# “Fragmentation” of a distribution (from paper)

I've been reading a paper by Robert Morris ("Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music") and came across a formula that I've found a bit tricky. He is referring to the "fragmentation" of a distribution and includes the formula below without derivation or reference. I'm pretty new to statistics, so this may be a standard formula that I'm just unaware of. However, I haven't been able to find it in the same format online.

One feature for use in ordering talas is fragmentation. We have already grouped talas into partition classes. All talas in a particular partition class have the same fragmentation. We use the partition P as the input to a function that yields the fragmentation of the partition. Fragmentation varies between 0 and 1 and is a measure of the uniformity of a distribution—the higher the fragmentation, the more even the distribution. We calculate the fragmentation of a partition of the number N into z parts using the following formula...:

$$FRAG(P)=1 - \frac{\sum_{k=1}^{z}{PAIRS(p_{k})}}{PAIRS(N)}$$ where $$PAIRS(s)=\frac{{s^2}-s}{2} \:, \: P=\{{p_{1},p_{2}, p_{3}},...p_{z}\},\\ N = sum(P), and \: z = card(p) .$$

I found the formula to be much more readable in this format:

$$Let \: P = \{p_{1}, p_{2}, p_{3},..., p_{z}\}, \: z = card(P),\: and \: N = sum(P).\\ FRAG(P)=1- \frac{\sum_{k=1}^{z} \frac{p_{k}^{2}-p_{k}}{2}}{\frac{N^{2}-N}{2}}=1-2\frac{\sum_{k=1}^{z}\frac{p_{k}^2-p_{k}}{2}}{N^2-N}$$

The author uses the formula with the example $$P=\{2, 2, 4\} \rightarrow N = 2 + 2 + 4 = 8$$ and $$z = 3.$$ This returns $$FRAG(P)=1-2(\frac{8}{56})=0.714285714...$$

Does this formula (or a similar one) have a name? Are there any places where I can find some further information? More generally, what does this mean?

Thanks for the help!

• Should we interpret this as $\sum_{k=1}^z \frac{p_k^2-p_k}{2} =\sum_{j \in B} b_j$ where $B$ is the bag where the integer $m \ge 0$ appears $a_m = \sum_{p_k > m} 1$ times, and $1-\frac{\sum_{k=1}^z \frac{p_k^2-p_k}{2}}{\frac{N^2 - N}{2}}$ is a measure of how $B$ differs from $\{ 0, \ldots, N-1\}$ where $N$ is the number of elements in $B$ – reuns Jan 7 at 4:56
• Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above? – Luke Poeppel Jan 9 at 15:17
• $p_k$ is a list from which I construct a multiset $B$ where the meaning of $\sum_{k=1}^z \frac{p_k^2-p_k}{2}$ and $1-\frac{\sum_{k=1}^z \frac{p_k^2-p_k}{2}}{\frac{N^2 - N}{2}}$ is obvious (using that $\frac{N^2 - N}{2} = \sum_{m=0}^{N-1} m$) – reuns Jan 9 at 15:40
• Does this kind of formula have a name? It seems to be similar to the R^2 measure (goodness-of-fit), but I can't say for certain. – Luke Poeppel Jan 10 at 18:56