K olmogorov– S mirnov Tests

While often confused, the Kolmogorov–Smirnov test and the Smirnov test are actually distinct. Specifically, the Kolmogorov–Smirnov test is used to test the goodness of fit of a given set of data to a theoretical distribution, making this a one‐sample test. In contrast, the Smirnov test is a two‐sample test, used to determine if two samples appear to follow the same distribution. The intuition behind the two tests is the same, however, in that both compare cumulative distribution functions, either two empirical cumulative distribution functions for the two‐sample Smirnov test, or one empirical cumulative distribution function and one known cumulative distribution function for the one‐sample Kolmogorov–Smirnov test.

<strong class="journal-contentHeaderColor">Abstract.</strong> 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. It is very sensitive to differences between the modes of two distributions, and relatively insensitive to differences between their tails. Here we introduce the Wasserstein-2 distance (<em>W</em>₂) as an alternative to address this issue. Whereas the KS-distance is defined as the maximum vertical distance between two empirical cumulative distribution functions, the <em>W</em>₂-distance is a function of the horizontal distances (i.e., age differences) between individual observations. Using a combination of synthetic examples and a published zircon U-Pb dataset, we show that the <em>W</em>₂ distance produces similar MDS results to the KS-distance in most cases, but significantly different results in some cases. Where the results differ, the <em>W</em>₂ results are geologically more sensible. For the case study, we find that the MDS map that is produced using <em>W</em>₂ can be readily interpreted in terms of the shape and average age of the age spectra. The <em>W</em>₂-distance has been added to the R package IsoplotR, for immediate use in detrital geochronology and other applications. The <em>W</em>₂ distance can be generalised to multiple dimensions, which opens opportunities beyond distributional data.

New methods are presented that utilize the optimization of goodness-of-fit statistics in order to estimate Weibull parameters from failure data. It is assumed that the underlying population is characterized by a three-parameter Weibull distribution. Goodness-of-fit tests are based on the empirical distribution function (EDF). The EDF is a step function, calculated using failure data, and represents an approximation of the cumulative distribution function for the underlying population. Statistics (such as the Kolmogorov-Smirnov statistic and the Anderson-Darling statistic) measure the discrepancy between the EDF and the cumulative distribution function (CDF). These statistics are minimized with respect to the three Weibull parameters. Due to nonlinearities encountered in the minimization process, Powell’s numerical optimization procedure is applied to obtain the optimum value of the EDF. Numerical examples show the applicability of these new estimation methods. The results are compared to the estimates obtained with Cooper’s nonlinear regression algorithm.

In this study, we quantify vegetation fire activity in India using the MODerate resolution Imaging Spectroradiometer (MODIS) active fire datasets. We assessed different fire regime attributes, i.e., fire frequency, seasonality, intensity and the type of vegetation burnt in diverse geographical regions. MODIS data from 2002–2010 revealed an average of 63696 fire counts per year with the highest during 2009. Fire season in India extends from October to June with the peak during March. The K-means algorithm identified hotspot regions of fire clusters in diverse regions of India. We examined fire radiative power (FRP) data in the hotspot regions to address which fires burn intensively than others based on the vegetation type. We first assessed the best statistical fit distributions for the FRP data using the probability density functions (PDFs) and ranked them based on Kolmogorov-Smirnov statistic. We then described the fire intensities using empirical cumulative distribution functions (CDFs). Results suggest diverse pdfs for the FRP data that included Burr, Dagum, Johnson as well as Pearson distribution and they varied based on the vegetation type burnt. Analysis from empirical CDFs suggested relatively high fire intensity for closed broadleaved evergreen/ semi-deciduous forests than the other vegetation types. Although, annual sum of FRP for agricultural fires was less than the closed broadleaved evergreen forests, the values were higher than the mosaic vegetation category and broadleaved deciduous forests. These results on fire hotspots and FRP will be useful to address the impact of vegetation fires on air pollution and climate in India.

Distribution test, a method to evaluate the goodness-of-fit, has been applied to automatic modulation clas-sification (AMC) in recent years. It compares the empirical cumulative distribution function (ECDF) of the sample signal with the theoretical cumulative distribution function (TCDF) under each candidate modulation format. Most existing works in MIMO systems assume the additive noise is Gaussian. However, additive noise often exhibits non-Gaussian characteristics in practical systems. The TCDF of each modulation developed for Gaussian noise can not perform well in practical system. To solve this issue, this paper proposes a practical MIMO system model, which assumes that the noise is Cauchy-Gaussian bi-parameter mixture and analyzes the TCDF corresponding to different modulation modes under this noise model. The Kolmogorov-Smirnov (KS) distribution test is used to AMC in flat-fading channel for both Quadrature Amplitude Modulation and Phase Shift Keying. Extensive simulation results demonstrate that, compared with two-sample KS test based classifier, one-sample KS test based classifier can achieve good recognition results under the non-Gaussian noise model.

