JSFHJSFHGeneration of correlated random numbers from univariate distributionsIgor HlivkaMUFJ Securities International, LONDONThis application is an extension of an earlier document on multivariate distributions and demonstrates how Maple can be used to generate random samples from such distribution. In a narrow sense, it presents the tool for generation of correlated samples. The sampling need for multi-factor random variables (RV) with a given correlation structure arises in many applications in economics, finance, but also in natural sciences such as genetics, physics etc. and here we show that such task can be accomplished with ease using Maple\342\200\231s Statistics and Linear Algebra packages. The multivariate RV are typically generated from univariate samples and translated into correlated ones with matrix transformation techniques. Here we will be presenting the two most common techniques: LU Decomposition SV DecompositionThe presented cases are generic - i.e. not restricted in dimensionality space so the user can create multivariate samples of the required size and dimension.
<Text-field style="Heading 1" layout="Heading 1">Generation of univariate samples</Text-field>We will first create univariate samples from a given distribution - in this particular case we set this top be the Standard Normal Distribution. For demonstration purposes, we limit the case to three dimensions: n= 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print(); # input placeholderWe create three independent vectors with 100,000 RV elements and create a RV matrixQzAtSSpSYW5kb21pemVHNiJGJSEiIj5JIk5HRiUiJysrNUYmPkkjUzFHRiUtSSQ8LD5HRiU2Iy1JJ1NhbXBsZUdGJTYkSSJYR0YlRihGJj5JI1MyR0YlRixGJj5JI1MzR0YlRixGJj5JIk1HRiUtRi02JUYrRjRGNkYmPkkjTTFHRiUtSSpUcmFuc3Bvc2VHRiU2I0Y4IiIiprint(); # input placeholderWe define the ccorrelation matrix CM [3 x 3]
QyQ+SSNDTUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjNyU3JSIiIiQiIiQhIiIkRjFGMTclRi9GLiQiIiZGMTclRjJGNEYuRi4=print(); # input placeholderJSFH
<Text-field style="Heading 1" layout="Heading 1">Correlated RV through LU Decomposition</Text-field>JSFHMatrix factorization is a general method to convert univariate RVs into a correlated counterparts when correlation matrix is defined.The task is to find a matrix U such that 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 of correlated RV is them obtained simply as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJVRidGL0YyRjk=JSFHQyQ+SSNDREc2Ii1JMExVRGVjb21wb3NpdGlvbkdGJTYlSSNDTUdGJS9JJ21ldGhvZEdGJS5JKUNob2xlc2t5R0YlL0knb3V0cHV0R0YlNyMuSSJVR0YlIiIiprint(); # input placeholderJSFHWe can test if this is the correct factorizationQyQtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiQtSSpUcmFuc3Bvc2VHRik2I0kjQ0RHRilGMSIiIg==print(); # input placeholderJSFHThis matches exactly the original input correlation matrixQyQ+JkknQ29yclJWRzYiNiNJJExVREdGJi1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiZJLFR5cGVzZXR0aW5nRzYkRixJKF9zeXNsaWJHRiY2JEkjTTFHRiZJI0NER0YmIiIiprint(); # input placeholder
We can verify if the three-dimensional matrix Hs been successfully transformed by computing the correlation coefficients from the computed result:QyQ+SShUZXN0TFVERzYiLUkyQ29ycmVsYXRpb25NYXRyaXhHRiU2IyZJJ0NvcnJSVkdGJTYjSSRMVURHRiUiIiI=print(); # input placeholderAs we can see, the returned correlation matrix closely matches the original CM. If better fit is required, the user will need to increase the sample size.JSFH
