Questions tagged [differential-equations]

A differential equation is an equation that contains at least one derivative.

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Is it possible to estimate parameters of a system of differential equations by having just initial state and final state data?

I have dataset that belongs to a dynamical system. It contains value of variables when the system is in initial state and when the system becomes stable after initializing (I call it final state). Is ...
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How can we modify SIR model to account multiple waves of infection?

We had a very nice discussion for modeling covid19 data with SIR model. If we monitor number of infected cases over time, most of these models will only have one wave. How can we modify the model to ...
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84 views

Neural ODEs gradient calculation for multiple time steps

I was reading the paper on Neural ODEs (here) and was wondering if anyone could offer some insight on calculation of the gradient of the loss function. If we are only considering 2 time points, $t_0,...
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Why “run the filter longer than needed and remove the initial values” will solve the issue of recursive solving equations?

Consider sequence of random variables $w_i$ iid normal(0,1). Given the equation, $x_t=x_{t-1}-0.9x_{t-2}+w_t$ with $t$ discrete, I want to solve for $x_t$ recursively by prescribing $x_1,x_2$. The ...
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Differential Equation Prediction Model for Future Revenue

I'm trying to predict time series with the physical model of the process. Simple heuristic model for the predicting the company's future revenue. The hypothesis for the model are: The company's ...
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1answer
83 views

Fixing convergence in SIR model using modified fit-model to fit COVID-19 data

I'm trying to model the data for covid-19 using SIR model in R. I followed the answer of the question, and the blog. I'm using the suggested code, However, the data does not converging. Any suggestion ...
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How to fit a time-delay variable to data in R?

I have two datasets that are known (or suspected) to be a similar shape, but with the second dataset delayed by time tau and scaled by a factor mu: F(t) C(t)=mu*F(t+tau) I have data for both C(t) ...
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Does this interpretation $\phi'(x)=-x\phi(x)$ of the normal distribution have any significance?

For the standard normal distribution $\phi(x)$, we can see that $\phi'(x)=-x\phi(x)$. Put differently, $\frac{\mathrm{d}\ln(\phi(x))}{\mathrm{d} x}= -x $. I see this as the fall in the value of the ...
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eigenstructure matching optimization

Is there any optimization loss functions that can approximately match the eigenstructure of the original samples and the transformed samples? For example, given a collection of samples $\mathbf{X}$ ...
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Fitting covid19 data with SIR model: question about the definitions of susceptible and recovered population in real world

We have some very nice discussions about SIR model fitting at CV. As I explore the model with different parameters, I have some questions on the definitions of susceptible and recovered population. ...
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What are the pros and cons to fit data with simple polynomial regression vs. complicated ODE model?

Suppose in a disease outbreak scenario and we want to estimate number of infected people based infections over time. Why we cannot simply fit the data with some polynomials (or some MLP neural ...
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Are the parameters $\beta$ and $\gamma$ in (Susceptible, Infected, Recovered) SIR model probability number? Can they larger than 1.0?

I am learning SIR model from this blog post. We also had a very good discussion in CV post The key parameters of the model are $\beta$ and $\gamma$, people usually describe them as the "infection ...
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194 views

Where does the logistic function come from?

I first learned the logistic function in machine learning course, where it is just a function that map a real number to 0 to 1. We can use calculus to get its derivative and use the derivative for ...
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87 views

Neural ODE's, Adjoint Method

I've been trying to understand the gist behind the Chen et. al paper on neural ODE's (https://arxiv.org/pdf/1806.07366.pdf). It seems like the main trick here is to be able to take derivatives of ...
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Uncertainty propagation in ODEs

I want to see the effect of parameter uncertainty in the Euler method for ODEs. For a differential equation: $dx/dt=f$ with initial condition $x(0)=xo$ and a function $f$ (that has uncertain ...
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Can neural ODEs “fit” an ODE from just measurements?

The neural ODE technique, to my knowledge, presents a neural network based way of solving ODEs efficiently, which implies it needs an ODE and an initial value in order to construct the evolution over ...
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Neural ODEs, augmentation and subspace “projection”

The answer to the neural ODE question, the Augmented neural ODEs paper is mentioned. There, the following process happens: 2D data is augmented by padding with 1 zero 3D data is augmented once again ...
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Fitting SIR model with 2019-nCoV data doesn't conververge

I am trying to calculate the basic reproduction number $R_0$ of the new 2019-nCoV virus by fitting a SIR model to the current data. My code is based on https://arxiv.org/pdf/1605.01931.pdf, p. 11ff: <...
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Correct way to write a stochastic differential equation

A few days ago, I disccussed my Phd thesis (thesis defense), one of the mathematical mistakes that the committe members alerted me to it, is how to write correctley an ordinary differential equation ...
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What are the practical uses of Neural ODEs?

"Neural Ordinary Differential Equations", by Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt and David Duvenaud, was awarded the best-paper award in NeurIPS in 2018 There, authors propose the ...
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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 ...
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311 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 ...
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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 ...
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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 (...
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610 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 ...
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131 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 ...
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1answer
84 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_\...
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1answer
169 views

How to numerically solve a matrix differential equation in R? [closed]

I have interest in using the R language and environment to numerically solve a system of linear ordinary differential equations. The numerical solver, deSolve, ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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91 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 ...
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1answer
23 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?
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68 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 ...
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136 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 ...
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1answer
369 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. ...
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solve a differential equation [closed]

I need to solve this differential equation where $F(x)$ is the distribution of $x$, with pdf $f(x)$: $$F(x) = 1 - x \cdot f(x) $$
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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 \...
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1answer
115 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 ...
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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 ...
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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: $$E = -\sum_jt_j\ln o_j$$ ...
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1answer
186 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: ...
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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)-...
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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 ...
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2answers
326 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 ...
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1answer
576 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 ...
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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....
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1k 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 ...
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2answers
239 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 ...