Numerical and Statistical Methods

Basic concepts of Mathematical Analysis and Linear Algebra.

Teaching method will be essentially expository in theoretical classes, in order to introduce concepts. Theoretical-practical and practical classes will be dedicated to problem solving, including their analysis, choice and/or implementation of the most appropriate technique/method, resolution and interpretation of results, under the guidance of the teacher. Problem solving will be initiated “by hand” to help understand the methods and will continue with the support of software appropriate to the contents of the CU, particularly in practical classes (operating in a laboratory environment).

With this Curricular Unit (CU) it is intended that the student knows how to apply some methods and techniques of Statistics and Numerical Analysis. The student will develop skills in statistical data analysis: data organization, statistical treatment, including Statistical Inference, reading and interpretation of results. The student also acquires skills in numerical problem solving, a fundamental approach in Engineering, when analytical solutions to problems are not possible, time-consuming or complex. The student will also develop his/her reasoning and problem solving skills with the support of computational tools.

Part I: Statistical Methods

1. Descriptive Statistics

Aims of descriptive statistics. Notions of population and sample. Data types and measurement scales. Univariate descriptive statistics: displaying and summarizing data; frequencies distribution; measures of central tendency, variability, symmetry and kurtosis. Bivariate descriptive statistics: contingency tables; scatter diagrams, linear correlation, coefficients; linear regression.

 2. Probability Distributions

Binomial, Poisson and Normal distributions. Sampling distributions (introduction). Central Limit Theorem.

 3. Introduction to Statistical Inference

Aims of statistical inference. Point and interval estimations. Confidence intervals for population parameters. Test of hypotheses.

Use of statistical software (MS Excel and/or SPSS) as a support tool in problem solving.

Part II: Numerical Methods

 4. Introduction

Introduction and motivation for the use of numerical methods in Bioengineering. Error estimation in iterative methods.

 5. Nonlinear Equations

Introduction. Location of roots. Graphical method. Bisection, false position, one-point iteration, Newton-Raphson and secant methods.

 6. Methods for Solving Linear Systems

Introduction. Iterative methods of Jacobi and Gauss-Seidel.

 7. Polynomial Interpolation

Introduction. Newton’s interpolating polynomial (divided differences). Lagrange interpolating polynomial.

 8. Numerical integration

Trapezoidal and Simpson rules.

Use of MS Excel and Matlab applications, including the implementation of some algorithms in VBA and Matlab, as support tools in problem solving.

Assessment can be either periodic or by final exam. In both cases, the topics of Part I (statistical methods) and the topics of Part II (numerical methods) will have the same weight in the final mark; that is, both parts will be evaluated through tests quoted for 10 values, and with a minimum of 3.0 values (no rounding). The student is only approved if the sum of  the two parts is greater than or equal to 10 values and meets the minimum requirements.

Periodic assessment consists of an intermediate test related with Part I, to take place during the semester (to be scheduled), and a final test related to Part II, to be carried out during the 1st exam period. It is assumed that the student chooses the periodic evaluation when taking the intermediate test.

Alternatively, the assessment can be carried out through final exam, meaning the simultaneous assessment of Part I and Part II during exam periods. At 2nd exam period, the student who has not yet been approved, can perform the assessment only in one of the parts, if he/she has already obtained the minimum required in previous assessments (3.0 values) in the other part. Improvements (approved student) will only be allowed by exam.

 Enrollment in the various evaluation moments will be mandatory.

NAO