On the Properties of the R2 Indicator

International audienceIn multiobjective optimization, set-based performance indicators are commonly used to assess the quality of a Pareto front approximation. Based on the scalarization obtained by these indicators, a performance comparison of multiobjective optimization algorithms becomes possible. The R2 and the Hypervolume (HV) indicator represent two recommended approaches which have shown a correlated behavior in recent empirical studies. Whereas the HV indicator has been comprehensively analyzed in the last years, almost no studies on the R2 indicator exist. In this paper, we thus perform a comprehensive investigation of the properties of the R2 indicator in a theoretical and empirical way. The influence of the number and distribution of the weight vectors on the optimal distribution of μ solutions is analyzed. Based on a comparative analysis, specific characteristics and differences of the R2 and HV indicator are presented

Preference Articulation by Means of the R2 Indicator

International audienceIn multi-objective optimization, set-based performance indicators have become the state of the art for assessing the quality of Pareto front approximations. As a consequence, they are also more and more used within the design of multi-objective optimization algorithms. The R2 and the Hypervolume (HV) indicator represent two popular examples. In order to understand the behavior and the approximations preferred by these indicators and algorithms, a comprehensive knowledge of the indicator's properties is required. Whereas this knowledge is available for the HV, we presented a first approach in this direction for the R2 indicator just recently. In this paper, we build upon this knowledge and enhance the considerations with respect to the integration of preferences into the R2 indicator. More specifically, we analyze the effect of the reference point, the domain of the weights, and the distribution of weight vectors on the optimization of μ solutions with respect to the R2 indicator. By means of theoretical findings and empirical evidence, we show the potentials of these three possibilities using the optimal distribution of μ solutions for exemplary setups

A Convergence indicator for Multi-Objective Optimisation Algorithms

The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.Comment: Submitted to TEM

Do probit models help in forecasting turning points of German business cycles?

In this paper we used a data set constructed for a companion paper (Fritsche/Stephan, 2000) where we explored the leading indicator properties of different time series for the German business cycle. Now we test for the ability of different indicator series to forecast recessions by using a probit approach as proposed by Estrella/Mishkin (1997). The dating procedure refers to the study by Artis et. al. (1997). We took into consideration the criticism made by Dueker (1997) who stated that in the probit model the fact that the economy is already in a state of recession must be controlled for. The results of our estimate are unsatisfactory on the whole. Only the ifo institute's business expectation of producers of intermediate inputs, the interest rate spread, the long-term interest rate, and money supply M2 show satisfactory leading properties.business cycle, probit model, modified McFadden's R2, recession, Germany

On the Construction of Pareto-Compliant Combined Indicators

The most relevant property that a quality indicator (QI) is expected to have is Pareto compliance, which means that every time an approximation set strictly dominates another in a Pareto sense, the indicator must reflect this. The hypervolume indicator and its variants are the only unary QIs known to be Pareto-compliant but there are many commonly used weakly Pareto-compliant indicators such as R2, IGD+,andɛ+. Currently, an open research area is related to finding new Pareto-compliant indicators whose preferences are different from those of the hypervolume indicator. In this article, we propose a theoretical basis to combine existing weakly Pareto-compliant indicators with at least one being Pareto-compliant, such that the resulting combined indicator is Pareto-compliant as well. Most importantly, we show that the combination of Paretocompliant QIs with weakly Pareto-compliant indicators leads to indicators that inherit properties of the weakly compliant indicators in terms of optimal point distributions. The consequences of these new combined indicators are threefold: (1) to increase the variety of available Pareto-compliant QIs by correcting weakly Pareto-compliant indicators, (2) to introduce a general framework for the combination of QIs, and (3) to generate new selection mechanisms for multiobjective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front

Mapping soil cation exchange capacity in a semiarid region through predictive models and covariates from remote sensing data.

Planning sustainable use of land resources requires reliable information about spatial distribution of soil physical and chemical properties related to environmental processes and ecosystemic functions. In this context, cation exchange capacity (CEC) is a fundamental soil quality indicator; however, it takes money and time to obtain this data. Although many studies have been conducted to spatially quantify soil properties on various scales and in different environments, not much is known about interactions between soil properties and environmental covariates in the Brazilian semiarid region. The goal of this study was to evaluate the efficiency of random forest and cokriging models applied to predict CEC in the Brazilian semiarid region. The covariates used to predict CEC consist of images from Landsat 5 TM and a legacy soil map (scale 1:10,000). The sample set comprises 499 samples from the topsoil layer (0.00-0.20 m), where 375 samples were used in training processes and 124 as validation samples. The cokriging model (R2= 0.57 and RMSE = 7.22 cmol c kg-1) performed better in predicting CEC than the random forest model (R2= 0.47 and RMSE = 7.89 cmol c kg-1). The approach used showed potential for estimating CEC content in the Brazilian semiarid region by using covariates obtained from orbital remote sensing and the legacy soil map

