More on Divisibility Criteria for Selected Primes

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].Naumowicz Adam - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandPiliszek Radosław - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. Veblen hierarchy. Formalized Mathematics, 19(2):83-92, 2011. doi:10.2478/v10037-011-0014-5.C.C. Briggs. Simple divisibility rules for the 1st 1000 prime numbers. arXiv preprint arXiv:math/0001012, 2000.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Magdalena Jastrz¸ebska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307-313, 2004.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y.Adam Naumowicz. On the representation of natural numbers in positional numeral systems. Formalized Mathematics, 14(4):221-223, 2006. doi:10.2478/v10037-006-0025-9.Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Computational Analysis Identifies Novel Biomarkers for High-Risk Bladder Cancer Patients

In the case of bladder cancer, carcinoma in situ (CIS) is known to have poor diagnosis. However, there are not enough studies that examine the biomarkers relevant to CIS development. Omics experiments generate data with tens of thousands of descriptive variables, e.g., gene expression levels. Often, many of these descriptive variables are identified as somehow relevant, resulting in hundreds or thousands of relevant variables for building models or for further data analysis. We analyze one such dataset describing patients with bladder cancer, mostly non-muscle-invasive (NMIBC), and propose a novel approach to feature selection. This approach returns high-quality features for prediction and yet allows interpretability as well as a certain level of insight into the analyzed data. As a result, we obtain a small set of seven of the most-useful biomarkers for diagnostics. They can also be used to build tests that avoid the costly and time-consuming existing methods. We summarize the current biological knowledge of the chosen biomarkers and contrast it with our findings

Integration of multiple types of genetic markers for neuroblastoma may contribute to improved prediction of the overall survival

Abstract Background Modern experimental techniques deliver data sets containing profiles of tens of thousands of potential molecular and genetic markers that can be used to improve medical diagnostics. Previous studies performed with three different experimental methods for the same set of neuroblastoma patients create opportunity to examine whether augmenting gene expression profiles with information on copy number variation can lead to improved predictions of patients survival. We propose methodology based on comprehensive cross-validation protocol, that includes feature selection within cross-validation loop and classification using machine learning. We also test dependence of results on the feature selection process using four different feature selection methods. Results The models utilising features selected based on information entropy are slightly, but significantly, better than those using features obtained with t-test. The synergy between data on genetic variation and gene expression is possible, but not confirmed. A slight, but statistically significant, increase of the predictive power of machine learning models has been observed for models built on combined data sets. It was found while using both out of bag estimate and in cross-validation performed on a single set of variables. However, the improvement was smaller and non-significant when models were built within full cross-validation procedure that included feature selection within cross-validation loop. Good correlation between performance of the models in the internal and external cross-validation was observed, confirming the robustness of the proposed protocol and results. Conclusions We have developed a protocol for building predictive machine learning models. The protocol can provide robust estimates of the model performance on unseen data. It is particularly well-suited for small data sets. We have applied this protocol to develop prognostic models for neuroblastoma, using data on copy number variation and gene expression. We have shown that combining these two sources of information may increase the quality of the models. Nevertheless, the increase is small and larger samples are required to reduce noise and bias arising due to overfitting. Reviewers This article was reviewed by Lan Hu, Tim Beissbarth and Dimitar Vassilev