<p> This thesis deals with linear combinations of a set of random variables uniformly distributed on a simplex. The exact joint distribution of two general linear combinations with real constant coefficients is considered and the results found in the form of the joint probability density function. Application of the result is also illustrated. </p> / Thesis / Master of Science (MSc)

Many important decisions must be made without full information. For example, a woman may need to make a treatment decision regarding breast cancer without full knowledge of important uncertainties, such as how well she might respond to treatment. In the financial domain, in the wake of the housing crisis, the government may need to monitor the credit market and decide whether to intervene. A key input in this case would be a model to describe the chance that one person (or company) will default given that others have defaulted. However, such a model requires addressing the lack of knowledge regarding the correlation between groups or individuals. How to model and make decisions in cases where only partial information is available is a significant challenge. In the past, researchers have made arbitrary assumptions regarding the missing information. In this research, we developed a modeling procedure that can be used to analyze many possible scenarios subject to strict conditions. Specifically, we developed a new Monte Carlo simulation procedure to create a collection of joint probability distributions, all of which match whatever information we have. Using this collection of distributions, we analyzed the accuracy of different approximations such as maximum entropy or copula-models. In addition, we proposed several new approximations that outperform previous methods. The objective of this research is four-fold. First, provide a new framework for approximation models. In particular, we presented four new models to approximate joint probability distributions based on geometric attributes and compared their performance to existing methods. Second, develop a new joint distribution simulation procedure (JDSIM) to sample joint distributions from the set of all possible distributions that match available information. This procedure can then be applied to different scenarios to analyze the sensitivity of a decision or to test the accuracy of an approximation method. Third, test the accuracy of seven approximation methods under a variety of circumstances. Specifically, we addressed the following questions within the context of multivariate discrete distributions: Are there new approximations that should be considered? Which approximation is the most accurate, according to different measures? How accurate are the approximations as the number of random variables increases? How accurate are they as we change the underlying dependence structure? How does accuracy improve as we add lower-order assessments? What are the implications of these findings for decision analysis practice and research? While the above questions are easy to pose, they are challenging to answer. For Decision Analysis, the answers open a new avenue to address partial information, which bing us to the last contribution. Fourth, propose a new approach to decision making with partial information. The exploration of old and new approximations and the capability of creating large collections of joint distributions that match expert assessments provide new tools that extend the field of decision analysis. In particular, we presented two sample cases that illustrate the scope of this work and its impact on uncertain decision making. / text

The dissertation focuses on the application of entropy theory in hydrologic analysis and simulation, namely, rainfall analysis, streamflow simulation and drought analysis. The extreme value distribution has been employed for modeling extreme rainfall values. Based on the analysis of changes in the frequency distribution of annual rainfall maxima in Texas with the changes in duration, climate zone and distance from the sea, an entropy-based distribution is proposed as an alternative distribution for modeling extreme rainfall values. The performance of the entropy based distribution is validated by comparing with the commonly used generalized extreme value (GEV) distribution based on synthetic and observed data and is shown to be preferable for extreme rainfall values with high skewness. An entropy based method is proposed for single-site monthly streamflow simulation. An entropy-copula method is also proposed to simplify the entropy based method and preserve the inter-annual dependence of monthly streamflow. Both methods are shown to preserve statistics, such as mean, standard deviation, skenwess and lag-one correlation, well for monthly streamflow in the Colorado River basin. The entropy and entropy-copula methods are also extended for multi-site annual streamflow simulation at four stations in the Colorado River basin. Simulation results show that both methods preserve the mean, standard deviation and skewness equally well but differ in preserving the dependence structure (e.g., Pearson linear correlation). An entropy based method is proposed for constructing the joint distribution of drought variables with different marginal distributions and is applied for drought analysis based on monthly streamflow of Brazos River at Waco, Texas. Coupling the entropy theory and copula theory, an entropy-copula method is also proposed for constructing the joint distribution for drought analysis, which is illustrated with a case study based on the Parmer drought severity index (PDSI) data in Climate Division 5 in Texas.

