## 21002063-3 - MATHEMATICAL AND STATISTICAL METHODS

The development of a project on an urban scale (masterplan) with particular attention to the themes of the resilience to the climate change and to the relationship between physical and social form. Among the topics discussed: use of space; temporality of movement of the inhabitants; open and built spaces; design of soil and infrastructure; places of social life, of living and working. In addition, they analyze the models of urban development and demographic, land use, traffic, food sustainability, social interactions and urban spaces, the economy and the metabolism of the city.
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Programme

The detection of statistical phenomena (1): introduction, characters, statistical units, collective, classification of statistical characters, subdivision into classes of a quantitative character, the different types of survey, total survey and sample survey, the questionnaire.
Distribution of a character and its representation (1): from the unitary distributions to the frequency distributions, relative frequencies and percentages, cumulative frequencies, graphic representation of simple distributions, bar or tape graphs, histograms, area charts, pie charts, graphs radar, cartograms, Cartesian diagrams.
Synthesis of the distribution of a character, the means (1): introduction, the arithmetic mean, the geometric mean, the trimmed mean, the median, the fashion, the percentiles.
Synthesis of the distribution of a character: variability (1): introduction, the variability of a distribution, indices based on the deviation from the arithmetic mean, Chebyshev's theorem, standardization, other variability indices, box plot, concentration, homogeneity and heterogeneity, asymmetry indices.
Index numbers, time series and statistical reports (1): introduction, measurement of the change in a historical series, simple index numbers, complex index numbers, statistical reports.
Analysis of the association between two characters (1): introduction, double frequency distributions, graphic representation of the character distribution, analysis of the association between two characters, dependence, independence, interdependence, study of the association between two characters in a double table of frequencies, association measure for disconnected qualitative characters, association measure for ordered qualitative characters, measure of the dependence of a quantitative character from a discrete qualitative or quantitative character, measure of the interdependence between two quantitative characters.
Probability (1): introduction, primitive concepts, events and event algebra, the postulates, probability measure in the classical approach, conditional probability and independence, Bayes theorem, the different concepts of probability.
Random variables and probability distributions (1): introduction, random variables, discrete random variables, continuous random variables, expected value and variance of a random variable, standardized random variables and Chebyshev theorem, Discrete uniform distribution, Bernoulli distribution, Binomial distribution , Poisson distribution, Uniform continuous distribution, Normal distribution, Chi-square distribution, Student t distribution, multiple random variables, central limit theorem.
Sampling and sampling distributions (1): introduction, population and population parameters, sampling from finite, simple, stratified, cluster and stage populations, sampling from infinite populations, sample statistics and sampling distributions, the distribution of the sample mean in populations infinite and finite.
Point estimate (1): introduction, point estimation and estimators, correct estimators, efficient estimators, minimum mean square error, consistent and asymptotically correct estimators, point estimate of the mean of a population, point estimate of the proportion of a population, point estimate of variance of the population, point estimation using the maximum likelihood method.
Estimation by interval (1): introduction, estimation by interval, confidence interval for the mean with known variance and unknown, confidence interval for the proportion, confidence interval for variance, determination of the sample size.
Statistical test theory (1): introduction, hypothesis formulation, acceptance region and rejection region, simple null hypothesis test, p-value, first and second type errors, power function, maximum ratio test likelihood, connection between confidence interval and test.
Tests for averages, proportions and variances (1): introduction, test for the average of a normal population with known and unknown variance, test for the average of a non-normal population, establish the sample size, test for proportion, test for the variance, test based on independent samples from two populations, difference between means, ratio between variances, difference between two proportions, independence test.
Inference in the Bayesian approach (1): introduction, Bayes theorem and a posteriori distribution, a priori distribution, point estimation, interval estimation, hypothesis test, predictive distribution.
The simple linear regression model (1): introduction, functional relationship and statistical relation between two variables, specification of the simple linear regression model, point estimate of regression coefficients, the decomposition of the total variance and the coefficient of determination, property of the estimators of coefficients and average response.
Inference in the linear regression model (1): introduction, assumption of error normality and parameter inference, analysis of variance and F test, inference for mean response and for prediction, residue analysis, outliers and detection methods.
Graph optimization (2,3): the minimal-cost transport problem, general and specialized transport models and their variants, minimum path problems, declaration of nodes and arcs, interactions and declaration of the objective function solving problems on graphs of linear type with AMPL.
Processing on PC (1,2,4,5,6): statistics with Excel, statistical package R, statistical analysis with R, QGIS, AMPL,SAS.
An application to a real case: A particular application of the statistical model for the estimation of the urban planning of the reorganization of the Tiburtina and Termini station with the use of QGIS. Possible extensions to the network model with AMPL.

Core Documentation

(1) Statistica, metodologie per le scienze economiche e sociali, S. Borra, A. Di Ciaccio, Mc Graw-Hill.
(2) AMPL, A Modeling Language for Mathematical Progamming, R. Fourer, D. Gay, B. Kernighan, The Scientific Press Series, boyd & fraser publishing company.
(3) Ricerca Operativa, M. Bruglieri, A. Colorni, Zanichelli.
(4) Analisi statistica con Excel, D. Giuliani, M. Dickson, Apogeo.
(5) Analisi esplorativa dei dati con R, G. Espa, R. Micciolo, Apogeo.
(6) A Handbook of Statistical Analyses using SAS, 3rd Edition, by Geoff Der, Brian S. Everitt, Chapman and Hall/CRC.

Reference Bibliography

(1) Statistica, metodologie per le scienze economiche e sociali, S. Borra, A. Di Ciaccio, Mc Graw-Hill. (2) AMPL, A Modeling Language for Mathematical Progamming, R. Fourer, D. Gay, B. Kernighan, The Scientific Press Series, boyd & fraser publishing company. (3) Ricerca Operativa, M. Bruglieri, A. Colorni, Zanichelli. (4) Analisi statistica con Excel, D. Giuliani, M. Dickson, Apogeo. (5) Analisi esplorativa dei dati con R, G. Espa, R. Micciolo, Apogeo. (6) A Handbook of Statistical Analyses using SAS, 3rd Edition, by Geoff Der, Brian S. Everitt, Chapman and Hall/CRC.

Attendance

attendance is mandatory for 75%

Type of evaluation

Oral exam and evaluation project. The student with the help of Teams will exhibit one or more topics covered in the course with application to the project.