The second derivative of a mean square error
$begingroup$
I'm not very familiar with multivariable calculus, I really hope someone could help me check whether or not the following computation is right.
Here's the mean square error of which I have to take the second derivative with respect to $phi$
$$L(phi)={1over N}sum_{n=1}^N{Vert V_phi(s_n)-hat V_nVert^2over 2sigma^2}$$
where $hat V_n$ and $sigma$ are constants, and $V_phi(s_n)$ is a function of $s_n$ and $phi$.
Here's how I do the math
$$
begin{align}
{partial^2L(phi)over partialphi^2}&={partialover partial phi}left({partial V_phi(s_n)over partial phi}{partial L(phi)over partial V_phi(s_n)}right)\
&={partial V_phi(s_n)over partial phi}{partial^2L(phi)over partial V_phi(s_n)^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{partial L(phi)overpartial V_phi(s_n)}\
&={partial V_phi(s_n)over partial phi}{1over sigma^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{1over N}sum_{n=1}^N{V_phi(s_n)-hat V_noversigma^2}
end{align}
$$
Did I do it right?
By the way, I'm not so sure whether I have to use $nabla_theta$ or $partial$ in this context. I frequently see both in papers and I've done some search -- it seems to me that $nabla_phi$ is more sensible, but I seldom see it used in the second derivative. Did I misuse symbols?
Thanks in advance :-)
multivariable-calculus derivatives partial-derivative
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
$endgroup$
add a comment |
$begingroup$
I'm not very familiar with multivariable calculus, I really hope someone could help me check whether or not the following computation is right.
Here's the mean square error of which I have to take the second derivative with respect to $phi$
$$L(phi)={1over N}sum_{n=1}^N{Vert V_phi(s_n)-hat V_nVert^2over 2sigma^2}$$
where $hat V_n$ and $sigma$ are constants, and $V_phi(s_n)$ is a function of $s_n$ and $phi$.
Here's how I do the math
$$
begin{align}
{partial^2L(phi)over partialphi^2}&={partialover partial phi}left({partial V_phi(s_n)over partial phi}{partial L(phi)over partial V_phi(s_n)}right)\
&={partial V_phi(s_n)over partial phi}{partial^2L(phi)over partial V_phi(s_n)^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{partial L(phi)overpartial V_phi(s_n)}\
&={partial V_phi(s_n)over partial phi}{1over sigma^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{1over N}sum_{n=1}^N{V_phi(s_n)-hat V_noversigma^2}
end{align}
$$
Did I do it right?
By the way, I'm not so sure whether I have to use $nabla_theta$ or $partial$ in this context. I frequently see both in papers and I've done some search -- it seems to me that $nabla_phi$ is more sensible, but I seldom see it used in the second derivative. Did I misuse symbols?
Thanks in advance :-)
multivariable-calculus derivatives partial-derivative
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
$endgroup$
add a comment |
$begingroup$
I'm not very familiar with multivariable calculus, I really hope someone could help me check whether or not the following computation is right.
Here's the mean square error of which I have to take the second derivative with respect to $phi$
$$L(phi)={1over N}sum_{n=1}^N{Vert V_phi(s_n)-hat V_nVert^2over 2sigma^2}$$
where $hat V_n$ and $sigma$ are constants, and $V_phi(s_n)$ is a function of $s_n$ and $phi$.
Here's how I do the math
$$
begin{align}
{partial^2L(phi)over partialphi^2}&={partialover partial phi}left({partial V_phi(s_n)over partial phi}{partial L(phi)over partial V_phi(s_n)}right)\
&={partial V_phi(s_n)over partial phi}{partial^2L(phi)over partial V_phi(s_n)^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{partial L(phi)overpartial V_phi(s_n)}\
&={partial V_phi(s_n)over partial phi}{1over sigma^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{1over N}sum_{n=1}^N{V_phi(s_n)-hat V_noversigma^2}
end{align}
$$
Did I do it right?
By the way, I'm not so sure whether I have to use $nabla_theta$ or $partial$ in this context. I frequently see both in papers and I've done some search -- it seems to me that $nabla_phi$ is more sensible, but I seldom see it used in the second derivative. Did I misuse symbols?
Thanks in advance :-)
multivariable-calculus derivatives partial-derivative
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
$endgroup$
I'm not very familiar with multivariable calculus, I really hope someone could help me check whether or not the following computation is right.
Here's the mean square error of which I have to take the second derivative with respect to $phi$
$$L(phi)={1over N}sum_{n=1}^N{Vert V_phi(s_n)-hat V_nVert^2over 2sigma^2}$$
where $hat V_n$ and $sigma$ are constants, and $V_phi(s_n)$ is a function of $s_n$ and $phi$.
Here's how I do the math
$$
begin{align}
{partial^2L(phi)over partialphi^2}&={partialover partial phi}left({partial V_phi(s_n)over partial phi}{partial L(phi)over partial V_phi(s_n)}right)\
&={partial V_phi(s_n)over partial phi}{partial^2L(phi)over partial V_phi(s_n)^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{partial L(phi)overpartial V_phi(s_n)}\
&={partial V_phi(s_n)over partial phi}{1over sigma^2}{partial V_phi(s_n)over partial phi}^T+{partial^2V_phi(s_n)over partialphi^2}{1over N}sum_{n=1}^N{V_phi(s_n)-hat V_noversigma^2}
end{align}
$$
Did I do it right?
By the way, I'm not so sure whether I have to use $nabla_theta$ or $partial$ in this context. I frequently see both in papers and I've done some search -- it seems to me that $nabla_phi$ is more sensible, but I seldom see it used in the second derivative. Did I misuse symbols?
Thanks in advance :-)
multivariable-calculus derivatives partial-derivative
multivariable-calculus derivatives partial-derivative
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
edited Dec 31 '18 at 0:40
Sherwin Chen
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
asked Dec 27 '18 at 2:42
Sherwin ChenSherwin Chen
1657
1657
add a comment |
add a comment |
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