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Pareto-jakauma on jatkuva paksuhäntäinen todennäköisyysjakauma, joka sai alkunsa italialaisen taloustieteilijän Vilfredo Pareton havainnosta, jonka mukaan 20 % Italian väestöstä omisti 80 % kaikesta omaisuudesta. Havainnon yleistys tunnetaan nykyään Pareton periaatteena, joka on Pareto-jakauman erityistapaus. Tässä tutkielmassa esitellään Pareto-jakauman keskeisiä ominaisuuksia ja sovelluksia sekä pohditaan Pareto-jakauman mahdollisuuksia lukio-opetuksessa. Tutkielman alussa esitellään Vilfredo Pareton henkilöhistoriaa sekä tutkielmassa tarvittavaa jatkuvien jakaumien ja paksuhäntäisten jakaumien teoriaa. Neljäs luku käsittelee jakauman keskeisiä ominaisuuksia ja historiaa sekä Pareton periaatetta. Luvussa esitellään myös yleistetty Pareto-jakauma ja sen eri tyypit, Pareto-jakauman parametrien estimointimenetelmiä sekä yhteyksiä muihin jakaumiin. Luku Pareto-jakauman keskeisistä ominaisuuksista perustuu pääosin Norman L. Johnsonin ja Samuel Kotzin kirjaan Continuous univariate distributions [1-2]: Distributions in Statistics. Viidennessä luvussa esitellään Pareto-jakauman sovelluksia eri aloilla. Pääpaino on tulojakaumien tutkimuksessa, joka oli Pareton keskeinen tutkimusala. Tulojen ja varallisuuden jakautumisen mallintamisesta Pareto-jakauman avulla on esimerkkejä Japanista, Norjasta ja Romaniasta. Muita esimerkkejä sovelluksista ovat tutkimukset järvien pinta-alojen ja määrien arvioinnista sekä parametrien estimoinnista vakuutusalalla. Kuudennessa luvussa pohditaan Pareto-jakauman käyttöä lukio-opetuksessa marraskuussa 2019 julkaistun lukion opetussuunnitelman pohjalta. Pareto jakauman mahdollisuuksia opetuksessa tarkastellaan tilastomatematiikan, laaja alaisen osaamisen sekä ainerajat ylittävän oppimisen näkökulmista.

Accessibility – what can be reached from a given point in space and how – is an essential field of study to measure the physical structure of cities, travel mode choices of residents, and the competitiveness of areas. Researchers increasingly acknowledge that accessibility is a fundamental concept on understanding how urban regions work and its position in future development of cities is paramount. Travel time is considered an intuitive measure to indicate accessibility and a strong predictor of mode choice, and usually, private car is the fastest mode of transport in urban environments. A central issue which stems from private cars and accessibility is the process of searching for parking. An understudied issue, the rather stressful activity is engaged in when arriving by car at the general area of desired parking, but no space is available. Motorists are then forced to continue search for parking, significantly contributing to urban congestion. In catering to mobility rather than accessibility, the modern urban planning has made it challenging to move away from private cars toward alternative, often more sustainable, modes of transport. Travel time studies, and more specifically, parking studies, can produce accurate data to aid in this transformation. In this thesis, a parking related research survey was developed and conducted in the Helsinki Capital Region, Finland. Adhering to the door-to-door approach, the survey respondents were enquired how long it took for them to find a parking place and park their car, and walk from the car to the destination in different postal code areas of Helsinki Capital Region. To explain a hypothetical variation in parking process durations (searching for parking, and walking to one's destination) in different areas, additional questions, such as the time of the day of parking, were presented. The invitation to respond to the survey was mostly spread on the social media platform Facebook. The survey, filled out with a web application specifically programmed for this thesis, received 5200 data rows from over 1000 unique visitors. The survey results indicate that there are spatial differences in parking process durations in different postal code areas of the Helsinki Capital Region. The inner city of Helsinki was experienced as the most difficult location to park in with regional subcenters such as Matinkylä, Espoo and Tikkurila, Vantaa, receiving relatively long parking process durations. Short parking process durations were reported from scarcely built areas but more often than not these areas had extreme values reported. Interestingly, area familiarity did not necessarily translate to faster parking process, while the type of the usual parking place was a better indicator. Out of the spatial explanatory variables added in the survey data processing, zones of urban structure (yhdyskuntarakenteen vyöhykkeet) could be used to find statistically significant differences in the parking process between variable groups and study area municipalities. Making use of the Helsinki Region Travel Time Matrix, a dataset developed by the research group Digital Geography Lab of the University of Helsinki, the thesis survey data was compared to total travel chain durations. The thesis survey data indicates that the proportion of time it takes to park one's car and walk to one's destination is a much larger part of the entire travel chain than previously estimated in the dataset. The parking process times are proportionally largest in the inner city of Helsinki, where the reported parking process duration exceeds that of the actual driving segment. This thesis, its entire version history, and all of the scripts developed for it have been made available at GitHub: https://github.com/sampoves/thesis-data-analysis.

