In regression, despite being both aimed at estimating the Mean Squared Prediction Error (MSPE), Akaike’s Final Prediction Error (FPE) and the Generalized Cross Validation (GCV) selection criteria are usually derived ...In regression, despite being both aimed at estimating the Mean Squared Prediction Error (MSPE), Akaike’s Final Prediction Error (FPE) and the Generalized Cross Validation (GCV) selection criteria are usually derived from two quite different perspectives. Here, settling on the most commonly accepted definition of the MSPE as the expectation of the squared prediction error loss, we provide theoretical expressions for it, valid for any linear model (LM) fitter, be it under random or non random designs. Specializing these MSPE expressions for each of them, we are able to derive closed formulas of the MSPE for some of the most popular LM fitters: Ordinary Least Squares (OLS), with or without a full column rank design matrix;Ordinary and Generalized Ridge regression, the latter embedding smoothing splines fitting. For each of these LM fitters, we then deduce a computable estimate of the MSPE which turns out to coincide with Akaike’s FPE. Using a slight variation, we similarly get a class of MSPE estimates coinciding with the classical GCV formula for those same LM fitters.展开更多
Because of global climate change, it is necessary to add forest biomass estimation to national forest resource monitoring. The biomass equations developed for forest biomass estimation should be compatible with volume...Because of global climate change, it is necessary to add forest biomass estimation to national forest resource monitoring. The biomass equations developed for forest biomass estimation should be compatible with volume equations. Based on the tree volume and aboveground biomass data of Masson pine （Pinus massoniana Lamb.） in southern China, we constructed one-, two- and three-variable aboveground biomass equations and biomass conversion functions compatible with tree volume equations by using error-in-variable simultaneous equations. The prediction precision of aboveground biomass estimates from one variable equa- tion exceeded 95%. The regressions of aboveground biomass equations were improved slightly when tree height and crown width were used together with diameter on breast height, although the contributions to regressions were statistically insignificant. For the biomass conversion function on one variable, the conversion factor decreased with increasing diameter, but for the conversion function on two variables, the conversion factor increased with increasing diameter but decreased with in- creasing tree height.展开更多
文摘In regression, despite being both aimed at estimating the Mean Squared Prediction Error (MSPE), Akaike’s Final Prediction Error (FPE) and the Generalized Cross Validation (GCV) selection criteria are usually derived from two quite different perspectives. Here, settling on the most commonly accepted definition of the MSPE as the expectation of the squared prediction error loss, we provide theoretical expressions for it, valid for any linear model (LM) fitter, be it under random or non random designs. Specializing these MSPE expressions for each of them, we are able to derive closed formulas of the MSPE for some of the most popular LM fitters: Ordinary Least Squares (OLS), with or without a full column rank design matrix;Ordinary and Generalized Ridge regression, the latter embedding smoothing splines fitting. For each of these LM fitters, we then deduce a computable estimate of the MSPE which turns out to coincide with Akaike’s FPE. Using a slight variation, we similarly get a class of MSPE estimates coinciding with the classical GCV formula for those same LM fitters.
基金the National Biomass Modeling Program for Continuous Forest Inventory(NBMP-CFI) funded by the State Forestry Administration of China
文摘Because of global climate change, it is necessary to add forest biomass estimation to national forest resource monitoring. The biomass equations developed for forest biomass estimation should be compatible with volume equations. Based on the tree volume and aboveground biomass data of Masson pine （Pinus massoniana Lamb.） in southern China, we constructed one-, two- and three-variable aboveground biomass equations and biomass conversion functions compatible with tree volume equations by using error-in-variable simultaneous equations. The prediction precision of aboveground biomass estimates from one variable equa- tion exceeded 95%. The regressions of aboveground biomass equations were improved slightly when tree height and crown width were used together with diameter on breast height, although the contributions to regressions were statistically insignificant. For the biomass conversion function on one variable, the conversion factor decreased with increasing diameter, but for the conversion function on two variables, the conversion factor increased with increasing diameter but decreased with in- creasing tree height.
