How would you visualize a multivariate Gaussian in high dimension (>3D)?












2












$begingroup$


Obviously, I can't do:



Plot3D[PDF[

  MultinormalDistribution[{0, 0, 

    0}, {{1, 0, 0}, {0, 1, 0} {0, 0, 1}}], {x, y}], {x, -20, 

  20}, {y, -20, 20}]


How would you suggest me to effectively plot this multivariate probability density function?






plotting visualization







share|improve this question











$endgroup$












  • $begingroup$
    What do you want out of the plot? Is it just informative, or would you plan to use it to actually determine some values of things?
    $endgroup$
    – MikeY
    2 hours ago










  • $begingroup$
    @MikeY, I have a 6D energy landscape I want to visualize.
    $endgroup$
    – 0x90
    1 hour ago
















2












$begingroup$


Obviously, I can't do:



Plot3D[PDF[

  MultinormalDistribution[{0, 0, 

    0}, {{1, 0, 0}, {0, 1, 0} {0, 0, 1}}], {x, y}], {x, -20, 

  20}, {y, -20, 20}]


How would you suggest me to effectively plot this multivariate probability density function?






plotting visualization







share|improve this question











$endgroup$












  • $begingroup$
    What do you want out of the plot? Is it just informative, or would you plan to use it to actually determine some values of things?
    $endgroup$
    – MikeY
    2 hours ago










  • $begingroup$
    @MikeY, I have a 6D energy landscape I want to visualize.
    $endgroup$
    – 0x90
    1 hour ago














2












2








2





$begingroup$


Obviously, I can't do:



Plot3D[PDF[

  MultinormalDistribution[{0, 0, 

    0}, {{1, 0, 0}, {0, 1, 0} {0, 0, 1}}], {x, y}], {x, -20, 

  20}, {y, -20, 20}]


How would you suggest me to effectively plot this multivariate probability density function?






plotting visualization







share|improve this question











$endgroup$




Obviously, I can't do:



Plot3D[PDF[

  MultinormalDistribution[{0, 0, 

    0}, {{1, 0, 0}, {0, 1, 0} {0, 0, 1}}], {x, y}], {x, -20, 

  20}, {y, -20, 20}]


How would you suggest me to effectively plot this multivariate probability density function?






plotting visualization




plotting visualization






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 1 hour ago







0x90

















asked 6 hours ago









0x900x90

24719




24719












  • $begingroup$
    What do you want out of the plot? Is it just informative, or would you plan to use it to actually determine some values of things?
    $endgroup$
    – MikeY
    2 hours ago










  • $begingroup$
    @MikeY, I have a 6D energy landscape I want to visualize.
    $endgroup$
    – 0x90
    1 hour ago


















  • $begingroup$
    What do you want out of the plot? Is it just informative, or would you plan to use it to actually determine some values of things?
    $endgroup$
    – MikeY
    2 hours ago










  • $begingroup$
    @MikeY, I have a 6D energy landscape I want to visualize.
    $endgroup$
    – 0x90
    1 hour ago
















$begingroup$
What do you want out of the plot? Is it just informative, or would you plan to use it to actually determine some values of things?
$endgroup$
– MikeY
2 hours ago




$begingroup$
What do you want out of the plot? Is it just informative, or would you plan to use it to actually determine some values of things?
$endgroup$
– MikeY
2 hours ago












$begingroup$
@MikeY, I have a 6D energy landscape I want to visualize.
$endgroup$
– 0x90
1 hour ago




$begingroup$
@MikeY, I have a 6D energy landscape I want to visualize.
$endgroup$
– 0x90
1 hour ago










2 Answers
2






active

oldest

votes


















4












$begingroup$

Full M-dimensional probability distributions are hard to display. However, the conditional distributions at the planes through the origin, showing the densities as cross-cut through the full distribution, can be plotted as contour plots. The conditional distribution of a multinomial Gaussian distribution is also a Gaussian distribution, and therefore the contours are ellipses.



Here are the conditional distributions for some 4-dimensional distributions (w0,w1,w2,w3).



enter image description here



I have used the ConditionalMultinormalDistribution function from Chris.






share|improve this answer









$endgroup$





















    3












    $begingroup$

    I am not sure what are the answers you are looking for. One way is to utilize color as a fourth dimension; another is to make multiple 3D plots over some grid for the variables non-visualized in those 3D plots.



