This thesis presents some methods of multivariate dimension reduction, with emphasis on methods guided by the work of R.A. Fisher. Some of the methods presented can be traced back to the 20th century, while some are much more recent. For the more recent methods, additional attention will paid to the foundational underpinnings. The presentation for each of the methods contains a brief introduction of its general philosophy, accompanied by some theorems and ends with the connection to the work of Fisher. / Den här kandidatuppsatsen presenterar ett antal metoder för dimensionsreducering, där betoning läggs på metoder some följer teori utvecklad av R.A. Fisher. En del av metoderna som presenteras utvecklades redan på tidigt 1900-tal, medan andra är utvecklade i närtid. För metoderna utvecklade i närtid, så kommer större vikt läggas vid den grundläggande teorin för metoden. Presentationen av varje metod består av en kortare beskrivning, följt av satser och slutligen beskrivs dess koppling to Fishers teorier.

Excessive concentrations of arsenic in soil- and groundwater constitute a global issue. The spread of arsenic is due to both natural and anthropogenic effects in the environment. Historically, the anthropogenic emissions have originated from several different industrial sectors e.g. wood impregnation. In Sweden, there are approximately 85 000 contaminated sites. Most of them are contaminated due to industrial activities. The emissions of arsenic in Sweden are mainly due to the wood impregnating agent CCA (copper, chromium and arsenic). Arsenic is a toxic metalloid that predominantly occurs as the inorganic compounds arsenite As[III] and arsenate As[V] in soil and groundwater. The mobilization of the arsenic compounds in soil water is affected by the redox conditions and by other conditions in the field, e.g. pH. Arsenate adsorbs stronger to iron- and aluminum hydroxides. Arsenite is the most toxic, mobile and soluble of the two compounds.   In a previously remediated impregnation plant in Hjältevad, Småland, arsenic has started to spread in the soil and groundwater. High dissolved concentrations of arsenic have been measured in the area. A hypothesis about the recent mobilization of arsenic is that pollutants below the groundwater table which were left behind after remediation started to dissolve due to changed redox conditions. This report aims to evaluate how the mobilization and adsorption of arsenic in Hjältevad is effected by pH and redox potential. This was addressed by leaching tests and geochemical modeling in Visual MINTEQ. Soil samples were collected during autumn 2017. Soil samples from seven different sampling points, taken from different depths were collected. Leaching tests were carried out for both dry and humid soil samples. Oxalate extractable arsenic was used together with measured dissolved concentrations of cat- and anions as input in Visual MINTEQ. The input data were used to evaluate the mobilization and adsorption of arsenite, arsenate and total arsenic due to pH, redox potential and the specific surface area of ferrihydrite. The modeling showed that the mobilization and adsorption of arsenic is dependent on pH, redox potential and reactive surfaces in the soil solution. Arsenate was adsorbed more strongly between pH 4.5 and 6.5 while arsenite was adsorbed at pH values greater than 7. The conclusion is that the mobilization and adsorption of arsenic were affected to some extent by the pH value. However the redox potential and the specific surface area of ferrihydrite were more influential. To verify the trends seen in the report, additional modeling is required. The report shows that most likely, arsenic started to mobilize in Hjältevad due to the changed redox conditions.

Fjärrvärmen utgör omkring hälften av all uppvärmning som sker i Sverige idag och har utöver det en viktig roll då den tar vara på mycket energi som annars hade gått till spillo. Att kunna vara ekonomiskt konkurrenskraftig gentemot andra uppvärmningsalternativ är en viktig del för fjärrvärmens fortsatta utveckling. Det finns flera alternativ och möjligheter att göra detta, ett av dem är att hålla nere driftkostnaderna. Det kan bland annat göras genom att minska tryckfallen i näten, som även kan ge en ökad expansionsmöjlighet. Idag används främst avstängningsventiler med reducerande genomlopp i näten, främst på grund av den billigare investeringen. Trots detta saknas en kunskap om hur dessa ventiler påverkar näten sett till tryckfall, drift- och investeringskostnader jämfört med ventiler med fullt genomlopp. För att undersöka detta har två metoder använts för att komplettera varandra och öka förståelsen för ventilerna. Den första metoden beräknade det generella fallet teoretiskt för att se vid vilka flöden som ventiler med fullt genomlopp är mer gynnsamma. Den andra metoden beräknade ett specifikt fall med simuleringar i ett verkligt nät med de olika ventilerna och beräkna vilken som är den mest ekonomiskt gynnsamma. Resultaten av det generella fallet visar att ventiler med fullt genomlopp blir mer gynnsamma när flödeshastigheten överstiger omkring 1 m/s, oavsett dimension, beräknat på under 30 år med ett elpris på 1,50 kr/kWh. Vid dimensionering av nät är det vanligt att använda 1,5 - 2,0 m/s, detta utgör ofta en liten del av nätets drifttid och är ofta lägre. Beräkning av det specifika fallet visade att ventiler med reducerande genomlopp är det ekonomiska alternativet då återbetalningstiden för ventiler med fullt genomlopp var över 600 år vid 1,50 kr/kWh. Utifrån detta kan slutsatsen dras att ventiler med fullt genomlopp inte bör ersätta reducerande genomlopp, utan att det i stället handlar om att identifiera de delar i nätet som utgör, eller kan komma att utgöra, en strypning eller förträngning i nätet där höga flödeshastigheter kan förekomma och möjligtvis byta till fullt genomlopp. / District heating makes up about half of all heating that takes place in Sweden today and, in addition to that, has an important role as it makes use of a lot of energy that would otherwise have been wasted. Being able to be economically competitive against other heating alternatives is an important part of the continued development of district heating. There are several options to do this, one of which is to keep operating costs down. This can be done by reducing the pressure drops in the networks, which can also provide an increased possibility of expansion. Today, shut-off valves with reducing throughput are mainly used in networks, often because of the cheaper investment. Despite this, there is a lack of knowledge about how these valves affect the networks, which is attributed to pressure drop, operating and investment costs compared to valves with full flow. To investigate this, two methods have been used to complement each other and increase the understanding of the valves. In the first method, the general case is calculated theoretically to see at which flows valves with full flow are advantageous. The second method calculates a specific case with simulations in a real network with the different valves and calculates which one is the most economically advantageous. The results of the general case show that valves with full flow become more advantageous when the flow rate exceeds about 1 m/s, regardless of dimension, calculated over 30 years with an electricity price of SEK 1.50 /kWh. When dimensioning networks, 1.5 - 2.0 m/s is a common value, this often constitutes a small part of the network's operating time and is often lower. Calculation of the specific case showed that valves with reducing throughput are the economic alternative as the payback period for valves with full throughput was over 600 years at SEK 1.50/kWh. Based on this, the conclusion can be drawn that valves with full throughput should not replace reducing throughput, but that it is instead a question of identifying the parts of the network that constitute a throttling or constriction in the network where high flow rates can occur and possibly switch to valves with full throughput.