In this thesis we consider extending the standard model of particle physics (SM) to include a fourth generation of elementary particles (SM4). The fourth generation would have to be sufficiently heavy to have escaped detection; specifically, its neutrino is required to be kinematically inaccessible to the Z boson in order to agree with the very precise LEP measurements of the Z width. This extension is appealing since the current theory (the SM) exhibits tension with some phenomena observed in nature. Such phenomena are, for example, the replication and number of fermion families, the ratio of matter to antimatter in the observable universe, charge-parity violation and the mixings between the fermions. Up to very recently the issue of the origin of mass, that is, the mechanism of electroweak symmetry breaking in the SM, was lacking experimental verification. However, during the writing of this thesis there have been some very exciting advances in this domain. In July 2012 the CMS and ATLAS collaborations at the LHC have reported the observation of a resonance at ∼ 125 GeV, an observation confirmed by experiments at another high-energy particle collider, the Tevatron. This resonance seems to correspond to the Higgs particle, the quantum of the scalar field responsible for the breaking of the electroweak symmetry. Besides verifying the answer to the theoretically fundamental question about the origin of mass, the experimental discovery also serves as a constraint for any theory of particle physics. The goodness of models describing particle physics are generally tested by performing global fits to the data, with the data set usually taken to be the most precisely measured quantities available — the electroweak precision observables. When the recent Higgs signal strengths are included in the data set, it is seen that the SM4 is not a correct theory of nature. Specifically, the Higgs signals predicted by the SM4 are not in agreement with the data, and the model has in September 2012 been quite decisively excluded at the statistical significance of 5.3σ. Following the developments in the field we next consider the phenomenological effects of adding another scalar doublet to the previously considered SM4. In the SM and SM4 there is just one scalar (Higgs) doublet: the models have a minimal scalar sector. The fermion sector, however, is not minimal: there are at least three replicas of a fermion family and so it is possible that the scalar sector is not minimal either. There are in fact arguments in favor of several Higgs doublets, for example supersymmetry and the baryon asymmetry of the universe. Two doublets give rise to five physical particles and so the phenomenology of such models is much richer than in the minimal scenario. Four family-models have received a great deal of interest in the last decade: some 500 articles are reported to have been published concerning their phenomenology during this time. This thesis is a review of the recent developments is this field.

