Simulating a random process with AWGN
$begingroup$
I am trying to simulate the following random process:
$$ frac{dy}{dt} = f(y) + AN(t) $$
where $N(t)$ is additive Gaussian white noise, which, if I remember right, is defined by $intlimits_0^t N(t) dt$ having zero mean and a variance of $t$.
Aside from the noise, I know I can simulate $frac{dy}{dt} = f(y)$ as
$$y[k+1] = y[k] + f(y[k])Delta t$$
as long as the dynamics of the system are much slower than $Delta t$, for example, if $f(y) = y/tau$ then the requirement is $Delta t ll tau$.
How does the simulation get changed when adding noise?
I think it scales so that if you halve the timestep, then you have to use random Gaussian samples with variance that halves, e.g. standard deviation divides by $sqrt{2}$, so I think this would work:
$$y[k+1] = y[k] + f(y[k])Delta t + An[k]sqrt{Delta t}$$
where $n[k]$ is a series of independent samples of a Gaussian distribution with zero mean and $sigma = 1$.
Is this correct?
probability ordinary-differential-equations noise
asked Jan 12 at 16:20
Jason SJason S
2,05311617
$endgroup$
add a comment |
$begingroup$
I am trying to simulate the following random process:
$$ frac{dy}{dt} = f(y) + AN(t) $$
where $N(t)$ is additive Gaussian white noise, which, if I remember right, is defined by $intlimits_0^t N(t) dt$ having zero mean and a variance of $t$.
Aside from the noise, I know I can simulate $frac{dy}{dt} = f(y)$ as
$$y[k+1] = y[k] + f(y[k])Delta t$$
as long as the dynamics of the system are much slower than $Delta t$, for example, if $f(y) = y/tau$ then the requirement is $Delta t ll tau$.
How does the simulation get changed when adding noise?
I think it scales so that if you halve the timestep, then you have to use random Gaussian samples with variance that halves, e.g. standard deviation divides by $sqrt{2}$, so I think this would work:
$$y[k+1] = y[k] + f(y[k])Delta t + An[k]sqrt{Delta t}$$
where $n[k]$ is a series of independent samples of a Gaussian distribution with zero mean and $sigma = 1$.
Is this correct?
probability ordinary-differential-equations noise
asked Jan 12 at 16:20
Jason SJason S
2,05311617
$endgroup$
$begingroup$
See Euler-Maruyama method to integrate a SDE. You might want to refine your understanding of the difference between white noise and Wiener process/Brownian motion.
$endgroup$
– LutzL
Jan 12 at 20:00
$begingroup$
I do understand the difference between white noise and a Wiener process which is the integral of white noise, I'm just trying to remember how to apply it to simulating a differential equation.
$endgroup$
– Jason S
Jan 13 at 2:31
$begingroup$
from en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method :def dW(delta_t): """Sample a random number at each call.""" return np.random.normal(loc = 0.0, scale = np.sqrt(delta_t))-- so it looks like I was right about $sqrt{Delta t}$
$endgroup$
– Jason S
Jan 13 at 2:32
$begingroup$
Yes, I just wanted to give a name to your method. "Additive white noise" seems like a handwaving construction using a non-function in $N$. The SDE formalism, Ito or Stratanovich, was developed to avoid using functions that take the value infinity almost everywhere.
$endgroup$
– LutzL
Jan 13 at 16:17
add a comment |
$begingroup$
I am trying to simulate the following random process:
$$ frac{dy}{dt} = f(y) + AN(t) $$
where $N(t)$ is additive Gaussian white noise, which, if I remember right, is defined by $intlimits_0^t N(t) dt$ having zero mean and a variance of $t$.
Aside from the noise, I know I can simulate $frac{dy}{dt} = f(y)$ as
$$y[k+1] = y[k] + f(y[k])Delta t$$
as long as the dynamics of the system are much slower than $Delta t$, for example, if $f(y) = y/tau$ then the requirement is $Delta t ll tau$.
How does the simulation get changed when adding noise?
I think it scales so that if you halve the timestep, then you have to use random Gaussian samples with variance that halves, e.g. standard deviation divides by $sqrt{2}$, so I think this would work:
$$y[k+1] = y[k] + f(y[k])Delta t + An[k]sqrt{Delta t}$$
where $n[k]$ is a series of independent samples of a Gaussian distribution with zero mean and $sigma = 1$.
Is this correct?
probability ordinary-differential-equations noise
asked Jan 12 at 16:20
Jason SJason S
2,05311617
$endgroup$
I am trying to simulate the following random process:
$$ frac{dy}{dt} = f(y) + AN(t) $$
where $N(t)$ is additive Gaussian white noise, which, if I remember right, is defined by $intlimits_0^t N(t) dt$ having zero mean and a variance of $t$.
Aside from the noise, I know I can simulate $frac{dy}{dt} = f(y)$ as
$$y[k+1] = y[k] + f(y[k])Delta t$$
as long as the dynamics of the system are much slower than $Delta t$, for example, if $f(y) = y/tau$ then the requirement is $Delta t ll tau$.
How does the simulation get changed when adding noise?
I think it scales so that if you halve the timestep, then you have to use random Gaussian samples with variance that halves, e.g. standard deviation divides by $sqrt{2}$, so I think this would work:
$$y[k+1] = y[k] + f(y[k])Delta t + An[k]sqrt{Delta t}$$
where $n[k]$ is a series of independent samples of a Gaussian distribution with zero mean and $sigma = 1$.
