Simulating a random process with AWGN












0












$begingroup$


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?










share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















0












$begingroup$


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?










share|cite|improve this question











$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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?










share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 13 at 2:33







Jason S

















asked Jan 12 at 16:20









Jason SJason S

2,04811617




2,04811617












  • $begingroup$
    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










  • $begingroup$
    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










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