Covariance matrices
0
Consider two matrices of random variables, X and W, of same dimension.
Now, consider the product X'W.
I've been told that this matrix gives us the covariances between the elements of X and W. However, this is not immediately apparent to me. You do get sums of products, but these aren't exactly covariances.
statistics
asked Dec 5 '18 at 2:44
StudentStudent
5811
|
show 1 more comment
0
Consider two matrices of random variables, X and W, of same dimension.
Now, consider the product X'W.
I've been told that this matrix gives us the covariances between the elements of X and W. However, this is not immediately apparent to me. You do get sums of products, but these aren't exactly covariances.
statistics
asked Dec 5 '18 at 2:44
StudentStudent
5811
1
Usually, the covariance matrix refers to the expectation of the outer product of two vectors of random variables, i.e. the $n times n$ real valued matrix $mathbb{E}[bf{X} bf{X}^t]$ where $bf{X}$ $= [X_1,...,X_n]$ is an $n times 1$ vector of random variables.
– maridia
Dec 5 '18 at 2:56
What you are referring to sounds like a $textit{random}$ sample covariance matrix. See Wishart matrix
– maridia
Dec 5 '18 at 2:59
Thanks, but does the matrix that I have mentioned give some sort of indication about covariances? Intuitively, I think it does, but am not convinced.
– Student
Dec 5 '18 at 3:06
It does if the columns of X and W are i.i.d. random variables. Then the $ab$ entry of $X^tW$ is $sum_i X_{ai} W_{bi}$, which can be thought of as a sample covariance between the random variables $X_a$ and $W_b$ where $X_a$ is distributed like the entries in column $a$ of $X$ and $W_b$ is distributed like the entries in column $b$ of $W$.
– maridia
Dec 5 '18 at 3:38
Not exactly the sample covariances right? Need to subtract sample averages.
– Student
Dec 5 '18 at 8:57
|
show 1 more comment
0
0
0
Consider two matrices of random variables, X and W, of same dimension.
Now, consider the product X'W.
I've been told that this matrix gives us the covariances between the elements of X and W. However, this is not immediately apparent to me. You do get sums of products, but these aren't exactly covariances.
statistics
asked Dec 5 '18 at 2:44
StudentStudent
5811
Consider two matrices of random variables, X and W, of same dimension.
Now, consider the product X'W.
I've been told that this matrix gives us the covariances between the elements of X and W. However, this is not immediately apparent to me. You do get sums of products, but these aren't exactly covariances.
statistics
statistics
asked Dec 5 '18 at 2:44
StudentStudent
5811
asked Dec 5 '18 at 2:44
StudentStudent
5811
asked Dec 5 '18 at 2:44
StudentStudent
5811
asked Dec 5 '18 at 2:44
StudentStudent
5811
asked Dec 5 '18 at 2:44
StudentStudent
5811
5811
1
Usually, the covariance matrix refers to the expectation of the outer product of two vectors of random variables, i.e. the $n times n$ real valued matrix $mathbb{E}[bf{X} bf{X}^t]$ where $bf{X}$ $= [X_1,...,X_n]$ is an $n times 1$ vector of random variables.
– maridia
Dec 5 '18 at 2:56
What you are referring to sounds like a $textit{random}$ sample covariance matrix. See Wishart matrix
– maridia
Dec 5 '18 at 2:59
Thanks, but does the matrix that I have mentioned give some sort of indication about covariances? Intuitively, I think it does, but am not convinced.
– Student
Dec 5 '18 at 3:06
It does if the columns of X and W are i.i.d. random variables. Then the $ab$ entry of $X^tW$ is $sum_i X_{ai} W_{bi}$, which can be thought of as a sample covariance between the random variables $X_a$ and $W_b$ where $X_a$ is distributed like the entries in column $a$ of $X$ and $W_b$ is distributed like the entries in column $b$ of $W$.
– maridia
Dec 5 '18 at 3:38
Not exactly the sample covariances right? Need to subtract sample averages.
– Student
Dec 5 '18 at 8:57
|
show 1 more comment
1
Usually, the covariance matrix refers to the expectation of the outer product of two vectors of random variables, i.e. the $n times n$ real valued matrix $mathbb{E}[bf{X} bf{X}^t]$ where $bf{X}$ $= [X_1,...,X_n]$ is an $n times 1$ vector of random variables.
