“Fragmentation” of a distribution (from paper)
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
$endgroup$
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
asked Jan 7 at 2:37
Luke PoeppelLuke Poeppel
314
314
$begingroup$
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
add a comment |
$begingroup$
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
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3064606%2ffragmentation-of-a-distribution-from-paper%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3064606%2ffragmentation-of-a-distribution-from-paper%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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