Читать книгу Planning and Executing Credible Experiments - Robert J. Moffat - Страница 38

The experimental approach requires gathering enough input–output datasets so that the form of the model function can be determined with acceptable uncertainty. This is, at best, an approximate process, as can be seen by a simple example. Consider the differences between the analytical and the experimental approaches to the function y = sin(x). Analytically, given that function and an input set of values of x, the corresponding values of y can be determined to within any desired accuracy, by using the known behavior of the function y = sin(x). Consider now a “black box” which, when fed values of x, produces values of y. With what certainty can we claim that the model function (inside the box) is really y = sin(x)? Obviously, the certainty is limited by the accuracy of the input and the output. What uncertainty must we acknowledge when we claim that the model function (inside the box) is y = sin(x)? That depends on the accuracy of the input and the output data points and the number and spacing of the points. With a set of data having some specified number of significant figures in the input and the output, we can say only that the model function, “evaluated at these data points, does not differ from y = sin(x) by more than …,” or alternatively, “y = sin(x) within the accuracy of this experiment, at the points measured.”

That is about all we can be sure of because our understanding of the model function can be affected by the choice of the input values. Suppose that we were unfortunate enough to have chosen a sampling rate that caused our input data points (the test rig set points) to exactly match values of nπ with n being an integer. Then all of the outputs would be zero, and we could not distinguish between the “aliased” model function y = 0 and the true model function y = sin(x).

In general, with randomly selected values of x, the “resolution” of the experiment is limited by the accuracy of the input and output data. Consider Figure 2.2. In this case, sin(x) may be indistinguishable from {sin(x) + 0.1 sin (10x)} if there is significant scatter in the data. In many cases, the scatter in data is, in reality, the trace of an unrecognized component of the model function that could be included. One of an experimenter's most challenging tasks is to interpret correctly small changes in the data: is this just “scatter,” or is the process trying to show me something?

Figure 2.2 Is this a single sine wave with some scatter in the data? Does it have a superposed signal or both signal and scatter?