Denis Allard

CART algorithm for spatial data: Application to environmental and ecological data

Most statistical learning techniques such as Classification And Regression Trees (CART) assume independent samples to compute classification rules. This assumption is very practical for estimating quantities involved in the algorithm and for assessing asymptotic properties of estimators. In many environmental or ecological applications, the data under study are a sample of some regionalized variables, which can be modeled as random fields with spatial dependence. When the sampling scheme is very irregular, a direct application of supervised classification algorithms leads to biased discriminant rules due, for example, to the possible oversampling of some areas. The CART algorithm is adapted to the case of spatially dependent samples, focusing on environmental and ecological applications. Two approaches are considered. The first one takes into account the irregularity of the sampling by weighting the data according to their spatial pattern using two existing methods based on Voronoi tessellation and regular grid, and one original method based on kriging. The second one uses spatial estimates of the quantities involved in the construction of the discriminant rule at each step of the algorithm. These methods are tested on simulations and on a classical dataset to highlight their advantages and drawbacks. They are then applied on an ecological data set to explore the relationship between pollen data and presence/absence of tree species, which is an important question for climate reconstruction based on paleoecological data.

Probability Aggregation Methods in Geoscience

The need for combining different sources of information in a probabilistic framework is a frequent task in earth sciences. This is a need that can be seen when modeling a reservoir using direct geological observations, geophysics, remote sensing, training images, and more. The probability of occurrence of a certain lithofacies at a certain location for example can easily be computed conditionally on the values observed at each source of information. The problem of aggregating these different conditional probability distributions into a single conditional distribution arises as an approximation to the inaccessible genuine conditional probability given all information. This paper makes a formal review of most aggregation methods proposed so far in the literature with a particular focus on their mathematical properties. Exact relationships relating the different methods is emphasized. The case of events with more than two possible outcomes, never explicitly studied in the literature, is treated in detail. It is shown that in this case, equivalence between different aggregation formulas is lost. The concepts of calibration, sharpness, and reliability, well known in the weather forecasting community for assessing the goodness-of-fit of the aggregation formulas, and a maximum likelihood estimation of the aggregation parameters are introduced. We then prove that parameters of calibrated log-linear pooling formulas are a solution of the maximum likelihood estimation equations. These results are illustrated on simulations from two common stochastic models for earth science: the truncated Gaussian model and the Boolean. It is found that the log-linear pooling provides the best prediction while the linear pooling provides the worst.

A new spatial skew-normal random field model

Skewness is often present in a wide range of geostatistical problems, and modeling it in the spatial context remains a challenging problem. In this article, we propose and study a new class of spatial skew-normal random fields, defined in terms of the closed multivariate skew-normal distribution. Such fields can be written as the sum of two independent fields: one Gaussian and the other truncated Gaussian. We derive theoretical expressions for the first- and second-order moments, and use them within a method of moments based procedure to estimate the parameters of the model. Data simulated from the model are used to illustrate the methodology developed.

Spatial interpolation of air temperature using environmental context: Application to a crop model

The air temperature is one of the main input data in models for water balance monitoring or crop models for yield prediction. The different phenological stages of plant growth are generally defined according to cumulated air temperature from the sowing date. When these crop models are used at the regional scale, the meteorological stations providing input climatic data are not spatially dense enough or in a similar environment to reflect the crop local climate. Hence spatial interpolation methods must be used. Climatic data, particularly air temperature, are influenced by local environment. Measurements show that the air above dry surfaces is warmer than above wet areas. We propose a method taking into account the environment of the meteorological stations in order to improve spatial interpolation of air temperature. The aim of this study is to assess the impact of these “corrected climatic data” in crop models. The proposed method is an external drift kriging where the Kriging system is modified to correct local environment effects. The environment of the meteorological stations was characterized using a land use map summarized in a small number of classes considered as a factor influencing local temperature. This method was applied to a region in south-east France (150×250 km) where daily temperatures were measured on 150 weather stations for two years. Environment classes were extracted from the CORINE Landcover map obtained from remote sensing data. Categorical external drift kriging was compared to ordinary kriging by a cross validation study. The gain in precision was assessed for different environment classes and for summer days. We then performed a sensitivity study of air temperature with the crop model STICS. The influence of interpolation corrections on the main outputs as yield or harvest date is discussed. We showed that the method works well for air temperature in summer and can lead to significant correction for yield prediction. For example, we observed by cross validation a bias reduction of 0.5 to 1.0°C (exceptionally 2.5°C for some class), which corresponds to differences in yield prediction from 0.6 to 1.5 t/ha.

