# 1.H. Driven IFS and Data Analysis

## Coarse-Graining the Data

We turn the data y1, y2, ..., yN into a sequence i1, i2, ..., iN of 1s, 2s, 3s, and 4s. This sequence is called the symbol string associated with the data.
The data values yi often are measured as decimals and because we are converting these to only four values, the process of turning the yk into ik is called coarse-graining.
The range of y values for corresponding to a symbol is the bin of that symbol.
Though there are others, we use five kinds of coarse-graining:
 equal-size bins Divide the range of values into four intervals of equal length. equal weight bins Arrange the bin boundaries so (approximately) the same number of points lie in each bin. zero-centered bins For data whose sign is important, take 0 as the boundary between bins 2 and 3; place the other boundaries symmetrically above and below 0. Unlike the first two cases, this is a family of coarse-grainings depending on the placements of the other two bin boundaries. mean-centered bins Take the mean of the data to be the boundary between bins 2 and 3; place the other boundaries symmetrically above and below the mean, usually expressed as a multiple of the standard deviation. median-centered bins Take the median of the data to be the boundary between bins 2 and 3; place the other boundaries symmetrically above and below the median, usually expressed as a multiple of the range. Note the equal-weight bins are a special case of this. Click on the small picture to see the first few points of a driven IFS using equal-size bins.
To illustrate the different kinds of coarse-graining, we use a data set consisting of successive differences of 1000 numbers generated by iterating the logistic map. Click the picture for each example.     Equal-size bins Equal-weight bins Zero-centered bins Mean-centerd bins Median-centerd bins
We will investigate some specific data sets in finance cartoons, for example.