En introduktion ges till portföljteori och tillämpning av tangentportföljen med en riskfri tillgång presenteras. Nyttofunktioner förklaras och hur tangentportföljen kan härledas med hjälp av en nyttofunktion. Annan bakgrundsinformation som är viktig för att förstå resultat från [1] om vikter i tangentportföljen då kovariansmatrisen inte är inverterbar presenteras också. Teorin tillämpas på historisk data från S&P. Resultaten visas i grafer och tabeller som analyseras.

While it is not the most abundant greenhouse gas, a significant portion of the greenhouse effect is caused by methane. The amount of methane in the atmosphere is increasing, indicating that there is a continuous source of methane to the atmosphere. One significant source of methane is freshwater lakes, even though they cover only a small portion of the Earth’s surface. Because of this, it is important to monitor methane fluxes from lakes in order to understand the processes which affect the magnitude of these fluxes. Methane is produced in the sediment at the bottom of the lake, and transported through the water by ebullition, diffusive flux, storage flux, or plant mediated emission. This study looked to examine the amount of methane transmitted to the atmosphere by these processes on Lake Erken in eastern Sweden. Using the eddy covariance method, we can study the methane flux with good spatial and temporal resolution. Regular sampling of lake water, both from the surface and depths of 5 and 10 meters, also helps us to understand the amount of methane dissolved in the lake. These measurements can help us to better understand the transfer velocity, or the efficiency of the exchange between water and air, as well as the amount of methane transported from lakes to the atmosphere. Water sampling showed that there is very little variation in methane concentration between different parts of the lake. Concentrations at four surface locations are nearly identical. These surface measurements are also similar to concentrations at different depths. Over time, the concentrations generally stayed the same, with isolated high and low concentration events. The amount of methane emitted by the lake was studied with the lake divided into a shallow water area, and a deep water area. The magnitude of fluxes from both areas was very similar, but the area of shallow water had a higher total flux. The fluxes were well correlated with wind speed; higher fluxes coming during times with higher wind speed. This relates well to the transfer velocity theory, and the bulk flux approximation. However, there was no clear diurnal cycle in methane fluxes. The fluxes during the night were similar to daytime fluxes. Atmospheric pressure also had an impact on fluxes, with greater fluxes coming at times of lower pressure. A large seasonal variation was clear. More methane escaped the water in autumn and winter than in spring or summer. This is due in part to the fluxes from when the lake freezes over/thaws and the water in the lake turns over, bringing methane rich water from the lake’s bottom to the surface. As expected, the waterside concentration of methane also had a strong correlation with the fluxes. The main conclusions of this study are: 1) Methane fluxes are variable with wind speed, waterside concentrations, and the seasons 2) Water depth and diurnal cycles do not affect methane fluxes as strongly. Keywords: methane, transfer velocity, flux, waterside concentration, eddy covariance

While many university students get introduced to the concept of statistics early in their education, random matrix theory (RMT) usually first arises (if at all) in graduate level classes. This thesis serves as a friendly introduction to RMT, which is the study of matrices with entries following some probability distribution. Fundamental results, such as Gaussian and Wishart ensembles, are introduced and a discussion of how their corresponding eigenvalues are distributed is presented. Two well-studied applications, namely neural networks and PCA, are discussed where we present how RMT can be applied / Medan många stöter på statistik och sannolikhetslära tidigt under sina universitetsstudier så är det sällan slumpmatristeori (RMT) dyker upp förän på forskarnivå. RMT handlar om att studera matriser där elementen följer någon sannolikhetsfördelning och den här uppsatsen presenterar den mest grundläggande teorin för slumpmatriser. Vi introducerar Gaussian ensembles, Wishart ensembles samt fördelningarna för dem tillhörande egenvärdena. Avslutningsvis så introducerar vi hur slumpmatriser kan användas i neruonnät och i PCA.