A simple permutation test for clusteredness

Hierarchical clustering is a popular method for finding structure in multivariate data,resulting in a binary tree constructed on the particular objects of the study, usually samplingunits. The user faces the decision where to cut the binary tree in order to determine the numberof clusters to interpret and there are various ad hoc rules for arriving at a decision. A simplepermutation test is presented that diagnoses whether non-random levels of clustering are presentin the set of objects and, if so, indicates the specific level at which the tree can be cut. The test isvalidated against random matrices to verify the type I error probability and a power study isperformed on data sets with known clusteredness to study the type II error.

Méthodologie intégrée de microgravimétrie en milieu urbain : test d'application au génie civil sur le métro M2 de Lausanne

RESUME Dès le printemps 2004, la construction d'une 2ème ligne de métro est entreprise dans la ville de Lausanne en Suisse. En reliant Ouchy, au bord du lac Léman (alt. 373 m) à Epalinges (alt. 711 m), le nouveau métro "M2" traversera dès 2008 l'agglomération lausannoise du Sud au Nord sur une distance de 6 km. Depuis l'avant-projet, en 1999, une grande quantité de données géologiques a été récolté et de nombreux forages exécutés sur le site. Ceci nous a donné une occasion unique d'entreprendre une étude de microgravimétrique urbaine de détail. Le mode de creusement du tunnel dépend fortement des matériaux à excaver et il est classiquement du domaine du géologue, avec ses connaissances de la géologie régionale et de la stratigraphie des forages, de fournir à l'ingénieur un modèle géologique. Ce modèle indiquera dans ce cas l'épaisseur des terrains meubles qui recouvrent le soubassement rocheux. La représentativité spatiale d'une information très localisée, comme celle d'un forage, est d'autant plus compliquée que le détail recherché est petit. C'est à ce moment là que la prospection géophysique, plus spécialement gravimétrique, peut apporter des informations complémentaires déterminantes pour régionaliser les données ponctuelles des forages. La microgravimétrie en milieu urbain implique de corriger avec soin les perturbations gravifiques sur la mesure de la pesanteur dues aux effets de la topographie, des bâtiments et des caves afin d'isoler l'effet gravifique dû exclusivement à l'épaisseur du remplissage des terrains meubles. Tenant compte de l'intensité des corrections topographiques en milieu urbain, nous avons donné une grande importance aux sous-sols, leurs effets gravifiques pouvant atteindre l'ordre du dixième de mGal. Nous avons donc intégré ces corrections celle de topographie et traité les effets des bâtiments de manière indépendante. Nous avons inclus dans le modèle numérique de terrain (MNT) la chaussée et les sous-sols afin de construire un modèle numérique de terrain urbain. Nous utiliserons un nouvel acronyme « MNTU »pour décrire ce modèle. Nous proposons d'établir des cartes de corrections topographiques préalables, basées sur les données à disposition fournies par le cadastre en faisant des hypothèses sur la profondeur des sous-sols et la hauteur des bâtiments. Les deux zones de test choisies sont caractéristiques des différents types d'urbanisation présente à Lausanne et se révèlent par conséquent très intéressantes pour élaborer une méthodologie globale de la microgravimétrie urbaine. Le but était d'évaluer l'épaisseur du remplissage morainique sur un fond rocheux molassique se situant à une profondeur variable de quelques mètres à une trentaine de mètres et d'en établir une coupe dans l'axe du futur tracé du métro. Les résultats des modélisations se sont révélés très convaincants en détectant des zones qui diffèrent sensiblement du modèle géologique d'avant projet. Nous avons également démontré que l'application de cette méthode géophysique, non destructive, est à même de limiter le nombre de sondages mécaniques lors de l'avant-projet et du projet définitif, ce qui peut limiter à la fois les coûts et le dérangement engendré par ces travaux de surface. L'adaptabilité de la technique gravimétrique permet d'intervenir dans toutes les différentes phases d'un projet de génie civil comme celui de la construction d'un métro en souterrain. KURZFASSUNG Seit dem Frühling 2004 ist in der Stadt Lausanne (Schweiz) die neue U-Bahn "M2" in Konstruktion. Diese soll auf 6 km Länge die Lausanner