Every year there is at least a couple of occasions when I have to simulate multivariate data that follow a given covariance matrix. For example, let’s say that we want to create an example of the effect of collinearity when fitting multiple linear regressions, so we want to create one variable (the response) that is correlated with a number of explanatory variables **and** the explanatory variables have different correlations with each other.

There is a matrix operation called Cholesky decomposition, sort of equivalent to taking a square root with scalars, that is useful to produce correlated data. If we have a covariance matrix `M`

, the Cholesky descomposition is a lower triangular matrix `L`

, such as that `M = L L'`

. How does this connect to our simulated data? Let’s assume that we generate a vector `z`

of random normally independently distributed numbers with mean zero and variance one (with length equal to the dimension of M), we can create a realization of our multivariate distribution using the product `L z`

.

The reason why this works is that the `Variance(L z) = L Variance(z) L'`

as `L`

is just a constant. The variance of `z`

is the identity matrix `I`

; remember that the random numbers have variance one and are independently distributed. Therefore `Variance(L z) = L I L' = L L` = M`

so, in fact, we are producing random data that follow the desired covariance matrix.

As an example, let’s simulate 100 observations with 4 variables. Because we want to simulate 100 realizations, rather than a single one, it pays to generate a matrix of random numbers with as many rows as variables to simulate and as many columns as observations to simulate.

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library(lattice) # for splom library(car) # for vif # number of observations to simulate nobs = 100 # Using a correlation matrix (let' assume that all variables # have unit variance M = matrix(c(1, 0.7, 0.7, 0.5, 0.7, 1, 0.95, 0.3, 0.7, 0.95, 1, 0.3, 0.5, 0.3, 0.3, 1), nrow=4, ncol=4) # Cholesky decomposition L = chol(M) nvars = dim(L)[1] # R chol function produces an upper triangular version of L # so we have to transpose it. # Just to be sure we can have a look at t(L) and the # product of the Cholesky decomposition by itself t(L) [,1] [,2] [,3] [,4] [1,] 1.0 0.0000000 0.00000000 0.0000000 [2,] 0.7 0.7141428 0.00000000 0.0000000 [3,] 0.7 0.6441288 0.30837970 0.0000000 [4,] 0.5 -0.0700140 -0.01589586 0.8630442 t(L) %*% L [,1] [,2] [,3] [,4] [1,] 1.0 0.70 0.70 0.5 [2,] 0.7 1.00 0.95 0.3 [3,] 0.7 0.95 1.00 0.3 [4,] 0.5 0.30 0.30 1.0 # Random variables that follow an M correlation matrix r = t(L) %*% matrix(rnorm(nvars*nobs), nrow=nvars, ncol=nobs) r = t(r) rdata = as.data.frame(r) names(rdata) = c('resp', 'pred1', 'pred2', 'pred3') # Plotting and basic stats splom(rdata) cor(rdata) # We are not that far from what we want to simulate! resp pred1 pred2 pred3 resp 1.0000000 0.7347572 0.7516808 0.3915817 pred1 0.7347572 1.0000000 0.9587386 0.2841598 pred2 0.7516808 0.9587386 1.0000000 0.2942844 pred3 0.3915817 0.2841598 0.2942844 1.0000000 |

Now we can use the simulated data to learn something about the effects of collinearity when fitting multiple linear regressions. We will first fit two models using two predictors with low correlation between them, and then fit a third model with three predictors where `pred1`

and `pred2`

are highly correlated with each other.