Is this correct?
probability ordinary-differential-equations noise
probability ordinary-differential-equations noise
asked Jan 12 at 16:20
Jason SJason S
2,05311617
asked Jan 12 at 16:20
Jason SJason S
2,05311617
edited Jan 13 at 2:33
Jason S
asked Jan 12 at 16:20
Jason SJason S
2,05311617
asked Jan 12 at 16:20
Jason SJason S
2,05311617
asked Jan 12 at 16:20
Jason SJason S
2,05311617
2,05311617
$begingroup$
See Euler-Maruyama method to integrate a SDE. You might want to refine your understanding of the difference between white noise and Wiener process/Brownian motion.
$endgroup$
– LutzL
Jan 12 at 20:00
$begingroup$
I do understand the difference between white noise and a Wiener process which is the integral of white noise, I'm just trying to remember how to apply it to simulating a differential equation.
$endgroup$
– Jason S
Jan 13 at 2:31
$begingroup$
from en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method :def dW(delta_t): """Sample a random number at each call.""" return np.random.normal(loc = 0.0, scale = np.sqrt(delta_t))-- so it looks like I was right about $sqrt{Delta t}$
$endgroup$
– Jason S
Jan 13 at 2:32
$begingroup$
Yes, I just wanted to give a name to your method. "Additive white noise" seems like a handwaving construction using a non-function in $N$. The SDE formalism, Ito or Stratanovich, was developed to avoid using functions that take the value infinity almost everywhere.
$endgroup$
– LutzL
Jan 13 at 16:17
add a comment |
$begingroup$
See Euler-Maruyama method to integrate a SDE. You might want to refine your understanding of the difference between white noise and Wiener process/Brownian motion.
$endgroup$
– LutzL
Jan 12 at 20:00
$begingroup$
I do understand the difference between white noise and a Wiener process which is the integral of white noise, I'm just trying to remember how to apply it to simulating a differential equation.
$endgroup$
– Jason S
Jan 13 at 2:31
$begingroup$
from en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method :def dW(delta_t): """Sample a random number at each call.""" return np.random.normal(loc = 0.0, scale = np.sqrt(delta_t))-- so it looks like I was right about $sqrt{Delta t}$
$endgroup$
– Jason S
Jan 13 at 2:32
$begingroup$
Yes, I just wanted to give a name to your method. "Additive white noise" seems like a handwaving construction using a non-function in $N$. The SDE formalism, Ito or Stratanovich, was developed to avoid using functions that take the value infinity almost everywhere.
$endgroup$
– LutzL
Jan 13 at 16:17
$begingroup$
See Euler-Maruyama method to integrate a SDE. You might want to refine your understanding of the difference between white noise and Wiener process/Brownian motion.
$endgroup$
– LutzL
Jan 12 at 20:00
$begingroup$
See Euler-Maruyama method to integrate a SDE. You might want to refine your understanding of the difference between white noise and Wiener process/Brownian motion.
$endgroup$
– LutzL
Jan 12 at 20:00
$begingroup$
I do understand the difference between white noise and a Wiener process which is the integral of white noise, I'm just trying to remember how to apply it to simulating a differential equation.
$endgroup$
– Jason S
Jan 13 at 2:31
$begingroup$
I do understand the difference between white noise and a Wiener process which is the integral of white noise, I'm just trying to remember how to apply it to simulating a differential equation.
$endgroup$
– Jason S
Jan 13 at 2:31
$begingroup$
from en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method :
def dW(delta_t): """Sample a random number at each call.""" return np.random.normal(loc = 0.0, scale = np.sqrt(delta_t)) -- so it looks like I was right about $sqrt{Delta t}$$endgroup$
– Jason S
Jan 13 at 2:32
$begingroup$
from en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method :
def dW(delta_t): """Sample a random number at each call.""" return np.random.normal(loc = 0.0, scale = np.sqrt(delta_t)) -- so it looks like I was right about $sqrt{Delta t}$$endgroup$
– Jason S
Jan 13 at 2:32
$begingroup$
Yes, I just wanted to give a name to your method. "Additive white noise" seems like a handwaving construction using a non-function in $N$. The SDE formalism, Ito or Stratanovich, was developed to avoid using functions that take the value infinity almost everywhere.
$endgroup$
– LutzL
Jan 13 at 16:17
$begingroup$
Yes, I just wanted to give a name to your method. "Additive white noise" seems like a handwaving construction using a non-function in $N$. The SDE formalism, Ito or Stratanovich, was developed to avoid using functions that take the value infinity almost everywhere.
$endgroup$
– LutzL
Jan 13 at 16:17
add a comment |
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$begingroup$
See Euler-Maruyama method to integrate a SDE. You might want to refine your understanding of the difference between white noise and Wiener process/Brownian motion.
$endgroup$
– LutzL
Jan 12 at 20:00
$begingroup$
I do understand the difference between white noise and a Wiener process which is the integral of white noise, I'm just trying to remember how to apply it to simulating a differential equation.
$endgroup$
– Jason S
Jan 13 at 2:31
$begingroup$
from en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method :
def dW(delta_t): """Sample a random number at each call.""" return np.random.normal(loc = 0.0, scale = np.sqrt(delta_t))-- so it looks like I was right about $sqrt{Delta t}$$endgroup$
– Jason S
Jan 13 at 2:32
$begingroup$
Yes, I just wanted to give a name to your method. "Additive white noise" seems like a handwaving construction using a non-function in $N$. The SDE formalism, Ito or Stratanovich, was developed to avoid using functions that take the value infinity almost everywhere.
$endgroup$
– LutzL
Jan 13 at 16:17