Error estimation must be used to find the accuracy of a designed classifier, an issue that is critical in biomarker discovery for disease diagnosis and prognosis in genomics and proteomics. This dissertation is concerned with the analytical formulation of the joint distribution of the true error of misclassification and two of its commonly used estimators, resubstitution and leave-one-out, as well as their marginal and mixed moments, in the context of the Linear Discriminant Analysis (LDA) classification rule. In the first part of this dissertation, we obtain the joint sampling distribution of the actual and estimated errors under a general parametric Gaussian assumption. Exact results are provided in the univariate case and an accurate approximation is obtained in the multivariate case. We show how these results can be applied in the computation of conditional bounds and the regression of the actual error, given the observed error estimate. In practice the unknown parameters of the Gaussian distributions, which figure in the expressions, are not known and need to be estimated. Using the usual maximum-likelihood estimates for such parameters and plugging them into the theoretical exact expressions provides a sample-based approximation to the joint distribution, and also sample-based methods to estimate upper conditional bounds. In the second part of this dissertation, exact analytical expressions for the bias, variance, and Root Mean Square (RMS) for the resubstitution and leave-one-out error estimators in the univariate Gaussian model are derived. All probabilistic characteristics of an error estimator are given by the knowledge of its joint distribution with the true error. Partial information is contained in their mixed moments, in particular, their second mixed moment. Marginal information regarding an error estimator is contained in its marginal moments, in particular, its mean and variance. Since we are interested in estimator accuracy and wish to use the RMS to measure that accuracy, we desire knowledge of the second-order moments, marginal and mixed, with the true error. In the multivariate case, using the double asymptotic approach with the assumption of knowing the common covariance matrix of the Gaussian model, analytical expressions for the first moments, second moments, and mixed moment with the actual error for the resubstitution and leave-one-out error estimators are derived. The results provide accurate small sample approximations and this is demonstrated in the present situation via numerical comparisons. Application of the results is discussed in the context of genomics.

As the cities are becoming more crowded, the urban distribution has become a major logistical challenge. Some commonproblems in the inner cities arelimited accessibility and idling. Many of Sweden's municipalities are above the limit values for PM10limit set by the EU.Sundsvall is one of these municipalities and has the vision to create a sustainable city and region. Sundsvall LogistikparkAB is part of an EU project, SMARTSET, which is collaborationbetween several cities in Europe. SMARTSETSundsvall is about to develop a market-based sustainable model for city logistics. Itsobjective has been to compare and assess the potential of two business models forcoordinated goods distribution, with a micro terminal, in to Sundsvall Stenstad,with the current status. To achieve the purpose of the survey is the mapping of the current situation made with the help of interviews with shippers and identification of criteria for a multi-criteria analysis. In the result, the interviews compiled and formed the basis of the data used in the MCA. To get the values of the different criteria, the calculations are made and then the dependency between the different criteria illustrated in the scatter plots. It has been found that business model 1 and 2 is better than the current statusofthe majority of the criteria. The study's main purpose has been fulfilled when the MCA has shown that both the business modelsare profitable for shippers and reduces emissions in Sundsvall. The study has also shown that there are more factors than just the profitability and environmental impacts that affect an imple-mentation of any of these business models. To further develop this survey is required that more shippers interviewed and that the stores specifications are compiled. To get a project with a longer lifethan the contribution period required financing method identified. / I och med att städerna blir allt mer befolkade har stadsdistributionen blivit en