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# Model 1: predictors 1 and 3 (correlation is 0.28) m1 = lm(resp ~ pred1 + pred3, rdata) summary(m1) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.07536 0.06812 1.106 0.27133 pred1 0.67316 0.06842 9.838 2.99e-16 *** pred3 0.20920 0.07253 2.884 0.00483 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.6809 on 97 degrees of freedom Multiple R-squared: 0.5762, Adjusted R-squared: 0.5675 F-statistic: 65.95 on 2 and 97 DF, p-value: < 2.2e-16 # Model 2: predictors 2 and 3 (correlation is 0.29) m2 = lm(resp ~ pred2 + pred3, rdata) summary(m2) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.06121 0.06649 0.921 0.35953 pred2 0.68513 0.06633 10.329 < 2e-16 *** pred3 0.19624 0.07097 2.765 0.00681 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.6641 on 97 degrees of freedom Multiple R-squared: 0.5968, Adjusted R-squared: 0.5885 F-statistic: 71.79 on 2 and 97 DF, p-value: < 2.2e-16 # Model 3: correlation between predictors 1 and 2 is 0.96 m3 = lm(resp ~ pred1 + pred2 + pred3, rdata) summary(m3) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.06421 0.06676 0.962 0.33856 pred1 0.16844 0.22560 0.747 0.45712 pred2 0.52525 0.22422 2.343 0.02122 * pred3 0.19584 0.07114 2.753 0.00706 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.6657 on 96 degrees of freedom Multiple R-squared: 0.5991, Adjusted R-squared: 0.5866 F-statistic: 47.83 on 3 and 96 DF, p-value: < 2.2e-16 # Variance inflation vif(m3) pred1 pred2 pred3 12.373826 12.453165 1.094875 |

In my example it is possible to see the huge increase for the standard error for `pred1`

and `pred2`

, when we use both highly correlated explanatory variables in model 3. In addition, model fit does not improve for model 3.

Unfortunately there are numerous problems when you use Cholesky decomposition for large matrices. You have a nice example of 4×4 matrix. When you try to do the same, for example, for images 1000×1000, very often it takes an hour and the matrix is not positive definite because of the rounding. There are some methods to escape computational difficulties, but they often do not work well. As usual, nice theory needs some changes to apply to real data 🙂

Hi Andiy, At least when working with the analysis of experiments we tend not to use such high dimensional matrices, particularly because we want to analyze the problem using multivariate mixed models. At the analysis level, anything beyond, say, five traits will make life miserable to achieve convergence when estimating a positive definite matrix. From an analysis point of view we would probably move to a factor analytic decomposition.

Hi Luis,

I am looking for this kind of R code. I want to generate data following a given covariance structure. I wonder whether I can use this R code for my research. If so, what is the correct citation for the code? Thanks a million!!

Yoonjeong

Sure you can use it; that’s why I put it here! Correct citation depends on the standard you are using; see examples here. If you want to reference a book search in google books for cholesky and simulation.

Hi Luis,

I came across your post while trying to solve a similar problem. I am trying to simulate a field of non-stationary, single variable spatial data in such a way that I can tweak various parameters and observe the effects.

Specifically, I am trying to recreate non-stationary data following a Matérn covariance structure based on equation (9) in this paper:

http://arxiv.org/pdf/1411.3174.pdf

Do you know if it would be correct to adapt your technique to the “Omega” covariates in the equation, for each specific site? And would it work to model a domain of, say, 100 x 100 points?

My apologies if this is confusing – I am confused myself! I am a geologist who has wandered into spatial statistics…

Thank you kindly,

-James

Sorry James, but I do not know how to do that from the top of my head (and can’t dedicate the time to figure it out right now).

Could you help me, please? How can I simulate data with your method if I would like to have negative correlation too? Like this one:

M=matrix(c(1, 0.6, 0.6, 0.6,

0.6, 1,-0.2,0,

0.6, -0.2,1,0,

0.6,0,0,1),nrow=4, ncol=4)

Why doesn’t it work with negative correlations?

The simple answer is that it works with any positive-definite covariance matrix, which includes matrices with negative correlations. One way to check is that the determinant of the matrix has to be positive, which is not for your matrix

`det(M)`

is -0.25.This suggests that the correlation structure represented by your matrix does not make sense. For example, you have a positive correlation between vars 1 & 2 and vars 1 & 3, but a negative correlation between vars 2 & 3.