Using First- and Second-Order Variograms for Characterizing Landscape Spatial Structures From Remote Sensing Imagery

The spatial structures displayed by remote sensing imagery are essential information characterizing the nature and the scale of spatial variation of Earth surface processes. This paper provides a new approach to characterize the spatial structures within remote sensing imagery using stochastic models and geostatistic metrics. Up to now, the second-order variogram has been widely used to describe the spatial variations within an image. In this paper, we demonstrate its limitation to discriminate distinct image spatial structures. We introduce a different geostatistic metric, the first-order variogram, which used in combination with the second-order variogram, will prove its efficiency to describe the image spatial structures. We then develop a method based on the simultaneous use of both first- and second-order variogram metrics to model the image spatial structures as the weighted linear combination of two stochastic models: a Poisson line mosaic model and a multi-Gaussian model. The image spatial structures are characterized by the variance weight and the variogram range related to each model. This method is applied to several SPOT-HRV Normalized Difference Vegetation Index (NDVI) images from the VALERI database in order to characterize the nature of the processes structuring different types of landscape. The mosaic model is an indicator of strong NDVI discontinuities within the image mainly generated by anthropogenic processes such as the mosaic pattern of crop sites. The multi-Gaussian model shows evidence of diffuse and continuous variation of NDVI generally engendered by ecological and environmental processes such as the fuzzy pattern observed over forest and natural vegetation sites

Disaggregating daily precipitations into hourly values with a transformed censored latent Gaussian process

A problem often encountered in agricultural and ecological modeling is to disaggregate daily precipitations into vectors of hourly precipitations used as input values by crop and plant models. A stochastic model for rainfall data, based on transformed censored latent Gaussian process is described. Compared to earlier similar work, our transform function provides an accurate fit for both the body and the heavy tail of the precipitation distribution. Simple empirical relationships between the parameters estimated at different time scales are established. These relationships are used for the disaggregation of daily values at stations where hourly values are not available. The method is illustrated on two stations located in the Paris basin.

Truncated skew-normal distributions: moments, estimation by weighted moments and application to climatic data

SummaryIn this paper we derive an expression of the mth order moments and some weighted moments of truncated skew-normal distributions. We link these formulas to previous results for truncated distributions and non truncated skew-normal distributions. Methods to estimate skew-normal parameters using weighted moments are proposed and compared to other classical techniques. In a second step we propose to model the distribution of relative humidity with a truncated skew-normal distribution.

Estimating and testing zones of abrupt change for spatial data

We propose a method for detecting the zones where a variable irregularly sampled in the plane changes abruptly. Our general model is that under the null hypothesis the variable is the realisation of a stationary Gaussian process with constant expectation. The alternative is that the mean function presents abrupt changes. We define potential Zones of Abrupt Change (ZACs) by the points where the gradient, estimated under the null hypothesis, exceeds a determined threshold. We then design a global test to assess the global significance of the potential ZACs, an issue missing in all existing methods. The theory that links the threshold and the global level is based on asymptotic distributions of excursion sets of non-stationary χ2 fields for which we provide new results. The method is evaluated by a simulation study and applied to a soil data set in the context of precision agriculture.

Modeling Skewness in Spatial Data Analyis without Data Transformation

Skewness is present in a large variety of spatial data sets (rainfalls, winds, etc) but integrating such a skewness still remains a challenge. Classically, the original variables are transformed into a Gaussian vector. Besides the problem of choosing the adequate transform, there are a few difficulties associated with this method. As an alternative, we propose a different way to introduce skewness. The skewness comes from the extension of the multivariate normal distribution to the multivariate skew-normal distribution. This strategy has many advantages. The spatial structure is still captured by the variogram and the classical empirical variogram has a known moment generating function. To illustrate the applicability of such this new approach, we present a variety of simulations.

A method to estimate plant density and plant spacing heterogeneity: application to wheat crops

BackgroundPlant density and its non-uniformity drive the competition among plants as well as with weeds. They need thus to be estimated with small uncertainties accuracy. An optimal sampling method is proposed to estimate the plant density in wheat crops from plant counting and reach a given precision.ResultsThree experiments were conducted in 2014 resulting in 14 plots across varied sowing density, cultivars and environmental conditions. The coordinates of the plants along the row were measured over RGB high resolution images taken from the ground level. Results show that the spacing between consecutive plants along the row direction are independent and follow a gamma distribution under the varied conditions experienced. A gamma count model was then derived to define the optimal sample size required to estimate plant density for a given precision. Results suggest that measuring the length of segments containing 90 plants will achieve a precision better than 10%, independently from the plant density. This approach appears more efficient than the usual method based on fixed length segments where the number of plants are counted: the optimal length for a given precision on the density estimation will depend on the actual plant density. The gamma count model parameters may also be used to quantify the heterogeneity of plant spacing along the row by exploiting the variability between replicated samples. Results show that to achieve a 10% precision on the estimates of the 2 parameters of the gamma model, 200 elementary samples corresponding to the spacing between 2 consecutive plants should be measured.ConclusionsThis method provides an optimal sampling strategy to estimate the plant density and quantify the plant spacing heterogeneity along the row.