    Here is an example:



    Clear[myPDF]
    
myPDF[x_, y_, z_] := 
    
  Evaluate[PDF[
    
    MultinormalDistribution[{-1, 1, 1}, {{1, 1/2, 0}, {1/2, 1, 0}, {0, 0, 1}}], {x, y, z}]];
    
myPDF[x, y, z]
    

    
(* E^(1/2 (-(1 + x) ((4 (1 + x))/3 - 
    
      2/3 (-1 + y)) - (-(2/3) (1 + x) + 4/3 (-1 + y)) (-1 + y) - (-1 +
    
      z)^2))/(Sqrt[6] [Pi]^(3/2)) *)
    

    
Multicolumn[
    
 Table[Plot3D[myPDF[x, y, z], {x, -3, 3}, {y, -6, 6}, 
    
   MeshFunctions -> {#3 &}, MeshShading -> {None, Red, None, Yellow}, 
    
   PlotRange -> {All, All, {0, 0.08}}, PlotPoints -> 25, 
    
   PlotLabel -> Row[{"z=", z}]], {z, -6, 6, 1}], 3]


    enter image description here






    share|improve this answer









    $endgroup$













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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      Full M-dimensional probability distributions are hard to display. However, the conditional distributions at the planes through the origin, showing the densities as cross-cut through the full distribution, can be plotted as contour plots. The conditional distribution of a multinomial Gaussian distribution is also a Gaussian distribution, and therefore the contours are ellipses.



      Here are the conditional distributions for some 4-dimensional distributions (w0,w1,w2,w3).



      enter image description here



      I have used the ConditionalMultinormalDistribution function from Chris.






      share|improve this answer









      $endgroup$


















        4












        $begingroup$

        Full M-dimensional probability distributions are hard to display. However, the conditional distributions at the planes through the origin, showing the densities as cross-cut through the full distribution, can be plotted as contour plots. The conditional distribution of a multinomial Gaussian distribution is also a Gaussian distribution, and therefore the contours are ellipses.



        Here are the conditional distributions for some 4-dimensional distributions (w0,w1,w2,w3).



        enter image description here



        I have used the ConditionalMultinormalDistribution function from Chris.






        share|improve this answer









        $endgroup$
















          4












          4








          4





          $begingroup$

          Full M-dimensional probability distributions are hard to display. However, the conditional distributions at the planes through the origin, showing the densities as cross-cut through the full distribution, can be plotted as contour plots. The conditional distribution of a multinomial Gaussian distribution is also a Gaussian distribution, and therefore the contours are ellipses.



          Here are the conditional distributions for some 4-dimensional distributions (w0,w1,w2,w3).



          enter image description here



          I have used the ConditionalMultinormalDistribution function from Chris.






          share|improve this answer









          $endgroup$



          Full M-dimensional probability distributions are hard to display. However, the conditional distributions at the planes through the origin, showing the densities as cross-cut through the full distribution, can be plotted as contour plots. The conditional distribution of a multinomial Gaussian distribution is also a Gaussian distribution, and therefore the contours are ellipses.



          Here are the conditional distributions for some 4-dimensional distributions (w0,w1,w2,w3).



          enter image description here



          I have used the ConditionalMultinormalDistribution function from Chris.







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered 5 hours ago









          Romke BontekoeRomke Bontekoe

          1,326818




          1,326818























              3












              $begingroup$

              I am not sure what are the answers you are looking for. One way is to utilize color as a fourth dimension; another is to make multiple 3D plots over some grid for the variables non-visualized in those 3D plots.



              Here is an example:



              Clear[myPDF]
              
myPDF[x_, y_, z_] := 
              
  Evaluate[PDF[
              
    MultinormalDistribution[{-1, 1, 1}, {{1, 1/2, 0}, {1/2, 1, 0}, {0, 0, 1}}], {x, y, z}]];
              
myPDF[x, y, z]
              

              
(* E^(1/2 (-(1 + x) ((4 (1 + x))/3 - 
              
      2/3 (-1 + y)) - (-(2/3) (1 + x) + 4/3 (-1 + y)) (-1 + y) - (-1 +
              
      z)^2))/(Sqrt[6] [Pi]^(3/2)) *)
              

              
Multicolumn[
              
 Table[Plot3D[myPDF[x, y, z], {x, -3, 3}, {y, -6, 6}, 
              
   MeshFunctions -> {#3 &}, MeshShading -> {None, Red, None, Yellow}, 
              
   PlotRange -> {All, All, {0, 0.08}}, PlotPoints -> 25, 
              
   PlotLabel -> Row[{"z=", z}]], {z, -6, 6, 1}], 3]


              enter image description here






              share|improve this answer









              $endgroup$


















                3












                $begingroup$

                I am not sure what are the answers you are looking for. One way is to utilize color as a fourth dimension; another is to make multiple 3D plots over some grid for the variables non-visualized in those 3D plots.