Despite a vast body of knowledge that has already been accumulated on particle transport at both theoretical and experimental level, a simple method for estimating particle source impact on plasma density profile peaking has been lacking. Fable et al. presented a parameter for calculating the source strength (Sstr, the S parameter) in [1]. The parameter is derived from particle flux continuity equation, and after approximations takes as input parameters only the information on neutral beam injection (NBI) power, beam ions injection energy, effective core heat transport diffusivity and plasma density, radius, and volume together with a fitted coefficient from an ASDEX Upgrade experiment. The formula was applied to a database of 165 pulses in both high and low confinement mode, mostly with neutral beam heating, in JET, Joint European Torus, fusion experiment. The results appear reasonable considering the fitted parameter and the approximations in the formula. In addition to the S parameter values, also normalised density gradient dependence on neutral beam heating power and collisionality were investigated, to compare the results with those obtained at ASDEX Upgrade in [1]. Detailed studies of six gas puff modulation shots [2, 3, 4] at JET are used as reference. In [2] the source contribution for the H-mode shots was 50-60% and low confinement mode shots 10-20%. This is further validated in [3] and the high confinement mode shots are compared to similar shots DIII-D fusion experiment in [4], where the source impact on density peaking was negligible. Observed differences are attributed to different dominant turbulent environments. The average calculated level of S parameter values suggest mostly non-negligible source contribution to density peaking, and the values differ for high and low confinement mode plasmas, in line with [2, 3, 4]. However, the results imply that the coefficient 2000 is not constant across the database, and while a scalar correction to fit the coefficient to JET may be possible for low confinement mode plasmas, the high confinement mode plasmas require further research.

Kaaottisessa järjestelmässä, kuten esimerkiksi ilmakehässä, ennustettavuus on rajoitettu. Yksittäinen numeerinen sääennuste tulee aina lopulta erkanemaan todellisesta ilmakehän tilasta. Numeeristen sääennusteiden parantamiseksi on kehitetty parviennusteet, missä yhden ennusteen sijasta tarkastellaan useamman ennusteen käyttäytymistä. Luotettavan parviennusteen luomiseksi on parven jäsenten alkutilat valittava tarkkaan. Parvi voidaan luoda niin, että arvioituun alkutilaan lisätään satunnaisia häiriöitä tai edeltävän parviennusteen pohjalta jalostettuja häiriöitä. Tässä pro gradu -työssä tarkasteltiin, onko satunnaishäiriö- ja jalostushäiriömenetelmän luotettavuudessa eroa Lorenz-96-mallissa. Luotettavuutta vertailtiin kolmella eri todentamismenetelmällä: sijoitushistogrammilla, hajaantuminen-osuvuus-tuloksella ja sijoitustodennnäköisyystuloksella. Parviennusteet luotiin sekä täydelliselle että stokastiselle mallille. Häiriöiden kokoa vaihdeltiin neljän eri skaalausparametrin avulla. Ennen varsinaisten parviennusteiden luontia mallin kaaottisuus varmistettiin selvittämällä ensimmäinen Lyapunovin eksponentti sekä tarkastelemalla, kääntyvätkö satunnaiset häiriöt Lyapunov-vektorin suuntaan. Tulosten mukaan häiriömenetelmien välillä oli eroa paikoin, riippuen käytetystä mallista ja skaalausparametristä. Täydellisessä mallissa jalostushäiriömenetelmä oli parviennusteen ajan hetkestä ja häiriön koosta riippuen luotettavampi kuin satunnaishäiriömenetelmä sijoitushistogrammin ja hajaantuminen-osuvuus-tuloksen osalta. Stokastisessa mallissa satunnaishäiriömenetelmä oli luotettavampi ainoastaan sijoitushistogrammin tulosten perusteella. Sijoitustodennnäköisyystulos ei antanut selviä viitteitä suuntaan tai toiseen kummankaan mallin osalta.

Optical frequency combs are broadband laser light sources that produce light consisting of equally separated narrow lines. The frequencies of these comb lines can be determined and stabilized accurately. A dual-comb spectrometer is based on two optical frequency combs, and it can measure spectra at high speed and with high spectral resolution and accuracy. This requires high mutual coherence between the two combs. In this thesis, a passively coherent dual-comb spectrometer was optimized for spectroscopic temperature measurements. Spectroscopic temperature measurements presented in this thesis are based on quantifying the temperature dependence of the molecular absorption lines. In the experimental part of this thesis, three different spectroscopic temperature measurement techniques were used to measure the temperature of acetylene gas and results were compared to the temperature value of a reference temperature sensor. The three methods that are demonstrated in this thesis are line-strength ratio thermometry (LRT), rotational-states distribution thermometry (RDT), and Doppler-broadening thermometry. The measured dual-comb spectra had high quality and the dual-comb figure of merit was determined to be 5.7×10^6 Hz^(-1/2), which is a typical value for a high-quality dual-comb spectrometer. All the temperature measurements were performed at room temperature (295 K). Line-strength ratio thermometry produced the most accurate temperature results, with an estimated uncertainty of approximately 1 K. Rotational-states distribution thermometry results had an estimated uncertainty of about 3 K. Doppler-broadening thermometry did not produce reliable results, most likely due to too high gas pressure. The possible future work should be performed with larger temperature and pressure ranges to assess the accuracy of the presented spectroscopic thermometry techniques more thoroughly.

