Data gathering is a constant in human history with ever increasing amounts in quantity and dimensionality. To get a feel for the data, make it interpretable, or find underlying laws it is necessary to fit a function to the finite and possibly noisy data. In this thesis we focus on a method achieving this, namely least squares approximation. Its discovery dates back to around 1800 and it has since then proven to be an indispensable tool which is efficient and has the capability to achieve optimal error when used right. Crucial for the least squares method are the ansatz functions and the sampling points. To discuss them, we gather tools from probability theory, frame subsampling, and $L_2$-Marcinkiewicz-Zygmund inequalities. With that we give results in the worst-case or minmax setting, when a set of points is sought for approximating a class of functions, which we model as a generic reproducing kernel Hilbert space. Further, we give error bounds in the statistical learning setting for approximating individual functions from possibly noisy samples. Here, we include the covariate-shift setting as a subfield of transfer learning. In a natural way a parameter choice question arises for balancing over- and underfitting effect. We tackle this by using the cross-validation score, for which we show a fast way of computing as well as prove the goodness thereof.:1 Introduction 2 Least squares approximation 3 Reproducing kernel Hilbert spaces (RKHS) 4 Concentration inequalities 5 Subsampling of finite frames 6 L2 -Marcinkiewicz-Zygmund (MZ) inequalities 7 Least squares in the worst-case setting 8 Least squares in statistical learning 9 Cross-validation 10 Outlook

Numerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however, has begun to outstrip efforts to reduce analysis overhead. As such, most commercially available software packages are now limited by the human effort required to prepare, develop, and initialize the necessary computational models. Primarily due to the mesh-based analysis methods utilized in these software packages, the dependence on model preparation greatly limits the accessibility of these analysis tools. In response, the so-called meshless or mesh-free methods have seen considerable interest as they promise to greatly reduce the necessary human interaction during model setup. However, despite the success of these methods in areas demanding high degrees of model adaptability (such as crack growth, multi-phase flow, and solid friction), meshless methods have yet to gain notoriety as a viable alternative to more traditional solution approaches in general solution domains. Although this may be due (at least in part) to the relative youth of the techniques, another potential cause is the lack of focus on developing robust methodologies. The failure to approach development from a practical perspective has prevented researchers from obtaining commercially relevant meshless methodologies which reach the full potential of the approach. The primary goal of this research is to present a novel meshless approach called MIMS (Model Integrated Meshless Solver) which establishes the method as a generalized solution technique capable of competing with more traditional PDE methodologies (such as the finite element and finite volume methods). This was accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input (small features, poorly formed faces, etc.) to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.

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Genom att analysera kortegenskaperna hos Magic the Gathering-kort harmodeller tagits fram för att bestämma deras inverkan på kortpriset. Tidigarestudier har inte fokuserat på spel-egenskaperna, vilket är vad som särskiljer dettaarbete från tidigare forskning. För att modellera effekten av spel-egenskapernahar dessa kvantifierats och undersökts med hjälp av Minsta-kvadratmetoden ochLasso-regression, med hjälp av programmeringsspråket R. Resultaten indikeraratt faktorer direkt kopplade till samlarbarhet och spelbarhet har den störstainverkan på priset för Magic the Gathering-kort. Dessa resultat har diskuteratsmed utgångspunkt från olika perspektiv, såsom Wizards of the Coast (utgivarenav Magic the Gathering), spelare, samlare och investerare. Genom att fokusera påspel-egenskaperna har denna studie bidragit till området på ett sätt som tidigareforskning inte har gjort, vilket ger en mer helhetsbild av Magic the Gathering-kortsvärde. / By analyzing the card properties of Magic the Gathering cards, models have beendeveloped to determine their impact on card prices. Previous studies have notfocused on gameplay properties, which distinguishes this work from previousresearch. To model the effect of gameplay properties, they have been quantifiedand examined using Least Squares Method and Lasso Regression, with the helpof the programming language R. The results indicate that factor directly relateradto collectability and playability have the greatest impact on the price of Magic theGathering cards. These results have been discussed from various perspectives,such as Wizards of the Coast (the publisher of Magic the Gathering), players,collectors, and investors. By focusing on gameplay properties, this study hascontributed to the field in a way that previous research has not, providing a morecomprehensive understanding of the value of Magic the Gathering cards.

<p>  </p> <p>Dicamba (3,6-dichloro-2-methoxybenzoic acid) and 2,4-D (2,4-dichlorophenoxyacetic acid) are two widely used herbicides for broadleaf weed control in soybeans. However, off-target application of dicamba and 2,4-D can cause severe damage to sensitive vegetation and crops. Early detection and assessment of off-target damage caused by these herbicides are necessary to help plant diagnostic labs and state regulatory agencies collect more information of the on-site conditions so to develop solutions to resolve the issue in the future. In 2021, the study was conducted to detect damage to soybean leaves caused by dicamba and 2,4-D by using LeafSpec, an accurate handheld hyperspectral leaf scanner. . High resolution single leaf hyperspectral images of 180 soybean plants in the greenhouse exposed to nine different herbicide treatments were taken 1, 7, 14, 21 and 28 days after herbicide spraying. Pairwise PLS-DA models based on spectral features were able to distinguish leaf damage caused by two different modes of action herbicides, specifically dicamba and 2,4-D, as early as 2 hours after herbicide spraying. In the spatial distribution analysis, texture and morphological features were selected for separating the dosages of herbicide treatments. Compared to the mean spectrum method, new models built upon the spectrum, texture, and morphological features, improved the overall accuracy to over 70% for all evaluation dates. The combined features are able to classify the correct dosage of the right herbicide as early as 7 days after herbicide sprays. Overall, this work has demonstrated the potential of using spectral and spatial features of LeafSpec hyperspectral images for early and accurate detection of dicamba and 2,4-D damage in soybean plants.</p> <p>   </p>