L1-Normed GM(1,1) Models and Reliability Analysis

Grey theory is a mathematical branch dealing with system dynamics having sparse data availability. Grey reliability analysis is thus advantageous because of the small sample size requirements, say, the first-order one-variable grey differential equation only needs as little as four data points. However, the grey estimation of state dynamic law uses least-square approach, i.e., parameter estimation under L
₂
norm. Problems associated with L
₂
norm grey estimation are the model accuracy specifications. The L
₂
norm grey modeling campus often borrows the model fitting criteria from statistical linear model analysis, for example, using mean-sum-of squared errors as model fitting criterion and even using probability bound for it. These exercises are putting themselves in controversy. In numerical analysis and approximation theory, relative error is a standard approximation criterion although the L
₂
norm grey modeling campus also uses relative error as model accuracy measure. In this paper, we propose a L
₁
norm based grey modeling and search the grey parameters in terms of simplex technique in linear programming. Grey reliability analysis under L
₁
norm based grey state dynamics will be briefly discussed.