This paper considers parameter estimation of regression models for spatial data. It is well-known that good prediction for the target variable needs incorporation of effects from neighbor cells. However, this is not an easy task in modeling and parameter estimation. We consider estimation for deforestation model, where forest areal ratios are regressed by human population densities and relief energies. The two-stage method proposed here is: 1) Find the best non-linear regression function of explanatory variables under independence assumption. 2) Predict target values by the estimated function. 3) Obtain the final prediction by correction based on linear regression with center's and neighbors' predicted values. The method is an extension of our pseudo maximum likelihood method reported at IGARSS 2019. By an application to actual data, it worked very well and successfully clarified the difference of neighbors' effects of explanatory variables. Note that the method is applicable in a general setting.

In this paper, we propose a two stage procedure to estimate regression parameters including spatial dependency. The proposed procedure without inversion task is carried out very fast with small memory. For the model estimation, the neighbor effect needs an inversion of a large matrix of size n × n where n denotes a sample size, which requires exhaustive calculations especially in geosciences. It has been attested thus far that in comparison with maximum likelihood estimation(MLE), our proposed method produced the same trend of relative appropriateness to real data by using BIC and AIC. That brings novel possibility to replace MLE in drastically shorter time to avoid elaborative effort which sometimes results in eratic results of local optimization also.

Deforestation is caused by various factors. In the literature, the impact of human activities as well as geographic circumstances on forests has been extensively discussed. Tanaka and Nishii have studied statistical models for prediction of forest area ratio by covariates: human population density and relief energy [1-3] observed in a grid-cell system. Parametric non-linear regression functions of the covariates were used for predicting forest coverage ratio [1], and cubic spline functions were also used for detection of small fluctuation of regression functions [2]. Furthermore, zero-one inflated distributions were proposed for classification of each site into one of three categories: completely-deforested, fully-forest-covered or partly-deforested areas [3]. These methods took the spatial dependency into the modeling, which is not an easy task.

This work reviews and straighten methodologies to explore functional forms of man-landcover interrelations. Tanaka and Nishii[7, 8, 9, 10, 12, 13], Nishii and Tanaka[4, 6], Nishii, Miyata, and Tanaka[5] have engaged in the analysis of deforestation by human population interactions. The basic strategy has been to use landcover data mashed up with cell-formed population density data.

This research prosecutes topographic attributes on forest acreage ratio (F) decrease by human population interactions (N), from the none to full coverage in grid cells. Tanaka and Nishii [9] verified that altitude as well as the slope steepness, i.e., topographic attributes, indicate strong influence to the deforestation in cells 0 <; F <; 1. Two new agenda were investigated in this research: zero-one inflated distribution of forest acreage, and effect of neighbor-cells. As for the former one, a classification problem of three forest coverage domains with F = 0, F = 1 and 0 <; F <; 1 is discussed. Then, statistical spatial models, as to the latter, are applied to the domain with 0 <; F <; 1, which take neighbor cell effects into account. This means that the methodology can analyze all terrestrial fields including deserts and climax forests.

Identification of the factors involved in deforestation could lead to a comprehensive understanding of deforestation on a broad scale, as well as a prediction capability. Tanaka and Nishii [7, 8] explored regression models with two explanatory variables - human population density (N) and relief energy (R), i.e., the difference between the maximum and minimum altitudes in a sampled area- as to whether they could elucidate aspects of deforestation. As relative appropriateness of the models, Akaike's Information Criterion was used to evaluate the models on real data. Although they suceeded in identifying the fuctional form of g(N) in Asian four test areas, the topographic term h(R) remained intact in terms of alternative possible variable forms. In this research, detailed verifications of the topographic feature were employed, and it was revealed that addition of mean altitude on the same cell will give great improvement to the model.

This paper considers statistical modeling of deforestation. Forest coverage ratio of grid-cell data was modeled by two covariates: human population density and relief energy. Conditional likelihood of the forest ratios given the covariates was decomposed by product of two likelihoods. The first one is due to trinomial logistic distributions on three classes: the ratios take zero, one or values between zero and one. The second one is due to a logistic-normal regression model for the ratios between zero and one. This model was applied to the real grid-cell data, and led remarkably interesting implications.

Quadtrees and linear quadtrees are well-known hierarchical data structures to represent square images of size 2(r) x 2(r). Finding the neighbors of a specific leaf node is a fundamental operation for many algorithms that manipulate quadtree data structures. In quadtrees, finding neighbors takes O(r) computational time for the worst case, where r is the resolution (or height) of a given quadtree. Schrack [1] proposed a constant-time algorithm for finding equal-sized neighbors in linear quadtrees. His algorithm calculates the location codes of equal-sized neighbors; it says nothing, however, about their existence. To ensure their existence, additional checking of the location codes is needed, which usually takes O(r) computational time. In this paper, a new algorithm to find the neighbors of a given leaf node in a quadtree is proposed which requires just O(1) (i.e., constant) computational time for the worst case. Moreover, the algorithm takes no notice of the existence or nonexistence of neighbors. Thus, no additional checking is needed. The new algorithm will greatly reduce the computational complexities of almost all algorithms based on quadtrees.

Identification of the factors involved in deforestation could lead to a comprehensive understanding of deforestation on a broad scale, as well as prediction capability. In this paper, regression models with two explanatory variables-human population and relief energy, i.e., the difference between the maximum and minimum altitudes in a sampled area-were verified as to whether they could elucidate aspects of deforestation. The functional forms of the nonlinear regression models were estimated by step functions analyzed with the use of high-precision Japanese data. Candidate smooth regression models were then derived from the obtained sigmoidal shapes by the step functions. Models with spatially dependent errors were also developed. Akaike's information criterion was used to evaluate the models on four data sets for the East Asia region. From the evaluation, we selected the best three models that systematically showed the best relative appropriateness to the real data.

Identification of limited number of factors shall provide comprehensive general understanding of deforestation at broad scale, as well as the projection for the future. Only two factors - human population and relief energy (difference of minimum altitude from the maximum in a sampled area) - were verified if they give sufficient elucidation of deforestation by a regression model, whose functional forms identified by linear combinations of dummy variables firstly explored with use of high-precision Japanese data. Likelihood with spatial dependency was derived and applied then to East-Asian data, with which our models systematically showed eminently good relative appropriateness to the real data.

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 broad scale, as well as projection for the future. Only two factors - human population and relief energy (difference of minimum altitude from the maximum in a sampled area) - were found to give sufficient elucidation of deforestation by a regression model, whose functional forms were verified by linear combinations of dummy variables firstly explored with use of Japanese data. Likelihood with spatial dependency was derived and applied then to East-Asian data, with which our models showed the eminently good relative appropriateness to the real data.

Several measures assessing accuracy of land-cover classification are available, e.g., overall and class-averaged accuracies. Also the kappa statistic is widely used for this purpose. The authors discuss properties of these criteria and point out that the kappa statistic has an unfavorable feature. They propose an alternative coefficient based on Kullback-Leibler information. A test statistic for significance difference between coefficients is also established. Further, the Bayes risk, which takes types of misclassifications into account, is discussed. Their assessment measures are examined through actual error matrices by Z. Ma and R. L. Redmond (1995).

Thermal infrared measurements of Band 6 acquired by Landsat TM sensor have lower spatial resolution than those of the other six bands. The authors propose a statistical approach to enhance the resolution of low spatial resolution image by using remaining bands. They employ a multivariate normal distribution for the seven band values. The values of Band 6 are predicted by the conditional expectations. Validity of their procedure is examined by mean squared errors based on actual images.