Survival analysis focuses on modeling the time until a specific event occurs, often in the presence of censored observations.While classical methods like the Cox model are widely used, modern machine learning (ML) approaches offer greater flexibility and predictive power.This paper compares classical and ML-based survival models on both real-world and simulated datasets.We demonstrate that techniques like CoxBoost and penalized Cox regression outperform tree-based models like Random Survival Forests in most settings.Explainable Artificial Intelligence (AI) tools are applied to improve the transparency and interpretability of model predictions.

Classical life tables assume fully observed lifespans and ignore censoring, which can bias survival estimates.In this study, we compare the classical life table with two censoringaware approaches: the Kaplan-Meier estimator applied to simulated censored lifetimes, and the actuarial estimator assuming uniform censoring within intervals.Using mortality data for the Czech Republic (from year 2021), we show that both the mentioned alternative methods yield systematically lower survival and life expectancy estimates, compared to life tables-based approach.The actuarial estimator is the most conservative, and we provide a formal proof that it underestimates survival relative to Kaplan-Meier.These differences have practical implications for actuarial pricing and longevity risk.We advocate incorporating survival analysis techniques into actuarial workflows when dealing with incomplete or censored data.

This paper investigates upper bounds on the length of the shortest watchman route in simple polygons using minimal structural information -specifically, the number and placement of reflex vertices -rather than full geometric detail.We show that the set of reflex vertices alone suffices to guarantee full visibility of the polygon: every interior or boundary point is visible from at least one reflex vertex.Moreover, these vertices can always be connected by a polyline within the polygon, enabling a constructive approach to route design.While it is known that the shortest watchman route is bounded above by the perimeter of the polygon, we prove that this bound is tight and cannot be improved in general.However, we identify several important classes of polygons where significantly better bounds are achievable.In particular, for polygons with exactly two reflex vertices, we establish that the shortest watchman route is strictly shorter than half the perimeter.We also introduce a novel triangulation method -successive dual-edge triangulation -in which each triangle is constructed by attaching to existing polygon edges.When such a triangulation is possible, the resulting watchman route has length strictly less than half the perimeter, regardless of the number of sides or reflex vertices.These results are not only theoretical but also constructive, offering efficient route designs when geometric data is available.The approach is particularly well suited to real-world environments such as industrial halls or private courtyards, where structural simplicity often permits route planning without requiring full geometric reconstruction.In such cases, our methods provide a significant and elegant reduction in route length compared to perimeter-based patrolling, making them applicable to scenarios such as autonomous drone navigation and indoor surveillance.

We propose a statistical hypothesis test for determining whether a given integer n ≥ 2 is prime.Under the null hypothesis H0, we assume that n is not a prime (i.e., composite).The test operates by randomly sampling integers from the candidate divisor set D = {2, 3, . . ., ⌊ √ n⌋} and checking whether any of them divide n.If a proper divisor is found, H0 is not rejected and n is declared composite.If no divisor is found among an initial set of k0 samples, additional k samples are drawn, and a p-value is computed based on the probability of missing all actual divisors under H0.This probability is calculated exactly via a hypergeometric distribution and approximated using an exponential bound.We derive a closed-form upper bound for the minimal number of trials k required to reject H0 at a given significance level α under the conservative assumption of only one true divisor (m = 1).The algorithm has worst-case time complexity O( √ n), matching that of classical trial division, but its expected runtime is substantially lower when n has multiple divisors.The proposed test is simple, statistically interpretable, and well-suited both as an educational tool and as a lightweight probabilistic pre-check in layered primality testing pipelines.

This paper explores a novel variation of the classical secretary problem, commonly referred to as the marriage or best choice problem.In this adaptation, a decision-maker sequentially dates n ∈ N candidates, each uniquely ranked without ties from 1 to n.The decision strategy involves a preliminary non-selection phase of the first d ∈ N candidates where, d < n, following which the decision-maker commits to the first subsequent candidate who surpasses all previously evaluated candidates in quality.The central focus of this study is the derivation and analysis of P (d, n, k), which denotes the probability that the selected candidate, under the prescribed strategy, ranks among the top k ∈ N overall candidates, where k ≤ n.This investigation employs combinatorial probability theory to formulate P (d, n, k) and explores its behavior across various parameter values of d, n, and k.Particularly, we seek to determine in what fraction of the entire decision process should a decision-maker stop the non-selection phase, i.e., we search for the optimal proportion d n , that maximizes the probability P (d, n, k), with a special focus on scenarios where k is in generally low.While for k = 1, the problem is simplified to the classical secretary problem with d n ≈ 1 e , our findings suggest that the strategy's effectiveness is optimized for portion d n decreasing below 1 e as k increases.Also, intuitively, probability P (d, n, k) increases as k increases, since the number of acceptable top candidates increases.These results not only extend the classical secretary problem but also provide strategic insights into decision-making processes involving ranked choices, sequential evaluation, and applications of searching not necessarily the best candidate, but one of the best candidates.

