Compositional hypotheses of subcompositional stability and specific perturbation change and their testing

In standard multivariate statistical analysis common hypotheses of interest concern changes in mean vectors and subvectors. In compositional data analysis it is now well established that compositional change is most readily described in terms of the simplicial operation of perturbation and that subcompositions replace the marginal concept of subvectors. To motivate the statistical developments of this paper we present two challenging compositional problems from food production processes.Against this background the relevance of perturbations and subcompositions can beclearly seen. Moreover we can identify a number of hypotheses of interest involvingthe specification of particular perturbations or differences between perturbations and also hypotheses of subcompositional stability. We identify the two problems as being the counterpart of the analysis of paired comparison or split plot experiments and of separate sample comparative experiments in the jargon of standard multivariate analysis. We then develop appropriate estimation and testing procedures for a complete lattice of relevant compositional hypotheses

Alternative ways to estimate change points in multinomial sequences. An application to an authorship attribution problem

The statistical analysis of literary style is the part of stylometry that compares measurable characteristicsin a text that are rarely controlled by the author, with those in other texts. When thegoal is to settle authorship questions, these characteristics should relate to the author’s style andnot to the genre, epoch or editor, and they should be such that their variation between authors islarger than the variation within comparable texts from the same author.For an overview of the literature on stylometry and some of the techniques involved, see for exampleMosteller and Wallace (1964, 82), Herdan (1964), Morton (1978), Holmes (1985), Oakes (1998) orLebart, Salem and Berry (1998).Tirant lo Blanc, a chivalry book, is the main work in catalan literature and it was hailed to be“the best book of its kind in the world” by Cervantes in Don Quixote. Considered by writterslike Vargas Llosa or Damaso Alonso to be the first modern novel in Europe, it has been translatedseveral times into Spanish, Italian and French, with modern English translations by Rosenthal(1996) and La Fontaine (1993). The main body of this book was written between 1460 and 1465,but it was not printed until 1490.There is an intense and long lasting debate around its authorship sprouting from its first edition,where its introduction states that the whole book is the work of Martorell (1413?-1468), while atthe end it is stated that the last one fourth of the book is by Galba (?-1490), after the death ofMartorell. Some of the authors that support the theory of single authorship are Riquer (1990),Chiner (1993) and Badia (1993), while some of those supporting the double authorship are Riquer(1947), Coromines (1956) and Ferrando (1995). For an overview of this debate, see Riquer (1990).Neither of the two candidate authors left any text comparable to the one under study, and thereforediscriminant analysis can not be used to help classify chapters by author. By using sample textsencompassing about ten percent of the book, and looking at word length and at the use of 44conjunctions, prepositions and articles, Ginebra and Cabos (1998) detect heterogeneities that mightindicate the existence of two authors. By analyzing the diversity of the vocabulary, Riba andGinebra (2000) estimates that stylistic boundary to be near chapter 383.Following the lead of the extensive literature, this paper looks into word length, the use of the mostfrequent words and into the use of vowels in each chapter of the book. Given that the featuresselected are categorical, that leads to three contingency tables of ordered rows and therefore tothree sequences of multinomial observations.Section 2 explores these sequences graphically, observing a clear shift in their distribution. Section 3describes the problem of the estimation of a suden change-point in those sequences, in the followingsections we propose various ways to estimate change-points in multinomial sequences; the methodin section 4 involves fitting models for polytomous data, the one in Section 5 fits gamma modelsonto the sequence of Chi-square distances between each row profiles and the average profile, theone in Section 6 fits models onto the sequence of values taken by the first component of thecorrespondence analysis as well as onto sequences of other summary measures like the averageword length. In Section 7 we fit models onto the marginal binomial sequences to identify thefeatures that distinguish the chapters before and after that boundary. Most methods rely heavilyon the use of generalized linear models

Inflammatory Markers and Incident Heart Failure Risk in Older Adults: The Health, Aging, and Body Composition Study

