Injecting cross-sectional and temporal correlations into a random matrix











up vote
0
down vote

favorite












Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



$$X=left(
begin{array}{cccc}
x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
end{array}
right).$$
I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
Rewriting
$$X=left(
begin{array}{cccc}
x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
end{array}
right)$$
and add cross-sectional correlation, rewriting
$X=left(
begin{array}{cccc}
x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
end{array}
right)$
Am I doing it right?






probability random-matrices







share|cite|improve this question


























    up vote
    0
    down vote

    favorite












    Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



    $$X=left(
    begin{array}{cccc}
    x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
    x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
    x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
    end{array}
    right).$$
    I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



    We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
    Rewriting
    $$X=left(
    begin{array}{cccc}
    x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
    rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
    rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
    end{array}
    right)$$
    and add cross-sectional correlation, rewriting
    $X=left(
    begin{array}{cccc}
    x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
    rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
    rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
    end{array}
    right)$
    Am I doing it right?






    probability random-matrices







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
      x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
      end{array}
      right).$$
      I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



      We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
      Rewriting
      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$$
      and add cross-sectional correlation, rewriting
      $X=left(
      begin{array}{cccc}
      x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$
      Am I doing it right?






      probability random-matrices







      share|cite|improve this question













      Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
      x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
      end{array}
      right).$$
      I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



      We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
      Rewriting
      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$$
      and add cross-sectional correlation, rewriting
      $X=left(
      begin{array}{cccc}
      x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$
      Am I doing it right?






      probability random-matrices




      probability random-matrices






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 15 at 18:51









      Nero

      1,91521678




      1,91521678



























          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',
          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
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3000115%2finjecting-cross-sectional-and-temporal-correlations-into-a-random-matrix%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown






























          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          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%2fmath.stackexchange.com%2fquestions%2f3000115%2finjecting-cross-sectional-and-temporal-correlations-into-a-random-matrix%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

          How to change which sound is reproduced for terminal bell?

          Image
          3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
          Read more

          Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

          Image
          up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
          Read more

          Can I use Tabulator js library in my java Spring + Thymeleaf project?

          Image
          0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
          Read more