A Bayesian Approach for Forecasting Heat Load in a District Heating System

The growing population in cities increases the energy demand and affects the environment by increasing carbon emissions. Information and communications technology solutions which enable energy optimization are needed to address this growing energy demand in cities and to reduce carbon emissions. District heating systems optimize the energy production by reusing waste energy with combined heat and power plants. Forecasting the heat load demand in residential buildings assists in optimizing energy production and consumption in a district heating system. However, the presence of a large number of factors such as weather forecast, district heating operational parameters and user behavioural parameters, make heat load forecasting a challenging task. This thesis proposes a probabilistic machine learning model using a Naive Bayes classifier, to forecast the hourly heat load demand for three residential buildings in the city of Skellefteå, Sweden over a period of winter and spring seasons. The district heating data collected from the sensors equipped at the residential buildings in Skellefteå, is utilized to build the Bayesian network to forecast the heat load demand for horizons of 1, 2, 3, 6 and 24 hours. The proposed model is validated by using four cases to study the influence of various parameters on the heat load forecast by carrying out trace driven analysis in Weka and GeNIe. Results show that current heat load consumption and outdoor temperature forecast are the two parameters with most influence on the heat load forecast. The proposed model achieves average accuracies of 81.23 % and 76.74 % for a forecast horizon of 1 hour in the three buildings for winter and spring seasons respectively. The model also achieves an average accuracy of 77.97 % for three buildings across both seasons for the forecast horizon of 1 hour by utilizing only 10 % of the training data. The results indicate that even a simple model like Naive Bayes classifier can forecast the heat load demand by utilizing less training data.

Bayesian methods for estimation, optimization and experimental design

Mathematical models often contain parameters that need to be calibrated from measured data. The emergence of efficient Markov Chain Monte Carlo (MCMC) methods has made the Bayesian approach a standard tool in quantifying the uncertainty in the parameters. With MCMC, the parameter estimation problem can be solved in a fully statistical manner, and the whole distribution of the parameters can be explored, instead of obtaining point estimates and using, e.g., Gaussian approximations. In this thesis, MCMC methods are applied to parameter estimation problems in chemical reaction engineering, population ecology, and climate modeling. Motivated by the climate model experiments, the methods are developed further to make them more suitable for problems where the model is computationally intensive. After the parameters are estimated, one can start to use the model for various tasks. Two such tasks are studied in this thesis: optimal design of experiments, where the task is to design the next measurements so that the parameter uncertainty is minimized, and model-based optimization, where a model-based quantity, such as the product yield in a chemical reaction model, is optimized. In this thesis, novel ways to perform these tasks are developed, based on the output of MCMC parameter estimation. A separate topic is dynamical state estimation, where the task is to estimate the dynamically changing model state, instead of static parameters. For example, in numerical weather prediction, an estimate of the state of the atmosphere must constantly be updated based on the recently obtained measurements. In this thesis, a novel hybrid state estimation method is developed, which combines elements from deterministic and random sampling methods.

Statistical Inversion Theory in X-ray Tomography

In this thesis the X-ray tomography is discussed from the Bayesian statistical viewpoint. The unknown parameters are assumed random variables and as opposite to traditional methods the solution is obtained as a large sample of the distribution of all possible solutions. As an introduction to tomography an inversion formula for Radon transform is presented on a plane. The vastly used filtered backprojection algorithm is derived. The traditional regularization methods are presented sufficiently to ground the Bayesian approach. The measurements are foton counts at the detector pixels. Thus the assumption of a Poisson distributed measurement error is justified. Often the error is assumed Gaussian, altough the electronic noise caused by the measurement device can change the error structure. The assumption of Gaussian measurement error is discussed. In the thesis the use of different prior distributions in X-ray tomography is discussed. Especially in severely ill-posed problems the use of a suitable prior is the main part of the whole solution process. In the empirical part the presented prior distributions are tested using simulated measurements. The effect of different prior distributions produce are shown in the empirical part of the thesis. The use of prior is shown obligatory in case of severely ill-posed problem.

Demographic Modelling of Human Population Growth

This work presents models and methods that have been used in producing forecasts of population growth. The work is intended to emphasize the reliability bounds of the model forecasts. Leslie model and various versions of logistic population models are presented. References to literature and several studies are given. A lot of relevant methodology has been developed in biological sciences. The Leslie modelling approach involves the use of current trends in mortality,fertility, migration and emigration. The model treats population divided in age groups and the model is given as a recursive system. Other group of models is based on straightforward extrapolation of census data. Trajectories of simple exponential growth function and logistic models are used to produce the forecast. The work presents the basics of Leslie type modelling and the logistic models, including multi- parameter logistic functions. The latter model is also analysed from model reliability point of view. Bayesian approach and MCMC method are used to create error bounds of the model predictions.

Communicative and Anticipatory Decision-Making Supported by Bayesian Networks

The purpose of this research is to draw up a clear construction of an anticipatory communicative decision-making process and a successful implementation of a Bayesian application that can be used as an anticipatory communicative decision-making support system. This study is a decision-oriented and constructive research project, and it includes examples of simulated situations. As a basis for further methodological discussion about different approaches to management research, in this research, a decision-oriented approach is used, which is based on mathematics and logic, and it is intended to develop problem solving methods. The approach is theoretical and characteristic of normative management science research. Also, the approach of this study is constructive. An essential part of the constructive approach is to tie the problem to its solution with theoretical knowledge. Firstly, the basic definitions and behaviours of an anticipatory management and managerial communication are provided. These descriptions include discussions of the research environment and formed management processes. These issues define and explain the background to further research. Secondly, it is processed to managerial communication and anticipatory decision-making based on preparation, problem solution, and solution search, which are also related to risk management analysis. After that, a solution to the decision-making support application is formed, using four different Bayesian methods, as follows: the Bayesian network, the influence diagram, the qualitative probabilistic network, and the time critical dynamic network. The purpose of the discussion is not to discuss different theories but to explain the theories which are being implemented. Finally, an application of Bayesian networks to the research problem is presented. The usefulness of the prepared model in examining a problem and the represented results of research is shown. The theoretical contribution includes definitions and a model of anticipatory decision-making. The main theoretical contribution of this study has been to develop a process for anticipatory decision-making that includes management with communication, problem-solving, and the improvement of knowledge. The practical contribution includes a Bayesian Decision Support Model, which is based on Bayesian influenced diagrams. The main contributions of this research are two developed processes, one for anticipatory decision-making, and the other to produce a model of a Bayesian network for anticipatory decision-making. In summary, this research contributes to decision-making support by being one of the few publicly available academic descriptions of the anticipatory decision support system, by representing a Bayesian model that is grounded on firm theoretical discussion, by publishing algorithms suitable for decision-making support, and by defining the idea of anticipatory decision-making for a parallel version. Finally, according to the results of research, an analysis of anticipatory management for planned decision-making is presented, which is based on observation of environment, analysis of weak signals, and alternatives to creative problem solving and communication.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.