Background: Inflammation is associated with heart failure (HF) risk factors and also directly affects myocardial function. However, the association between inflammation and HF risk in older adults has not been adequately evaluated. Methods: The association of baseline serum concentrations of interleukin-6 (IL-6), tumor necrosis factor alpha (TNF- ), and C-reactive protein (CRP) with incident HF was assessed with Cox proportional hazards models among 2610 older persons without prevalent HF enrolled in the Health, Aging, and Body Composition (Health ABC) Study (age, 73.6±2.9 years; 48.3% men; 59.6% white). Results: Median (interquartile range) baseline concentrations of IL-6, TNF- , and CRP were 1.80 (1.23, 2.76) pg/mL, 3.14 (2.41, 4.06) pg/mL, and 1.64 (0.99, 3.04) µg/mL, respectively. On follow-up (median, 9.4 years), 311 participants (11.9%) developed HF. In models controlling for clinical predictors of HF and incident coronary heart disease, doubling of IL-6, TNF- , and CRP concentrations was associated with 34% (95% CI, 18 -52%; P<.001), 33% (95% CI, 9 - 63%; P=.006), and 13% (95% CI, 3-24%; P=.01) increase in HF risk, respectively. In models including all 3 markers, IL-6 and TNF- , but not CRP, remained significant. Findings were similar across sex and race. Post-HF ejection fraction (EF) was available in 239 (76.8%) cases. When only cases with preserved EF were considered (n=105), IL-6 (HR per doubling, 1.57; 95% CI, 1.28 -1.94; P<.001), TNF- (HR per doubling, 1.59; 95% CI, 1.12-2.26; P=.01), and CRP (HR per doubling, 1.23; 95% CI, 1.05-1.44; P=.01) were all associated with HF risk in adjusted models. In contrast, when only cases with reduced EF (n=134) were considered, only IL-6 attained marginal significance in adjusted models (HR per doubling, 1.20; 95% CI, 0.99 -1.46; P=.06). Participants with 2 or 3 markers above median had pronounced HF risk in adjusted models (HR, 1.66; 95% CI, 1.12-2.46; P=.01; and HR, 1.76; 95% CI, 1.16 -2.65; P=.007, respectively). Addition of IL-6 to the clinical Health ABC HF model improved discrimination (C index from 0.717 to 0.734; P=.001) and fit (decreased Bayes information criterion by 17.8; P<.001). Conclusions: Inflammatory markers are associated with HF risk among older adults and may improve HF risk stratification.

Compositional analysis of bivariate discrete probabilities

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table hasn rows and m columns and all probabilities are non-null. This kind of table can beseen as an element in the simplex of n · m parts. In this context, the marginals areidentified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclideanelements of the Aitchison geometry of the simplex can also be translated into the tableof probabilities: subspaces, orthogonal projections, distances.Two important questions are addressed: a) given a table of probabilities, which isthe nearest independent table to the initial one? b) which is the largest orthogonalprojection of a row onto a column? or, equivalently, which is the information in arow explained by a column, thus explaining the interaction? To answer these questionsthree orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independenttwo-way tables and fully dependent tables representing row-column interaction. Animportant result is that the nearest independent table is the product of the two (rowand column)-wise geometric marginal tables. A corollary is that, in an independenttable, the geometric marginals conform with the traditional (arithmetic) marginals.These decompositions can be compared with standard log-linear models.Key words: balance, compositional data, simplex, Aitchison geometry, composition,orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure,contingency table

A new perspective on the zoogeography of the sibling mouse-eared bat species Myotis myotis and Myotis blythii: Morphological, genetical and ecological evidence

The actual geographic distribution of the two sibling mouse-eared bat species Myotis myotis and Myotis blythii, which occur widely sympatrically in the western Palaearctic region, remains largely controversial. This concerns particularly the specific attribution of marginal populations from the Mediterranean islands and from adjacent areas of North Africa and Asia, which are morphologically intermediate between continental M. myotis and M. blythii from Europe. This study attempts to clarify this question by using four different approaches: cranial morphology, external morphology, genetics and trophic ecology. The three latter methods show unambiguously that North Africa, Malta, Sardinia and Corsica are presently inhabited by monospecific populations of M. myotis. In contrast, cranial morphometrics do not yield conclusive results. These results contradict all recent studies, which attribute North African and Maltese mouse-eared bats to M. blythii and consider that Sardinia and Corsica harbour sympatric populations of the two species. As concerns south-eastern populations, doubts are also expressed about the attribution of the subspecific taxon omari which may actually refer to M. myotis instead of M. blythii. Protein electrophoresis is presently the only absolute method available for determining M. myotis and M. blythii throughout their distribution ranges. However, species identification may be approached by relying on less sophisticated morphometrical methods as presented in this study. Species-specific habitat specializations are probably responsible for the differences observed between the geographic distributions of M. myotis and M. blythii, as they provide a logical groundwork for a coherent model of speciation for these two bat species.

