“Fragmentation” of a distribution (from paper)












2












$begingroup$


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:



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



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!






statistics uniform-distribution music-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    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 $B$
    $endgroup$
    – reuns
    Jan 7 at 4:56












  • $begingroup$
    Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above?
    $endgroup$
    – Luke Poeppel
    Jan 9 at 15:17










  • $begingroup$
    $p_k$ 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 $frac{N^2 - N}{2} = sum_{m=0}^{N-1} m$)
    $endgroup$
    – reuns
    Jan 9 at 15:40










  • $begingroup$
    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.
    $endgroup$
    – Luke Poeppel
    Jan 10 at 18:56
















2












$begingroup$


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:



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



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!






statistics uniform-distribution music-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    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 $B$
    $endgroup$
    – reuns
    Jan 7 at 4:56












  • $begingroup$
    Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above?
    $endgroup$
    – Luke Poeppel
    Jan 9 at 15:17










  • $begingroup$
    $p_k$ 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 $frac{N^2 - N}{2} = sum_{m=0}^{N-1} m$)
    $endgroup$
    – reuns
    Jan 9 at 15:40










  • $begingroup$
    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.
    $endgroup$
    – Luke Poeppel
    Jan 10 at 18:56














2












2








2





$begingroup$


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:



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



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!






statistics uniform-distribution music-theory







share|cite|improve this question









$endgroup$




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:



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



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!






statistics uniform-distribution music-theory




statistics uniform-distribution music-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 7 at 2:37









Luke PoeppelLuke Poeppel

212




212












  • $begingroup$
    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 $B$
    $endgroup$
    – reuns
    Jan 7 at 4:56












  • $begingroup$
    Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above?
    $endgroup$
    – Luke Poeppel
    Jan 9 at 15:17










  • $begingroup$
    $p_k$ 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 $frac{N^2 - N}{2} = sum_{m=0}^{N-1} m$)
    $endgroup$
    – reuns
    Jan 9 at 15:40










  • $begingroup$
    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.
    $endgroup$
    – Luke Poeppel
    Jan 10 at 18:56


















  • $begingroup$
    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 $B$
    $endgroup$
    – reuns
    Jan 7 at 4:56












  • $begingroup$
    Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above?
    $endgroup$
    – Luke Poeppel
    Jan 9 at 15:17










  • $begingroup$
    $p_k$ 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 $frac{N^2 - N}{2} = sum_{m=0}^{N-1} m$)
    $endgroup$
    – reuns
    Jan 9 at 15:40










  • $begingroup$
    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.
    $endgroup$
    – Luke Poeppel
    Jan 10 at 18:56
















$begingroup$
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 $B$
$endgroup$
– reuns
Jan 7 at 4:56






$begingroup$
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 $B$
$endgroup$
– reuns
Jan 7 at 4:56














$begingroup$
Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above?
$endgroup$
– Luke Poeppel
Jan 9 at 15:17




$begingroup$
Thanks for your reply! When you refer to B as the "bag," are you referring to the multiset P above?
$endgroup$
– Luke Poeppel
Jan 9 at 15:17












$begingroup$
$p_k$ 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 $frac{N^2 - N}{2} = sum_{m=0}^{N-1} m$)
$endgroup$
– reuns
Jan 9 at 15:40




$begingroup$
$p_k$ 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 $frac{N^2 - N}{2} = sum_{m=0}^{N-1} m$)
$endgroup$
– reuns
Jan 9 at 15:40












$begingroup$
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.
$endgroup$
– Luke Poeppel
Jan 10 at 18:56




$begingroup$
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.
$endgroup$
– Luke Poeppel
Jan 10 at 18:56










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