Smirnov Statistic Research Articles

Quality control of DEMs using check surfaces

ABSTRACT This paper proposes the use of check surfaces or ‘patches’ for the assessment of Digital Elevation Model (DEM) elevations and derived variables (e.g. slope and aspect). We work with grid-type DEMs, and the patches are implemented through square windows of different sizes. Since the data included in a patch are autocorrelated, a trial is performed under controlled conditions of simulation, and then the same simulation method is applied to a DEM data set. Using the Kolmogorov–Smirnov statistic, the observed distribution functions of the results for different sample sizes (20, 50, 100) and different patch sizes (3 × 3, 5 × 5, 9 × 9, 11 × 11, 15 × 15, 19 × 19) and for a sample of points are compared against the true population distribution function. The trial based on autocorrelated synthetic data created following a geostatistical model (exponential law) outputs a clear result regarding the influence of intra-patch autocorrelation. The lower this autocorrelation is, the better the patch performs in terms of sampling efficiency. The same simulation process applied to synthetic data has also been applied to the case of a DEM product. The results obtained allow us to establish an equivalence between sample size when using control patches and control points. The main result of this study is that it allows us to understand the behaviour of samples based on patches and offers us a guide for determining the size of patch-based samples. The most relevant contribution of this proposal is that it opens the possibility of analysing variables derived from elevation data, such as slope and aspect.

Dynamic robustness evaluation for automated model selection in operation

Context:The increasing use of artificial neural network (ANN) classifiers in systems, especially safety-critical systems (SCSs), requires ensuring their robustness against out-of-distribution (OOD) shifts in operation, which are changes in the underlying data distribution from the data training the classifier. However, measuring the robustness of classifiers in operation with only unlabeled data is challenging. Additionally, machine learning engineers may need to compare different models or versions of the same model and switch to an optimal version based on their robustness. Objective:This paper explores the problem of dynamic robustness evaluation for automated model selection. We aim to find efficient and effective metrics for evaluating and comparing the robustness of multiple ANN classifiers using unlabeled operational data. Methods:To quantitatively measure the differences between the model outputs and assess robustness under OOD shifts using unlabeled data, we choose distance-based metrics. An empirical comparison of five such metrics, suitable for higher-dimensional data like images, is performed. The selected metrics include Wasserstein distance (WD), maximum mean discrepancy (MMD), Hellinger distance (HL), Kolmogorov–Smirnov statistic (KS), and Kullback–Leibler divergence (KL), known for their efficacy in quantifying distribution differences. We evaluate these metrics on 20 state-of-the-art models (ten CIFAR10-based models, five CIFAR100-based models, and five ImageNet-based models) from a widely used robustness benchmark (RobustBench) using data perturbed with various types and magnitudes of corruptions to mimic real-world OOD shifts. Results:Our findings reveal that the WD metric outperforms others when ranking multiple ANN models for CIFAR10- and CIFAR100-based models, while the KS metric demonstrates superior performance for ImageNet-based models. MMD can be used as a reliable second option for both datasets. Conclusion:This study highlights the effectiveness of distance-based metrics in ranking models’ robustness for automated model selection. It also emphasizes the significance of advancing research in dynamic robustness evaluation.

Synthesis of engineering surfaces using representative elementary patterns of roughness

Synthesis of surface roughness is a long-standing problem that has many practical applications. Here novel algorithms for the synthesis of rough surfaces at nano/micro scales are proposed. The algorithms are based on introduction and development of two new concepts, namely the representative elementary pattern of roughness (REPR) and the statistically representative pattern of surface roughness (SRPSR). From the statistical point of view, the REPR is the smallest interval (or area) over which a measurement can be made that represents statistically the whole surface. However, synthesis of surfaces by the direct use of the REPR may cause some artificial singularities. To avoid this drawback and to incorporate the synthetic surface in a numerical scheme of the contact solver, one needs to extend the REPR to a non-singular SRPSR that satisfies additional conditions of the scheme used. Our findings indicate that specific time-series analysis techniques, such as the moving window approach, can be effectively utilized to extract the REPR from experimental data. The representativeness may be justified by the use of the Kolmogorov–Smirnov statistic. Extraction of REPRs of surfaces and constructions of appropriate SRPSRs are demonstrated on experimental data obtained by stylus and atomic-force microscopy at micro and atomic/nano scales, respectively.

