ch/psycho/ibmlab/ ).