From discrete equation to ode












2














I was wondering in how to convert a moving average of the form:



$$x_t=alpha x_{t-1} + (1-alpha)y_t$$



into an ODE. I thought I could do something like:



$$x_t-x_{t-1} = (alpha-1)(x_t-y_t)$$



and then divide by $Delta_t$ to take limits. But this does not seem to help as it results on an unwanted $Delta_t$ on the rhs before taking the limit ($Delta_t rightarrow 0$) to convert the lfh into a derivative.






differential-equations limits







share|cite|improve this question
























  • Put $beta=frac{(alpha-1)}{Delta_t}$
    – user121049
    Nov 21 '18 at 12:05
















2














I was wondering in how to convert a moving average of the form:



$$x_t=alpha x_{t-1} + (1-alpha)y_t$$



into an ODE. I thought I could do something like:



$$x_t-x_{t-1} = (alpha-1)(x_t-y_t)$$



and then divide by $Delta_t$ to take limits. But this does not seem to help as it results on an unwanted $Delta_t$ on the rhs before taking the limit ($Delta_t rightarrow 0$) to convert the lfh into a derivative.






differential-equations limits







share|cite|improve this question
























  • Put $beta=frac{(alpha-1)}{Delta_t}$
    – user121049
    Nov 21 '18 at 12:05














2












2








2







I was wondering in how to convert a moving average of the form:



$$x_t=alpha x_{t-1} + (1-alpha)y_t$$



into an ODE. I thought I could do something like:



$$x_t-x_{t-1} = (alpha-1)(x_t-y_t)$$



and then divide by $Delta_t$ to take limits. But this does not seem to help as it results on an unwanted $Delta_t$ on the rhs before taking the limit ($Delta_t rightarrow 0$) to convert the lfh into a derivative.






differential-equations limits







share|cite|improve this question















I was wondering in how to convert a moving average of the form:



$$x_t=alpha x_{t-1} + (1-alpha)y_t$$



into an ODE. I thought I could do something like:



$$x_t-x_{t-1} = (alpha-1)(x_t-y_t)$$



and then divide by $Delta_t$ to take limits. But this does not seem to help as it results on an unwanted $Delta_t$ on the rhs before taking the limit ($Delta_t rightarrow 0$) to convert the lfh into a derivative.






differential-equations limits




differential-equations limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 21 '18 at 11:44

























asked Nov 21 '18 at 11:22









Ambesh

1,4131039




1,4131039












  • Put $beta=frac{(alpha-1)}{Delta_t}$
    – user121049
    Nov 21 '18 at 12:05


















  • Put $beta=frac{(alpha-1)}{Delta_t}$
    – user121049
    Nov 21 '18 at 12:05
















Put $beta=frac{(alpha-1)}{Delta_t}$
– user121049
Nov 21 '18 at 12:05




Put $beta=frac{(alpha-1)}{Delta_t}$
– user121049
Nov 21 '18 at 12:05










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