Agglomeration von Süd nach Nord durchqueren. Die dem Projekt zu Grunde liegende technische Planung sieht vor, daß die Bahnlinie hauptsächlich in der Molasse angesiedelt sein wird. Seit dem Vorentwurf (1999) ist eine große Anzahl geologischer Angaben gesammelt worden. Daraus ergab sich die einmalige Gelegenheit, die Informationen aus den damit verbundenen zahlreichen Bohrungen zu einer detaillierten mikrogravimetrischen Studie der Stadt Lausanne zu erweitern und zu vervollständigen. Das Ziel bestand darin, die Mächtigkeit der die Molasseüberdeckenden Moräneablagerung abzuschätzen, um eine entsprechendes geologisches Profile entlang der künftigen Bahnlinie zu erstellen. Weiterhin sollte gezeigt werden, daß die Anwendung dieser nicht-invasiven geophysikalischen Methode es ermöglicht, die Anzahl der benötigten Bohrungen sowohl in der Pilotphase wie auch im endgültigen Projekt zu reduzieren, was zu wesentlichen finanziellen Einsparungen in der Ausführung des Werkes beitragen würde. Die beiden in dieser Studie bearbeiteten Testzonen befinden sich im Nordteil und im Stadtzentrum von Lausanne und sind durch eine unterschiedliche Urbanisierung charakterisiert. Das anstehende Gestein liegt in verschiedenen Tiefen: von einigen Metern bis zu etwa dreißig Metern. Diese Zonen weisen alle Schwierigkeiten einer urbanen Bebauung mit hoher Verkehrsdichte auf und waren daher massgebend bei der Ausarbeitung einer globalen mikrogravimetrischen Methodologie für die Stadt Lausanne. Die so entwickelte Technik ermöglicht, die störenden Auswirkungen der Topographie, der Gebäude, der Keller und der Öffentlichen Infrastrukturen sorgfältig zu korrigieren, um so die ausschließlich auf die Mächtigkeit des Lockergesteins zurückzuführenden Effekte zu isolieren. In Bezug auf die Intensität der Auswirkungen der topographischen Korrekturen im Stadtgebiet wurde den Untergeschossen eine besonders grosse Bedeutung zugemessen da die entsprechenden Schwerkrafteffekte eine Grösse von rund einem Zehntel mGal erreichen können. Wir schlagen deshalb vor, vorläufige Karten der topographischen Korrekturen zu erstellen. Diese Korrekturen basieren auf den uns vom Katasterplan gelieferten Daten und einigen Hypothesen bezüglich der Tiefe der Untergeschosse und der Höhe der Gebäude. Die Verfügbarkeit einer derartigen Karte vor der eigentlichen gravimetrischen Messkampagne würde uns erlauben, die Position der Meßstationen besser zu wählen. Wir sahen zudem, daß ein entsprechenden a priori Filter benutzt werden kann, wenn die Form und die Intensität der Anomalie offensichtlich dem entsprechenden Gebäude zugeordnet werden können. Diese Strategie muß jedoch mit Vorsicht angewandt werden, denn falls weitere Anomalien dazukommen, können bedeutende Verschiebungen durch Übèrlagerungen der Schwerewirkung verschiedener Strukturen entstehen. Die Ergebnisse der Modellierung haben sich als sehr überzeugend erwiesen, da sie im Voraus unbekannte sensible Zonen korrekt identifiziert haben. Die Anwendbarkeit der in dieser Arbeit entwickelten gravimetrischen Technik ermöglicht es, während allen Phasen eines Grossbauprojekts, wie zum Beispiel bei der Konstruktion einer unterirdischen U-Bahn, einzugreifen. ABSTRACT Since Spring of 2004 a new metro line has been under construction in the city of Lausanne in Switzerland. The new line, the M2, will be 6 km long and will traverse the city from south to north. The civil engineering project determined that the line would be located primarily in the Molasse. Since the preparatory project in 1999, a great quantity of geological data has been collected, and the many drillings made on the site have proved to be a unique opportunity to undertake a study of urban microgravimetry. The goal was to evaluate the thickness of the morainic filling over the molassic bedrock, and to establish a section along the axis of the future line. It then had to be shown that the application of this nondestructive geophysical method could reduce the number of mechanical surveys required both for a preparatory and a definitive project, which would lead to real savings in the realization of a civil engineering project. The two test zones chosen, one in the northern part of the city and one in the city centre, are characterised by various types of urbanisation. Bedrock is at a depth varying from a few metres to about thirty metres. These zones well exemplify the various difficulties encountered in an urban environment and are therefore very interesting