1-D random landscapes and non-random data series

We study the simplest random landscape, the curve formed by joining consecutive data points $f_{1},\ldots,f_{N+1}$ with line segments, where the fi are i.i.d. random numbers and $f_{i}\ne f_{j}$. We label each segment increasing (+) or decreasing (-) and call this string of +'s and -'s the up-down signature $\sigma $. We calculate the probability $P(\sigma (f))$ for a random curve and use it to bound the algorithmic information content of f. We show that f can be compressed by $k = \log_2 {1/P(\sigma)} - N $ bits, where k is a universal currency for comparing the amount of pattern in different curves. By applying our results to microarray time series data, we blindly identify regulatory genes.