Pulsar Population Synthesis with Magnetorotational Evolution: Constraining the Decay of the Magnetic Field

We present a population synthesis model for normal radio pulsars in the Galaxy incorporating the latest developments in the field and the magnetorotational evolution processes. Our model considers spin-down with a force-free magnetosphere and the decay of the magnetic field strength and its inclination angle. The simulated pulsar population is fit to a large observation sample that covers the majority of radio surveys using the Markov Chain Monte Carlo technique. We compare the distributions of four major observables—spin period (P), spin-down rate ( Ṗ ), dispersion measure, and radio flux density—using accurate high-dimensional Kolmogorov–Smirnov statistics. We test two B-field decay scenarios, an exponential model motivated by ohmic dissipation and a power-law model motivated by the Hall effect. The former clearly provides a better fit, and it can successfully reproduce the observed pulsar distributions with a decay timescale of 8.3−3.0+3.9 Myr. The result suggests that significant B-field decay in aged pulsars and ohmic dissipation could be the dominant process.

An assessment of global land susceptibility to wind erosion based on deep-active learning modelling and interpretation techniques

Spatial accurate mapping of land susceptibility to wind erosion is necessary to mitigate its destructive consequences. In this research, for the first time, we developed a novel methodology based on deep learning (DL) and active learning (AL) models, their combination (e.g., recurrent neural network (RNN), RNN-AL, gated recurrent units (GRU), and GRU-AL) and three interpretation techniques (e.g., synergy matrix, SHapley Additive exPlanations (SHAP) decision plot, and accumulated local effects (ALE) plot) to map global land susceptibility to wind erosion. In this respect, 13 variables were explored as controlling factors to wind erosion, and eight of them (e.g., wind speed, topsoil carbon content, topsoil clay content, elevation, topsoil gravel fragment, precipitation, topsoil sand content and soil moisture) were selected as important factors via the Harris Hawk Optimization (HHO) feature selection algorithm. The four models were applied to map land susceptibility to wind erosion, and their performance was assessed by three measures consisting of area under of receiver operating characteristic (AUROC) curve, cumulative gain and Kolmogorov Smirnov (KS) statistic plots. The results revealed that GRU-AL model was considered as the most accurate, revealing that 38.5%, 12.6%, 10.3%, 12.5% and 26.1% of the global lands are grouped at very low, low, moderate, high and very high susceptibility classes to wind erosion hazard, respectively. Interpretation techniques were applied to interpret the contribution and impact of the eight input variables on the model’s output. Synergy plot revealed that the soil carbon content exhibited high synergy with DEM and soil moisture on the model’s predictions. ALE plot showed that soil carbon content and precipitation had negative feedback on the prediction of land susceptibility to wind erosion. Based on SHAP decision plot, soil moisture and DEM presented the highest contribution on the model’s output. Results highlighted new regions at high latitudes (southern Greenland coast, hotspots in Alaska and Siberia), which exhibited high and very high land susceptibility to wind erosion.

A Multiscale Statistical Analysis of Rough Surfaces and Applications to Tribology

Mathematical modelling of surface roughness is of significant interest for a variety of modern applications, including, but not limited to, tribology and optics. The most popular approaches to modelling rough surfaces are reviewed and critically examined. By providing counterexamples, it is shown that approaches based solely on the use of the fractal geometry or power spectral density have many drawbacks. It is recommended to avoid these approaches. It is argued that the surfaces that cannot be distinguished from the original rough surfaces can be synthesised by employing the concept of the representative elementary pattern of roughness (REPR), i.e., the smallest interval (or area) of a rough surface that statistically represents the whole surface. The REPR may be extracted from surface measurement data by the use of the “moving window” technique in combination with the Kolmogorov–Smirnov statistic.