La nappe du Lebendun entre Alte Kaserne et le Val Cairasca (massif du Simplon): nouvelles observations et interprétations

The lithostratigraphic description of the covers of three Lower Penninic nappes (Monte Leone, Lebendun and Antigorio) allows the comparison of their sedimentary content and their thickness. It has been established that the Lebendun nappe is formed by an ante-Triassic paragneissic core (Valgrande gneiss), and a Mesozoic sedimentary cover in reversed position. The cover series shows a continuous detritic sedimentation, off which the material comes from a continental erosion related to the early Lias rifting phase of the Alpine Tethys. The erosion has reached the basement, resedimented as pebbles and sandstones. This can be observed in both Lebendun and Antigorio covers. The definition of a unit named <<serie intermediaire>> between the Lebendun and the Antigorio covers has important palinspastic implications for both nappes. The unit is composed of a banded marble, a garnet bearing gneiss and a calcschist with great blocks. The comparison between the thickness of Antigorio and Lebendun covers suggests a shoulder position for Antigorio. and a proximal rift basin position tor Lebendun. The general thickness decrease of the series towards the SW points to a NE origin for the Lebendun clastics, taking into account the increase of tectonic deformation in the region trending from east to west. The detritic sedimentation ends with the basin drowning during the Malm, represented by a pure marble sealing the erosive disconformity of the Antigorio cover, and the clastic deposits of Lebendun. Three hypotheses are proposed for the calcschists age and attribution of the <<serie intermediaire>>: A: they belong entirely or partially to the Lebendun cover and correspond to a conglomeratic deposit of Cretaceous-Tertiary Niesen flysch type, of proximal facies. The tectonic limit could be situated in the middle of the calcschists at the level of the huge blocks encountered. B: they belong to Antigorio and correspond to an upper Lias-Dogger synrift deposit, then the marble is liassic. C: they belong to Antigorio and have been deposited following the Lebendun basin inversion (Cretaceous-Tertiary). that generates Tertiary wildflysch deposits, coming from the South for the ultrahelvetic and from the North for the Niesen.

Clinical performance of cervical restorations: a meta-analysis.

OBJECTIVES: To carry out a meta-analysis in order to assess the influencing factors on retention loss and marginal discoloration of cervical restorations made of composites and glass ionomer (derivates). METHODS: The literature was searched for prospective clinical studies on cervical restorations with an observation period of at least 18 months. RESULTS: Fifty clinical studies involving 40 adhesive systems matched the inclusion criteria. On average, 10% of the cervical fillings were lost and 24% exhibited marginal discoloration after 3 years. The variability ranged from 0% to 50% for retention loss and from 0% to 74% for marginal discoloration. Hardly any secondary caries was detected. When linear mixed models with a study and experiment effect were used, the analysis revealed that the adhesive/restorative class had the most significant influence, with 2-step self-etching adhesive systems performing best and 1-step self-etching adhesive systems performing worst; 3-step etch-and-rinse systems, glass ionomers/resin-modified glass ionomers, 2-step etch-and-rinse systems and polyacid-modified resin composites were ranked in between. Restorations placed in teeth whose dentin/enamel had been prepared/roughened showed a statistically significant higher retention rate than those placed in teeth with unprepared dentin (p<0.05). Beveling of the enamel and the type of isolation used (rubberdam/cotton rolls) had no significant influence. SIGNIFICANCE: The clinical performance of cervical restorations is significantly influenced by the type of adhesive system used and/or the adhesive class to which the system belonged and whether the dentin/enamel is prepared or not. 2-Step self-etching- and 3-step etch&rinse systems shall be chosen over 1-step self-etching systems and glass ionomer derivates. The dentin (and enamel) surface shall be roughened before placement of the restoration.

The effect of standard and high-fluence corneal cross-linking (CXL) on cornea and limbus.