for the development of an overall methodology of urban microgravimetry. Microgravimetry in an urban environment requires careful corrections for gravific disturbances due to the effects of topography, buildings, cellars, and the infrastructure of distribution networks, in order to isolate the gravific effect due exclusively to the thickness of loose soil filling. Bearing in mind the intensity of the topographic corrections in an urban environment, we gave particular importance to basements. Their gravific effects can reach the order of one tenth of one meal, and can influence above all the precision of the Bouguer anomaly. We propose to establish preliminary topographic correction charts based on data provided to us by the land register, by making assumptions on the depths of basements and the heights of buildings. Availability of this chart previous to a gravimetry campaign would enable us to choose optimum measuring sites. We have also seen that an a priori filter can be used when the form and the intensity of the anomaly correspond visually to the corresponding building. This strategy must be used with caution because if other anomalies are to be associated, important shifts can be generated by the superposition of the effects of different structures. The results of the model have proved to be very convincing in detecting previously unknown sensitive zones. The adaptability of the gravimetry technique allows for application in all phases of a civil engineering project such as the construction of an underground metro line. RIASSUNTO Dalla primavera 2004 una nuova linea metropolitana é in costruzione nella città di Losanna in Svizzera. La nuova metropolitana "M2" traverserà per la lunghezza di 6 km il centro urbano di Losanna da sud a nord. II progetto d'ingegneria civile prevedeva un tracciato situato essenzialmente nel fondo roccioso arenaceo terziario (molassa). Dalla redazione del progetto preliminare, avvenuta nel 1999, una grande quantità di dati geologici sono stati raccolti e sono stati eseguiti numerosi sondaggi. Questo sì é presentato come un'occasione unica per mettere a punto uno studio microgravimetrico in ambiente urbano con lo scopo di valutare lo spessore dei terreni sciolti di origine glaciale che ricoprono il fondo roccioso di molassa e di mettere in evidenza come l'applicazione di questo metodo geofisico non distruttivo possa limitare il numero di sondaggi meccanici nella fase di progetto preliminare ed esecutivo con conseguente reale risparmio economico nella realizzazione di una tale opera. Le due zone di test sono situate una nella zona nord e la seconda nel centro storico di Losanna e sono caratterizzate da stili architettonici differenti. II fondo roccioso é situato ad una profondità variabile da qualche metro ad una trentina. Queste due zone sembrano ben rappresentare tutte le difficoltà di un ambiente urbano e ben si prestano per elaborare una metodologia globale per la microgravimetria in ambiente urbano. L'applicazione di questa tecnica nell'ambiente suddetto implica la correzione attenta delle perturbazioni sulla misura dell'accelerazione gravitazionale, causate dalla topografia, gli edifici, le cantine e le infrastrutture dei sottoservizi, per ben isolare il segnale esclusivamente causato dallo spessore dei terreni sciolti. Tenuto conto, dell'intensità delle correzioni topografiche, abbiamo dato grande importanza alle cantine, poiché il loro effetto sulle misure può raggiungere il decimo di mGal. Proponiamo quindi di redigere una carta delle correzioni topografiche preliminare all'acquisizione, facendo delle ipotesi sulla profondità delle cantine e sull'altezza degli edifici, sulla base delle planimetrie catastali. L'analisi di questa carta permetterà di scegliere le posizioni più adatte per le stazioni gravimetriche. Abbiamo anche osservato che un filtro a priori, qualora la forma e l'intensità dell'anomalia fosse facilmente riconducibile in maniera visuale ad un edificio, possa essere efficace. Tuttavia questa strategia deve essere utilizzata con precauzione, poiché può introdurre uno scarto, qualora più anomalie, dovute a differenti strutture, si sovrappongano. I risultati delle modellizzazioni si sono rivelati convincenti, evidenziando zone sensibili non conosciute preventivamente. L'adattabilità della tecnica gravimetrica ha mostrato di poter intervenire in differenti fasi di un progetto di ingegneria civile, quale è quella di un'opera in sotterraneo.