The Impact of the Experiential Learning Model on Increasing Digital Skills for Employees

Objective: This experimental study aims to find out which training model is more appropriate to use in preparing employees who have digital skills in a shorter. The purpose of this research are knowing a more appropriate training model for the preparation of employee digital skills and knowing the impact of the experiential learning model on increasing the digital skills for employee, namely Enterprise Architecture and Database & Programming Knowledge. Methods: This research was conducted using experiments study, conducted in 2 (two) groups where group A will receive learning treatment using conventional learning models as the control and group B will receive learning using experiential learning models as the experiment. The findings for the research are that there are differences in all aspects of digital skills between group experiment and group control. Experiential Learning learning models have an impact on increasing employee digital skills, especially on aspects of Database & Programming Knowledge Results: The dependent variable in this study was the gain score differences in the aspects of Enterprise Architecture and Database & Programming Knowledge of the employees. The calculation results of the central size (mean and standard deviation) for gain scores data in both aspects of Enterprise Architecture and Database & Programming Knowledge. To test the research hypothesis, the assumption test was first carried out as a prerequisite for hypothesis testing, namely the data distribution normality test, the covariance variance homogeneity test, and the dependent variable intercorrelation test. The normality test in this study was conducted using Kolmogorov Smirnov statistics. Conclusion: There are differences in all aspects of digital skills between employees who learn with the experiential learning model and employees who learn with the conventional learning model. There is no difference in the aspects of digital skills Enterprise Architecture between employees who learn with experiential learning models and employees who learn with conventional learning models.

Estimation of time-varying kernel densities and chronology of the impact of COVID-19 on financial markets

The time-varying kernel density estimation relies on two free parameters: the bandwidth and the discount factor. We propose to select these parameters so as to minimize a criterion consistent with the traditional requirements of the validation of a probability density forecast. These requirements are both the uniformity and the independence of the so-called probability integral transforms, which are the forecast time-varying cumulated distributions applied to the observations. We thus build a new numerical criterion incorporating both the uniformity and independence properties by the mean of an adapted Kolmogorov–Smirnov statistic. We apply this method to financial markets during the onset of the COVID-19 crisis. We determine the time-varying density of daily price returns of several stock indices and, using various divergence statistics, we are able to describe the chronology of the crisis as well as regional disparities. For instance, we observe a more limited impact of COVID-19 on financial markets in China, a strong impact in the US, and a slow recovery in Europe.

Threshold selection for extremal index estimation

ABSTRACT We propose a new threshold selection method for nonparametric estimation of the extremal index of stochastic processes. The discrepancy method was proposed as a data-driven smoothing tool for estimation of a probability density function. Now it is modified to select a threshold parameter of an extremal index estimator. A modification of the discrepancy statistic based on the Cramér–von Mises–Smirnov statistic ω 2 is calculated by k largest order statistics instead of an entire sample. Its asymptotic distribution as k → ∞ is proved to coincide with the ω 2 -distribution. Its quantiles are used as discrepancy values. The convergence rate of an extremal index estimate coupled with the discrepancy method is derived. The discrepancy method is used as an automatic threshold selection for the intervals and K-gaps estimators. It may be applied to other estimators of the extremal index. The performance of our method is evaluated by simulated and real data examples.

A new extension of linear failure rate distribution with estimation, simulation, and applications

In this article, we provide a new three-parameter model derived from the newly reduced Cauchy power-G family and linked to the linear failure rate model. The truncated Cauchy power linear failure rate (TCPLFR) is the name given to this distribution. The TCPLFR distribution also contains the truncated Cauchy power Rayleigh distribution and the truncated Cauchy power exponential distribution as sub-models. The TCPLFR distribution has rising, falling, and U-shaped hazard rate functions. The distribution characteristics of the TCPLFR are presented. To compute the population parameters’ point and estimated confidence intervals, the maximum likelihood approach is employed. We explore the behavior of the maximum likelihood estimates as well as the estimated confidence intervals for the model parameters using Monte Carlo simulation. To demonstrate the significance and flexibility of the TCPLFR distribution, the Akaike information criterion (D1), Bayesian information criterion (D2), consistent Akaike information criterion (D3), Hannan–Quinn information criterion (D4), and Kolmogorov–Smirnov (D5) statistic with its p-value (D6) were employed. According to a real-world data analysis, the truncated Cauchy power linear failure rate distribution outperforms alternative models with two, three, and four parameters.

