The fundamental mathematical tools needed to understand machine learning include linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. These topics are traditionally taught in disparate courses, making it hard for data science or computer science students, or professionals, to efficiently learn the mathematics. This self-contained textbook bridges the gap between mathematical and machine learning texts, introducing the mathematical concepts with a minimum of prerequisites. It uses these concepts to derive four central machine learning methods: linear regression, principal component analysis, Gaussian mixture models and support vector machines. For students and others with a mathematical background, these derivations provide a starting point to machine learning texts. For those learning the mathematics for the first time, the methods help build intuition and practical experience with applying mathematical concepts. Every chapter includes worked examples and exercises to test understanding. Programming tutorials are offered on the book's web site. This textbook considers statistical learning applications when interest centers on the conditional distribution of a response variable, given a set of predictors, and in the absence of a credible model that can be specified before the data analysis begins. Consistent with modern data analytics, it emphasizes that a proper statistical learning data analysis depends in an integrated fashion on sound data collection, intelligent data management, appropriate statistical procedures, and an
Used by hundreds of thousands of students since its first edition, INTRODUCTION TO PROBABILITY AND STATISTICS continues to blend the best of its proven coverage with new innovations. While retaining the straightforward presentation and traditional outline for descriptive and inferential statistics, the Twelfth Edition incorporates exciting new learning aids like MyPersonal Trainer, MyApplet, and MyTip to ensure that students learn and understand the relevance of the material. The book takes advantage of modern technology, including computational software and interactive visual tools, to facilitate statistical reasoning as well as the understanding and interpretation of statistical results. In addition to showing how to apply statistical procedures, the authors explain how to meaningfully describe real sets of data, what the statistical tests mean in terms of their practical applications, how to evaluate the validity of the assumptions behind statistical tests, and what to do when statistical assumptions have been violated. This new edition retains the statistical integrity, examples, exercises and exposition that have made it a market leader, and builds upon this tradition of excellence with new technology integration.
D. Arvidsson-Shukur, William F Braasch Jr, Stephan De Bievre
et al.
There are several mathematical formulations of quantum mechanics. The Schrödinger picture expresses quantum states in terms of wavefunctions over, e.g. position or momentum. Alternatively, phase-space formulations represent states with quasi-probability distributions over, e.g. position and momentum. A quasi-probability distribution resembles a probability distribution but may have negative and non-real entries. The most famous quasi-probability distribution, the Wigner function, has played a pivotal role in the development of a continuous-variable quantum theory that has clear analogues of position and momentum. However, the Wigner function is ill-suited for much modern quantum-information research, which is focused on finite-dimensional systems and general observables. Instead, recent years have seen the Kirkwood–Dirac (KD) distribution come to the forefront as a powerful quasi-probability distribution for analysing quantum mechanics. The KD distribution allows tools from statistics and probability theory to be applied to problems in quantum-information processing. A notable difference to the Wigner function is that the KD distribution can represent a quantum state in terms of arbitrary observables. This paper reviews the KD distribution, in three parts. First, we present definitions and basic properties of the KD distribution and its generalisations. Second, we summarise the KD distribution’s extensive usage in the study or development of measurement disturbance; quantum metrology; weak values; direct measurements of quantum states; quantum thermodynamics; quantum scrambling and out-of-time-ordered correlators; and the foundations of quantum mechanics, including Leggett–Garg inequalities, the consistent-histories interpretation and contextuality. We emphasise connections between operational quantum advantages and negative or non-real KD quasi-probabilities. Third, we delve into the KD distribution’s mathematical structure. We summarise the current knowledge regarding the geometry of KD-positive states (the states for which the KD distribution is a classical probability distribution), describe how to witness and quantify KD non-positivity, and outline relationships between KD non-positivity, coherence and observables’ incompatibility.
In this article, we study the spectrum, fine spectrum and boundedness property of second order quantum difference operator ∆2q (0 < q < 1) over the class of sequence lp (1 < p < ∞), the pth summable sequence space. The second order quantum difference operator ∆2q is a lower triangular triple band matrix ∆2q(1,−(1+ q),q). We also determine the approximate point spectrum, defect spectrum, compression spectrum, and Goldberg classification of the operator on the class of sequence. We obtained the results by solving an infinite system of linear equations and computing the inverse of a lower triangular infinite matrix. We also provide appropriate examples along with graphical representations where necessary.
Forecasting is essential for improving aviation safety, with air humidity being a critical factor influenced by air temperature. This study analyzes daily humidity data from I Gusti Ngurah Rai Airport, one of Indonesia’s busiest air stations, using two time series modeling approaches: Autoregressive (AR) and high-order fuzzy modeling. The objective is to evaluate and compare their forecasting accuracy. Historical daily data from the Meteorology, Climatology, and Geophysics Agency of Indonesia were used to build the forecasting models. The optimal linear AR model served as the foundation for constructing the AR high-order fuzzy model, which incorporates linguistic rules to capture nonlinear patterns. Both models were implemented and evaluated using the Mean Squared Error (MSE) metric. Results show that the AR(2) model outperforms the AR high-order fuzzy model, achieving a lower MSE of 13.23. This suggests that the AR(2) model provides more accurate humidity forecasts over the observed period. These findings offer practical insights for policymakers and decision-makers in forecasting daily humidity levels and supporting aviation operations. While the study confirms the effectiveness of traditional AR modeling, it also highlights limitations of the fuzzy approach, particularly its sensitivity to parameter tuning and data sparsity. The integration of high-order fuzzy modeling represents a novel contribution to this domain, though further refinement is needed to enhance its forecasting performance.