stor logistisk utmaning. Några vanliga problem i innerstäderna är begränsad framkomlighet och tomgångskörning. Många av Sveriges kommuner ligger överde gränsvärden för PM10som bestämts av EU. Sundsvall är en av dessa kommuner och har som vision att skapa en hållbar stad och region. Sundsvall Logiskpark AB ingår i ett EU-projekt, SMARTSET, som är ett samarbete mellan flera städer i Europa. SMARTSET Sundsvall handlar om att fram en marknadsmässigt hållbar modell för citylogistik. Undersökningens syfte äratt jämförasamt bedöma potentialen hostvå logistikmodellerför samordnad varudistribution,med en mikroterminal,in till Sundsvalls Stenstadmed nuläget.För att uppnå undersökningens syfte har kartläggning av nuläget gjorts med hjälp av interjuver med speditörer samt identifiering av kriterier för en multikriterieanalys. I resultatet har intervjuerna sammanställts och legat till grund för dedata som har använts i multikriterieanalysen. För att få värden på de olika kriterierna har beräkningar gjorts och sedan har beroendet mellan de olika kriterierna illustrerats i scatter plots. Det har visat sig att logistikmodell1 och 2 är bättre än nuläget på majoriteten av kriterierna.Undersökningens huvudsakliga syfte har uppfyllts då multikriterieanalysen harvisat att båda logistikmodellernaär lönsam för speditörerna samt minskar utsläppen i Sundsvall. Undersökningen har också visat att det är fler faktorer än bara lönsamhet och miljöpåverkan som påverkar en implementation av någon av dessa logistikmodeller.För att vidareutveckla denna undersökning krävs det att fler speditörer intervjuas samt att butikernas kravspecifikation sammanställs. För att få ett projekt med längre livslängd än bidragsperioden behövs finansieringssätt identifieras.

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給定聯合分配，可以容易地導出對應的條件分配。反之，給定條件分配的資訊，是否能導出對應的聯合分配呢？例如根據O. Paul et al.（1963，1968）對造成心血管疾病因素之追蹤研究，可得出咖啡量、吸菸量及是否有心血管疾病三者間的條件機率模型資料，是否能找到對應的聯合機率模型，以便可以更深入地研究三者之關係，是一個重要的議題。在選定參考點下，Chen（2010）提出以勝算比法找條件密度函數相容的充要條件，以及在相容性成立時，如何求得聯合分配。在二維中，當兩正值條件機率矩陣不相容時，郭俊佑（2013）以幾何平均法修正勝算比矩陣，並導出近似聯合分配，同時利用幾何平均法之特性，提出最佳參考點之選擇法則。本研究以二維的勝算比法為基礎，探討三維離散的相容性問題，獲得下列幾項結果：一、證明了三個三維條件機率矩陣相容的充要條件就是兩兩相容。二、當三維條件機率矩陣不相容時，利用幾何平均法導出近似聯合分配。三、利用兩兩相容的充要條件，導出三維條件機率矩陣相容的充要條件，並證明該充要條件與Chen的結果一致。四、在幾何平均法下，提出最少點法，有效率地找出最佳參考點，以產生總誤差最小的近似聯合分配。五、設計出程式檢驗三維條件機率矩陣是否相容，並找出最佳參考點，同時比較最少點法與窮舉法之間效率的差異。 / Given a joint distribution, we can easily derive the corresponding fully conditional distributions. Conversely, given fully conditional distributions, can we find out the corresponding joint distribution? For example, according to a longitudinal study of coronary heart disease risk factors by O. Paul et al. (1963, 1968), we obtain conditional probability model data among coffee intake, the number of cigarettes smoked and whether he/she has coronary heart disease or not. Whether we can find out the corresponding joint distribution is an important issue as the joint distribution may be used to do further analyses. Chen (2010) used odds ratio method to find a necessary and sufficient condition for their compatibility and also gave the corresponding joint distribution for compatible situations. When two positive discrete conditional distributions in two dimensions are incompatible, Kuo (2013) used a geometric mean method to modify odds ratio matrices and derived an approximate joint distribution. Kuo also provided a rule to find the best reference point when the geometric mean method is used. In this research, based on odds ratio method in two dimensions, we discuss their compatibility problems and obtain the following results on three-dimensional discrete cases. Firstly, we prove that a necessary and sufficient condition for the compatibility of three conditional probability matrices in three dimensions is pairwise compatible. Secondly, we extend Kuo’s method on two-dimensional cases to derive three-dimensional approximate joint distributions for incompatible situations. Thirdly, we derive a necessary and sufficient condition for the compatibility of three conditional probability matrices in three dimensions in terms of pairwise compatibility and also prove that this condition is consistent with Chen’s results. Fourthly, we provide a minimum-points method to efficiently find the best reference point and yield an approximate joint distribution such that total error is the smallest. Fifthly, we design a computer program to run three-dimensional discrete conditional probability matrices problems for compatibility and also compare the efficiency between minimum-points method and exhausting method.