<Text-field style="Heading 1" layout="Heading 1">Correlated RV through SV Decomposition</Text-field>The second method resides on Singular value decomposition technique (SVD) that uses eigenvectors and eigenvalues approach.We will achieve this in following steps:Compute SV decomposition of the correlation matrix CMObtain eigenvalues of CMCreate Diagonal matrix of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNiQtSSVtc3ViR0YkNiUtRiw2JVEnJiM5NDc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUYsNiZRImlGJy9GO1EldHJ1ZUYnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0ZGL0Y+USdpdGFsaWNGJ0ZFRkdGSS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GPQ== where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NDc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRImlGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 is the i-the eigenvalue of CMObtain the resultant matrix 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 RV matrix = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJXRidGL0YyLUYjNiUtRiw2JVEiVEYnRi9GMkYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjk=JSFHQyw+SSNTVkc2Ii1JL1Npbmd1bGFyVmFsdWVzR0YlNiRJI0NNR0YlL0knb3V0cHV0R0YlLkkiVUdGJSEiIj5JI1N2R0YlLUYnNiRGKS9GKy5JIlNHRiVGLj5JI0RNR0YlLUkvRGlhZ29uYWxNYXRyaXhHRiU2I0YwIiIiPkkkRE0yR0YlLUksTWF0cml4UG93ZXJHRiU2JEY3I0Y7IiIjRjs+SSVTVkRNR0YlLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHNiRGSEkoX3N5c2xpYkdGJTYkRiRGPUY7print(); # input placeholderSVDM is the resultant matrix we will need to transform uncorrelated RV into correlated ones:QyQ+JkknQ29yclJWRzYiNiNJJFNWREdGJi1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiZJLFR5cGVzZXR0aW5nRzYkRixJKF9zeXNsaWJHRiY2JEkjTTFHRiYtSSpUcmFuc3Bvc2VHRiY2I0klU1ZETUdGJiIiIg==print(); # input placeholderWe can check again if the generated multivariate sample is consistent with the input:QyQtSTJDb3JyZWxhdGlvbk1hdHJpeEc2IjYjJkknQ29yclJWR0YlNiNJJFNWREdGJSIiIg==print(); # input placeholderAs in the previous case, the returned correlation matrix closely resembles the original CM
<Text-field style="Heading 1" layout="Heading 1">Visualization examples</Text-field>We will present few examples in bi-varaite space to show how correlation changes the behavior of original RV:We will use "high" correlation coefficient to highlight the correlation effect:
<Text-field style="Heading 2" layout="Heading 2">Normal distribution</Text-field>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%;JSFH
<Text-field style="Heading 2" layout="Heading 2">LogNormal distribution</Text-field>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%;
<Text-field style="Heading 2" layout="Heading 2">Gamma distribution</Text-field>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%;JSFH
<Text-field style="Heading 2" layout="Heading 2">Student T distribution</Text-field>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%;JSFH
<Text-field style="Heading 2" layout="Heading 2">Trinomial Normal</Text-field>We will use the correlation matrix CM [3x3] defined above in this example
QzY+SSJRRzYiIiQrJiEiIj5JI0ExR0YlLUknU2FtcGxlR0YlNiRJIlhHRiVGJEYnPkkjQTJHRiVGKkYnPkkjQTNHRiVGKkYnPkkjTWFHRiUtSSQ8LD5HSShfc3lzbGliR0YlNiUtRjU2I0YpLUY1NiNGLy1GNTYjRjFGJz5JI01BR0YlLUkqVHJhbnNwb3NlR0YlNiNGM0YnPkkkQ2QyR0YlLUkwTFVEZWNvbXBvc2l0aW9uR0YlNiVJI0NNR0YlL0knbWV0aG9kR0YlLkkpQ2hvbGVza3lHRiUvSSdvdXRwdXRHRiU3Iy5JIlVHRiVGJz5JJFJzMkdGJS1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nRzYkRldGNjYkRj9GRCIiIi1JMUFnZ2xvbWVyYXRlZFBsb3RHRiU2KEYpRi9GMS9JIm5HRiUiIiQvSSdsZW5ndGhHRlckIiNOISIjL0kmdGl0bGVHRiU3JFEuUmF3fk5vcm1hbH5SVkYlL0klZm9udEdGJTclSSZUSU1FU0dGJUklQk9MREdGJSIjN0Znbi1GaW42KC1JJ0NvbHVtbkdGJTYkRlNGZ24tRmBwNiRGUyIiIy1GYHA2JEZTRl1vRltvRl5vL0ZkbzckUTVDb3JyZWxhdGVkfk5vcm1hbH5SVkYlRmdvRmdu%;JSFHJSFH