                Here is an example:



                Clear[myPDF]
                
myPDF[x_, y_, z_] := 
                
  Evaluate[PDF[
                
    MultinormalDistribution[{-1, 1, 1}, {{1, 1/2, 0}, {1/2, 1, 0}, {0, 0, 1}}], {x, y, z}]];
                
myPDF[x, y, z]
                

                
(* E^(1/2 (-(1 + x) ((4 (1 + x))/3 - 
                
      2/3 (-1 + y)) - (-(2/3) (1 + x) + 4/3 (-1 + y)) (-1 + y) - (-1 +
                
      z)^2))/(Sqrt[6] [Pi]^(3/2)) *)
                

                
Multicolumn[
                
 Table[Plot3D[myPDF[x, y, z], {x, -3, 3}, {y, -6, 6}, 
                
   MeshFunctions -> {#3 &}, MeshShading -> {None, Red, None, Yellow}, 
                
   PlotRange -> {All, All, {0, 0.08}}, PlotPoints -> 25, 
                
   PlotLabel -> Row[{"z=", z}]], {z, -6, 6, 1}], 3]


                enter image description here






                share|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  I am not sure what are the answers you are looking for. One way is to utilize color as a fourth dimension; another is to make multiple 3D plots over some grid for the variables non-visualized in those 3D plots.



                  Here is an example:



                  Clear[myPDF]
                  
myPDF[x_, y_, z_] := 
                  
  Evaluate[PDF[
                  
    MultinormalDistribution[{-1, 1, 1}, {{1, 1/2, 0}, {1/2, 1, 0}, {0, 0, 1}}], {x, y, z}]];
                  
myPDF[x, y, z]
                  

                  
(* E^(1/2 (-(1 + x) ((4 (1 + x))/3 - 
                  
      2/3 (-1 + y)) - (-(2/3) (1 + x) + 4/3 (-1 + y)) (-1 + y) - (-1 +
                  
      z)^2))/(Sqrt[6] [Pi]^(3/2)) *)
                  

                  
Multicolumn[
                  
 Table[Plot3D[myPDF[x, y, z], {x, -3, 3}, {y, -6, 6}, 
                  
   MeshFunctions -> {#3 &}, MeshShading -> {None, Red, None, Yellow}, 
                  
   PlotRange -> {All, All, {0, 0.08}}, PlotPoints -> 25, 
                  
   PlotLabel -> Row[{"z=", z}]], {z, -6, 6, 1}], 3]


                  enter image description here






                  share|improve this answer









                  $endgroup$



                  I am not sure what are the answers you are looking for. One way is to utilize color as a fourth dimension; another is to make multiple 3D plots over some grid for the variables non-visualized in those 3D plots.



                  Here is an example:



                  Clear[myPDF]
                  
myPDF[x_, y_, z_] := 
                  
  Evaluate[PDF[
                  
    MultinormalDistribution[{-1, 1, 1}, {{1, 1/2, 0}, {1/2, 1, 0}, {0, 0, 1}}], {x, y, z}]];
                  
myPDF[x, y, z]
                  

                  
(* E^(1/2 (-(1 + x) ((4 (1 + x))/3 - 
                  
      2/3 (-1 + y)) - (-(2/3) (1 + x) + 4/3 (-1 + y)) (-1 + y) - (-1 +
                  
      z)^2))/(Sqrt[6] [Pi]^(3/2)) *)
                  

                  
Multicolumn[
                  
 Table[Plot3D[myPDF[x, y, z], {x, -3, 3}, {y, -6, 6}, 
                  
   MeshFunctions -> {#3 &}, MeshShading -> {None, Red, None, Yellow}, 
                  
   PlotRange -> {All, All, {0, 0.08}}, PlotPoints -> 25, 
                  
   PlotLabel -> Row[{"z=", z}]], {z, -6, 6, 1}], 3]


                  enter image description here







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered 5 hours ago









                  Anton AntonovAnton Antonov

                  23.5k167114




                  23.5k167114






























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