KEYWORDS: Data modeling, Autoregressive models, Statistical analysis, Statistical modeling, Fourier transforms, Data analysis, 3D modeling, Mathematical modeling, Mathematics, Data conversion
This paper establishes a statistical framework of forest coverage models for spatio-temporal data. The forest coverage ratio of grid-cell data is modeled by taking human population density and relief energy as explanatory variables. The likelihood of the forest ratios is decomposed by the product of two likelihoods. The first likelihood discussed by Nishii and Tanaka (2010) is due to trinomial logistic distributions on three categories: the ratios take zero, one, or values between zero and one. We consider a precise modeling to the second likelihood for partlydeforested ratios by considering a) spline functions to the additive mean structure, b) wide spatial dependency of normal error terms, and c) an extended logistic type transform to the forest ratio. For spatio-temporal data, we implement auto-regressive terms based on the ratios observed in past. The proposed model was applied to real grid-cell data and resulted significant improvement compared to our previous model.
Tanaka and Nishii (2005) figured out that deforestation can be elucidated quantitatively by nonlinear logit regression models in four East Asian test fields: forest areal rate F as a target variable, and human population size (N) and relief energy (R: difference of minimum altitude from the maximum in a sampled area) as explanatory variables, whose functional forms had been suggested by step functions fitted to one-kilometer square high precision grid-cell data firstly in Japan (n=6825): log(F/(1 - F)) = β0 + g(N) + h(R) + error, where g(N) and h(R) are regression functions of explanatory variables N and R, respectively. Likelihood functions with spatial dependency were derived, and several deforestation models were selected for the application to four regions in East Asia by calculating relative appropriateness to data. For the measure of appropriateness, Akaike's Information Criterion (AIC) was used. To formulate East-Asian dataset, landcover dataset estimated from NOAA observations available at UNEP, Tsukuba for F, gridded population of the world of CIESIN, US for N, and GTOPO30 of USGS for R, were used. The resolutions were matched by taking their common multiple of 20 minute square. Tanaka and Nishii (ibid.) omitted the data with F = 0.0 and F = 1.0 to employ the logit models. Unfortunately the reduction of the data size for regression led to instability of parameter estimation. As for the test field in Harbin, China, n = 76 for 0.0 < F < 1.0, but n = 504 for 0.0 less than or equal to F less than or equal to 1.0. In this study, we therefore compare the models based on all data, especially with F = 1.0, by the following extended logit transformation with two additional positive parameters of κ and λ: log((F + κ)/(1 - F + λ)) = β0 + g(N) + h(R) + error. Obvious improvements in terms of relative appropriateness to data are observed in extended logit models.
Deforestation is a result of complex causality chains in most cases. But identification of limited number of factors shall provide comprehensive general understanding of the vital phenomenon at a broad scale, as well as projection for the future. Only two factors -- human population size (N) and relief energy (R: difference of minimum altitude from the maximum in a sampled area) -- were found to give sufficient elucidation of deforestation by nonlinear logit regression models, whose functional forms were suggested by step functions fitted to one-kilometer square high precision grid-cell data in Japan (n=6825). Likelihood with spatial dependency was derived, and several deforestation models were selected for the application to East Asia by calculating relative appropriateness to data. For the measure of appropriateness, Akaike's Information Criterion (AIC) was used. Logit model is employed so as to avoid anomaly in asymptotic lower and upper bounds. Therefore the forest areal rate, 0 < F < 1. To formulate East-Asian dataset, landcover dataset estimated from NOAA observations available at UNEP, Tsukuba for F, gridded population of the world of CIESIN, US for N, and GTOPO30 of USGS for R, were used. The resolutions were matched by taking their common multiple of 20 minutes square. It was suggested that data of full forest coverage, F=1.0, which were not dealt in calculations due to logit transformation this time, should give important role in stabilizing parameter estimations.
Several measures assessing the accuracy of land-cover classification are available, e.g., overall and class- averaged accuracies. Also kappa statistic is widely used for this purpose. In this article, we discuss the properties of these criteria, and point out that the kappa statistic has an unfavorable feature. We propose alternative coefficients based on Kullback-Leibler information. Further, significance tests for the difference between the coefficients derived by classification results are established.
A multistep method for segmentation of feature space using triplet decision tree is developed, and another approach to cope with uncertain samples by extended Bayesian discriminant function is introduced. The latter has the lower limit for posterior probability of classification. The triplet-decision tree includes a division-wait mechanism that postpone the decision about uncertain samples which are in marginal area and not able to be classified to any categories definitely. The third node is generated for such samples. Improvement of the triplet tree method is made by introducing linearly-combined variables related to principal components. Flexible and effective segmentation is accomplished by this refinement. Results of experiments by simulation data and real remotely-sensed data are compared by the two methods in the viewpoint of cutting of feature space and classification accuracy. When the normality or representability of sample is hold, classifier with extended quadratic discrimination function has the best performance. The advantage of triplet tree appears when categories are diversified in nature or training samples have poor representabilities.
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