Questions tagged [differential-equations]
A differential equation is an equation that contains at least one derivative.
45
questions
0
votes
0answers
3 views
How to convert GEKKO solutions into a float [migrated]
I am using GEKKO to solve a system of non-linear differential equations. It is able to solve the equations and give me solutions, however the the solutions seem to be saved in some GEKKO object kind ...
0
votes
0answers
17 views
How to show CDF uniquely determined from equation
I have the following equation for a CDF that I would like to show is uniquely determined:
$$\frac{h_1(x)}{f(x)}=\int_x^a h_2(z)(F(z)-F(x))^k\;dF(z)$$
Here $F$ is the CDF and $f$ is the PDF, which I ...
2
votes
1answer
60 views
derivative of mathematical expectation [closed]
As we know,if x is a random variable, we could write mathematical expectation based on cumulative distribution function $(F)$ as follow:
$E(X)=\int[1-F(x)]d(x) $
In my problem, t is a random variable ...
1
vote
1answer
20 views
I would like to find Goodness-of-fit between my differential equation simulation result and observed field data. How to find it?
I modelled a ecological system using ordinary differential equation. Simulated the model for 12 months. I also have 12 months of field data. Now I want to find the goodness of fit between my model ...
0
votes
0answers
11 views
How to do cross validation in ODE models with more predicted than measured time courses?
I have an ODE model of biochemical reactions with 37 state variables and 88 strictly positive parameters. Unfortunately, I can only expect to get time course measurements of about 10 state variables (...
0
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2answers
129 views
What does it mean L1 loss is not differentiable?
I was looking through this lecture
https://davidrosenberg.github.io/ml2015/docs/3a.loss-functions.pdf
Slide 3: Absolute or Laplace or L1 loss not differentiable
What does it mean ...
2
votes
1answer
78 views
Fitting flexible spline using ODEs
I'm fitting a series of ordinary differential equations (describing movement through disease states: susceptible, infected, recovered) to weekly counts of a disease through time. I'm solving the ODEs ...
0
votes
1answer
64 views
Calculating the AIC based on histograms for selection of stochastic models
I am modelling a nonlinear stochastic process and have data to compare model output against. My aim is to obtain an evolution equation of the form,
$$\frac{du}{dt} = f(u,\theta_f)+\alpha(u,\theta_\...
0
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0answers
12 views
Practicing Understanding Stochastic Differential Equations using R
Is there a book or set of notes that I can use to practice differential equations using R-Studio or Python. I don't want a solver, I want a way of visualizing them and understanding their properties. ...
0
votes
0answers
36 views
parameter estimation using censored data by fitting a Maximum likelihood to a differential equation
I have a population data ($N$) measured over certain time points ($t$). The rate of change of the population is modelled as a ODE as,
${dN\over dt}=-\lambda N$
My intention is to estimate the ...
1
vote
0answers
63 views
How to numerically solve a matrix differential equation in R?
I have interest in using the R language and environment to numerically solve a system of linear ordinary differential equations. The numerical solver, deSolve, ...
2
votes
0answers
14 views
Common distributions as the solutions of differential equations
All of the Pearson and Burr distributions are solutions to particular differential equations, of the pdf and CDF, respectively. Why was this a good strategy for generating distributions? Why do so ...
0
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0answers
9 views
Threshold optimization with two parameters where first parameter need to be minimum and second value to be maximum
We have threshold values ranging from 1 to 10 where attributes p1, p2 increase with increase in threshold. Our intention is to find threshold with minimum p1 and maximum p2. It would be of great help ...
0
votes
0answers
19 views
Derive taylor series expansion of df
I was trying to understand ito's lemma. When I came across the taylor series expansion of df(x).
df(x) = f'(x) dx + (1/2!) f''(x) (dx)^2 + ...
I searched everywhere for the derivation of this but ...
0
votes
0answers
20 views
ANN for Boundary Value Problem
I have a question regarding solving Boundary Value Problems (BVP) using ANNs.
My understanding is that this is currently a challenging task. Most scientific literature on the subject is interested in ...
