We turn the data y_{1}, y_{2}, ..., y_{N} into a sequence
i_{1}, i_{2}, ..., i_{N}
of 1s, 2s, 3s, and 4s. This sequence is called the symbol
string associated with the data. 
The data values y_{i} often are measured
as decimals and because we are converting these to only four values, the process of
turning the
y_{k} into i_{k} is called coarsegraining. 
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 coarsegraining: 
equalsize 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. 
zerocentered 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
coarsegrainings depending on the placements of the other two bin boundaries. 
meancentered 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. 
mediancentered 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 equalweight bins are a special case of this. 


Click on the small picture to see the first few points of a driven IFS using equalsize bins. 

To illustrate the different kinds of coarsegraining, we use a data set consisting of
successive differences of 1000 numbers generated by iterating the
logistic map.
Click the picture for each example. 





Equalsize bins 
Equalweight bins 
Zerocentered bins 
Meancenterd bins 
Mediancenterd bins 

We will investigate some specific data sets in
finance cartoons,
for example.
