Approximation of an Exponential Correlation Matrix with a Constant Correlation Matrix
Clash Royale CLAN TAG#URR8PPP
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I am given with a $N times N$ matrix
$$
beginalign*
A =
beginbmatrix
1 & rho_h & ldots & rho_h^N - 1 \
rho_h & 1 & ldots & rho_h^N - 2 \
vdots & vdots & ddots & vdots \
rho_h^N - 1 & rho_h^N - 2 & ldots & 1
endbmatrix,
endalign*
$$
where $0 leq rho_h leq 1$ and $N geq 2$. I would like to approimate A with another $N times N$ matrix given as:
$$
B= beginbmatrix
1 & rho & ldots & rho \
rho & 1 & ldots & rho \
vdots & vdots & ddots & vdots \
rho & rho & ldots & 1
endbmatrix.
$$
This basically means, finding a relation between $rho$ and $rho_h$ such that $A approx B$.
Any thoughts on, how can this be achieved?
linear-algebra matrices approximation
asked Sep 4 at 5:35
Kamal K Garg
12
add a comment |Â
up vote
0
down vote
favorite
I am given with a $N times N$ matrix
$$
beginalign*
A =
beginbmatrix
1 & rho_h & ldots & rho_h^N - 1 \
rho_h & 1 & ldots & rho_h^N - 2 \
vdots & vdots & ddots & vdots \
rho_h^N - 1 & rho_h^N - 2 & ldots & 1
endbmatrix,
endalign*
$$
where $0 leq rho_h leq 1$ and $N geq 2$. I would like to approimate A with another $N times N$ matrix given as:
$$
B= beginbmatrix
1 & rho & ldots & rho \
rho & 1 & ldots & rho \
vdots & vdots & ddots & vdots \
rho & rho & ldots & 1
endbmatrix.
$$
This basically means, finding a relation between $rho$ and $rho_h$ such that $A approx B$.
Any thoughts on, how can this be achieved?
linear-algebra matrices approximation
asked Sep 4 at 5:35
Kamal K Garg
12
Do you want a 1 in the lower right corners?
– AlgebraicsAnonymous
Sep 4 at 5:51
Yes... made correction. Thanks!
– Kamal K Garg
Sep 4 at 8:36
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am given with a $N times N$ matrix
$$
beginalign*
A =
beginbmatrix
1 & rho_h & ldots & rho_h^N - 1 \
rho_h & 1 & ldots & rho_h^N - 2 \
vdots & vdots & ddots & vdots \
rho_h^N - 1 & rho_h^N - 2 & ldots & 1
endbmatrix,
endalign*
$$
where $0 leq rho_h leq 1$ and $N geq 2$. I would like to approimate A with another $N times N$ matrix given as:
$$
B= beginbmatrix
1 & rho & ldots & rho \
rho & 1 & ldots & rho \
vdots & vdots & ddots & vdots \
rho & rho & ldots & 1
endbmatrix.
$$
This basically means, finding a relation between $rho$ and $rho_h$ such that $A approx B$.
Any thoughts on, how can this be achieved?
linear-algebra matrices approximation
asked Sep 4 at 5:35
Kamal K Garg
12
I am given with a $N times N$ matrix
$$
beginalign*
A =
beginbmatrix
1 & rho_h & ldots & rho_h^N - 1 \
rho_h & 1 & ldots & rho_h^N - 2 \
vdots & vdots & ddots & vdots \
rho_h^N - 1 & rho_h^N - 2 & ldots & 1
endbmatrix,
endalign*
$$
where $0 leq rho_h leq 1$ and $N geq 2$. I would like to approimate A with another $N times N$ matrix given as:
$$
B= beginbmatrix
1 & rho & ldots & rho \
rho & 1 & ldots & rho \
vdots & vdots & ddots & vdots \
rho & rho & ldots & 1
endbmatrix.
$$
This basically means, finding a relation between $rho$ and $rho_h$ such that $A approx B$.
Any thoughts on, how can this be achieved?
linear-algebra matrices approximation
linear-algebra matrices approximation
asked Sep 4 at 5:35
Kamal K Garg
12
asked Sep 4 at 5:35
Kamal K Garg
12
edited Sep 4 at 8:41
asked Sep 4 at 5:35
Kamal K Garg
12
asked Sep 4 at 5:35
Kamal K Garg
12
asked Sep 4 at 5:35
Kamal K Garg
12
12
Do you want a 1 in the lower right corners?
– AlgebraicsAnonymous
Sep 4 at 5:51
Yes... made correction. Thanks!
– Kamal K Garg
Sep 4 at 8:36
add a comment |Â
Do you want a 1 in the lower right corners?
– AlgebraicsAnonymous
Sep 4 at 5:51
Yes... made correction. Thanks!