This thesis strives to familiarize the ideas behind the success of patch-based image representations in image processing applications in recent years. Furthermore we show how to restore images using the idea of patch-based dictionary learning and the k-means clustering algorithm. In chapter 1 we introduce the notion of patch-based image processing and take a look at why dictionary learning using sparsity is a hot topic and useful in processing natural images. The second chapter aims to formulate the different methods and approaches used in this thesis mathematically. Dictionary learning, the k-means algorithm and the Structural similarity index (SSIM) are in the main focus. Chapter 3 goes into the details of the experiments. We present and discuss the results as well. The fourth and final chapter summarizes the main ideas of the thesis and introduces development suggestions for further investigation based on the methods used. Using a fairly simplistic patch-based image processing method we manage to reconstruct images from a set of similar images to a reasonable extent. As the main result we see how the size of the patches as well as the size of the learned dictionary effects the quality of the restored image. We also detect the limitations and problems of this approach such as the appearance of patch artifacts which is an issue to attack and resolve in following studies.

Tässä tutkielmassa esitellään lyhyesti PCP-teoreema, minkä jälkeen tutkielmassa pala palalta käydään läpi teoreeman todistamiseen tarvittavia työkaluja. Tutkielman lopussa todistetaan PCP-teoreema. Vaativuusluokka PCP[O(log n), O(1)] sisältää ne ongelmat, joilla on olemassa todistus, josta vakio määrän bittejä lukien probabilistinen Turingin kone kykenee ratkaisemaan ongelman käyttäen samalla vain logaritmisen määrän satunnaisuutta suhteessa syötteen kokoon. PCP-teoreema väittää vaativuusluokan NP kuuluvan vaativuusluokkaan PCP[O(log n), O(1)]. Väritys on funktio, joka yhdistää kuhunkin joukon muuttujaan jonkin symbolin. Rajoite joillekin muuttujille on lista symboleista, joita rajoite sallii asetettavan muuttujille. Jos väritys asettaa muuttujille vain rajoitteen sallimia symboleja, rajoite on tyytyväinen väritykseen. Optimointi-ongelmat koskevat sellaisten väritysten etsimistä, että mahdollisimman moni rajoite joukosta rajoitteita on tyytyväinen väritykseen. PCP-teoreemalla on yhteys optimointi-ongelmiin, ja tätä yhteyttä hyödyntäen tutkielmassa todistetaan PCP-teoreema. Tutkielma seuraa I. Dinurin vastaavaa todistusta vuoden 2007 artikkelista The PCP Theorem by Gap Amplification. Rajoiteverkko on verkko, jonka kuhunkin kaareen liittyy jokin rajoite. Rajoiteverkkoon liittyy lisäksi aakkosto, joka sisältää ne symbolit, joita voi esiintyä verkon rajoitteissa ja värityksissä. Tutkielman päälauseen avulla kyetään kasvattamaan rajoiteverkossa olevien värityksiin tyytymättömien rajoitteiden suhteellista osuutta. Päälause takaa, että verkon koko säilyy samassa kokoluokassa, ja että verkon aakkoston koko ei muutu. Lisäksi jos verkon kaikki rajoitteet ovat tyytyväisiä johonkin väritykseen, päälauseen tuottaman verkon kaikki rajoitteet ovat edelleen tyytyväisiä johonkin väritykseen. Päälause koostetaan kolmessa vaiheessa, joita kutakin vastaa tutkielmassa yksi osio. Näistä ensimmäisessä, tutkielman osiossa 4, verkon rakenteesta muovataan sovelias seuraavia osioita varten. Toisessa vaiheessa, jota vastaa osio 6, verkon kävelyitä hyödyntäen kasvatetaan tyytymättömien rajoitteiden lukumäärää, mutta samalla verkon aakkosto kasvaa. Kolmannessa vaiheessa, osiossa 5, aakkoston koko saadaan pudotettua kolmeen sopivan algoritmin avulla. Osiossa 7 kootaan päälause ja todistetaan lausetta toistaen PCP-teoreema.

