Generating correlated normal vectors with observationsWhat is the correlation function in multivariable/vectoral case?Random point distribtionCorrelation Coefficient Distribution Function: An Apparent Discrepancy?Whether or not $X_1$,…,$X_n$ are independent and exchangeableAnalytical form of a joint PDF of two uniformly distributed random variableswhat do these odds ratios represent?Affine transformation does not preserve normal vectorsLinear transformation of random variable to get correlation 0Use Pearson Correlation to Assess Nonlinear Association?Conditional density in hierarchical model

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Generating correlated normal vectors with observations


What is the correlation function in multivariable/vectoral case?Random point distribtionCorrelation Coefficient Distribution Function: An Apparent Discrepancy?Whether or not $X_1$,…,$X_n$ are independent and exchangeableAnalytical form of a joint PDF of two uniformly distributed random variableswhat do these odds ratios represent?Affine transformation does not preserve normal vectorsLinear transformation of random variable to get correlation 0Use Pearson Correlation to Assess Nonlinear Association?Conditional density in hierarchical model













0












$begingroup$


Assume we are given two normally distributed random variables, $X_1$ and $X_2$, with $X_i sim mathcal N (0, sigma_x_i^2)$, with correlation coefficient $rho_x$. Assume further that we need to generate another two normally distributed random variables $Y_1$ and $Y_2$ with $Y_j sim mathcal N (0,sigma_y_j^2)$, with correlation coefficient $rho_y$ and cross correlation



$$rho_i,j =fracmathbbEX_i Y_jsigma_x_i sigma_y_i$$



I.e., we need to generate $[Y_1, Y_2]$ taken into consideration the given $[X_1, X_2]$. I know how to generate all of them together $[X_1,X_2, Y_1, Y_2]$ using covariance matrix



$$M_4times4 =
beginbmatrix
sigma_x_1^2 &rho_x sigma_x_1sigma_x_2 &rho_1,1sigma_x_1 sigma_y_1 &rho_1,2sigma_x_1sigma_y_2\
rho_x sigma_x_1sigma_x_2 &sigma_x_2^2 &rho_2,1sigma_x_2 sigma_y_1 & rho_2,2sigma_x_2 sigma_y_2\
rho_1,1 sigma_x_1 sigma_y_1 & rho_1,2 sigma_x_2 sigma_y_1 & sigma_y_1^2 & rho_ysigma_y_1sigma_y_2\
rho_1,2 sigma_x_1 sigma_y_2 & rho_2,2 sigma_x_2sigma_y_2 & rho_ysigma_y_1sigma_y_2 & sigma_y_2^2 \
endbmatrix = beginbmatrix A_2times2 & B_2times2 \ B^top_2times2 & C_2times2 \ endbmatrix $$



I think we can generate $[Y_1, Y_2]$ by conditioning on $[X_1, X_2]$ and calculate the new conditional covariance, $tildeC = beginbmatrix C - B^top A^-1B endbmatrix$. Assuming that $A,B$ and $C$ are positive semidefinite (PSD), but not $M$, is it still possible to generate $[Y_1,Y_2]$?! or are we going to get always a non-PSD $tildeC$.










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    Assume we are given two normally distributed random variables, $X_1$ and $X_2$, with $X_i sim mathcal N (0, sigma_x_i^2)$, with correlation coefficient $rho_x$. Assume further that we need to generate another two normally distributed random variables $Y_1$ and $Y_2$ with $Y_j sim mathcal N (0,sigma_y_j^2)$, with correlation coefficient $rho_y$ and cross correlation



    $$rho_i,j =fracmathbbEX_i Y_jsigma_x_i sigma_y_i$$



    I.e., we need to generate $[Y_1, Y_2]$ taken into consideration the given $[X_1, X_2]$. I know how to generate all of them together $[X_1,X_2, Y_1, Y_2]$ using covariance matrix



    $$M_4times4 =
    beginbmatrix
    sigma_x_1^2 &rho_x sigma_x_1sigma_x_2 &rho_1,1sigma_x_1 sigma_y_1 &rho_1,2sigma_x_1sigma_y_2\
    rho_x sigma_x_1sigma_x_2 &sigma_x_2^2 &rho_2,1sigma_x_2 sigma_y_1 & rho_2,2sigma_x_2 sigma_y_2\
    rho_1,1 sigma_x_1 sigma_y_1 & rho_1,2 sigma_x_2 sigma_y_1 & sigma_y_1^2 & rho_ysigma_y_1sigma_y_2\
    rho_1,2 sigma_x_1 sigma_y_2 & rho_2,2 sigma_x_2sigma_y_2 & rho_ysigma_y_1sigma_y_2 & sigma_y_2^2 \
    endbmatrix = beginbmatrix A_2times2 & B_2times2 \ B^top_2times2 & C_2times2 \ endbmatrix $$



    I think we can generate $[Y_1, Y_2]$ by conditioning on $[X_1, X_2]$ and calculate the new conditional covariance, $tildeC = beginbmatrix C - B^top A^-1B endbmatrix$. Assuming that $A,B$ and $C$ are positive semidefinite (PSD), but not $M$, is it still possible to generate $[Y_1,Y_2]$?! or are we going to get always a non-PSD $tildeC$.