PURPOSE: When treating peripheral ectatic disease-like pellucid marginal degeneration (PMD), corneal cross-linking with UV-A and riboflavin (CXL) must be applied eccentrically to the periphery of the lower cornea, partly irradiating the corneal limbus. Here, we investigated the effect of standard and double-standard fluence corneal cross-linking with riboflavin and UV-A (CXL) on cornea and corneal limbus in the rabbit eye in vivo. METHODS: Epithelium-off CXL was performed in male New Zealand White rabbits with two irradiation diameters (7 mm central cornea, 13 mm cornea and limbus), using standard fluence (5.4 J/cm(2)) and double-standard fluence (10.8 J/cm(2)) settings. Controls were subjected to epithelial removal and riboflavin instillation, but were not irradiated with UV-A. Following CXL, animals were examined daily until complete closure of the epithelium, and at 7, 14, 21, and 28 days. Animals were killed and a corneoscleral button was excised and processed for light microscopy and immunohistochemistry. RESULTS: For both irradiation diameters and fluences tested, no signs of endothelial damage or limbal vessel thrombosis were observed, and time to re-epithelialization was similar to untreated controls. Histological and immunohistochemical analysis revealed no differences in the p63 putative stem cell marker expression pattern. CONCLUSIONS: Even when using fluence twice as high as the one used in current clinical CXL settings, circumferential UV-A irradiation of the corneal limbus does not alter the regenerative capacity of the limbal epithelial cells, and the expression pattern of the putative stem cell marker p63 remains unchanged. This suggests that eccentric CXL may be performed safely in PMD.

Geochemical evidence for mixing of magmatic fluids with seawater, Nisyros hydrothermal system, Greece

The chemical and isotopic compositions (deltaD(H2O), delta(18)O(H2O), delta(18)O(CO2), delta(13)C(CO2), delta(34)S, and He/N-2 and He/Ar ratios) of fumarolic gases from Nisyros, Greece, indicate that both arc-type magmatic water and local seawater feed the hydrothermal system. Isotopic composition of the deep fluid is estimated to be +4.9+/-0.5parts per thousand for delta(18)O and -11+/-5parts per thousand for deltaD corresponding to a magmatic water fraction of 0.7. Interpretation of the stable water isotopes was based on liquid-vapor separation conditions obtained through gas geothermometry. The H-2-Ar, H-2-N-2, and H-2-H2O geothermometers suggest reservoir temperatures of 345+/-15 degreesC, in agreement with temperatures measured in deep geothermal wells, whereas a vapor/liquid separation temperature of 260+/-30 degreesC is indicated by gas equilibria in the H2O-H-2-CO2-CO-CH4 system. The largest magmatic inputs seem to occur below the Stephanos-Polybotes Micros crater, whereas the marginal fumarolic areas of Phlegeton-Polybotes Megalos craters receive a smaller contribution of magmatic gases.

Efficient in vivo induction of CTL by cell-associated covalent H-2Kd-peptide complexes.

A novel procedure is presented describing the induction of antigen-specific cytolytic T lymphocytes (CTL) in vivo, that uses as immunogen syngeneic Concanavalin A stimulated spleen cells expressing H-2Kd (Kd) molecules photocrosslinked with a photoreactive peptide derivative. The Kd restricted Plasmodium berghei circumsporozoite (PbCS) peptide 253-260 (YIPSAEKI) was conjugated with photoreactive iodo-4-azidosalicylic acid (IASA) at the NH2-terminus and with 4-azidobenzoic acid (ABA) at the TCR contact residue Lys259 to make IASA-YIPSAEK(ABA)I. Selective photoactivation of the IASA group allowed specific photoaffinity labeling of cell-associated Kd molecules. Optimal peptide derivative binding to Kd molecules of concanavalin A stimulated spleen cells was obtained upon 4-6 h incubation at 26 degrees C in the presence of human beta 2 microglobulin. Photocrosslinking prevented the rapid dissociation of cell-associated Kd-peptide derivative complexes at 37 degrees C. The photoaffinity labeled cells were injected i.p. into syngeneic recipients. After 10 days, the peritoneal exudate lymphocytes were harvested and in vitro stimulated with peptide derivative pulsed P815 mastocytoma cells. The resulting bulk cultures displayed high cytolytic activity that was specific for IASA-YIPSAEK(ABA)I and YIPSAEK(ABA)I. In contrast, peritoneal exudate lymphocytes from mice inoculated with concanavalin A blasts that were pulsed, but not photocrosslinked, with IASA-YIPSAEK(ABA)I expressed only marginal levels of IASA-YIPSAEK(ABA)I-specific cytolytic activity. This immunization strategy, using neither adjuvants nor potentially hazardous transfected/transformed cells, is safe and should be universally applicable.