Interpretability of simple RNN and GRU deep learning models used to map land susceptibility to gully erosion

Gully erosion possess a serious hazard to critical resources such as soil, water, and vegetation cover within watersheds. Therefore, spatial maps of gully erosion hazards can be instrumental in mitigating its negative consequences. Among the various methods used to explore and map gully erosion, advanced learning techniques, especially deep learning (DL) models, are highly capable of spatial mapping and can provide accurate predictions for generating spatial maps of gully erosion at different scales (e.g., local, regional, continental, and global). In this paper, we applied two DL models, namely a simple recurrent neural network (RNN) and a gated recurrent unit (GRU), to map land susceptibility to gully erosion in the Shamil-Minab plain, Hormozgan province, southern Iran. To address the inherent black box nature of DL models, we applied three novel interpretability methods consisting of SHaply Additive explanation (SHAP), ceteris paribus and partial dependence (CP-PD) profiles and permutation feature importance (PFI). Using the Boruta algorithm, we identified seven important features that control gully erosion: soil bulk density, clay content, elevation, land use type, vegetation cover, sand content, and silt content. These features, along with an inventory map of gully erosion (based on a 70 % training dataset and 30 % test dataset), were used to generate spatial maps of gully erosion using DL models. According to the Kolmogorov–Smirnov (KS) statistic performance assessment measure, the simple RNN model (with KS = 91.6) outperformed the GRU model (with KS = 66.6). Based on the results from the simple RNN model, 7.4 %, 14.5 %, 18.9 %, 31.2 % and 28 % of total area of the plain were classified as very-low, low, moderate, high and very-high hazard classes, respectively. According to SHAP plots, CP-PD profiles, and PFI measures, soil silt content, vegetation cover (NDVI) and land use type had the highest impact on the model's output. Overall, the DL modelling techniques and interpretation methods used in this study proved to be helpful in generating spatial maps of soil erosion hazard, especially gully erosion. Their interpretability can support watershed sustainable management.

Uncovering the fractal nature of water vapor distribution above the surface of the earth

Fractals have been at the heart of geophysical and geospatial studies in the recent past. We explore the fractal character of water vapor distributions above the surface of the Earth as a function of both image resolution (number of pixels) and moisture content percentile and study the emergent scaling properties. The percolation phenomena associated with the vapor distribution clusters has been studied for both the cases. For our analysis, computational methods and algorithms have been used to derive the physically relevant quantities such as fractal dimension, number of clusters, and size of the largest cluster. The box counting method, used to calculate the fractal dimension, unravels a potential multi-fractal character of the data. We also test the applicability of Korcak’s law on our system and determine the quality of the fit using the Kolmogorov–Smirnov statistic. We show that the fractal character of the distribution is exact as a function of image resolution and approximate in some regimes as a function of the vapor percentile.

Is housing price distribution across cities, scale invariant? Fractal distribution of settlements' house prices as signature of self-organized complexity

Locations and sizes of settlements within regions often exhibit a self-organized central place pattern, where larger settlements are encircled by smaller ones, with the former more set apart. This hierarchical spatial pattern is displayed by the fractal signature of the power law typical of the rank-size distribution of settlements. Is a similar pattern also revealed in the housing prices across settlements? An empirical analysis on the almost 8000 Italian municipalities, whose residential population varies between 33 and ∼3 million, shows that, for 15 out of 20 regions, the upper tails rank-house price distribution of settlements is plausibly following a power law. This result discloses a universality in the function behaviour: a scale invariance. The average scaling parameters have been estimated via Maximum Likelihood Estimator and Kolmogorov–Smirnov statistics, and the goodness-of-fit assessed via bootstrapping. Ordinary Least Squares regressions have also been used for comparative purposes. In average, Maximum Likelihood Estimator found a scaling parameter of 4.4. Namely, doubling the rank of the settlement (in terms of housing price), its housing price decreases by 15 %: between the settlement with the highest average house prices (ranked 1), and the settlement with the second highest average house prices (ranked 2), there is a 15 % difference in the average house prices. Idem between the ranked 5 and 10, 10 and 20 … it does not matter; it is a universal behavioural pattern whose scale invariance suggests a self-organized complexity in the phenomena behind it.