Test of four colon cancer risk-scores in formalin fixed paraffin embedded microarray gene expression data.

BACKGROUND: Prognosis prediction for resected primary colon cancer is based on the T-stage Node Metastasis (TNM) staging system. We investigated if four well-documented gene expression risk scores can improve patient stratification. METHODS: Microarray-based versions of risk-scores were applied to a large independent cohort of 688 stage II/III tumors from the PETACC-3 trial. Prognostic value for relapse-free survival (RFS), survival after relapse (SAR), and overall survival (OS) was assessed by regression analysis. To assess improvement over a reference, prognostic model was assessed with the area under curve (AUC) of receiver operating characteristic (ROC) curves. All statistical tests were two-sided, except the AUC increase. RESULTS: All four risk scores (RSs) showed a statistically significant association (single-test, P < .0167) with OS or RFS in univariate models, but with HRs below 1.38 per interquartile range. Three scores were predictors of shorter RFS, one of shorter SAR. Each RS could only marginally improve an RFS or OS model with the known factors T-stage, N-stage, and microsatellite instability (MSI) status (AUC gains < 0.025 units). The pairwise interscore discordance was never high (maximal Spearman correlation = 0.563) A combined score showed a trend to higher prognostic value and higher AUC increase for OS (HR = 1.74, 95% confidence interval [CI] = 1.44 to 2.10, P < .001, AUC from 0.6918 to 0.7321) and RFS (HR = 1.56, 95% CI = 1.33 to 1.84, P < .001, AUC from 0.6723 to 0.6945) than any single score. CONCLUSIONS: The four tested gene expression-based risk scores provide prognostic information but contribute only marginally to improving models based on established risk factors. A combination of the risk scores might provide more robust information. Predictors of RFS and SAR might need to be different.

Inferring transport characteristics in a fractured rock aquifer by combining single-hole ground-penetrating radar reflection monitoring and tracer test data

Investigations of solute transport in fractured rock aquifers often rely on tracer test data acquired at a limited number of observation points. Such data do not, by themselves, allow detailed assessments of the spreading of the injected tracer plume. To better understand the transport behavior in a granitic aquifer, we combine tracer test data with single-hole ground-penetrating radar (GPR) reflection monitoring data. Five successful tracer tests were performed under various experimental conditions between two boreholes 6 m apart. For each experiment, saline tracer was injected into a previously identified packed-off transmissive fracture while repeatedly acquiring single-hole GPR reflection profiles together with electrical conductivity logs in the pumping borehole. By analyzing depth-migrated GPR difference images together with tracer breakthrough curves and associated simplified flow and transport modeling, we estimate (1) the number, the connectivity, and the geometry of fractures that contribute to tracer transport, (2) the velocity and the mass of tracer that was carried along each flow path, and (3) the effective transport parameters of the identified flow paths. We find a qualitative agreement when comparing the time evolution of GPR reflectivity strengths at strategic locations in the formation with those arising from simulated transport. The discrepancies are on the same order as those between observed and simulated breakthrough curves at the outflow locations. The rather subtle and repeatable GPR signals provide useful and complementary information to tracer test data acquired at the outflow locations and may help us to characterize transport phenomena in fractured rock aquifers.

