定义一个计算相关矩阵的协方差矩阵的函数 [英] Defining a function that calculates the covariance-matrix of a correlation-matrix
问题描述
我在变换矩阵以及行和列的名称时遇到一些问题。
我的问题如下：
作为输入矩阵，我有一个（对称）相关矩阵，就像这样：
相关矢量由下三角矩阵的值给出：
方差-协方差矩阵：
方差可以用
-> N是样本大小（在此示例中N = 66）
协方差可以近似为
例如，r_02和r_13之间的协方差由
$给出b$ b
现在，我想在R中定义一个函数，该函数将相关矩阵作为输入并返回方差-协方差矩阵。但是，我在实现协方差的计算方面遇到问题。我的想法是给相关向量的元素命名，如上所示（r_01，r_02 ...）。然后，我想创建一个空的方差-协方差矩阵，它的长度为correlation_vector。行和列应与correlation_vector具有相同的名称，因此我可以通过[01] [03]来调用它们。然后，我要实现一个for循环，该循环将i和j以及k和l的值设置为协方差公式中所示的相关值，以作为我作为协方差公式输入的相关列和行。这些值必须始终是六个不同的值（ij； ik； il； jk； jl； lk）。这是我的主意，但是我现在不知道如何在R中实现它。
这是我的代码（不计算协方差）：
require（corpcor）
related_matrix_input<-matrix（data = c（1.00,0.561,0.393,0.561， 0.561,1.00,0.286,0.549,0.393,0.286,1.00,0.286,0.561,0.549,0.286,1.00），ncol = 4，byrow = T）
N <-66＃样本量
vector_of_correlations <-sm2vec（correlation_matrix_input，diag = F）#correlation_matrix_input的下三角矩阵
variance_covariance_matrix <-matrix（nrow = length（length（vector_of_correlations），ncol = length（ vector_of_correlations））＃创建空方差-协方差矩阵
＃函数通过计算方差和协方差
variance_covariances <-函数来填充矩阵vector_of_correlations_input，sample_size）{
for（i in（seq（along = vector_of_correlations_input）））{
for（j in（seq（along = vector_ of_correlations_input）））{
＃如果（i == j）{
variance_covariance_matrix [i，j] =（（（1-vector_of_correlations_input [i ] ** 2）** 2）/ sample_size
}
＃如果（i！= j）{
variance_covariance_matrix [ i，j] = ???
}
}
}
return（variance_covariance_matrix）;
}
任何人都有一个想法，如何使用来实现协方差的计算。上面显示的公式？
对于这个问题，我将不胜感激！
如果将 r 保留为矩阵，并使用此辅助函数使情况更清晰，则会更容易：
covr<-function（r，i，j，k，l，n）{
if（i == k& j = = l）
return（（1-r [i，j] ^ 2）^ 2 / n）
（0.5 * r [i，j] * r [k，l] *（r [ i，k] ^ 2 + r [i，l] ^ 2 + r [j，k] ^ 2 + r [j，l] ^ 2）+
r [i，k] * r [j，l ] + r [i，l] * r [j，k]-（r [i，j] * r [i，k] * r [i，l] +
r [j，i] * r [ j，k] * r [j，l] + r [k，i] * r [k，j] * r [k，l] + r [l，i] * r [l，j] * r [l ，k]））/ n
}
现在定义第二个函数：
vcovr<-函数（r，n）{
p<-combn（nrow（r），2）
q<-seq（ncol（p））
外（q，q，Vectorize（函数（x，y）covr（r，p [1，x ]，p [2，x]，p [1，y]，p [2，y]，n）））
}
然后瞧：
> vcovr（correlation_matrix_input，66）
[，1] [，2] [，3] [，4] [，5] [，6]
[1，] 0.007115262 0.001550264 0.002917481 0.003047666 0.003101602 0.001705781
[2，] 0.001550264 0.010832674 0.001550264 0.006109565 0.001127916 0.006109565
[3，] 0.002917481 0.001550264 0.007115262 0.001705781 0.003101602 0.003047666
[4，] 0.003047666 0.006109565 0.001705781 0.012774221 0.002036422 0.006625868 $ b $ 0.003101602 0.002036422 0.007394554 0.002036422
[6，] 0.001705781 0.006109565 0.003047666 0.006625868 0.002036422 0.012774221
EDIT：
对于转换后的Z值，如您的注释中所示，您可以使用以下代码：
covrZ<-function（r，i，j，k，l，n）{
if（i == k&& j == l）
return（1 / （n-3））
covr（r，i，j，k，l，n）/（（1-r [i，j] ^ 2）*（1-r [k，l] ^ 2 ））
}
只需将其替换为 vcovr ：
vcovrZ <-function（r，n）{
p< -combn（nrow（r），2）
q<-seq（ncol（p））
external（q，q，Vectorize（function（x，y）covrZ（r，p [1， x]，p [2，x]，p [1，y]，p [2，y]，n）））
}
新结果：
> vcovrZ（correlation_matrix_input，66）
[，1] [，2] [，3] [，4] [，5] [，6]
[1，] 0.015873016 0.002675460 0.006212598 0.004843517 0.006478743 0.002710920
[2，] 0.002675460 0.015873016 0.002675460 0.007869213 0.001909452 0.007869213
[3，] 0.006212598 0.002675460 0.015873016 0.002710920 0.006478743 0.004843517
[4，] 0.004843517 0.007869213 0.002710920 0.015873016 0.003174685 0.003858909 $ 897 0.006478743 0.003174685 0.015873016 0.003174685
[6，] 0.002710920 0.007869213 0.004843517 0.007858948 0.003174685 0.015873016
I have some problems with the transformation of a matrix and the names of the rows and columns.
