電算機を用いた数値地形解析による山岳地域の地形変動の計量化について (III) : AIC最小化推定法 (MAICE) による地形計量モデル化への可能性

Abstract

計量モデルが離散的な有限個のデータから構造と誤差とを分離する過程を通して構成されると考えるならば (統計的誤差処理), これには 〔1〕 データ特性に関するアプリオリな知識に基づいてモデルが的確に構成されること, 〔2〕 想定されるモデルの中から適切なモデルを比較・評価するための客観的基準が提示されること, 〔3〕 得られたモデルによって何らかの予知・予測が可能になること, が不可欠な条件となる。これらのいずれかの条件が不備であれば, そのモデルの有効性は損われる。著者等は, 数値地形データに対して多項式あるいは2重フーリエ級数をモデルとして採用し 〔1〕, 統計的仮説検定によって多項式の次数およびフーリエ級数の有限項の打切り項数を決定し 〔2〕, これらのモデルを通して, クラスター分析による小地形単位の地形形状の計量的区分法, 残差分布と相対開析量・卓越要素分布の対応関係に基づく開析量を指標とした地形特性の把握法, 林道開設における地形と土工量の問題等 〔3〕, を論じてきた。しかしながら, 前報で指摘した様に 〔2〕 の条件については適合度判定上の問題として課題を残しつつ, これにかわる誤差尺度の導入あるいは開発の必要性を説いてきた。何らの統計数値表も主観的判断さえも必要としないAIC最小化推定法 (Minimum AIC Estimation MAICE) は, この問題に対する極めて実用的で有効な接近法である。このととは, 数値実験および芦生の数値地形データ解析の結果から明らかになった。

Fourier series provide convenient means for interpolation, and limited extrapolation of numerical data those are oscillatory. A possible general use of double Fourier series would produce numerical indexes that are useful in classification. For example, an objective numerical system for classifying the configuration of landforms could be developed by means of double Fourier series analysis. However, additional consideration needs to be done which includes (1) assessment of the desirability of leveling of the data prior to Fourier analysis, (2) establishing the guides for grid point spacing and for the number of columns and rows in the grid, and (3) arriving at suitable guides for determing the "cut-off" point with respect to the numbers of terms (index of harmonic degree of terms) which are used in the Fourier series. The Fourier model fails to accord with the actual landform at places along the edge of the map. This is because the map-edge values of the Fourier model, prior to retilting, are the same at opposite points on any two edges. Computation of coefficients of a model is a part of Fourier analysis. In addition it is essential to compute measures that express the goodness of fit of the model to the data and then to determine whether the model are statistically significant or not (statistical hypothesis testing procedure as a method of statistical model construction or identification). In a practical situation, it was suggested that there was some gap between the theoretical results and the practical procedures of adequate model constructing through the analysis of the digital terrain data. From the information criterion's point of view, Akaike (1973a) introduced the MAICE (Minimum AIC Estimation) procedure which selects the model whose structure with its associated parameter values gives the minimum of AIC. Akaike's Information Criterion (AIC) is defined as follows: AIC=(-2) log (maximum likelihood)+2(number of free parameter). AIC examines both the fit of a model and the unreliability of a mode; the former is measured by the first term of the definition, which decreases as the number of parameters within the model is increased, and the latter by the second term, which increases with the number of parameters. AIC, which is an estimate of the Kullback-Leibler information quantity, provides a powerful and almost automatic procedure for the identification of models free from the ambiguities inherent in the application of conventional hypothesis testing procedure.