Purpose: Continously growing cities and increasing population sets new demands inthe building industry. Contractors are facing new challenges regarding constructionsite-logistics in densely populated areas where the use of large transports is limited.Many of the actors involved in construction logistics don’t seem to recognize thepotential in early-stage cooperation and sharing of information. Large improvementsare available in the construction logistics which seem to lack structure and goodplanning. Large demands are going to be put on contractor corporations where smartlogisticsolutions can be a deciding factor for competition and thus, important toidentify. Method: In order to supply answer to this thesis question formulation a case study wasperformed at Skanska AB Sverige. The methods used are literature-review, documentanalysisand interviews. The literature-review helped gathering knowledge in the fieldas well as providing a framework in chapter four. The document-analysis where usedto identify the guidelines and tools that Skanska AB Sverige uses today. The interviewstook place at Skanska AB Sverige where employees on influential positions suppliedtheir view on the construction logistics. Findings: The thesis examined four problem areas within construction logistics:material flow issues, internal communication, external communication and complexity.The study showed that the largest problem was material flow issues in citylogisticswhere it was shown that the problems included was uncertainty of time of delivery,transport issues, uncertainty of space requirement and space for waste storage. Theproblems may be solved through thorough planning and inclusion of more actors in theplanning. An increased use of join-distribution may also prove beneficial to the projectsas more space and time is created. Implications: The thesis conclude that information of solutions exist today but that itmust find a way to reach the project in a better way. Problems regarding the externalcommunication with subcontractors was found in the interviews which could beprevented through a better understanding between different actors through inclusion inthe early stage planning and continuous involvement. As a recommendation to furtherresearch we recommend to examine the profitability of joint-distribution and supplychain management. Limitations: The the thesis is limited to construction enterprise where logistics isexamined where it is in direct contact with the construction site. The supply chain ofsubcontractors and their handling of materials is not studied in this report. / Syfte: Ständigt växande stadskärnor och ökad befolkning ställer nya krav påbyggmarknaden. Entreprenörer ställs inför nya utmaningar gällandearbetsplatslogistiken på trånga innerstadprojekt där stora transporter är begränsade.Många inblandade som berörs av logistikfrågor ser inte den stora potentialen isamarbete och utbyte av information. Stora förbättringsmöjligheter finns gällandelogistiken inom byggbranschen vilken saknar struktur och bra planering. Stora kravkommer ställas på entreprenadföretag där smarta logistiklösningar kan vara enavgörande konkurrensfaktor därför är det viktigt att identifiera dessa möjligheter. Metod: För att besvara rapporten frågeställningar utgick examensarbetet från enfallstudie. Metoderna som användes för datainsamling var litteraturstudie,dokumentanalys och intervjuer. Litteraturstudier hjälpte med att införskaffa kunskap iämnet samt gav grund i empirikapitlet. Därefter gjordes dokumentanalys för attidentifiera Skanskas Sverige AB riktlinjer och hjälpmedel som finns idag.Avslutningsvis intervjuades personer på företaget för att se deras syn kringlogistiksituationen. Resultat: Rapporten undersökte fyra problematiska områden inom bygglogistiken:materialflöde, intern och extern kommunikation samt komplexitet. Studien visade attproblem kring materialflöde var största bekymmer utifrån logistiksynpunkt. Inom detområdet fastställdes problematik kring leveransreliabilitet, transportproblem påbyggarbetsplats, osäkerhet om utrymmeskrav och hantering av avfall. Problemen kanlösas genom noggrannare planering och inkludering av flera aktörer i planeringen.