Endoscopic off-pump aortic valve replacement: does the pericardial cuff improve the sutureless closure of left ventricular access?

OBJECTIVE: Off-pump trans left ventricular approach provides more precise deployment of stented aortic valve of any size with respect to the endovascular replacement. One of the key steps of this procedure is the ventricle repair after catheter withdrawing. We designed an animal study to compare the consistency of a sutureless repair of the left ventricle access using nitinol occluder with and without pericardial cuff on the ventricular side. METHODS: Material description: The Amplatz-nitinol occluder consists of two square heads squeezing ventricle wall in between them, sealing the defect. To improve its sealing property, a pericardial patch was sutured to the ventricular head of the occluder. Animal study setup: In adult pigs, a 30F sheath was inserted into the epigastric area through the cardiac apex, up to the left ventricle, simulating the approach for off-pump aortic valve replacement. The sheath was then removed and the ventricle closed with standard occluder in half of the animals, and cuffed occluder in the other half. Animals were followed-up for 3h, collecting haemodynamics data and pericardial bleeding. RESULTS: Device was successfully deployed in 12 animals in less than 1min. In the group where the standard occluder was used, bleeding during the deployment was 80+/-20ml and after the deployment was 800+/-20ml over 3h. In the group where the cuffed occluder was used, bleeding during the deployment was 85+/-20ml and after the deployment was 100+/-5ml over 3h. In the cuffed group, bleeding was significantly lower than the standard group, p-value being <0.001. CONCLUSIONS: The occluder is easy to use and the pericardial cuff dramatically increases its efficacy as demonstrated by a significant reduction of blood loss. The cuffed occluder opens the way for endoscopic, off-pump, transventricular aortic valve replacement.

Selective use of postoperative neck radiotherapy in oral cavity and oropharynx cancer: a prospective clinical study.

BACKGROUND In cervical postoperative radiotherapy, the target volume is usually the same as the extension of the previous dissection. We evaluated a protocol of selective irradiation according to the risk estimated for each dissected lymph node level. METHODS Eighty patients with oral/oropharyngeal cancer were included in this prospective clinical study between 2005 and 2008. Patients underwent surgery of the primary tumor and cervical dissection, with identification of positive nodal levels, followed by selective postoperative radiotherapy. Three types of selective nodal clinical target volume (CTV) were defined: CTV0, CTV1, and CTV2, with a subclinical disease risk of <10%, 10-25%, and 25% and a prescribed radiation dose of <35 Gy, 50 Gy, and 66-70 Gy, respectively. The localization of node failure was categorized as field, marginal, or outside the irradiated field. RESULTS A consistent pattern of cervical infiltration was observed in 97% of positive dissections. Lymph node failure occurred within a high-risk irradiated area (CTV1-CTV2) in 12 patients, marginal area (CTV1/CTVO) in 1 patient, and non-irradiated low-risk area (CTV0) in 2 patients. The volume of selective lymph node irradiation was below the standard radiation volume in 33 patients (mean of 118.6 cc per patient). This decrease in irradiated volume was associated with greater treatment compliance and reduced secondary toxicity. The three-year actuarial nodal control rate was 80%. CONCLUSION This selective postoperative neck irradiation protocol was associated with a similar failure pattern to that observed after standard neck irradiation and achieved a significant reduction in target volume and secondary toxicity.