Probability distribution for the shear strength of seafloor sediment in the KR5 area for the development of manganese nodule miner

Let X 1,…, X n be i.i.d. random variables from an unknown cumulative distribution function F defined on the real line ℝ. No assumptions are made on the unknown F. The problem is to predict the empirical distribution function of a future sample Y = (Y 1,…,Y m ) from the distribution function F on the basis of the observations X 1,…, X n . To solve the problem, a decision theoretic approach is used. In certain situations, when for example a weighted quadratic loss function is used in the prediction problem, the problem of finding the minimax predictor of the empirical distribution function can be reduced to the problem of minimax estimation of the cumulative distribution function F under a composite weighted quadratic loss function. In such a case, a general method of finding minimax predictors of is presented. The method is also useful when no equalizer rule is available. The problem of prediction considered refers also to prediction of the empirical lifetime distribution function and may have some practical significance. For example, the results enable to predict, under the minimax criterion, the number of devices (from among all the m ones) which will fail up to any fixed time t 0.

The human body generates more proteins than it has genes that code for proteins. The diversity of proteins stems from the alternative ways in which RNA can be spliced and reassembled. Each alternative version of RNA produces a different protein, providing a way for our bodies to produce a wide range of proteins with a single gene. Some alternative RNA transcripts, however, have splicing errors and produce faulty proteins involved in genetic diseases. Understanding splicing patterns and profiles has wide implications for our understanding of healthy and diseased tissue states. Currently little is known regarding the splicing profiles of healthy tissue which vary across individuals and within individuals by tissue type. Therefore, this project explored the use of RNA splicing data from the first chromosome to predict the tissue type of non-cancerous samples using distribution analysis and supervised learning methods. The Kolmogorov-Smirnov test was used to classify the samples based on empirical cumulative distribution functions and was not able to reliably distinguish between tissue types. Our SVM model was run using both the quantity of splice junctions observed and their presence, and had a high prediction accuracy for both data sets. The performance between the two SVM model outcomes were not significantly different. Overall, the findings suggest the utility of using splice junction data in biological classification and sets the foundation for future work of mapping splicing patterns with phenotype.

The kernel distribution estimator (KDE) is proposed based on residuals of the innovation distribution in the autoregressive moving-average (ARMA) time series. The deviation between KDE and the innovation distribution function is shown to converge to the Brownian bridge, leading to the construction of a Kolmogorov–Smirnov smooth simultaneous confidence band for the innovation distribution function. Additionally, an empirical cumulative distribution function (CDF) based on prediction residuals is introduced for the multi-step-ahead prediction error distribution function. This empirical process weakly converges to a Gaussian process with a specific covariance function. Furthermore, a quantile estimator is derived from the empirical CDF of prediction residuals, and multi-step-ahead prediction intervals (PIs) for future observations are established using these estimated quantiles. The PIs achieve the nominal prediction level asymptotically for the finite variance ARMA model. Simulation studies and analysis of crude oil price data confirm the validity of the asymptotic theory.

There are many methods for measuring the difference between two location parameters. In this paper, statistics are proposed in order to estimate the difference of two location parameters. The statistics are designed not using the means, variances, signs and ranks, but with the cumulative distribution functions. Hence these are measured as the differences in the area between two univariate cumulative distribution functions. It is found that the difference in the area between two empirical cumulative distribution functions is the difference of two sample means, and its integral is also the difference of two population means.

The Calculation of the Spare Parts in the Auto-service Enterprise on the Base of Real Demand

Impact of mapping strategies on rockfall frequency-size distributions

This paper proposes a data-driven approximation of the Cumulative Distribution Function using the Finite Mixtures of the Cumulative Distribution Function of Logistic distribution. Since it is not possible to solve the logistic mixture model using the Maximum likelihood method, the mixture model is modeled to approximate the empirical cumulative distribution function using the computational intelligence algorithms. The Probability Density Function is obtained by differentiating the estimate of the Cumulative Distribution Function. The proposed technique estimates the Cumulative Distribution Function of different benchmark distributions. Also, the performance of the proposed technique is compared with the state-of-the-art kernel density estimator and the Gaussian Mixture Model. Experimental results on κ−μ distribution show that the proposed technique performs equally well in estimating the probability density function. In contrast, the proposed technique outperforms in estimating the cumulative distribution function. Also, it is evident from the experimental results that the proposed technique outperforms the state-of-the-art Gaussian Mixture model and kernel density estimation techniques with less training data.

An explainable unsupervised risk early warning framework based on the empirical cumulative distribution function: Application to dairy safety

In this paper, we present a new copy-move forgery detection method based on singular value decomposition (SVD) and the Kolmogorov Smirnov (KS) test. This work introduces a new method of detecting copy-move forgery in images with accuracy up to the pixel level using only 4 features per image block. The proposed method consists of three steps. First, an image is partitioned into blocks of size $$16 \times 16$$ . Second, image features are extracted from each block using steerable pyramid and SVD transforms. Finally, the extracted features are sorted lexicographically and matched using the KS test. The performance of the proposed method is evaluated using the CoMoFoD database. Four post-processing techniques are considered, namely brightness change, contrast adjustment, color reduction, and image blurring. This method achieved a high precision of more than 95% for 3 of the 4 post-processing techniques. The fourth post-processing (i.e., image blurring), we achieved a precision of 75% which is considerably high for such forgery. In addition, the proposed method outperformed published methods when the images were subjected to brightness change, contrast adjustment, color reduction and image blurring. Finally, the performance of the proposed algorithm shown to provide better precision using fewer features compared to several well-known techniques in the literature.