Mathematically, gas sensor arrays map each gas composition vector x to an equilibrium response vector m. For gravimetric, MOFbased gas sensor arrays, this mapping is characterized by a mixed-gas adsorption model m = a(x). An observation of the [total] mass of gas adsorbed in a MOF generally places one independent constraint m i = a i (x) on the gas composition. We considered when there are fewer MOFs in the array than degrees of freedom in the gas composition (the number of components minus one, given fixed temperature and pressure). As a result, the mapping a : x → m is non-injective (manyto-one). Then, some gas composition changes from x op to x do not cause changes in the The mapping from gas composition space (left) to sensor response space (right) under a(x) for the sensor array in (a). The small sphere of radius = 0.04 in composition space (blue) is mapped to the solid ellipse-like shape (not an exact ellipse like in Fig. because a(x) is non-linear) in response space (blue). Under the linear approximant of a(x), ã(x), the vectors v 2 , v 2 , v 3 are mapped to σ 1 u 1 , σ 2 u 2 , 0, respectively. The unresponsive direction is v 3 (red) and the sensitive direction is v 1 (green).