Northern peatlands are a valuable, volatile carbon stock and hold 30 % global terrestrial organic carbon. Climate is the most important control on peatland ecology. Positive and negative feedback loops between peatlands and climate make this complex relationship. Climate change affects high northern latitudes in particular, making the northern peatlands prone to experience massive changes in ecology. It is unclear exactly how long-term climate change will affect such an integral part of the carbon cycle. My intent was to produce a reliable study of what kind of response did bogs develop after the Little Ice Age (LIA) and how climate warming has affected these important carbon stocks. This study uses high frequency, multi-proxy, post-LIA peatland response data to map the ecological response of a boreal ombrotrophic peatland ecosystems. Study sites are located in Southern-Finland and Estonia. I concentrate on three distinct micro-habitats. I use low-frequency data from previous studies to compare with my novel data and to give the study more spatial scope. My analysis is structured around a chronology consisting of 210 Pb-dates. Plant macrofossil data, present vegetation data, Sphagnum mosses as the most important group, and modern water-table data are used to model past plant composition and hydrology. Age-depth models and water-table reconstructions have been created on this basis. Bulk density and C/N ratio were also analysed. Large and fast paced shifts in accumulation rates and changes in vegetation composition were revealed. After LIA, peatland surfaces have dried and dry-habitat vegetation has increased. I identified a two-step pattern in response to post-LIA climate shifts. A wet period ended LIA, followed by a two-step warming identified from different proxies and models. The pattern of change coincides with the known changes in climate. This suggests that after the LIA, changes in climate have been the driving force behind changes in peatland ecology and carbon sequestration in it. The results show that the way northern bogs respond to changes in climate can on the short term have huge effects for the vegetation, and on long term threaten the whole peatland and its carbon stocks. These changes are manifested through changes in the relation of primary production to decomposition and local hydrology. Different microhabitats are more vulnerable to climate shifts than others. In future climate warming will continue to influence northern peatlands. Depending on the scale of change, peatlands can act as a sink of atmospheric carbon, or if a watershed is reached, release large amounts of carbon to the atmosphere. Most likely this would not only destroy peatlands in large quantities, but also further enhance the positive feedback between carbon release and peatland drying.

The soft soil sediments in Southern Finland were originally deposited during different stages of the Baltic Sea. There are many problems concerning construction on these soft soils. These include, among others, sinking and stability problems. Demands for construction work on soft soil areas are growing all the time and it is therefore important to study the geological and geophysical properties of the soil. This thesis examines the possibility of drawing conclusions on soft soil stratigraphy with the help of in situ measurements. In situ measurements are performed on-site and they speed up working because separate sampling and laboratory analysis are not required. Measurements were carried out in the Helsinki Metropolitan Area. Sediment depositions' apparent resistivity, temperature and magnetic susceptibility were measured with probes installed on the tip of bog drills. The results were compared to an existing stratigraphic data and other information. Research also included method testing with susceptibility measuring instrument purchased by Geological Survey of Finland. In situ measurements of apparent resistivity provides reliable information on soft soil stratigraphy. Resistivity profiles can be used to distinguish topsoil and different clays. In addition, sandy intermediate layers are well distinguished in these high resolution measurements. The apparent resistivity varies similarly in every research area. In situ measurements of magnetic susceptibility can be used to estimate the incidence of sulphide clays. In this work no conclusions could be drawn regarding the variability of susceptibility due to the weaknesses of the apparatus. Use of the apparatus revealed several problems. The most notable is the lack of a temperature sensor.