Calibration of high-resolution geophysical data with tracer test measurements to improve hydrological predictions

Relationships between porosity and hydraulic conductivity tend to be strongly scale- and site-dependent and are thus very difficult to establish. As a result, hydraulic conductivity distributions inferred from geophysically derived porosity models must be calibrated using some measurement of aquifer response. This type of calibration is potentially very valuable as it may allow for transport predictions within the considered hydrological unit at locations where only geophysical measurements are available, thus reducing the number of well tests required and thereby the costs of management and remediation. Here, we explore this concept through a series of numerical experiments. Considering the case of porosity characterization in saturated heterogeneous aquifers using crosshole ground-penetrating radar and borehole porosity log data, we use tracer test measurements to calibrate a relationship between porosity and hydraulic conductivity that allows the best prediction of the observed hydrological behavior. To examine the validity and effectiveness of the obtained relationship, we examine its performance at alternate locations not used in the calibration procedure. Our results indicate that this methodology allows us to obtain remarkably reliable hydrological predictions throughout the considered hydrological unit based on the geophysical data only. This was also found to be the case when significant uncertainty was considered in the underlying relationship between porosity and hydraulic conductivity.

Level shifts in a panel data based unit root test : an application to the rate of unemployment

Several unit root tests in panel data have recently been proposed. The test developed by Harris and Tzavalis (1999 JoE) performs particularly well when the time dimension is moderate in relation to the cross-section dimension. However, in common with the traditional tests designed for the unidimensional case, it was found to perform poorly when there is a structural break in the time series under the alternative. Here we derive the asymptotic distribution of the test allowing for a shift in the mean, and assess the small sample performance. We apply this new test to show how the hypothesis of (perfect) hysteresis in Spanish unemployment is rejected in favour of the alternative of the natural unemployment rate, when the possibility of a change in the latter is considered.

Level shifts in a panel data based unit root test : an application to the rate of unemployment

Several unit root tests in panel data have recently been proposed. The test developed by Harris and Tzavalis (1999 JoE) performs particularly well when the time dimension is moderate in relation to the cross-section dimension. However, in common with the traditional tests designed for the unidimensional case, it was found to perform poorly when there is a structural break in the time series under the alternative. Here we derive the asymptotic distribution of the test allowing for a shift in the mean, and assess the small sample performance. We apply this new test to show how the hypothesis of (perfect) hysteresis in Spanish unemployment is rejected in favour of the alternative of the natural unemployment rate, when the possibility of a change in the latter is considered.

Stochastic inversion of tracer test and electrical geophysical data to estimate hydraulic conductivities.

Quantifying the spatial configuration of hydraulic conductivity (K) in heterogeneous geological environments is essential for accurate predictions of contaminant transport, but is difficult because of the inherent limitations in resolution and coverage associated with traditional hydrological measurements. To address this issue, we consider crosshole and surface-based electrical resistivity geophysical measurements, collected in time during a saline tracer experiment. We use a Bayesian Markov-chain-Monte-Carlo (McMC) methodology to jointly invert the dynamic resistivity data, together with borehole tracer concentration data, to generate multiple posterior realizations of K that are consistent with all available information. We do this within a coupled inversion framework, whereby the geophysical and hydrological forward models are linked through an uncertain relationship between electrical resistivity and concentration. To minimize computational expense, a facies-based subsurface parameterization is developed. The Bayesian-McMC methodology allows us to explore the potential benefits of including the geophysical data into the inverse problem by examining their effect on our ability to identify fast flowpaths in the subsurface, and their impact on hydrological prediction uncertainty. Using a complex, geostatistically generated, two-dimensional numerical example representative of a fluvial environment, we demonstrate that flow model calibration is improved and prediction error is decreased when the electrical resistivity data are included. The worth of the geophysical data is found to be greatest for long spatial correlation lengths of subsurface heterogeneity with respect to wellbore separation, where flow and transport are largely controlled by highly connected flowpaths.

