A review of spatial causal inference methods for environmental and epidemiological applications

The scientific rigor and computational methods of causal inference have had great impacts on many disciplines, but have only recently begun to take hold in spatial applications. Spatial casual inference poses analytic challenges due to complex correlation structures and interference between the treatment at one location and the outcomes at others. In this paper, we review the current literature on spatial causal inference and identify areas of future work. We first discuss methods that exploit spatial structure to account for unmeasured confounding variables. We then discuss causal analysis in the presence of spatial interference including several common assumptions used to reduce the complexity of the interference patterns under consideration. These methods are extended to the spatiotemporal case where we compare and contrast the potential outcomes framework with Granger causality, and to geostatistical analyses involving spatial random fields of treatments and responses. The methods are introduced in the context of observational environmental and epidemiological studies, and are compared using both a simulation study and analysis of the effect of ambient air pollution on COVID-19 mortality rate. Code to implement many of the methods using the popular Bayesian software OpenBUGS is provided

Jack knifing for semivariogram validation

The semivariogram function fitting is the most important aspect of geostatistics and because of this the model chosen must be validated. Jack knifing may be one the most efficient ways for this validation purpose. The objective of this study was to show the use of the jack knifing technique to validate geostatistical hypothesis and semivariogram models. For that purpose, topographical heights data obtained from six distinct field scales and sampling densities were analyzed. Because the topographical data showed very strong trend for all fields as it was verified by the absence of a sill in the experimental semivariograms, the trend was removed with a trend surface fitted by minimum square deviation. Semivariogram models were fitted with different techniques and the results of the jack knifing with them were compared. The jack knifing parameters analyzed were the intercept, slope and correlation coefficient between measured and estimated values, and the mean and variance of the errors calculated by the difference between measured and estimated values, divided by the square root of the estimation variances. The ideal numbers of neighbors used in each estimation was also studied using the jack knifing procedure. The jack knifing results were useful in the judgment of the adequate models fitted independent of the scale and sampling densities. It was concluded that the manual fitted semivariogram models produced better jack knifing parameters because the user has the freedom to choose a better fit in distinct regions of the semivariogram

Optimum size in grid soil sampling for variable rate application in site-specific management Tamanho ideal em grades de amostragem de solos para aplicação em taxa variável em manejo localizado

The importance of understanding spatial variability of soils is connected to crop management planning. This understanding makes it possible to treat soil not as a uniform, but a variable entity, and it enables site-specific management to increase production efficiency, which is the target of precision agriculture. Questions remain as the optimum soil sampling interval needed to make site-specific fertilizer recommendations in Brazil. The objectives of this study were: i) to evaluate the spatial variability of the main attributes that influence fertilization recommendations, using georeferenced soil samples arranged in grid patterns of different resolutions; ii) to compare the spatial maps generated with those obtained with the standard sampling of 1 sample ha-1, in order to verify the appropriateness of the spatial resolution. The attributes evaluated were phosphorus (P), potassium (K), organic matter (OM), base saturation (V%) and clay. Soil samples were collected in a 100 × 100 m georeferenced grid. Thinning was performed in order to create a grid with one sample every 2.07, 2.88, 3.75 and 7.20 ha. Geostatistical techniques, such as semivariogram and interpolation using kriging, were used to analyze the attributes at the different grid resolutions. This analysis was performed with the Vesper software package. The maps created by this method were compared using the kappa statistics. Additionally, correlation graphs were drawn by plotting the observed values against the estimated values using cross-validation. P, K and V%, a finer sampling resolution than the one using 1 sample ha-1 is required, while for OM and clay coarser resolutions of one sample every two and three hectares, respectively, may be acceptable.<br>A importância de compreender a variabilidade espacial do solo está conectada ao planejamento do manejo das culturas. Este entendimento faz com que seja possível tratar o solo não como uma entidade uniforme, mas variável, e permite o gerenciamento de sítios específicos para aumentar a eficiência de produção, que é o objetivo da agricultura de precisão. Questões relacionadas com a otimização do intervalo de amostragem do solo se faz necessário para a realização das recomendações de adubações no Brasil. Os objetivos deste estudo foram: i) avaliar a variabilidade espacial dos principais atributos que influenciam as recomendações de adubação, usando amostras de solos georreferenciadas dispostas em padrões de grades de diferentes resoluções; ii) comparar os mapas espaciais gerados com o mapa padrão obtido com amostragem de 1 amostra ha-1, a fim de verificar a adequação da resolução espacial. Os atributos avaliados foram fósforo (P), potássio (K), matéria orgânica (MO), saturação por bases (V%) e argila. As amostras de solos foram coletadas numa grade de 100 × 100 m e georreferenciadas. Um desbaste foi realizado, criando-se uma grade com 1 amostra a cada 2,07, 2,88, 3,75 e 7,20 ha. Técnicas de geoestatística, como semivariograma e interpolação usando krigagem, foram utilizadas para analisar os atributos nas grades com diferentes resoluções. Esta análise foi realizada com o programa computacional Vesper. Os mapas criados por este método foram comparados utilizando-se a estatística kappa. Além disso, gráficos de correlação foram construídos plotando-se os valores observados pelos valores estimados utilizando-se a validação cruzada. Para P, K e V%, uma amostragem de resolução mais fina do que aquela usando 1 amostra ha-1 foi necessária, enquanto que para MO e argila, resoluções mais grosseiras de uma amostra ou dois em dois e três hectares, respectivamente, pode ser aceitável

Modelos matemáticos para predição da chuva de projeto para regiões do Estado de Minas Gerais Mathematical models for the estimation of rainfall in selected regions of Minas Gerais State, Brazil

O uso de modelos matemáticos para predição da chuva é uma forma prática e precisa para determinação do valor a ser aplicado em projetos, sendo útil para localidades desprovidas de informações pluviométricas. Objetivou-se ajustar o método de Bell, que possui características de regionalização para a chuva de projeto, com base em equações de chuvas intensas e modelos de probabilidade de Gumbel de estações meteorológicas do Estado de Minas Gerais ajustando, também, um modelo para cada região do estado. Avaliaram-se os modelos considerando-se o coeficiente de determinação e os erros médios em relação aos dados originais. Para validação, trabalhou-se com três estações meteorológicas da região Norte não usadas para ajuste do respectivo modelo. Foram analisadas três metodologias para estimativa da chuva intensa padrão (h(60,2)), que pondera o método usado, ressaltando-se a média aritmética, a média ponderada pelo inverso do quadrado da distância e a predição geoestatística (krigagem). Observou-se que os modelos possuem bons indicadores estatísticos e a validação produziu erros baixos, mostrando que os modelos podem ser aplicados, especialmente se a krigagem for usada para estimativa do parâmetro h(60,2).<br>The use of mathematical models for predicting rainfall is a practical and accurate way of determining this parameter to be applied to regions which do not have any precipitation data. Based on the intense rainfall equations and Gumbel's probability model for maximum daily precipitation of meteorological stations in Minas Gerais State, the objective of this work was to adjust the Bell's Method, with regional features, for rainfall, adjusting one model for each region. The regional parameters were estimated by non-linear multiple regression, using Gauss-Newton's method. The goodness of the models was evaluated by the coefficient of determination and mean errors of prediction as compared to the original data. Data from three meteorological stations in the Northern region, which were not used to adjust the respective model, were used for validation purposes. The most frequent precipitation was tested by the arithmetic mean, the weighted mean by the inverse-square-distance and the geo-statistical prediction (kriging). The models produced good statistical parameters, with low mean errors, showing their accuracy, specially when the kriging method for estimating the most frequent precipitation was used