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      Assume we are given two normally distributed random variables, $X_1$ and $X_2$, with $X_i sim mathcal N (0, sigma_x_i^2)$, with correlation coefficient $rho_x$. Assume further that we need to generate another two normally distributed random variables $Y_1$ and $Y_2$ with $Y_j sim mathcal N (0,sigma_y_j^2)$, with correlation coefficient $rho_y$ and cross correlation



      $$rho_i,j =fracmathbbEX_i Y_jsigma_x_i sigma_y_i$$



      I.e., we need to generate $[Y_1, Y_2]$ taken into consideration the given $[X_1, X_2]$. I know how to generate all of them together $[X_1,X_2, Y_1, Y_2]$ using covariance matrix



      $$M_4times4 =
      beginbmatrix
      sigma_x_1^2 &rho_x sigma_x_1sigma_x_2 &rho_1,1sigma_x_1 sigma_y_1 &rho_1,2sigma_x_1sigma_y_2\
      rho_x sigma_x_1sigma_x_2 &sigma_x_2^2 &rho_2,1sigma_x_2 sigma_y_1 & rho_2,2sigma_x_2 sigma_y_2\
      rho_1,1 sigma_x_1 sigma_y_1 & rho_1,2 sigma_x_2 sigma_y_1 & sigma_y_1^2 & rho_ysigma_y_1sigma_y_2\
      rho_1,2 sigma_x_1 sigma_y_2 & rho_2,2 sigma_x_2sigma_y_2 & rho_ysigma_y_1sigma_y_2 & sigma_y_2^2 \
      endbmatrix = beginbmatrix A_2times2 & B_2times2 \ B^top_2times2 & C_2times2 \ endbmatrix $$



      I think we can generate $[Y_1, Y_2]$ by conditioning on $[X_1, X_2]$ and calculate the new conditional covariance, $tildeC = beginbmatrix C - B^top A^-1B endbmatrix$. Assuming that $A,B$ and $C$ are positive semidefinite (PSD), but not $M$, is it still possible to generate $[Y_1,Y_2]$?! or are we going to get always a non-PSD $tildeC$.










      share|cite|improve this question











      $endgroup$




      Assume we are given two normally distributed random variables, $X_1$ and $X_2$, with $X_i sim mathcal N (0, sigma_x_i^2)$, with correlation coefficient $rho_x$. Assume further that we need to generate another two normally distributed random variables $Y_1$ and $Y_2$ with $Y_j sim mathcal N (0,sigma_y_j^2)$, with correlation coefficient $rho_y$ and cross correlation



      $$rho_i,j =fracmathbbEX_i Y_jsigma_x_i sigma_y_i$$



      I.e., we need to generate $[Y_1, Y_2]$ taken into consideration the given $[X_1, X_2]$. I know how to generate all of them together $[X_1,X_2, Y_1, Y_2]$ using covariance matrix



      $$M_4times4 =
      beginbmatrix
      sigma_x_1^2 &rho_x sigma_x_1sigma_x_2 &rho_1,1sigma_x_1 sigma_y_1 &rho_1,2sigma_x_1sigma_y_2\
      rho_x sigma_x_1sigma_x_2 &sigma_x_2^2 &rho_2,1sigma_x_2 sigma_y_1 & rho_2,2sigma_x_2 sigma_y_2\
      rho_1,1 sigma_x_1 sigma_y_1 & rho_1,2 sigma_x_2 sigma_y_1 & sigma_y_1^2 & rho_ysigma_y_1sigma_y_2\
      rho_1,2 sigma_x_1 sigma_y_2 & rho_2,2 sigma_x_2sigma_y_2 & rho_ysigma_y_1sigma_y_2 & sigma_y_2^2 \
      endbmatrix = beginbmatrix A_2times2 & B_2times2 \ B^top_2times2 & C_2times2 \ endbmatrix $$



      I think we can generate $[Y_1, Y_2]$ by conditioning on $[X_1, X_2]$ and calculate the new conditional covariance, $tildeC = beginbmatrix C - B^top A^-1B endbmatrix$. Assuming that $A,B$ and $C$ are positive semidefinite (PSD), but not $M$, is it still possible to generate $[Y_1,Y_2]$?! or are we going to get always a non-PSD $tildeC$.







      statistics numerical-linear-algebra conditional-probability






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 19 hours ago







      MrX

















      asked yesterday









      MrXMrX

      302313




      302313




















          1 Answer
          1






          active

          oldest

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          0












          $begingroup$

          You have a small typo: the conditional covariance is $C - B^top A^-1 B$.