Optimisation of data analysis method for a fluidized bed combustion test rig

Nowadays the used fuel variety in power boilers is widening and new boiler constructions and running models have to be developed. This research and development is done in small pilot plants where more faster analyse about the boiler mass and heat balance is needed to be able to find and do the right decisions already during the test run. The barrier on determining boiler balance during test runs is the long process of chemical analyses of collected input and outputmatter samples. The present work is concentrating on finding a way to determinethe boiler balance without chemical analyses and optimise the test rig to get the best possible accuracy for heat and mass balance of the boiler. The purpose of this work was to create an automatic boiler balance calculation method for 4 MW CFB/BFB pilot boiler of Kvaerner Pulping Oy located in Messukylä in Tampere. The calculation was created in the data management computer of pilot plants automation system. The calculation is made in Microsoft Excel environment, which gives a good base and functions for handling large databases and calculations without any delicate programming. The automation system in pilot plant was reconstructed und updated by Metso Automation Oy during year 2001 and the new system MetsoDNA has good data management properties, which is necessary for big calculations as boiler balance calculation. Two possible methods for calculating boiler balance during test run were found. Either the fuel flow is determined, which is usedto calculate the boiler's mass balance, or the unburned carbon loss is estimated and the mass balance of the boiler is calculated on the basis of boiler's heat balance. Both of the methods have their own weaknesses, so they were constructed parallel in the calculation and the decision of the used method was left to user. User also needs to define the used fuels and some solid mass flowsthat aren't measured automatically by the automation system. With sensitivity analysis was found that the most essential values for accurate boiler balance determination are flue gas oxygen content, the boiler's measured heat output and lower heating value of the fuel. The theoretical part of this work concentrates in the error management of these measurements and analyses and on measurement accuracy and boiler balance calculation in theory. The empirical part of this work concentrates on the creation of the balance calculation for the boiler in issue and on describing the work environment.

Selection of patients for myocardial perfusion scintigraphy based on fuzzy sets theory applied to clinical-epidemiological data and treadmill test results

Coronary artery disease (CAD) is a worldwide leading cause of death. The standard method for evaluating critical partial occlusions is coronary arteriography, a catheterization technique which is invasive, time consuming, and costly. There are noninvasive approaches for the early detection of CAD. The basis for the noninvasive diagnosis of CAD has been laid in a sequential analysis of the risk factors, and the results of the treadmill test and myocardial perfusion scintigraphy (MPS). Many investigators have demonstrated that the diagnostic applications of MPS are appropriate for patients who have an intermediate likelihood of disease. Although this information is useful, it is only partially utilized in clinical practice due to the difficulty to properly classify the patients. Since the seminal work of Lotfi Zadeh, fuzzy logic has been applied in numerous areas. In the present study, we proposed and tested a model to select patients for MPS based on fuzzy sets theory. A group of 1053 patients was used to develop the model and another group of 1045 patients was used to test it. Receiver operating characteristic curves were used to compare the performance of the fuzzy model against expert physician opinions, and showed that the performance of the fuzzy model was equal or superior to that of the physicians. Therefore, we conclude that the fuzzy model could be a useful tool to assist the general practitioner in the selection of patients for MPS.