Short communication: The Wasserstein distance as a dissimilarity metric for comparing detrital age spectra and other geological distributions

Abstract. Distributional data such as detrital age populations or grain size distributions are common in the geological sciences. As analytical techniques become more sophisticated, increasingly large amounts of distributional data are being gathered. These advances require quantitative and objective methods, such as multidimensional scaling (MDS), to analyse large numbers of samples. Crucial to such methods is choosing a sensible measure of dissimilarity between samples. At present, the Kolmogorov–Smirnov (KS) statistic is the most widely used of these dissimilarity measures. However, the KS statistic has some limitations such as high sensitivity to differences between the modes of two distributions and insensitivity to their tails. Here, we propose the Wasserstein-2 distance (W2) as an additional and alternative metric for use in geochronology. Whereas the KS distance is defined as the maximum vertical distance between two empirical cumulative distribution functions, the W2 distance is a function of the horizontal distances (i.e. age differences) between observations. Using a variety of synthetic and real datasets, we explore scenarios where the W2 may provide greater geological insight than the KS statistic. We find that in cases where absolute time differences are not relevant (e.g. mixing of known, discrete age peaks), the KS statistic can be more intuitive. However, in scenarios where absolute age differences are important (e.g. temporally and/or spatially evolving sources, thermochronology, and overcoming laboratory biases), W2 is preferable. The W2 distance has been added to the R package, IsoplotR, for immediate use in detrital geochronology and other applications. The W2 distance can be generalized to multiple dimensions, which opens opportunities beyond distributional data.

Evaluation of global climate models for the simulation of precipitation and maximum and minimum temperatures at coarser and finer resolutions based on temporal and spatial assessment metrics in mainland of China

Abstract Many studies have evaluated the performance of multiple global climate models (GCMs) from a temporal or spatial perspective at finer resolution, but no study has evaluated the performance of individual GCMs at different resolutions and before and after bias correction from both temporal and spatial perspectives. The goal of this study is to evaluate the performance of 21 Coupled Model Inter-comparison Project 6 (CMIP6) GCMs at the raw (coarser) and downscaled (finer) resolutions and after bias correction in relation to their skills in the simulation of daily precipitation and maximum and minimum temperatures over China for the period 1961–2014 using state-of-the-art temporal and spatial metrics, Kolmogorov–Smirnov statistic and SPAtial EFficiency. The results indicated some differences in the ranks of GCMs between temporal and spatial metrics at different resolutions. The overall ranking shows that the simulations at the raw resolution of GCMs are more similar to the observations than the simulations after inverse distance weighted interpolation in SPAtial EFficiency. Three variables from bias-corrected GCMs ranked from 1 to 21 show similar good performance in spatial patterns but the poorest trend in empirical Cumulative Distribution Functions (ECDFs) except daily precipitation.

Distribution of Computerized Datasets To Fit Weibull, Log-Logistics and Gompertz Survival Models

The lack of detailed and analytic study on the risk factors of Breast cancer in Nigerian presents us with the challenge of not knowing the risk factors distinct to the Nigerian settings. Effectively handling the breast cancer treatment in co-infected patients is delicate. Some challenges to the treatment of breast cancer infection include drug-to-drug interactions between procedures required to treat the cancer cells, thus the hormonal drugs burdens patient’s inability to comply with dosage intake thereby making women with breast cancer prone to developing other forms of cancer. The aim of the research was to analyzed distributed dataset to fit Weibull, Log-logistics and Gompertz survival models. The research methodology involved the collection of data cases on oncological study analyzed using descriptive statistics and parametric survival models. The result shows that the Cramer-von Misses (W), the Anderson Darling (A) and the Kolmogorov Smirnov (D) statistics, Log-logistic distribution has greater p-value than other distributions, hence it is the best fit for survival times of a group of patients given chemotherapy treatment. The research concluded that Gompertz model was the best fitted model to survival times of a group of patients given chemotherapy treatment data, while log-logistic distribution was the best fit distribution for the data. It also concluded that Weibull and log-logistic models perform better than Gompertz model in fatigue fracture data, while log-logistic distribution was the best fit for the data and Gompertz model and the distribution was the best fit to nicotine measurements.

