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
    8 hours ago










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










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







0x90

















asked 11 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
    8 hours ago










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










  • $begingroup$
    @MikeY, I have a 6D energy landscape I want to visualize.
    $endgroup$
    – 0x90
    6 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
8 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
8 hours ago












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




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









    $endgroup$













      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: "387"
      };
      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: false,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: null,
      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
      },
      onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathematica.stackexchange.com%2fquestions%2f191164%2fhow-would-you-visualize-a-multivariate-gaussian-in-high-dimension-3d%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      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 10 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 11 hours ago









                  Anton AntonovAnton Antonov

                  23.5k167114




                  23.5k167114






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematica 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.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathematica.stackexchange.com%2fquestions%2f191164%2fhow-would-you-visualize-a-multivariate-gaussian-in-high-dimension-3d%23new-answer', 'question_page');
                      }
                      );

                      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







                      -

                      Popular posts from this blog

                      Усть-Каменогорск

                      Image
                      У этого термина существуют и другие значения, см. Усть-Каменогорск. Город Усть-Каменогорск каз. Өскемен Флаг Герб 49°57′ с. ш. 82°37′ в. д. H G Я O L Страна Казахстан   Казахстан Статус областной центр Область Восточно-Казахстанская Аким Жаксылык Омар [1] История и география Основан 1720 Город с 1868 Площадь 303,15 [2] км² Высота центра 283 ± 1 м Тип климата резко-континентальный Часовой пояс UTC+6 Население Население 329 090 [3]  человек ( 2018 ) Национальности русские 53,56 % казахи 42,99 % немцы 0,82 % украинцы 0,57 % татары 0,79 % белорусы 0,14 % азербайджанцы 0,23 % корейцы 0,20 % прочие 1,08 % -1 шт [4] Этнохороним усть-каменогорцы, усть-каменогорец Цифровые идентификаторы Телефонный код +7 (7232) Почтовый индекс F0***** [5] Автомобильный код 16 (ранее F) oskemen.kz Усть-Каменогорск Усть-Каменогорск ...
                      Read more

                      Халкинская богословская школа

                      Image
                      Халкинская богословская школа тур. Rum Ortodoks Ruhban Okulu греч. Ιερά Θεολογική Σχολή της Χάλκης Главное здание Халкинской школы Международное название Theological School of Halki Год основания 1844 Конфессия православие Церковь Константинопольская Ректор Елпидифор (Ламбриниадис) Расположение о. Хейбелиада, Турция Юридический адрес Ümit tepesi No 5, 34973 Heybeliada, Istanbul, Turkey Сайт www.greece.org  [[commons:Category:Halki seminary|Халкинская богословская школа тур. Rum Ortodoks Ruhban Okulu греч. Ιερά Θεολογική Σχολή της Χάλκης на Викискладе]] Ха́лкинская богосло́вская шко́ла (или Духовная семинария на Халки , тур. Rum Ortodoks Ruhban Okulu , греч. Ιερά Θεολογική Σχολή της Χάλκης ) — недействующее среднее специальное учебное заведение Константинопольской православной церкви, расположенное на территории Свято-Троицкого мужского монастыря на вершине холма Надежды (тур. Ümit Tepesi ) острова Хейбелиада, в Турции....
                      Read more

                      Высокополье (Харьковская область)

                      Image
                      У этого термина существуют и другие значения, см. Высокополье. Село Высокополье укр. Високопілля Герб 49°55′07″ с. ш. 35°25′13″ в. д. H G Я O L Страна Украина   Украина Область Харьковская Район Валковский Сельский совет Высокопольский История и география Основан 1670 Прежние названия Хмелевое Площадь 55,56 км² Часовой пояс UTC+2, летом UTC+3 Население Население 1957 человек ( 2001 ) Плотность 35,220 чел./км² Цифровые идентификаторы Телефонный код +380 5753 Почтовый индекс 63020 Автомобильный код AX, КХ / 21 КОАТУУ 6321281501 Высокополье Высокополье Высокополье  Аудио, фото и видео на Викискладе Высокопо́лье (укр. Високопілля ) — село, Высокопольский сельский совет, Валковский район, Харьковская область, Украина. Код КОАТУУ — 6321281501. Население по переписи 2001 г. составляет 1957 (866/1091 м/ж) человек. Является административным центр...
                      Read more