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- histogram, frequency distribution, frequency density distribution
- cumulative frequency distribution
- box-plot
- graphical representations of a time series
- empirical quantile plot
- Q-Q plot
- Statistical descriptors of data:
- mean, root mean square, median
- quantiles
- variance, st. dev., mean dev., coefficient of variation, moments
- coeff. of skewness, coeff. of kurtosis
3. Probability theory:
- Elementary probability theory
- probability laws
- probability mass function, probability density function, cum. distribution function
- moments of distributions of random variables
- Probability distributions
- model of sums: normal; model of products: lognormal
- model of time between events: exponential (Poisson process), Gamma, Pearson III
- Pareto, Weibull, beta, uniform prob. distributions
- normal related or sampling distributions: t-, Chi-square, F- distributions
- modified distributions: truncated and compound distributions
- multivariate distributions
- Estimation of parameters
- method of moments
- maximum likelihood method
- confidence intervals
- Testing statistical hypotheses
incl. trend tests, goodness-of-fit tests, serial correlation tests
- Frequency analysis / Extreme value analysis
- periodic maxima method vs. peak-over-threshold method
- GEV vs. GPD distributions
- extreme value index, distribution classes
- return period
4. Regression and correlation
incl. discussion on different types of uncertainty sources in modelling
and the calculation of parameter uncertainties and prediction uncertainties
5. Introduction to time series analysis and stochastic modelling
- autocorrelation, autocovariance
- (semi-)variogram
- ARMA model
- Kalman filter
- Random simulations
The practical work consists of the application of these techniques for a number of datasets (time series of river
discharges, simultaneous measurements of water levels and discharges in a river, BOD concentrations at the
influent and the effluent of a wastewater treatment plant, …). The following techniques are applied to these
datasets:
- selection, calibration and plotting of probability distributions
- regression and correlation + error analysis + statistical hypothesis tests
- confidence limits for model parameters, model prediction uncertainty
- extreme value analysis, return period calculation
Complementary studies in Water Resources Engineering / 8