Two-stage credit scoring using Bayesian approach

Commercial banks are required to explain the credit evaluation results to their customers. Therefore, banks attempt to improve the performance of their credit scoring models while ensuring the interpretability of the results. However, there is a tradeoff between the logistic regression model and machine learning-based techniques regarding interpretability and model performance because machine learning-based models are a black box. To deal with the tradeoff, in this study, we present a two-stage logistic regression method based on the Bayesian approach. In the first stage, we generate the derivative variables by linearly combining the original features with their explanatory powers based on the Bayesian inference. The second stage involves developing a credit scoring model through logistic regression using these derivative variables. Through this process, the explanatory power of a large number of original features can be utilized for default prediction, and the use of logistic regression maintains the model's interpretability. In the empirical analysis, the independent sample t-test reveals that our proposed approach significantly improves the model’s performance compared to that based on the conventional single-stage approach, i.e., the baseline model. The Kolmogorov–Smirnov statistics show a 3.42 percentage points (%p) increase, and the area under the receiver operating characteristic shows a 2.61%p increase. Given that our two-stage modeling approach has the advantages of interpretability and enhanced performance of the credit scoring model, our proposed method is essential for those in charge of banking who must explain credit evaluation results and find ways to improve the performance of credit scoring models.

Robust online detection on highly censored data using a semi-parametric EWMA chart

For time and cost considerations, high censoring rate is quite common in life tests, which is a critical issue in lifetime monitoring. Conventional control charts designed for highly censored data are commonly based on the Weibull distribution. However, the distribution assumption may not be valid in practice, which brings challenges to the monitoring procedures. Motivated by this, a semi-parametric exponential weighted moving average (EWMA) control charting procedure is developed for highly censored lifetime data of any distribution. The control scheme uses the Kaplan–Meier estimator to construct the cumulative distribution function (CDF) and generalized Pareto distribution to improve the tail estimation. Then a Kolmogorov–Smirnov statistic defined by the differences between the in-control CDF and the empirical CDF is integrated into an EWMA charting scheme to monitor the Type I right-censored sample. We use simulation studies and a real-data analysis to show the efficiency of the proposed control chart.

Modeling the Incomes of the Upper-Class Group in Malaysia using New Pareto-Type Distribution

The new Pareto-type distribution has been previously introduced as an alternative to the conventional Pareto distribution in modeling income distribution. It is claimed to provide better flexibility for mathematical simplicity of probability functions and has a more straightforward mathematical form. In this study, the new Pareto-type distribution is used to model the income of the Malaysian upper-class group. The threshold is determined using the fixed proportion technique and the maximum likelihood estimator method is used to estimate the shape parameter. Then, the goodness-of-fit of the fitted new Pareto model is measured using the coefficient of determination, R2 and Kolmogorov–Smirnov statistics. We also measure the income inequality among the Malaysian top income earners using the Lorenz curve, Gini and Theil indices based on the fitted new Pareto model. Finally, the new Pareto distribution is compared to alternative distributions to analyze which model can give the best fit for the data. Our analysis shows that the Pareto type-1 and the new Pareto models are well fitted to the top income data for all years considered. However, the new Pareto model provides better flexibility which covering more incomes in the upper tail of the distribution than the Pareto type-1 model.

A new family of compound exponentiated logarithmic distributions with applications to lifetime data

Abstract The logarithmic distribution and a given lifetime distribution are compounded to construct a new family of lifetime distributions. The compounding is performed with respect to maxima. Expressions are derived for lifetime properties like moments and the behavior of extreme values. Estimation procedures for the method of maximum likelihood are also derived and their performance assessed by a simulation study. Three real data (including two lifetime data) applications are described that show superior performance (assessed with respect to Kolmogorov Smirnov statistics, likelihood values, AIC values, BIC values, probability-probability plots and density plots) versus at least five known lifetime models, with each model having the same number of parameters as the model it is compared to.