Measurement-Free Tests of Linearity in Biological Systems

It is desirable to test the linearity of a system without interval-scale measurement of its input and output. Four tests are proposed. The first is a generalization of the superposition principle. If it is satisfied, the system can be regarded as an integral operator, with variable input distortion and variable but monotonic output distortion. The second test concerns invariance of the input distortion over the input and output domains. It is based on a method of reconstructing an unknown variable input distortion by constructing standard sequences (i.e., sequences of input values that are equally spaced with respect to the integral operator). Test three concerns invariance of the output distortion; it generalizes the superposition principle further, to the case of comparing outputs across different points in the output domain. The fourth test is simply commutativity with translations; if it is satisfied, as well as the first two tests, then the system also satisfies the third test and, in addition, it can be written as a convolution integral.