Abstrakt: Vänster förmaksstorlek har prognostisk betydelse inom kardiologi. Det finns generellt enighet om att vänster förmaksvolymmätning (VFV) är det bästa mätmåttet av vänster förmaksstorlek. För närvarande används S5-1 som är en phased array givare, som första val, för att avbilda 2-dimensionella två- och fyrkammarvyer. Matrix array givaren X5-1 är ett praktiskt kliniskt alternativ för insamling av samma 2D projektioner men den har inte utvärderats på ett adekvat sätt mot S5-1 givare avseende VFV. Syftet med studien är att undersöka om det föreligger någon statistisk signifikant skillnad vid bestämning av VFV i apikal tvåkammarvy beroende på val av givare. Studien omfattade 50 patienter som var remitterade för en ekokardografisk undersökning på avdelningen för klinisk fysiologi och nuklearmedicin på Skånes Universitetssjukhus i Malmö. Ekokardiografiska bilder, insamlades med båda givarna, från både patienter med normal och abnormal vänster förmaksstorlek som hade sinusrytm. VFV mättes med Simpsons biplanmetod efter gällande amerikanska och europeiska riktlinjer. Det fanns en signifikant korrelation mellan medelvärdet av VFV, mätta på bilder som erhållits med de två olika insamlingsmetoderna (r =0,98, P 0,0001). Den utförda Bland-Altmananalysen visade också en statistiskt signifikant överensstämmelse i VFV mätning mellan de två insamlingsmetoderna. Studien visade att X5-1 givaren kan vara ett praktiskt alternativ för att erhålla 2D tvåkammarprojektion på ett mer tidseffektivt sätt jämfört med S5-1 givaren. / Abstract: Left atrial size has a prognostic significance in cardiology.There is ecumenical agreement that measurement of left atrial volume (LAV) is the best way to evaluate the left atrial size. Currently, S5-1, a phased array transducer, is used as the first choice to depict the 2-dimensional (2D) two- and four apical chamber views. X5-1, a matrix array transducer, is a practical clinical option for collecting the same 2D projections, but it has not been adequately assessed against the S5-1 transducer with LAV in consideration. The purpose of the present study was to investigate whether there is any statistically significant difference in the determination of LAV in apical two chamber views depending on the choice of transducer. The study included 50 patients who were referred for an echocardiographic examination at the Department of Clinical Physiology and Nuclear Medicine at Skåne University Hospital in Malmö. Echocardiographic images collected with both transducers, from patients with both normal and abnormal left atrial sizes and with sinus rhythm. LAV was measured using Simpson's biplane method according to the current American Society of Echocardiography (ASE) and European Association of Cardiovascular Imaging (EACVI) guidelines. There was a significant correlation between the mean of LAV, measured in images obtained by the two different transducers (r = 0.98, P 0.0001). The Bland-Altman analysis showed a statistically significant agreement in LAV measurement between the two methods. The X5-1 transducer is possibly a practical alternative to obtain 2D apical two-chamber projection in a more time efficient manner compared to the S5-1 transducer.

The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale.

Introduktion: Vänsterkammarens ejektionsfraktion (VKEF) är ett central mått på systolisk funktion i vänster kammare och är en av de mest betydelsefulla parametrar vid ekokardiografiska undersökningar. Idag är Simpson biplan metoden den mest använda metoden för bestämning av ejektionsfraktionen. Vid ekokardiografiska undersökningar sänder givaren med piezoelektriska kristaller ut ultraljudsvågor med en frekvens över 20 000 Hz. Ljudvågorna som skickas ut i kroppen reflekteras och sedan återvänder de till givaren för att skapa en bild. Denna studie jämför den tvådimensionella (2D) ultraljudsmetoden Simpsons biplan med Philips Heart Model som är en automatiserad tredimensionella (3D) funktion för bedömning av VKEF.  Material och metod: I studien inkluderades 31 hjärtfriska försökspersoner mellan åldrarna 21-64. Det samlades in bilder på apikala 4- och 2 kammarbilder från alla försökspersoner där Simpsons biplan metoden användes för att beräkna ejektionsfraktion. Apikala 4-kammarbilder samlades in för att beräkna ejektionsfraktionen med Philips Heart Model 3D funktion.  Resultat: Resultatet från denna studie visade att det inte föreligger någon signifikant skillnad mellan Simpsons biplan metoden och Philips Heart Model metoden för bestämning av ejektionsfraktion. Båda metoderna visade likvärdiga mätresultat.  Diskussion: Philips Heart Model metoden är en relativ ny funktion som använder sig av artificiell intelligens för att analysera 3D bilder. Philips Heart Model metoden är en säker funktion att använda då de flesta studier bevisar likvärdiga och säkra mätresultat i jämförelse med andra metoder.  Konklusion: Enligt denna studie ger Philips Heart Model funktionen likvärdiga mätresultat av vänsterkammarens ejektionsfraktion i jämförelse med Simpsons biplan. / Introduction: Simpson’s biplane method is the most used method for determining the left ventricular ejection fraction (LVEF) in echocardiographic examinations. Ejection fraction is a central measurement of the heart's global systolic function. The probe with piezoelectric crystals emits ultrasound waves with a frequency above 20,000 Hz. The sound waves that are sent out into the body are reflected and then return to the probe to create an image. This study compares the two-dimensional (2D) ultrasound Simpson's biplane method with the Philips Heart Model method, which is an automated three-dimensional (3D) function for assessment of LVEF.                                                                                                                          Material and method: 31 subjects with no recorded heart pathologies between the ages of 21-64 were included in the study. Apical 4- and 2-chamber images were collected from the test subjects, where the Simpson's biplane method was applied to calculate the ejection fraction. 2D apical 4-chamber images were collected to convert to 3D and used to calculate the ejection fraction with the Philips Heart Model.    Results: The results of this study showed that there is no significant difference between the Simpson’s biplane method and the Philips Heart Model method for determining ejection fraction.    Discussion: The Philips Heart Model method is a relatively new feature that uses artificial intelligence to analyze 3D images. The Philips Heart Model method is a reliable feature to use as most studies have proven similar and reliable measurements when comparing it with other methods for determining LVEF.    Conclusion: According to this study, the Philips Heart Model feature provides equivalent measurements in comparison with the manual method Simpson's biplane.

This diploma thesis is focused on the analysis of small samples where it is not possible to obtain more data. It can be especially due to the capital intensity or time demandingness. Where the production have not a wherewithall for the realization more data or absence of the financial resources. Of course, analysis of small samples is very uncertain, because inferences are always encumbered with the level of uncertainty.

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In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)