We are all accustomed to the idea of growth to a limit, for example, the number of people becoming ill in an epidemic. In fact, observers have recorded thousands of examples of such S-shaped growth in settings as diverse as animal populations [10], energy and transport infrastructures[1,7,15], language acquisition[9], and technological performance[6,12,16]. Often the measured quantity (population of a species, height of a plant, power of an engine) grows exponentially at the outset. However, natural systems cannot sustain exponential growth indefinitely. Rather, negative feedback mechanisms or signals from the environment slow the growth, producing the S-shaped curve. Thus, for a single growth process, a single sigmoidal curve is often a useful model.
However, many systems exhibit complex growth, with multiple processes occurring sequentially or simultaneously. Kindleberger [8], in an economic history of the world since 1500 wrote: ``In the real world there are many wiggles, speedups, and setbacks, new S-curves growing out of old, separate curves for different sectors and regions of a national economy, all of which present difficulties when an attempt is made to aggregate them on a weighted basis.'' Many such phenomena, it turns out, can be explained elegantly with a simple mathematical model.
Humans create technologies, some of which are selected and diffused into society, much like a servers on a network creating waves of packets and sending them out onto the Internet. Aggregating all these wavelets 1creates much of the apparent complexity we observe; in this case, the problem of decoding complexity is essentially a problem of deconvolution. Generally speaking, the human eye is not well-suited to perform this task with a reasonable degree of precision; Loglet Lab was designed to assist us in this endeavor.
We propose the development of loglet analysis for the the analysis, decomposition, and prediction of complex growth processes. The term ``loglet'', coined at The Rockefeller University in 1994, joins ``logistic'' and ``wavelet.'' Two main objectives of loglet analysis are to analyze existing time-series growth data sets in order to decompose the growth process into sub-processes and to elucidate information on carrying capacities and other aspects (``top-down'' approach); and to analyze individual sub-processes in order to determine macro or envelope system behavior (``bottom-up'' approach). At the heart of loglet analysis is the three-parameter S-shaped logistic growth model. The logistic is attractive for modeling S-shaped growth because it is a parsimonious model where the three parameters have clear, physical interpretations.
The Loglet Lab software package allows users to perform loglet analysis on any suitable time-series data set. The user interface is easy and informative for the casual user and the common case of a single logistic in isolation; at the same time, Loglet Lab has an advanced fitting engine to analyze complex data and compound phenomena. The following sections describe the mathematics behind the Loglet Lab software package.