This paper describes an algorithm that takes a trace (i.e., a sequence of numbers or vectors of numbers) as input, and from that produces a sequence of loop nests that, when run, produces exactly the original sequence. The input format is suitable for any kind of program execution trace, and the output conforms to standard models of loop nests. The first, most obvious, use of such an algorithm is for program behavior modeling for any measured quantity (memory accesses, number of cache misses, etc.). Finding loops amounts to detecting periodic behavior and provides an explanatory model. The second application is trace compression, i.e., storing the loop nests instead of the original trace. Decompression consists of running the loops, which is easy and fast. A third application is value prediction. Since the algorithm forms loops while reading input, it is able to extrapolate the loop under construction to predict further incoming values. Throughout the paper, we provide examples that explain our algorithms. Moreover, we evaluate trace compression and value prediction on a subset of the SPEC2000 benchmarks.