My problem is as follows:
As input-matrix I have a (symmetric) correlation matrix like this one:
The correlation-vector is given by the values of the lower triangular matrix:
Now, I want to compute the variance-covariance-matrix of the these correlations, which are approximately normally distributed with the variance-covariance-matrix:
The variances can be approximated by
-> N is the sample size (in this example N = 66)
The covariances can be approximated by
For example the covariance between r_02 and r_13 is given by
Now, I want to define a function in R which gets the correlation matrix as input and returns the variance-covariance matrix. However, I have problems to implement the calculation of the covariances. My idea is to give names to the elements of the correlation_vector as shown above (r_01, r_02...). Then I want to create the empty variance-cocariance matrix, which has the length of the correlation_vector. The rows and the columns should have the same names as the correlation_vector, so I can call them for example by [01][03]. Then I want to implement a for-loop which sets the value of i and j as well as k and l as shown in the formula for the covariance to the columns and rows of the correlations that I need as input for the covariance-formula. These must always be six different values (ij; ik; il; jk; jl; lk). This is my idea, but I don't now how to implement this in R.
This is my code (without the calculation of the covariances):
require(corpcor)
correlation_matrix_input <- matrix(data=c(1.00,0.561,0.393,0.561,0.561,1.00,0.286,0.549,0.393,0.286,1.00,0.286,0.561,0.549,0.286,1.00),ncol=4,byrow=T)
N <- 66 # Sample Size
vector_of_correlations <- sm2vec(correlation_matrix_input, diag=F) # lower triangular matrix of correlation_matrix_input
variance_covariance_matrix <- matrix(nrow = length(vector_of_correlations), ncol = length(vector_of_correlations)) # creates the empty variance-covariance matrix
# function to fill the matrix by calculating the variance and the covariances
variances_covariances <- function(vector_of_correlations_input, sample_size) {
for (i in (seq(along = vector_of_correlations_input))) {
for (j in (seq(along = vector_of_correlations_input))) {
# calculate the variances for the diagonale
if (i == j) {
variance_covariance_matrix[i,j] = ((1-vector_of_correlations_input[i]**2)**2)/sample_size
}
# calculate the covariances
if (i != j) {
variance_covariance_matrix[i,j] = ???
}
}
}
return(variance_covariance_matrix);
}
Does anyone have an idea, how to implement the calculation of the covariances using the formula shown above?
I would be grateful for any kind of help regarding this problem!!!
It's easier if you keep r as a matrix and use this helper function to make things clearer:
covr <- function(r, i, j, k, l, n){
if(i==k && j==l)
return((1-r[i,j]^2)^2/n)
( 0.5 * r[i,j]*r[k,l]*(r[i,k]^2 + r[i,l]^2 + r[j,k]^2 + r[j,l]^2) +
r[i,k]*r[j,l] + r[i,l]*r[j,k] - (r[i,j]*r[i,k]*r[i,l] +
r[j,i]*r[j,k]*r[j,l] + r[k,i]*r[k,j]*r[k,l] + r[l,i]*r[l,j]*r[l,k]) )/n
}
Now define this second function:
vcovr <- function(r, n){
p <- combn(nrow(r), 2)
q <- seq(ncol(p))
outer(q, q, Vectorize(function(x,y) covr(r, p[1,x], p[2,x], p[1,y], p[2,y], n)))
}
And voila:
> vcovr(correlation_matrix_input, 66)
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.007115262 0.001550264 0.002917481 0.003047666 0.003101602 0.001705781
[2,] 0.001550264 0.010832674 0.001550264 0.006109565 0.001127916 0.006109565
[3,] 0.002917481 0.001550264 0.007115262 0.001705781 0.003101602 0.003047666
[4,] 0.003047666 0.006109565 0.001705781 0.012774221 0.002036422 0.006625868
[5,] 0.003101602 0.001127916 0.003101602 0.002036422 0.007394554 0.002036422
[6,] 0.001705781 0.006109565 0.003047666 0.006625868 0.002036422 0.012774221
EDIT:
For the transformed Z values, as in your comment, you can use this:
covrZ <- function(r, i, j, k, l, n){
if(i==k && j==l)
return(1/(n-3))
covr(r, i, j, k, l, n) / ((1-r[i,j]^2)*(1-r[k,l]^2))
}
And simply replace it in vcovr:
vcovrZ <- function(r, n){
p <- combn(nrow(r), 2)
q <- seq(ncol(p))
outer(q, q, Vectorize(function(x,y) covrZ(r, p[1,x], p[2,x], p[1,y], p[2,y], n)))
}
New result:
> vcovrZ(correlation_matrix_input,66)
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.015873016 0.002675460 0.006212598 0.004843517 0.006478743 0.002710920
[2,] 0.002675460 0.015873016 0.002675460 0.007869213 0.001909452 0.007869213
[3,] 0.006212598 0.002675460 0.015873016 0.002710920 0.006478743 0.004843517
[4,] 0.004843517 0.007869213 0.002710920 0.015873016 0.003174685 0.007858948
[5,] 0.006478743 0.001909452 0.006478743 0.003174685 0.015873016 0.003174685
[6,] 0.002710920 0.007869213 0.004843517 0.007858948 0.003174685 0.015873016
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