Linear regression, an introductory statistics topic, typically involves finding and evaluating the line of best fit as well as making predictions based on it. However, the subtopic of influential points is often neglected. We present and report on findings of a series of four interactive student-centered activities in Desmos that facilitate student discovery of potential influential points and determine the conditions that would result in them having a large effect on the position and strength of a linear relationship. Placing these activities in the Desmos environment enables students to test their conjectures about leverage points and points with large residuals as they receive immediate feedback that promotes the development of understanding of influential points. Supplementary materials for this article are available online.
Probabilities. Mathematical statistics, Special aspects of education
Statistical inference with non-probability survey samples is an emerging topic in survey sampling and official statistics and has gained increased attention from researchers and practitioners in the field. Much of the existing literature, however, assumes that the participation mechanism for non-probability samples is ignorable. In this paper, we develop a pseudo-likelihood approach to estimate participation probabilities for nonignorable non-probability samples when auxiliary information is available from an existing reference probability sample. We further construct three estimators for the finite population mean using regression-based prediction, inverse probability weighting (IPW), and augmented IPW estimators, and study their asymptotic properties. Variance estimation for the proposed methods is considered within the same framework. The efficiency of our proposed methods is demonstrated through simulation studies and a real data analysis using the ESPACOV survey on the effects of the COVID-19 pandemic in Spain.
This work is concerned with coupled semi-linear pseudo-parabolic equations with memory terms in both equations, associated with the homogeneous Dirichlet boundary condition. We show that the solution grows exponentially under specific conditions
regarding the relaxation functions and initial energy. In order to prove the result, we use the energy method based on the construction of a suitable Lyapunov function.
The most important behavior of the evolution system is the exponential growth phenomena because of its wide range of applications in modern science, such as chemistry, biology, ecology, and other areas of engineering and physical sciences.
Probabilities. Mathematical statistics, Instruments and machines
For a connected graph ???? = (????, ????) of order a set is called a monophonic set of ????if every vertex of ????is contained in a monophonic path joining some pair of vertices in ????. The monophonic number (????) of is the minimum cardinality of its monophonic sets. If or the subgraph is connected, then a detour monophonic set of a connected graph is said to be an outer connected detour monophonic setof .The outer connecteddetourmonophonic number of , indicated by the symbol , is the minimum cardinality of an outer connected detour monophonic set of . The outer connected detour monophonic number of some standard graphs are determined. It is shown that for positive integers , and ???? ≥ 2 with ,there exists a connected graph ????with???????????????????? = , ????????????m???????? = and = ????. Also, it is shown that for every pair of integers ????and b with 2 ≤ ???? ≤ ????, there exists a connected graph with and .
A fundamental problem in analysis of complex systems is getting a reliable estimate of entropy of their probability distributions over the state space. This is difficult because unsampled states can contribute substantially to the entropy, while they do not contribute to the Maximum Likelihood estimator of entropy, which replaces probabilities by the observed frequencies. Bayesian estimators overcome this obstacle by introducing a model of the low-probability tail of the probability distribution. Which statistical features of the observed data determine the model of the tail, and hence the output of such estimators, remains unclear. Here we show that well-known entropy estimators for probability distributions on discrete state spaces model the structure of the low probability tail based largely on few statistics of the data: the sample size, the Maximum Likelihood estimate, the number of coincidences among the samples, the dispersion of the coincidences. We derive approximate analytical entropy estimators for undersampled distributions based on these statistics, and we use the results to propose an intuitive understanding of how the Bayesian entropy estimators work.
This study intends to determine the statistical literacy levels of pre-service mathematics teachers and to evaluate the contribution of the statistics courses in the elementary mathematics education curriculum to statistical literacy. A mixed methods research design was adopted. The study group consisted of 202 pre-service mathematics teachers enrolled in the Statistics and Probability course. In the data collection process, a pre-test and post-test was administered to determine the pre-service teachers’ statistical literacy before and after the statistical course, and classroom observations were performed to identify the contribution of the statistics course to statistical literacy. The Rasch model was used for validity-reliability analyses, and one-way ANOVA tests were used to analyze the quantitative data. Content analysis was utilized in the analysis of qualitative data, which revealed that statistical literacy levels of pre-service teachers are generally low, generally influencing the competence of pre-service teachers. The pre-service teachers failed in the sample selection component in the pre-test and data interpretation in the post-test, while they were more successful with table and graphs in the pre-test and sample selection in the post-test. The comparative analysis of revealed statistically significant differences in favor of U4 in the pre-test, but in favor of U1 in the post-test. It was concluded that practices included in the statistics lessons could be effective on these differences.