Bioactive Components and Antioxidant Activity of Moroccan Paprika (Capsicum annuum L.) at Different Period of Harvesting and Processing

In the present study the total phenols, flavonoids, flavonols, carotenoids content and ASTA of extracts of paprika powder produced at four periods of harvesting and processing were investigated for their antioxidant activity. A different composition between the four periods was evidenced. Paprika produced in November presented a major content of phenols, carotenoids, and ASTA (1360 mg/100g DW, 3727.54 mg/kg DW, 167.15 unit) respectively. Also it showed the highest radical scavenging activity in DPPH assay (IC50 of 260?g/ml). Total flavonol and total flavonoid contents show a little variation depending on the time of harvesting and processing. The total phenolic and carotenoid contents were highly correlated with DPPH values (R2= 0.95 and R2= 0.96) respectively. Therefore, the total phenolic and carotenoid contents can serve as a useful indicator for the antioxidant activity of paprika. The obtained results suggest that Moroccan paprika could be used as valuable flavor with functional properties for foods. Keywords: Niora, ground paprika, ASTA, carotenoids, phenolics, flavonoids, flavonols, antioxidant activity.

Interactive Hyperparameter Optimization in Multi-Objective Problems via Preference Learning

Hyperparameter optimization (HPO) is important to leverage the full potential of machine learning (ML). In practice, users are often interested in multi-objective (MO) problems, i.e., optimizing potentially conflicting objectives, like accuracy and energy consumption. To tackle this, the vast majority of MO-ML algorithms return a Pareto front of non-dominated machine learning models to the user. Optimizing the hyperparameters of such algorithms is non-trivial as evaluating a hyperparameter configuration entails evaluating the quality of the resulting Pareto front. In literature, there are known indicators that assess the quality of a Pareto front (e.g., hypervolume, R2) by quantifying different properties (e.g., volume, proximity to a reference point). However, choosing the indicator that leads to the desired Pareto front might be a hard task for a user. In this paper, we propose a human-centered interactive HPO approach tailored towards multi-objective ML leveraging preference learning to extract desiderata from users that guide the optimization. Instead of relying on the user guessing the most suitable indicator for their needs, our approach automatically learns an appropriate indicator. Concretely, we leverage pairwise comparisons of distinct Pareto fronts to learn such an appropriate quality indicator. Then, we optimize the hyperparameters of the underlying MO-ML algorithm towards this learned indicator using a state-of-the-art HPO approach. In an experimental study targeting the environmental impact of ML, we demonstrate that our approach leads to substantially better Pareto fronts compared to optimizing based on a wrong indicator pre-selected by the user, and performs comparable in the case of an advanced user knowing which indicator to pick

LAND –USE CHANGE EFFECTS ON SOIL QUALITY AT UMUDIKE AREA, ABIA STATE, SOUTH-EAST NIGERIA

Land use change affects soil quality, but the extent of the effect at Umudike area of Abia State Nigeria, remainsunclear. The study examined the impact of seven land use types: Natural Forest Reserved land (NFRL), ArableLand (AL), Recreational Land (RL), Pasture Grass Land (PGL), Reforestation Land (RFL), Short Period FallowLand (SPFL), and Oil Palm Plantation Land (OPPL) on some selected soil properties. The aim of the study were:to investigate what extent, if any, deterioration of the soil as a result of change from forest to other forms of use;and if soil's organic matter was related to the soil's other properties. Soils were sampled under forest and six otherforms of land use mentioned. The soil properties determined were; pH, soil organic matter, total nitrogen, bulkdensity, total porosity, moisture content, particle size distribution, and mean weight diameter. Relationshipsbetween the soil properties and soil organic matter, which is assumed as an indicator of soil quality in the studyarea, were also estimated. Results show that soil organic matter increased significantly (p<0.05) under NFRL to43.47 g kg-1, and decreased to 0.52 g kg-1 under AL. Bulk density value was lowest ( 1.25 g cm-3) for NFRL,followed by the PGL (1.30 g cm-3). Results of the percentage soil deterioration index indicated that RL had thehighest value of 24.6%. There was a linear and strong negative relationship (R2 = 0.8778 and r = -0.83***)between soil organic matter and bulk density. Conversely, between soil organic matter and mean weight diameter,strong positive relationship (R2 = 0.7428 and r = -0.83***) was estimated. We conclude therefore, that theadoption of good soil management practices that would improve the soil properties of the other land use apartfrom NFRL should be encouraged