A parameter specifying the number of clusters in an unsupervised clustering algorithm is often unknown. Different cluster validity indices proposed in the past have attempted to address this issue, and their performance is directly related to the accuracy of a clustering algorithm. Toe gap statistic proposed by Tibshirani (2001) was applied to k-means and hierarchical clustering algorithms for estimating the number of clusters and is shown to outperform other cluster validity measures, especially in the null model case. In our experiments, the gap statistic is applied to the Fuzzy c-Means (FCM) algorithm and compared to existing FCM cluster validity indices examined by Pal (1995). A comparison is also made between two initialization methods where centers are randomly assigned to data points or initialized using the furthest first algorithm (Hochbaum, 1985). Toe gap statistic can be applied using the FCM algorithm as long as the fuzzy partition matrix can be employed in computing the gap statistic metric, Wk . Three new methodologies are examined for computing this metric in order to apply the gap statistic to the FCM algorithm. Toe fuzzy partition matrix generated by FCM can also be thresholded based upon the maximum membership to allow computation similar to the kmeans algorithm. This is assumed to be the current method for employing the gap statistic with the FCM algorithm and is compared to the three proposed methods. In our results, the gap statistic outperformed the cluster validity indices for FCM, and one of the new methodologies introduced for computing the metric, based upon the FCM objective function, out performed the threshold method for m=2.

The increasing usage of algorithms and extensive collections of data have changed the discipline of finance and created new possibilities for analyzing the financial markets. To further explore the potential of developing new methods for understanding financial market behaviour, this thesis examines the first digit probability distribution of Benford's Law and its applicability within the financial markets. The research investigates various indices', equities', and technical analysis tools' conformity to Benford's Law by using relative price changes and volume traded. It was found that both indices and equities exhibit resemblance with Benford's Law, whereas technical analysis tools did not. In addition, the relevance of data frequency was explored, but it was deemed not to have any effect on conformity found. In an attempt to apply the findings, a regression analysis was conducted to forecast volatility. However, even though correlation was found, the regression model failed to predict future volatility accurately. / Den ökade användningen av algoritmer och omfattande datainsamling har förändrat det finansiella spelrummet och skapat nya möjligheter för analys av finansmarknaden. För att ytterligare undersöka potentialen i att utveckla nya metoder för att förstå finansmarknadens beteende utforskar denna avhandling Benfords lag och dess tillämpbarhet på den finansiella marknaden. Studien testar olika index, aktiers och tekniska analysverktygs överensstämmelse med Benfords lag genom att använda relativa prisförändringar och handlad volym. Det visade sig att både index och aktier följer Benfords lag medan tekniska analysverktyg inte gjorde det. Dessutom undersöktes datafrekvensens relevans, men detta ansågs inte ha någon effekt på överensstämmelsen med fördelningen. I ett försök att tillämpa resultaten genomfördes en regressionsanalys för att prognosticera volatilitet. Korrelation hittades men regressionsmodellen gav inte ett tillförlitligt resultat.