– Kamal K Garg
Sep 4 at 8:36
Do you want a 1 in the lower right corners?
– AlgebraicsAnonymous
Sep 4 at 5:51
Do you want a 1 in the lower right corners?
– AlgebraicsAnonymous
Sep 4 at 5:51
Yes... made correction. Thanks!
– Kamal K Garg
Sep 4 at 8:36
Yes... made correction. Thanks!
– Kamal K Garg
Sep 4 at 8:36
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
You might want to use the least squares method, that is, to find a $rho in [0,1]$ that minimizes
$$
sum_k=1^N 2sum_j=1^k-1 (rho - rho_h^j)^2;
$$
note that the sum sums for each $k$ a boomerang-shaped or edge-shaped part of the matrix that is basically a rectangle that starts in $c_N,k$, goes to $c_k,k$ and makes an edge there and continues to $c_k,N$, where $c_k,k$ is left out, where $C$ is an arbitrary matrix.
Moreover, whenever $| cdot |$ is a matrix norm (submultiplicative or not), you can think about minimizing $|A - B|$ as a function of $rho$. In the above example, the norm was just the Euclidean norm on $mathbb R^n^2$.
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You might want to use the least squares method, that is, to find a $rho in [0,1]$ that minimizes
$$
sum_k=1^N 2sum_j=1^k-1 (rho - rho_h^j)^2;
$$
note that the sum sums for each $k$ a boomerang-shaped or edge-shaped part of the matrix that is basically a rectangle that starts in $c_N,k$, goes to $c_k,k$ and makes an edge there and continues to $c_k,N$, where $c_k,k$ is left out, where $C$ is an arbitrary matrix.
Moreover, whenever $| cdot |$ is a matrix norm (submultiplicative or not), you can think about minimizing $|A - B|$ as a function of $rho$. In the above example, the norm was just the Euclidean norm on $mathbb R^n^2$.
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
add a comment |Â
up vote
0
down vote
You might want to use the least squares method, that is, to find a $rho in [0,1]$ that minimizes
$$
sum_k=1^N 2sum_j=1^k-1 (rho - rho_h^j)^2;
$$
note that the sum sums for each $k$ a boomerang-shaped or edge-shaped part of the matrix that is basically a rectangle that starts in $c_N,k$, goes to $c_k,k$ and makes an edge there and continues to $c_k,N$, where $c_k,k$ is left out, where $C$ is an arbitrary matrix.
Moreover, whenever $| cdot |$ is a matrix norm (submultiplicative or not), you can think about minimizing $|A - B|$ as a function of $rho$. In the above example, the norm was just the Euclidean norm on $mathbb R^n^2$.
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You might want to use the least squares method, that is, to find a $rho in [0,1]$ that minimizes
$$
sum_k=1^N 2sum_j=1^k-1 (rho - rho_h^j)^2;
$$
note that the sum sums for each $k$ a boomerang-shaped or edge-shaped part of the matrix that is basically a rectangle that starts in $c_N,k$, goes to $c_k,k$ and makes an edge there and continues to $c_k,N$, where $c_k,k$ is left out, where $C$ is an arbitrary matrix.
Moreover, whenever $| cdot |$ is a matrix norm (submultiplicative or not), you can think about minimizing $|A - B|$ as a function of $rho$. In the above example, the norm was just the Euclidean norm on $mathbb R^n^2$.
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
You might want to use the least squares method, that is, to find a $rho in [0,1]$ that minimizes
$$
sum_k=1^N 2sum_j=1^k-1 (rho - rho_h^j)^2;
$$
note that the sum sums for each $k$ a boomerang-shaped or edge-shaped part of the matrix that is basically a rectangle that starts in $c_N,k$, goes to $c_k,k$ and makes an edge there and continues to $c_k,N$, where $c_k,k$ is left out, where $C$ is an arbitrary matrix.
Moreover, whenever $| cdot |$ is a matrix norm (submultiplicative or not), you can think about minimizing $|A - B|$ as a function of $rho$. In the above example, the norm was just the Euclidean norm on $mathbb R^n^2$.
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
answered Sep 4 at 5:58
AlgebraicsAnonymous
1,04512
1,04512
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
add a comment |Â
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
The similar problem is addressed in link. The snapshot from the paper is available at link. The paper used the technique presented in [17, Section IV] which is available at link. Just wondering, whether the approach you offered is different from or similar to this. Can you please have a look at the links and advise?
– Kamal K Garg
Sep 5 at 2:50
add a comment |Â
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Do you want a 1 in the lower right corners?
– AlgebraicsAnonymous
Sep 4 at 5:51
Yes... made correction. Thanks!
– Kamal K Garg
Sep 4 at 8:36