Questions tagged [differential-equations]
A differential equation is an equation that contains at least one derivative.
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questions
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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 ...
0
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0answers
16 views
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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1answer
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,...
0
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1answer
17 views
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 ...
2
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0answers
25 views
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 ...
0
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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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0answers
12 views
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) ...
8
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2answers
124 views
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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0answers
13 views
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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0answers
47 views
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. ...
2
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2answers
184 views
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 ...
2
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2answers
153 views
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 ...
8
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4answers
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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1answer
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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0answers
28 views
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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0answers
74 views
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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5answers
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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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0answers
39 views
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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1answer
2k views
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 ...
2
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1answer
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 ...
1
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1answer
31 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 ...
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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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3answers
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 ...
3
votes
1answer
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 ...
0
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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_\...
2
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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, ...
2
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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 ...
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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 ...
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0answers
26 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 ...
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1answer
31 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 ...
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0answers
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 ...
0
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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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1answer
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 ...
3
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1answer
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 ...
2
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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}$$ with $Y_t$ being some (mean-zero) white noise process. ...
0
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1answer
89 views
solve a differential equation [closed]
I need to solve this differential equation where $F(x)$ is the distribution of $x$, with pdf $f(x)$:
$$F(x) = 1 - x \cdot f(x) $$
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0answers
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Do continuous time “algorithms” that converge to measuring entropy exist?
Assume we have a system of ODE's, $ \dot{x} = \xi(x) $ where $ \xi : \mathbb{R}^N \rightarrow \mathbb{R}^N $, that generate a stationary p.d.f. $ \rho(x_t) dx_t = \rho(x_s) dx_s ~~ \forall t, s \in \...
1
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1answer
115 views
Best estimate for Stochastic difference equation
On the subject of Stochastic differential equations. If we consider the difference equation $$\Delta x(t_n) = x(t_n) \Delta t + f(t_n) \Delta t$$ where we consider $f(t_n) \Delta t$, the driving term ...
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0answers
167 views
How to estimate a continuous analog of the (discrete) vector autoregression (VAR) model
I have some ten to 100 thousand observations on each of around 500 entities. I have good reason to believe that these observations all mutually influence one another, in possibly complicated ways, or ...
9
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1answer
11k views
Differentiation of Cross Entropy
I have been trying to create a program for training Neural Networks on my computer. For the Network in question, I have decided to use the Cross Entropy Error function:
$$E = -\sum_jt_j\ln o_j$$
...
3
votes
1answer
186 views
How can I differentiate the equation with respect to $\theta$?
I want to differentiate the following equation by taking $\log$ with respect to $\theta$.
$\log (\theta^{ a_H+\alpha-1}(1-\theta)^{a_T+\beta-1})$
and have the result of the differentiation as below:
...
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0answers
81 views
Setting up maximum likelihood estimation with multi-response data
I was trying to fit the parameters of a time-dependent system coupled of ODES related to a kinetic experiment with multi response data.
Example:
A->B+H
A+H->C+H
A->D
dcA(t)/dt=-k1Ca(t)-k2Ca(t)*Ch(t)-...
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0answers
43 views
Nonlinear mathematical modeling using differential equations
$$ \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
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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 ...
1
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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 ...
2
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0answers
42 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....
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2answers
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 ...
0
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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 ...