Ökning av samlastning tycks också gynna projekten där mer plats och tid skapas.Medarbetarnas okunskap gällande samlastning skapar rädslan för extra kostnaderistället för att se möjligheter. Konsekvenser: Rapportens slutsats är att information kring lösningar som finnstillgängliga i dagsläge måste nå projekten på ett bättre sätt. Problem kring den externakommunikationen med underentreprenörer konstaterades. Bättre förståelse förvarandras arbetssätt kan skapas genom täta avstämningar och bättre involvering. Somförslag till vidare studier kan lönsamhet av samlastning respektive supply chainmanagement för olika typer entreprenader undersökas. Begränsningar: Studien är avgränsad till ett byggentreprenadföretag på husbyggnaddär logistiken undersöks endast i direktkontakt med byggarbetsplats.Underentreprenörernas leveranskedja och hantering av material beaktas inte i dettaarbete.

Considere um meio caracterizado por $N$ pontos cujas coordenadas são geradas aleatoriamente de maneira uniforme nas arestas unitárias de um hipercubo $d$-dimensional. Um caminhante parte de cada ponto deste meio desordenado e se movimenta obedecendo à regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos $mu$ passos. Este processo foi denominado de caminhada determinista do turista. Cada trajetória gerada por esta dinâmica possui uma parte inicial não-periódica de $t$ passos (transiente) e uma parte final periódica de $p$ passos (atrator). As probabilidades de vizinhança são expressas através da fórmula de Cox, que é parametrizada pela função beta incompleta normalizada $I_d = I_{1/4}[1/2,(d+1)/2]$. Enfati-zamos aqui que a distribuição relevante é $S_{mu,d}^{(N)}(t,p)$, a distribuição conjunta de $t$ e $p$, que tem como casos particulares as distribuições marginais, previamente estudadas. O objetivo deste estudo é obter analiticamente estas distribuições para a caminhada determinista do turista sem memória no espaço euclideano, no modelo de distâncias aleatórias (que corresponde ao limite $d ightarrow infty$) e no modelo de mapeamento aleatório (que é um caso limite das redes de Kauffman). As distribuições analíticas obtidas foram validadas através de experimentos numéricos. A distribuição conjunta de tempos de transiente e período de atratores, no limite termodinâmico para uma dimensionalidade arbitrária vale: $S_{1,d}^{(infty)}(t,p) = [Gamma(1+I_d^{-1}) cdot (t+I_d^{-1})/Gamma(t+p+I_d^{-1})] cdot delta_{p,2}$, onde $t=0,1,2,ldots,infty$; $Gamma(z)$ é a função gama e $delta_{i,j}$ é o delta de Kronecker. A caminhada determinista do turista sem memória no modelo de mapeamento aleatório produz uma distribuição de períodos não-trivial ($S_{0,rm}^{(N)}(p) propto p^{-1}$), que é obtida de $S_{0,rm}^{(N)}(t,p) = Gamma(N)/{Gamma[N+1-(t+p)]N^{t+p}}$, onde enfatizamos que o número de pontos explorados $n_e=t+p$ é a grandeza fundamental nos problemas considerados. / Consider a medium characterized by $N$ points whose coordinates are randomly generated by a uniform distribution along the unitary edges of a $d$-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go the nearest point which has not been visited in the preceding $mu$ steps. This process has been called the deterministic tourist walk. Each trajectory generated by this dynamics has an initial non-periodic part of $t$ steps (transient) and a final periodic part of $p$ steps (attractor). The neighborhood probabilities are given by the Cox formula, which is parameterized by the normalized incomplete beta function $I_d = I_{1/4}[1/2,(d+1)/2]$. Here we stress that the relevant distribution is the joint $t$ and $p$ distribution $S_{mu,d}^{(N)}(t,p)$, which has as particular cases, the marginal distributions previously studied. The objective of this study is to obtain