1
vote
0answers
24 views
Reference Request: How to Solve PDE with ANN
I've recently heard that it's possible to approximate the solutions of PDEs with ANNs. However, upon looking it up on the Google, it seems that I can't find a detailed example in R (or even a ...
1
vote
1answer
30 views
Why are mixed effect methods more effective when data are limited
In the study in here, it is said that mixed effects models are better in estimating parameters of a ODE system when there is only very small number of data to estimate the parameters. So, in a ...
0
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0answers
60 views
Is an ITO diffusion time slice always Normally distributed?
As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed?
This seems like a useful property for computer generalization and ...
0
votes
1answer
21 views
Degree Dynamics Preferential Attachment Model
I am trying to follow http://barabasi.com/f/622.pdf.
In page 11-12:
Solving this, I could not reach (5.7). After 5.6, I reach only up to:
Any hints on how to proceed?
1
vote
1answer
60 views
Derivation Harvey (1984) Logistic Curve
Given a logistic function of the form.
\begin{align*}
f(t) = \frac{\alpha}{1 + \beta e^{\gamma t}}
\end{align*}
Harvey (1984) differentiates this and takes logs to yield:
\begin{align*}
\ln f' = 2 ...
1
vote
0answers
91 views
Exponential distribution as a differential equation
I'm trying to interpret the following situation. In an economy, let $T$ denote the remaining lifetime (a stochastic variable) with exponential distribution and a Cumulative distribution function ...
2
votes
1answer
232 views
Example Transforming A time series using the Backshift operator
I am trying to understand the idea of differencing (using the Backshift operator): Let's look at this times series: $$X_t = at^2+bt+c+Y_{t-1}$$ with $Y_t$ being some (mean-zero) white noise process. ...
0
votes
1answer
88 views
solve a differential equation [closed]
I need to solve this differential equation where $F(x)$ is the distribution of $x$, with pdf $f(x)$:
$$F(x) = 1 - x \cdot f(x) $$
1
vote
0answers
17 views
Do continuous time “algorithms” that converge to measuring entropy exist?
Assume we have a system of ODE's, $ \dot{x} = \xi(x) $ where $ \xi : \mathbb{R}^N \rightarrow \mathbb{R}^N $, that generate a stationary p.d.f. $ \rho(x_t) dx_t = \rho(x_s) dx_s ~~ \forall t, s \in \...
1
vote
1answer
109 views
Best estimate for Stochastic difference equation
On the subject of Stochastic differential equations. If we consider the difference equation $$\Delta x(t_n) = x(t_n) \Delta t + f(t_n) \Delta t$$ where we consider $f(t_n) \Delta t$, the driving term ...
4
votes
0answers
130 views
How to estimate a continuous analog of the (discrete) vector autoregression (VAR) model
I have some ten to 100 thousand observations on each of around 500 entities. I have good reason to believe that these observations all mutually influence one another, in possibly complicated ways, or ...
8
votes
1answer
9k views
Differentiation of Cross Entropy
I have been trying to create a program for training Neural Networks on my computer. For the Network in question, I have decided to use the Cross Entropy Error function:
$$E = -\sum_jt_j\ln o_j$$
...
3
votes
1answer
172 views
How can I differentiate the equation with respect to $\theta$?
I want to differentiate the following equation by taking $\log$ with respect to $\theta$.
$\log (\theta^{ a_H+\alpha-1}(1-\theta)^{a_T+\beta-1})$
and have the result of the differentiation as below:
...
0
votes
0answers
76 views
Setting up maximum likelihood estimation with multi-response data
I was trying to fit the parameters of a time-dependent system coupled of ODES related to a kinetic experiment with multi response data.
Example:
A->B+H
A+H->C+H
A->D
dcA(t)/dt=-k1Ca(t)-k2Ca(t)*Ch(t)-...