Comparison of two survival functions, which describe the probability of not experiencing an event of interest by a given time point in two different groups, is a typical task in survival analysis.There are several well-established methods for comparing survival functions, such as the log-rank test and its variants.However, these methods often come with rigid statistical assumptions.In this work, we introduce a non-parametric alternative for comparing survival functions that is nearly free of assumptions.Unlike the log-rank test, which requires the estimation of hazard functions derived from (or facilitating the derivation of) survival functions and assumes a minimum number of observations to ensure asymptotic properties, our method models all possible scenarios based on observed data.These scenarios include those in which the compared survival functions differ in the same way or even more significantly, thus allowing us to calculate the p-value directly.Individuals in these groups may experience an event of interest at specific time points or may be censored, i.e., they might experience the event outside the observed time points.Focusing on all scenarios where survival probabilities differ at least as much as observed usually requires computationally intensive calculations.Censoring is treated as a form of noise, increasing the range of scenarios that need to be calculated and evaluated.Therefore, to estimate the p-value, we compare a computationally exhaustive approach that computes all possible scenarios in which groups' survival functions differ as observed or more, with a Monte Carlo simulation of these scenarios, alongside a traditional approach based on the log-rank test.Our proposed method reduces the first type error rate, enhancing its utility in studies where robustness against false positives is critical.We also analyze the asymptotic time complexity of both proposed approaches.

Regarding the parameter estimation task, besides the time effectiveness of the simulation, parameter estimates are required to be precise enough.Usually, the estimates are Monte Carlo-simulated using a prior estimated variability within a small sample.However, the problem with pre-estimated variability is that it can be estimated imprecisely or, even worse, underestimated, resulting in estimation bias.In this work, we address the abovementioned issue and suggest estimating all parameters as probabilities.Since the probability is not only finite but has its theoretical maximum as 1, using outcomes of Bernoulli and binomial distribution's upper-bounded variance and Chebyshev's inequality, the estimator's variability is theoretically upperbounded within the Monte Carlo simulation and estimation process.It cannot be underestimated or estimated inaccurately; thus, its precision is ensured till a given decimal digit, with very high probability.If there is a known process that treats the parameter of interest in terms of probability, we can estimate how many iterations of the Monte Carlo simulation are needed to ensure parameter estimate on a given level of precision.Also, we analyze the asymptotic time complexity of the proposed estimation strategy and illustrate the approach on a short case study of π constant estimation.

Assuming a bounded polygon and a point inside the polygon or on its boundary, the visibility polygon, also called the visibility region, is a polygon reachable, i.e., visible by straight lines from the point without hitting the polygon's edges or vertices.If the polygon is bounded, then the visibility polygon is bounded, and the proportion of the visibility polygon's surface area to the given polygon's surface area could be enumerated.Many papers investigate applications of the visibility polygons in robotics and computer graphics or focus on computationally effective finding the visibility region for a given polygon.However, surprisingly, there seems to be no work estimating the proportion of a visibility polygon's surface to an entire polygon's surface or its bounds.Thus, in this paper, we search for a lower bound of the surface proportion of a visibility polygon to a given one.Assuming n-sided simple polygon, i.e., a polygon without holes and edge intersections, we apply the well-known art gallery problem and derive there is always a point inside the polygon or on its boundary that guarantees the proportion of the visibility polygon's surface to the entire polygon's surface is at least

When testing whether a continuous variable differs between categories of a factor variable or their combinations, taking into account other continuous covariates, one may use an analysis of covariance.Several post-hoc methods, such as Tukey's honestly significant difference test, Scheffé's, Dunn's, or Nemenyi's test are well-established when the analysis of covariance rejects the hypothesis there is no difference between any categories.However, these methods are statistically rigid and usually require meeting statistical assumptions.In this work, we address the issue using a random forest-based algorithm, practically assumption-free, classifying individual observations into the factor's categories using the dependent continuous variable and covariates on input.The higher the proportion of trees classifying the observations into two different categories is, the more likely a statistical difference between the categories is.To adjust the method's first-type error rate, we change random forest trees' complexity by pruning to modify the proportions of highly complex trees.Besides simulations that demonstrate a relationship between the tree pruning level, tree complexity, and first-type error rate, we analyze the asymptotic time complexity of the proposed random forest-based method compared to established techniques.

The log-rank test and Cox's proportional hazard model can be used to compare survival curves but are limited by strict statistical assumptions.In this study, we introduce a novel, assumption-free method based on a random forest algorithm able to compare two or more survival curves.A proportion of the random forest's trees with sufficient complexity is close to the test's p-value estimate.The pruning of trees in the model modifies trees' complexity and, thus, both the method's robustness and statistical power.The discussed results are confirmed using a simulation study, varying the survival curves and the tree pruning level.