analytically these distributions for the memoryless deterministic tourist walk in the euclidean space, random link model (which corresponds to $d ightarrow infty$ limit) and random map model (which is a limiting case of the Kauffman model). The obtained distributions have been validated by numerical experiments. The joint transient time and attractor period distribution in the thermodynamic limit for an arbitrary dimensionality is: $S_{1,d}^{(infty)}(t,p) = [Gamma(1+I_d^{-1}) cdot (t+I_d^{-1})/Gamma(t+p+I_d^{-1})] cdot delta_{p,2}$, where $t=0,1,2,ldots,infty$; $Gamma(z)$ is the gamma function and $delta_{i,j}$ is the Kronecker's delta. The memoryless deterministic tourist walk in the random map leads to a non-trivial cycle distribution ($S_{0,rm}^{(N)}(p) propto p^{-1}$), which is obtained from $S_{0,rm}^{(N)}(t,p) = Gamma(N)/{Gamma[N+1-(t+p)]N^{t+p}}$, where we stress that the number of explored points $n_e=t+p$ is the fundamental quantity in the considered problems.

Considere um meio desordenado constituído por $N$ pontos cujas coordenadas são geradas aleatoriamente de maneira uniforme e independente nas arestas unitárias de um hipercubo $d$-dimensional. As probabilidades de vizinhança entre os pares de pontos deste meio são expressas através da fórmula de Cox. Um caminhante parte de um dado ponto deste meio desordenado e se movimenta obedecendo à regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos $\\mu$ passos. Este processo foi denominado de caminhada determinista do turista. Cada trajetória gerada por esta dinâmica possui uma parte inicial não-periódica de $t$ passos (transiente) e uma parte final periódica de $p$ passos (atrator). Neste trabalho, obtemos analiticamente algumas distribuições estatísticas para a caminhada determinista do turista com memória $\\mu$ arbitrária em sistemas unidimensionais e com memória $\\mu=2$ no modelo Random Link (que corresponde ao limite $d ightarrow 1$). Estes resultados nos permitiram compreender o papel da memória no comportamento exploratório do turista e explicar a equivalência não-trivial entre o modelo Random Link e o modelo Random Map (que é um caso limite das redes de Kauffman). Enfatizamos que o número de pontos explorados pelo turista é a grandeza fundamental nos problemas considerados. As distribuições analíticas obtidas foram validadas através de experimentos numéricos. Também obtivemos uma dedução alternativa para a fórmula de Cox, apresentando os resultados finais em termos de distribuições estatísticas elementares. / Consider a medium characterized by $N$ points whose coordinates are randomly and independently generated by a uniform distribution along the unitary edges of a $d$-dimensional hypercube. The neighborhood probabilities between any pair of points in this medium are given by the Cox formula. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go the nearest point which has not been visited in the preceding $\\mu$ steps. This process has been called the deterministic tourist walk. Each trajectory generated by this dynamics has an initial non-periodic part of $t$ steps (transient) and a final periodic part of $p$ steps (attractor). In this work, we obtain analytically some statistical distributions for the deterministic tourist walk with arbitrary memory $\\mu$ in one-dimensional systems and with memory $\\mu=2$ in the random link model (which corresponds to $d ightarrow 1$ limit). These results enable us to understand the main role played by the memory on the tourist\'s exploratory behavior and explain the non-trivial equivalence between the random link model and the random map model (which is a limiting case of the Kauffman model). We stress that the number of explored points is the fundamental quantity in the considered problems. The obtained distributions have been validated by numerical experiments. We also obtain an alternative derivation for the Cox formula, writing the final results in terms of known statistical distributions.