Ecological Footprint Inequality: A methodological review and some results

Scarcities of environmental services are no longer merely a remote hypothesis. Consequently, analysis of their inequalities between nations becomes of paramount importance for the achievement of sustainability in terms either of international policy, or of Universalist ethical principles of equity. This paper aims, on the one hand, at revising methodological aspects of the inequality measurement of certain environmental data and, on the other, at extending the scarce empirical evidence relating to the international distribution of Ecological Footprint (EF), by using a longer EF time series. Most of the techniques currently important in the literature are revised and then tested on EF data with interesting results. We look in depth at Lorenz dominance analyses and consider the underlying properties of different inequality indices. Those indices which fit best with environmental inequality measurements are CV2 and GE(2) because of their neutrality property, however a trade-off may occur when subgroup decompositions are performed. A weighting factor decomposition method is proposed in order to isolate weighting factor changes in inequality growth rates. Finally, the only non-ambiguous way of decomposing inequality by source is the natural decomposition of CV2, which additionally allows the interpretation of marginal term contributions. Empirically, this paper contributes to the environmental inequality measurement of EF: this inequality has been quite stable and its change over time is due to per capita vector changes rather than population changes. Almost the entirety of the EF inequality is explainable by differences in the means between the countries of the World Bank group. This finding suggests that international environmental agreements should be attempted on a regional basis in an attempt to achieve greater consensus between the parties involved. Additionally, source decomposition warns of the dangers of confining CO2 emissions reduction to crop-based energies because of the implications for basic needs satisfaction.

Effect of expiratory positive airway pressure on sleep disordered breathing.

STUDY OBJECTIVES: We sought to determine the effect of expiratory positive airway pressure on end expiratory lung volume (EELV) and sleep disordered breathing in obstructive sleep apnea patients. DESIGN: Observational physiology study PARTICIPANTS: We studied 10 OSA patients during sleep wearing a facial mask. We recorded 1 hour of NREM sleep without treatment (baseline) and 1 hour with 10 cm H2O EPAP in random order, while measuring EELV and breathing pattern. RESULTS: The mean EELV change between baseline and EPAP was only 13.3 mL (range 2-25 mL). Expiratory time was significantly increased with EPAP compared to baseline 2.64 +/- 0.54 vs 2.16 +/- 0.64 sec (P = 0.002). Total respiratory time was longer with EPAP than at baseline 4.44 +/- 1.47 sec vs 3.73 +/- 0.88 sec (P = 0.3), and minute ventilation was lower with EPAP vs baseline 7.9 +/- 4.17 L/min vs 9.05 +/- 2.85 L/min (P = 0.3). For baseline (no treatment) and EPAP respectively, the mean apnea+hypopnea index (AHI) was 62.6 +/- 28.7 and 56.8 +/- 30.3 events per hour (P = 0.4). CONCLUSION: In OSA patients during sleep, the application of 10 cm H2O EPAP led to prolongation of expiratory time with only marginal increases in FRC. These findings suggest important mechanisms exist to avoid hyperinflation during sleep.

Ecological Footprint Inequality: A methodological review and some results

Scarcities of environmental services are no longer merely a remote hypothesis. Consequently, analysis of their inequalities between nations becomes of paramount importance for the achievement of sustainability in terms either of international policy, or of Universalist ethical principles of equity. This paper aims, on the one hand, at revising methodological aspects of the inequality measurement of certain environmental data and, on the other, at extending the scarce empirical evidence relating to the international distribution of Ecological Footprint (EF), by using a longer EF time series. Most of the techniques currently important in the literature are revised and then tested on EF data with interesting results. We look in depth at Lorenz dominance analyses and consider the underlying properties of different inequality indices. Those indices which fit best with environmental inequality measurements are CV2 and GE(2) because of their neutrality property, however a trade-off may occur when subgroup decompositions are performed. A weighting factor decomposition method is proposed in order to isolate weighting factor changes in inequality growth rates. Finally, the only non-ambiguous way of decomposing inequality by source is the natural decomposition of CV2, which additionally allows the interpretation of marginal term contributions. Empirically, this paper contributes to the environmental inequality measurement of EF: this inequality has been quite stable and its change over time is due to per capita vector changes rather than population changes. Almost the entirety of the EF inequality is explainable by differences in the means between the countries of the World Bank group. This finding suggests that international environmental agreements should be attempted on a regional basis in an attempt to achieve greater consensus between the parties involved. Additionally, source decomposition warns of the dangers of confining CO2 emissions reduction to crop-based energies because of the implications for basic needs satisfaction. Keywords: ecological footprint; ecological inequality measurement, inequality decomposition.