The assessment of knowledge and skills acquired by the student at each academic stage is crucial for every educational process. This paper proposes and tests an approach based on a structured assessment test for mathematical competencies in higher education and methods for statistical evaluation of the test. A case study is presented for the assessment of knowledge and skills for solving linear algebra and analytic geometry problems by first-year university students. The test includes three main parts—a multiple-choice test with four selectable answers, a solution of two problems with and without the use of specialized mathematical software, and a survey with four questions for each problem. The processing of data is performed mainly by the classification and regression tree (CART) method. Comparative analysis, cross-tables, and reliability statistics were also used. Regression tree models are built to assess the achievements of students and classification tree models for competency assessment on a three-categorical scale. The influence of 31 variables and groups of them on the assessment of achievements and grading of competencies is determined. Regression models show over 94% fit with data and classification ones—up to 92% correct classifications. The models can be used to predict students’ grades and assess their mathematical competency.

The aim of this study is to evaluate students’ achievements in mathematics using three machine learning regression methods: classification and regression trees (CART), CART ensembles and bagging (CART-EB) and multivariate adaptive regression splines (MARS). A novel ensemble methodology is proposed based on the combination of CART and CART-EB models in a new ensemble to regress the actual data using MARS. Results of a final exam test, control and home assignments, and other learning activities to assess students’ knowledge and competencies in applied mathematics are examined. The exam test combines problems on elements of mathematical analysis, statistics and a small practical project. The project is the new competence-oriented element, which requires students to formulate problems themselves, to choose different solutions and to use or not use specialized software. Initially, empirical data are statistically modeled using six CART and six CART-EB competing models. The models achieve a goodness-of-fit up to 96% to actual data. The impact of the examined factors on the students’ success at the final exam is determined. Using the best of these models and proposed novel ensemble procedure, final MARS models are built that outperform the other models for predicting the achievements of students in applied mathematics.

A novel framework for stacked regression based on machine learning was developed to predict the daily average concentrations of particulate matter (PM₁₀), one of Bulgaria’s primary health concerns. The measurements of nine meteorological parameters were introduced as independent variables. The goal was to carefully study a limited number of initial predictors and extract stochastic information from them to build an extended set of data that allowed the creation of highly efficient predictive models. Four base models using random forest, CART ensemble and bagging, and their rotation variants, were built and evaluated. The heterogeneity of these base models was achieved by introducing five types of diversities, including a new simplified selective ensemble algorithm. The predictions from the four base models were then used as predictors in multivariate adaptive regression splines (MARS) models. All models were statistically tested using out-of-bag or with 5-fold and 10-fold cross-validation. In addition, a variable importance analysis was conducted. The proposed framework was used for short-term forecasting of out-of-sample data for seven days. It was shown that the stacked models outperformed all single base models. An index of agreement IA = 0.986 and a coefficient of determination of about 95% were achieved.

Two different predictive modelling approaches – classical SARIMA time series methodology and the new Generalised PathSeeker (GPS) regularised regression method, supported by stochastic gradient boosting trees, RuleLearner and other data mining techniques - are used to examine the concentration of particulate matter PM10 in the town of Kardzhali, Bulgaria. Empirical models are developed to simulate and forecast pollution levels based on hourly PM10 data from 1 January 2011 to 28 February 2014 in dependence on six meteorological variables. The constructed models have been used for 5-days-ahead hourly forecasts, compared to actual data from 1 to 5 March 2014. The obtained SARIMA and GPS models fit very well to historical data with coefficients of determination