Statistical inference and random network simulation
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Doctoral thesis (monograph)
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Helsinki University of Technology Laboratory of Computational Engineering publications. Report B, 44
AbstractThe scope of this dissertation is twofold, in the sense that it deals on one hand with statistical inference and on the other hand with random graphs. Due to inherent randomness in both areas the scope can also be seen as onefold, which is further united methodologically by the attempt to build models of random processes involved and by simulating their behaviour. The statistical part of the thesis follows the Bayesian theory of probability, and applies it to a fault diagnostic setting. This part also contains an exploration of metrics on probability distributions, in which the introduction of a new metric is one of the main contributions. This new metric is constructed from utilities of the samples instead of the more conventional entropy-based metrics. In Bayesian methods the simulation of samples from distributions is an integral part of the analysis. It also becomes the leading principle in the evaluation of the proposed metrics. This metric is shown to be useful in statistical inference in some cases where the probabilities are difficult to compute. The problem of uncomputable likelihoods is analysed also from the Bayesian perspective and two branches emerge: the kernel estimate and the indirect inference. In the analysis of random graphs the attention is on the small-world property, requiring that any two sites in the network are joined by only a short path with a relatively small average number of connections per site. Again one of the main tools in analysing complex graphs is by simulation of random dynamics on the graphs. The first dynamic property that is analysed is the spreading phenomenon. Spreading means the number of unique sites a random walker on the graphs goes through. This number is shown to have transition points relative to the small-world control parameter. Apart from the spreading phenomenon the thesis also studies the self-organised criticality properties through the so called sandpile model on the one dimensional small-world networks. In this setting of self-organised criticality there are interesting behaviours that are absent in the standard 1-dimensional sandpile model. Both the spreading and the sandpile model are analysed with two forms of disorder: quenched and annealed. The quenched case corresponds to a simulation setting on an ensemble of random graphs, whereas in the case of annealed disorder the simulation is performed on a regular graph but the dynamics also allow random moves to other sites. The annealed form allows simpler analytic tools to be used, but the quenched form corresponds more closely to natural systems. Even though these forms of disorder are different it is shown that the annealed systems can be made to behave in a qualitatively similar fashion as the quenched case.
Bayesian inference, indirect inference, metrics of distributions, random graphs, self-organised criticality, Bayesilainen päättely, epäsuora päättely, metriikka, satunnaisverkot, itse-organisoituva kriittisyys