– maridia
Dec 5 '18 at 2:56
What you are referring to sounds like a $textit{random}$ sample covariance matrix. See Wishart matrix
– maridia
Dec 5 '18 at 2:59
Thanks, but does the matrix that I have mentioned give some sort of indication about covariances? Intuitively, I think it does, but am not convinced.
– Student
Dec 5 '18 at 3:06
It does if the columns of X and W are i.i.d. random variables. Then the $ab$ entry of $X^tW$ is $sum_i X_{ai} W_{bi}$, which can be thought of as a sample covariance between the random variables $X_a$ and $W_b$ where $X_a$ is distributed like the entries in column $a$ of $X$ and $W_b$ is distributed like the entries in column $b$ of $W$.
– maridia
Dec 5 '18 at 3:38
Not exactly the sample covariances right? Need to subtract sample averages.
– Student
Dec 5 '18 at 8:57
1
1
Usually, the covariance matrix refers to the expectation of the outer product of two vectors of random variables, i.e. the $n times n$ real valued matrix $mathbb{E}[bf{X} bf{X}^t]$ where $bf{X}$ $= [X_1,...,X_n]$ is an $n times 1$ vector of random variables.
– maridia
Dec 5 '18 at 2:56
Usually, the covariance matrix refers to the expectation of the outer product of two vectors of random variables, i.e. the $n times n$ real valued matrix $mathbb{E}[bf{X} bf{X}^t]$ where $bf{X}$ $= [X_1,...,X_n]$ is an $n times 1$ vector of random variables.
– maridia
Dec 5 '18 at 2:56
What you are referring to sounds like a $textit{random}$ sample covariance matrix. See Wishart matrix
– maridia
Dec 5 '18 at 2:59
What you are referring to sounds like a $textit{random}$ sample covariance matrix. See Wishart matrix
– maridia
Dec 5 '18 at 2:59
Thanks, but does the matrix that I have mentioned give some sort of indication about covariances? Intuitively, I think it does, but am not convinced.
– Student
Dec 5 '18 at 3:06
Thanks, but does the matrix that I have mentioned give some sort of indication about covariances? Intuitively, I think it does, but am not convinced.
– Student
Dec 5 '18 at 3:06
It does if the columns of X and W are i.i.d. random variables. Then the $ab$ entry of $X^tW$ is $sum_i X_{ai} W_{bi}$, which can be thought of as a sample covariance between the random variables $X_a$ and $W_b$ where $X_a$ is distributed like the entries in column $a$ of $X$ and $W_b$ is distributed like the entries in column $b$ of $W$.
– maridia
Dec 5 '18 at 3:38
It does if the columns of X and W are i.i.d. random variables. Then the $ab$ entry of $X^tW$ is $sum_i X_{ai} W_{bi}$, which can be thought of as a sample covariance between the random variables $X_a$ and $W_b$ where $X_a$ is distributed like the entries in column $a$ of $X$ and $W_b$ is distributed like the entries in column $b$ of $W$.
– maridia
Dec 5 '18 at 3:38
Not exactly the sample covariances right? Need to subtract sample averages.
– Student
Dec 5 '18 at 8:57
Not exactly the sample covariances right? Need to subtract sample averages.
– Student
Dec 5 '18 at 8:57
|
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1
Usually, the covariance matrix refers to the expectation of the outer product of two vectors of random variables, i.e. the $n times n$ real valued matrix $mathbb{E}[bf{X} bf{X}^t]$ where $bf{X}$ $= [X_1,...,X_n]$ is an $n times 1$ vector of random variables.
– maridia
Dec 5 '18 at 2:56
What you are referring to sounds like a $textit{random}$ sample covariance matrix. See Wishart matrix
– maridia
Dec 5 '18 at 2:59
Thanks, but does the matrix that I have mentioned give some sort of indication about covariances? Intuitively, I think it does, but am not convinced.
– Student
Dec 5 '18 at 3:06
It does if the columns of X and W are i.i.d. random variables. Then the $ab$ entry of $X^tW$ is $sum_i X_{ai} W_{bi}$, which can be thought of as a sample covariance between the random variables $X_a$ and $W_b$ where $X_a$ is distributed like the entries in column $a$ of $X$ and $W_b$ is distributed like the entries in column $b$ of $W$.
– maridia
Dec 5 '18 at 3:38
Not exactly the sample covariances right? Need to subtract sample averages.
– Student
Dec 5 '18 at 8:57