1
vote
0answers
42 views
Nonlinear mathematical modeling using differential equations
$$ \left\{ \begin{array}{ll}
\dot{N_1}=r_1N_1\left(1 - \frac{N_1}{K+b_{12}N_2}\right)\\
\dot{N_2}=r_2N_2\left(1 - \frac{N_2}{K+b_{21}N_1}\right)
\end{array} \right. $$
I would like to ask you, how ...
0
votes
2answers
195 views
What is a good method for predicting temperature and precipitation
I was wondering what is according to you the best method for predicting temperature and precipitation in say 10 days period. So far I have tried ARIMA models, which make sense, but I'm not satisfied ...
1
vote
1answer
526 views
Using AICc distributions to assess goodness-of-fit and model selection
I have a couple of ordinary differential equation models that I'm trying to fit to time-dependent biological data ($y_n$). One model is more complex than then other as it has more free parameters.
I ...
2
votes
0answers
41 views
Best way to measure how well stochastic models fits a system of differential equations?
I have a system of differential equations which can be easily solved using ode45 in matlab. The equations represent a biochemical pathway.
I have simulated the same biochemical pathway stochastically....
7
votes
2answers
951 views
Maximum Likelihood Estimate of Infection Model Parameters
I'm using the standard infection model on some data I am working with.
$ dS = -\beta SI $
$ dI = \beta SI - \gamma I $
$ dR = \gamma I $
Where $S$ is the number of susceptible subjects, $I$ is the ...
0
votes
2answers
224 views
Least Squares Estimate of Infection Model Parameters
I'm using the standard infection model on some data I am working with.
$ dS = -\beta SI $
$ dI = \beta SI - \gamma I $
$ dR = \gamma I $
Where $S$ is the number of susceptible subjects, $I$ is the ...
1
vote
2answers
407 views
what is the biological meaning of R nought equal 1 ? endemicity?
From an epidemiological model with differential equation, we can compute the basic reproductive number R0 (the number of expected secondary case per primary case in a disease free population).
...
4
votes
1answer
623 views
Least squares with exponential model
I'm trying to fit values from this model
$$y(x)=ae^{−bx}+c$$
where a, b and c are 3 different parameters that I want to find with least squares. So using least squares I want to find the value of a, ...
6
votes
3answers
680 views
Stochastic Differential Equations - A Few General Questions
I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
0
votes
1answer
61 views
Total Variation Denoising help
I am trying to work through the "Mathematical exposition for 1D digital signals" in the wikipedia entry for Total Variation Denoising (TVD). I am familiar with Lagrange multipliers. However, I cant ...
0
votes
0answers
542 views
Textbooks on stochastic calculus and stochastic differential equations
I am looking for key reference books in stochastic calculus, Stochastic Differential Equations (SDEs) as well as Stochastic Partial Differential Equations (SPDEs), from the most theoretical to the ...
4
votes
0answers
1k views
Fitting data in differential equation
I have a sampled data for two variables y and x, where y is the dependent and x is the independent variable. The two variables are related as
$A\frac{dy}{dt} + y = B\frac{d^2x}{dt^2}+C\frac{dx}{dt} +...
5
votes
1answer
328 views
Does the noise term in a SDE need to be Gaussian?
Most of the examples I've seen for stochastic differential equations are of the form:
$$
dX_t = \mu(X_t, t)dt + \sigma(X_t, t) dW_t
$$
where $dW_t$ is a Wiener process, i.e., the independent ...
2
votes
0answers
258 views
Survival models and differential equations
I have a question regarding survival models and differential equations. Is it possible to translate survival models ( in survival analysis) into differential equations? For example can we write the ...
3
votes
1answer
219 views
Numerical solution of varying coefficients ODE
I have a set of observed raw data and use 2nd order ODE to fit the data
$$y''+b_1(t)y'+b_0(t)y = 0$$
The $b_1(t)$ and $b_0(t)$ are time-dependent and I use principal differential analysis(PDA) (R-...
2
votes
0answers
58 views
Sampling from elliptic pde solution in high dimensions
What is known about sampling from solutions to elliptic PDE's in high dimensions, where it is computationally infeasible to construct or store the actual solution?
For example, let $u$ solve the ...