Comparison of two time-event survival curves representing two groups of individuals' evolution in time is relatively usual in applied biostatistics.Although the log-rank test is the suggested tool how to face the above-mentioned problem, there is a rich statistical toolbox used to overcome some of the properties of the log-rank test.However, all of these methods are limited by relatively rigorous statistical assumptions.In this study, we introduce a new robust method for comparing two time-event survival curves.We briefly discuss selected issues of the robustness of the log-rank test and analyse a bit more some of the properties and mostly asymptotic time complexity of the proposed method.The new method models individual time-event survival curves in a discrete combinatorial way as orthogonal monotonic paths, which enables direct estimation of the p-value as it was originally defined.We also gently investigate how the surface of an area, bounded by two survival curves plotted onto a plane chart, is related to the test's p-value.Finally, using simulated time-event data, we check the robustness of the introduced method in comparison with the log-rank test.Based on the theoretical analysis and simulations, the introduced method seems to be a promising and valid alternative to the log-rank test, particularly in case on how to compare two time-event curves regardless of any statistical assumptions.

Computerized adaptive testing (CAT) is a modern alternative to classical paper and pencil testing.CAT is based on an automated selection of optimal item corresponding to current estimate of test-taker's ability, which is in contrast to fixed predefined items assigned in linear test.Advantages of CAT include lowered test anxiety and shortened test length, increased precision of estimates of test-takers' abilities, and lowered level of item exposure thus better security.Challenges are high technical demands on the whole test work-flow and need of large item banks.In this study, we analyze feasibility and advantages of computerized adaptive testing using a Monte-Carlo simulation and posthoc analysis based on a real linear admission test administrated at a medical college.We compare various settings of the adaptive test in terms of precision of ability estimates and test length.We find out that with adaptive item selection, the test length can be reduced to 40 out of 100 items while keeping the precision of ability estimates within the prescribed range and obtaining ability estimates highly correlated to estimates based on complete linear test (Pearson's ρ .= 0.96).We also demonstrate positive effect of content balancing and item exposure rate control on item composition.

Since the plastic surgery should consider that facial impression is always dependent on current facial emotion, it came to be verified how precise classification of facial images into sets of defined facial emotions is.Multivariate regression was performed using R language to identify indicators increasing facial attractiveness after undergoing rhinoplasty.Bayesian naive classifiers, decision trees (CART) and neural networks, respectively, were applied to assign a landmarked facial image data into one of the facial emotions, based on Ekman-Friesen FACS scale.Enlargement of nasolabial and nasofrontal angle within rhinoplasty significantly predicts facial attractiveness increasing (p < 0.05).Decision trees showed the geometry of a mouth, then eyebrows and finally eyes affect in this descending order an impact on classified emotion.Neural networks proved the highest accuracy of the classification.Performed machine-learning analyses pointed out which geometric facial features increase facial attractiveness the most and should be consequently treated by plastic surgeries.

Many current studies conclude that facial attractiveness perception is data-based and irrespective of the perceiver. However, analyses of facial geometric image data and its visual impact always exceeded power of classical statistical methods. In this study, we have applied machine-learning methods to identify geometric features of a face associated with an increase of facial attractiveness after undergoing rhinoplasty. Furthermore, we explored how accurate classification of faces into sets of facial emotions and their facial manifestations is, since categorization of human faces into emotions manifestation should take into consideration the fact that total face impression is also dependent on expressed facial emotion. Both profile and portrait facial image data were collected for each patient (n = 42), processed, landmarked and analysed using R language. Multivariate linear regression was performed to select predictors increasing facial attractiveness after undergoing rhinoplasty. The sets of used facial emotions originate from Ekman-Friesen FACS scale, but was improved substantially. Bayesian naive classifiers, decision trees (CART) and neural networks were learned to allow assigning a new face image data into one of facial emotions. Enlargements of both a nasolabial and nasofrontal angle within rhinoplasty were determined as significant predictors increasing facial attractiveness (p <; 0.05). Neural networks manifested the highest predictive accuracy of a new face classification into facial emotions. Geometrical shape of a mouth, then eyebrows and finally eyes affect in descending order final classified emotion, as was identified using decision trees. We performed machine-learning analyses to point out which facial geometric features, based on large data evidence, affect facial attractiveness the most, and therefore should preferentially be treated within plastic surgeries.

University admission exams belong to so-called highstakes tests, i. e. tests with important consequences for the exam taker.Given the importance of the admission process for the applicant and the institution, routine evaluation of the admission tests and their items is desirable.In this work, we introduce a quick and efficient methodology and on-line tool for semi-real-time evaluation of admission exams and their items based on classical test theory (CTT) and item response theory (IRT) models.We generalize some of the traditional item analysis concepts to tailor them for specific purposes of the admission test.On example of medical school admission test we demonstrate how R-based web application may simplify admissions evaluation work-flow and may guarantee quick accessibility of the psychometric measures.We conclude that the presented tool is convenient for analysis of any admission or educational test in general.