          If you take for granted that the formula for conditional covariance is correct, then you know it must be PSD simply because it is a covariance matrix. (Similarly, you know $A$ and $C$ must be PSD since they are covariance matrices for $(X_1, X_2)$ and $(Y_1, Y_2)$).



          For general matrices that don't have any context as covariance matrices, you can refer to these results on the Schur complement.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            @MrX But why do you think $M$ is not PSD?
            $endgroup$
            – angryavian
            19 hours ago










          • $begingroup$
            I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
            $endgroup$
            – MrX
            19 hours ago











          Your Answer





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          1 Answer
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          0












          $begingroup$

          You have a small typo: the conditional covariance is $C - B^top A^-1 B$.



          If you take for granted that the formula for conditional covariance is correct, then you know it must be PSD simply because it is a covariance matrix. (Similarly, you know $A$ and $C$ must be PSD since they are covariance matrices for $(X_1, X_2)$ and $(Y_1, Y_2)$).



          For general matrices that don't have any context as covariance matrices, you can refer to these results on the Schur complement.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            @MrX But why do you think $M$ is not PSD?
            $endgroup$
            – angryavian
            19 hours ago










          • $begingroup$
            I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
            $endgroup$
            – MrX
            19 hours ago
















          0












          $begingroup$

          You have a small typo: the conditional covariance is $C - B^top A^-1 B$.



          If you take for granted that the formula for conditional covariance is correct, then you know it must be PSD simply because it is a covariance matrix. (Similarly, you know $A$ and $C$ must be PSD since they are covariance matrices for $(X_1, X_2)$ and $(Y_1, Y_2)$).



          For general matrices that don't have any context as covariance matrices, you can refer to these results on the Schur complement.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            @MrX But why do you think $M$ is not PSD?
            $endgroup$
            – angryavian
            19 hours ago










          • $begingroup$
            I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
            $endgroup$
            – MrX
            19 hours ago














          0












          0








          0





          $begingroup$

          You have a small typo: the conditional covariance is $C - B^top A^-1 B$.



          If you take for granted that the formula for conditional covariance is correct, then you know it must be PSD simply because it is a covariance matrix. (Similarly, you know $A$ and $C$ must be PSD since they are covariance matrices for $(X_1, X_2)$ and $(Y_1, Y_2)$).



          For general matrices that don't have any context as covariance matrices, you can refer to these results on the Schur complement.






          share|cite|improve this answer









          $endgroup$



          You have a small typo: the conditional covariance is $C - B^top A^-1 B$.



          If you take for granted that the formula for conditional covariance is correct, then you know it must be PSD simply because it is a covariance matrix. (Similarly, you know $A$ and $C$ must be PSD since they are covariance matrices for $(X_1, X_2)$ and $(Y_1, Y_2)$).



          For general matrices that don't have any context as covariance matrices, you can refer to these results on the Schur complement.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered yesterday









          angryavianangryavian

          42k23381




          42k23381











          • $begingroup$
            I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            @MrX But why do you think $M$ is not PSD?
            $endgroup$
            – angryavian
            19 hours ago










          • $begingroup$
            I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
            $endgroup$
            – MrX
            19 hours ago

















          • $begingroup$
            I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
            $endgroup$
            – MrX
            19 hours ago










          • $begingroup$
            @MrX But why do you think $M$ is not PSD?
            $endgroup$
            – angryavian
            19 hours ago










          • $begingroup$
            I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
            $endgroup$
            – MrX
            19 hours ago
















          $begingroup$
          I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
          $endgroup$
          – MrX
          19 hours ago




          $begingroup$
          I corrected the typo, it is due to the fact I was initially using $sigma_x_i = sigma_x$, i.e.,$B$ would be symmetric.
          $endgroup$
          – MrX
          19 hours ago












          $begingroup$
          But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
          $endgroup$
          – MrX
          19 hours ago




          $begingroup$
          But the core of my question boils down to the following: I think from the "if and only of" when $M$ is not PSD then $tildeC$ is not PSD, is this correct? and thus we cannot generate $[Y_1, Y_2]$ (unless we use the generalized inverse)
          $endgroup$
          – MrX
          19 hours ago












          $begingroup$
          @MrX But why do you think $M$ is not PSD?
          $endgroup$
          – angryavian
          19 hours ago




          $begingroup$
          @MrX But why do you think $M$ is not PSD?
          $endgroup$
          – angryavian
          19 hours ago












          $begingroup$
          I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
          $endgroup$
          – MrX
          19 hours ago





          $begingroup$
          I am assuming it is given that $M$ is not a PSD, I was thinking if there is a work around to generate $[X_1, X_2, Y_1,Y_2]$ when $M$ is not PSD, and then thought of this iterative method (first $[X_1, X_2]$ then $[Y_1,Y_2]$)... frankly it would make sense that $tildeC$ is not PSD if $M$ is not.
          $endgroup$
          – MrX
          19 hours ago


















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          Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee