Mixing
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Projecting 6D cartesian coordinates to lower dimension
Clash Royale CLAN TAG#URR8PPP
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I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the points (See below). For 2D and 3D this is simple, but is there a way to do this for higher dimensions other than a histogram?
In other words, is there a way to project 6 cartesian coordinates $x = (x_1,x_2,x_3,x_4,x_5,x_6)$ with length 1 onto a unit circle/3-sphere?
geometry visualization
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
asked Jul 21 at 9:29
Ortix92
1499
add a comment |Â
up vote
2
down vote
favorite
I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the points (See below). For 2D and 3D this is simple, but is there a way to do this for higher dimensions other than a histogram?
In other words, is there a way to project 6 cartesian coordinates $x = (x_1,x_2,x_3,x_4,x_5,x_6)$ with length 1 onto a unit circle/3-sphere?
geometry visualization
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
asked Jul 21 at 9:29
Ortix92
1499
The norm histogram does present the distribution of norms quite well. What else you can/may want to show probably depends on the details of the distribution. For example, if the directions of points are truly uniformly distrubuted (say, w.r.t. to a uniform measure on $S^5$) then there may be not much else to show. If the data has preferred directions, say they tend to concentrate on a certain 4-dimensional cross section of the sphere, then you can find such subspaces with statistical methods, and then plan a visualization that shows such a tendency.
– Jyrki Lahtonen
Jul 21 at 9:58
I'm afraid I don't know for sure, but IIRC finding a Karhunen-Loéve basis would reveal such tendencies (if they exist).
– Jyrki Lahtonen
Jul 21 at 10:00
1
You can, of course, do things like $$(x_1,x_2,ldots,x_6)mapsto(sqrtx_1^2+x_2^2,sqrtx_3^2+x_4^2,sqrtx_5^2+x_6^2),$$ or pair up the coordinates some other way. That will map the 6D unit sphere to the first octant of the 3D unit sphere. But this distorts a number of things. Oh, and KL may be overkill here, all you probably need is the covariance matrix - I plead guilty of a bit of name dropping. Ask someone well versed in stats.
– Jyrki Lahtonen
Jul 21 at 10:12
@JyrkiLahtonen I know all those names ;) Covariance matrix is indeed something I did not think of. That is indeed smart. KL is also a good measure since the points are indeed normally distributed.
– Ortix92
Jul 21 at 10:27
Icosahedral quasilattice, maybe?
– Oscar Lanzi
Jul 22 at 23:38
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the points (See below). For 2D and 3D this is simple, but is there a way to do this for higher dimensions other than a histogram?
In other words, is there a way to project 6 cartesian coordinates $x = (x_1,x_2,x_3,x_4,x_5,x_6)$ with length 1 onto a unit circle/3-sphere?
geometry visualization
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
asked Jul 21 at 9:29
Ortix92
1499
I've got a noisy set of points that lie on/near a unit 6-sphere. I want to visualise their proximity to the sphere in some way. The only way I can come up with now is a histogram of the norm of the points (See below). For 2D and 3D this is simple, but is there a way to do this for higher dimensions other than a histogram?
In other words, is there a way to project 6 cartesian coordinates $x = (x_1,x_2,x_3,x_4,x_5,x_6)$ with length 1 onto a unit circle/3-sphere?
geometry visualization
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
asked Jul 21 at 9:29
Ortix92
1499
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
edited Jul 21 at 9:50
Jyrki Lahtonen
105k12161355
105k12161355
asked Jul 21 at 9:29
Ortix92
1499
asked Jul 21 at 9:29
Ortix92
1499
asked Jul 21 at 9:29
Ortix92
1499
1499
The norm histogram does present the distribution of norms quite well. What else you can/may want to show probably depends on the details of the distribution. For example, if the directions of points are truly uniformly distrubuted (say, w.r.t. to a uniform measure on $S^5$) then there may be not much else to show. If the data has preferred directions, say they tend to concentrate on a certain 4-dimensional cross section of the sphere, then you can find such subspaces with statistical methods, and then plan a visualization that shows such a tendency.
– Jyrki Lahtonen
Jul 21 at 9:58
I'm afraid I don't know for sure, but IIRC finding a Karhunen-Loéve basis would reveal such tendencies (if they exist).
– Jyrki Lahtonen
Jul 21 at 10:00
1
You can, of course, do things like $$(x_1,x_2,ldots,x_6)mapsto(sqrtx_1^2+x_2^2,sqrtx_3^2+x_4^2,sqrtx_5^2+x_6^2),$$ or pair up the coordinates some other way. That will map the 6D unit sphere to the first octant of the 3D unit sphere. But this distorts a number of things. Oh, and KL may be overkill here, all you probably need is the covariance matrix - I plead guilty of a bit of name dropping. Ask someone well versed in stats.
– Jyrki Lahtonen
Jul 21 at 10:12
@JyrkiLahtonen I know all those names ;) Covariance matrix is indeed something I did not think of. That is indeed smart. KL is also a good measure since the points are indeed normally distributed.
– Ortix92
Jul 21 at 10:27
Icosahedral quasilattice, maybe?
– Oscar Lanzi
Jul 22 at 23:38
add a comment |Â
The norm histogram does present the distribution of norms quite well. What else you can/may want to show probably depends on the details of the distribution. For example, if the directions of points are truly uniformly distrubuted (say, w.r.t. to a uniform measure on $S^5$) then there may be not much else to show. If the data has preferred directions, say they tend to concentrate on a certain 4-dimensional cross section of the sphere, then you can find such subspaces with statistical methods, and then plan a visualization that shows such a tendency.
– Jyrki Lahtonen
Jul 21 at 9:58
I'm afraid I don't know for sure, but IIRC finding a Karhunen-Loéve basis would reveal such tendencies (if they exist).
– Jyrki Lahtonen
Jul 21 at 10:00
1
You can, of course, do things like $$(x_1,x_2,ldots,x_6)mapsto(sqrtx_1^2+x_2^2,sqrtx_3^2+x_4^2,sqrtx_5^2+x_6^2),$$ or pair up the coordinates some other way. That will map the 6D unit sphere to the first octant of the 3D unit sphere. But this distorts a number of things. Oh, and KL may be overkill here, all you probably need is the covariance matrix - I plead guilty of a bit of name dropping. Ask someone well versed in stats.
– Jyrki Lahtonen
Jul 21 at 10:12
@JyrkiLahtonen I know all those names ;) Covariance matrix is indeed something I did not think of. That is indeed smart. KL is also a good measure since the points are indeed normally distributed.
– Ortix92
Jul 21 at 10:27
Icosahedral quasilattice, maybe?
– Oscar Lanzi
Jul 22 at 23:38
The norm histogram does present the distribution of norms quite well. What else you can/may want to show probably depends on the details of the distribution. For example, if the directions of points are truly uniformly distrubuted (say, w.r.t. to a uniform measure on $S^5$) then there may be not much else to show. If the data has preferred directions, say they tend to concentrate on a certain 4-dimensional cross section of the sphere, then you can find such subspaces with statistical methods, and then plan a visualization that shows such a tendency.
– Jyrki Lahtonen
Jul 21 at 9:58
The norm histogram does present the distribution of norms quite well. What else you can/may want to show probably depends on the details of the distribution. For example, if the directions of points are truly uniformly distrubuted (say, w.r.t. to a uniform measure on $S^5$) then there may be not much else to show. If the data has preferred directions, say they tend to concentrate on a certain 4-dimensional cross section of the sphere, then you can find such subspaces with statistical methods, and then plan a visualization that shows such a tendency.
– Jyrki Lahtonen
Jul 21 at 9:58
I'm afraid I don't know for sure, but IIRC finding a Karhunen-Loéve basis would reveal such tendencies (if they exist).
– Jyrki Lahtonen
Jul 21 at 10:00
I'm afraid I don't know for sure, but IIRC finding a Karhunen-Loéve basis would reveal such tendencies (if they exist).
– Jyrki Lahtonen
Jul 21 at 10:00
1
1
You can, of course, do things like $$(x_1,x_2,ldots,x_6)mapsto(sqrtx_1^2+x_2^2,sqrtx_3^2+x_4^2,sqrtx_5^2+x_6^2),$$ or pair up the coordinates some other way. That will map the 6D unit sphere to the first octant of the 3D unit sphere. But this distorts a number of things. Oh, and KL may be overkill here, all you probably need is the covariance matrix - I plead guilty of a bit of name dropping. Ask someone well versed in stats.
– Jyrki Lahtonen
Jul 21 at 10:12
You can, of course, do things like $$(x_1,x_2,ldots,x_6)mapsto(sqrtx_1^2+x_2^2,sqrtx_3^2+x_4^2,sqrtx_5^2+x_6^2),$$ or pair up the coordinates some other way. That will map the 6D unit sphere to the first octant of the 3D unit sphere. But this distorts a number of things. Oh, and KL may be overkill here, all you probably need is the covariance matrix - I plead guilty of a bit of name dropping. Ask someone well versed in stats.
– Jyrki Lahtonen
Jul 21 at 10:12
@JyrkiLahtonen I know all those names ;) Covariance matrix is indeed something I did not think of. That is indeed smart. KL is also a good measure since the points are indeed normally distributed.
– Ortix92
Jul 21 at 10:27
@JyrkiLahtonen I know all those names ;) Covariance matrix is indeed something I did not think of. That is indeed smart. KL is also a good measure since the points are indeed normally distributed.
– Ortix92
Jul 21 at 10:27
Icosahedral quasilattice, maybe?
– Oscar Lanzi
Jul 22 at 23:38
Icosahedral quasilattice, maybe?
– Oscar Lanzi
Jul 22 at 23:38
add a comment |Â
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The norm histogram does present the distribution of norms quite well. What else you can/may want to show probably depends on the details of the distribution. For example, if the directions of points are truly uniformly distrubuted (say, w.r.t. to a uniform measure on $S^5$) then there may be not much else to show. If the data has preferred directions, say they tend to concentrate on a certain 4-dimensional cross section of the sphere, then you can find such subspaces with statistical methods, and then plan a visualization that shows such a tendency.
– Jyrki Lahtonen
Jul 21 at 9:58
I'm afraid I don't know for sure, but IIRC finding a Karhunen-Loéve basis would reveal such tendencies (if they exist).
– Jyrki Lahtonen
Jul 21 at 10:00
1
You can, of course, do things like $$(x_1,x_2,ldots,x_6)mapsto(sqrtx_1^2+x_2^2,sqrtx_3^2+x_4^2,sqrtx_5^2+x_6^2),$$ or pair up the coordinates some other way. That will map the 6D unit sphere to the first octant of the 3D unit sphere. But this distorts a number of things. Oh, and KL may be overkill here, all you probably need is the covariance matrix - I plead guilty of a bit of name dropping. Ask someone well versed in stats.
– Jyrki Lahtonen
Jul 21 at 10:12
@JyrkiLahtonen I know all those names ;) Covariance matrix is indeed something I did not think of. That is indeed smart. KL is also a good measure since the points are indeed normally distributed.
– Ortix92
Jul 21 at 10:27
Icosahedral quasilattice, maybe?
– Oscar Lanzi
Jul 22 at 23:38