The six-minute walk test and body weight-walk distance product in healthy Brazilian subjects

We assessed the 6-min walk distance (6MWD) and body weight x distance product (6MWw) in healthy Brazilian subjects and compared measured 6MWD with values predicted in five reference equations developed for other populations. Anthropometry, spirometry, reported physical activity, and two walk tests in a 30-m corridor were evaluated in 134 subjects (73 females, 13-84 years). Mean 6MWD and 6MWw were significantly greater in males than in females (622 ± 80 m, 46,322 ± 10,539 kg.m vs 551 ± 71 m, 36,356 ± 8,289 kg.m, P < 0.05). Four equations significantly overestimated measured 6MWD (range, 32 ± 71 to 137 ± 74 m; P < 0.001), and one significantly underestimated it (-36 ± 86 m; P < 0.001). 6MWD significantly correlated with age (r = -0.39), height (r = 0.44), body mass index (r = -0.24), and reported physical activity (r = 0.25). 6MWw significantly correlated with age (r = -0.21), height (r = 0.66) and reported physical activity (r = 0.25). The reference equation devised for walk distance was 6MWDm = 622.461 - (1.846 x Ageyears) + (61.503 x Gendermales = 1; females = 0); r2 = 0.300. In an additional group of 85 subjects prospectively studied, the difference between measured and the 6MWD predicted with the equation proposed here was not significant (-3 ± 68 m; P = 0.938). The measured 6MWD represented 99.6 ± 11.9% of the predicted value. We conclude that 6MWD and 6MWw variances were adequately explained by demographic and anthropometric attributes. This reference equation is probably most appropriate for evaluating the exercise capacity of Brazilian patients with chronic diseases.

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocks based on compositional statistics is introduced. It improves and generalizes the common joins-count statistics known from map analysis in geographic information systems. Assigning phases independently to objects in RD is modelled by a single-trial multinomial random function Z(x), where the probabilities of phases add to one and are explicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistencies of the tests based on the conventional joins{count statistics and their possibly contradictory interpretations are avoided. In practical applications we assume that the probabilities of phases do not depend on the location but are identical everywhere in the domain of de nition. Thus, the model involves the sum of r independent identical multinomial distributed 1-trial random variables which is an r-trial multinomial distributed random variable. The probabilities of the distribution of the r counts can be considered as a composition in the Q-part simplex SQ. They span the so called Hardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This is a generalisation of the well-known Hardy-Weinberg law of genetics. If the assignment of phases accounts for some kind of spatial dependence, then the r-trial probabilities do not remain on H. This suggests the use of the Aitchison distance between observed probabilities to H to test dependence. Moreover, when there is a spatial uctuation of the multinomial probabilities, the observed r-trial probabilities move on H. This shift can be used as to check for these uctuations. A practical procedure and an algorithm to perform the test have been developed. Some cases applied to simulated and real data are presented. Key words: Spatial distribution of crystals in rocks, spatial distribution of phases, joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinberg manifold, Aitchison geometry

Coherent forecasting of multiple-decrement life tables: a test using Japanese cause of death data

Planners in public and private institutions would like coherent forecasts of the components of age-specic mortality, such as causes of death. This has been di cult to achieve because the relative values of the forecast components often fail to behave in a way that is coherent with historical experience. In addition, when the group forecasts are combined the result is often incompatible with an all-groups forecast. It has been shown that cause-specic mortality forecasts are pessimistic when compared with all-cause forecasts (Wilmoth, 1995). This paper abandons the conventional approach of using log mortality rates and forecasts the density of deaths in the life table. Since these values obey a unit sum constraint for both conventional single-decrement life tables (only one absorbing state) and multiple-decrement tables (more than one absorbing state), they are intrinsically relative rather than absolute values across decrements as well as ages. Using the methods of Compositional Data Analysis pioneered by Aitchison (1986), death densities are transformed into the real space so that the full range of multivariate statistics can be applied, then back-transformed to positive values so that the unit sum constraint is honoured. The structure of the best-known, single-decrement mortality-rate forecasting model, devised by Lee and Carter (1992), is expressed in compositional form and the results from the two models are compared. The compositional model is extended to a multiple-decrement form and used to forecast mortality by cause of death for Japan