Questions tagged [differential-equations]

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

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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 ...
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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 ...
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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 ...
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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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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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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 ...
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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_\...
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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. ...
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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 ...
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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, ...
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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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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 ...
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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 ...
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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
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 ...
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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 ...
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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?
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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 ...
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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 ...
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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. ...
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88 views

solve a differential equation [closed]

I need to solve this differential equation where $F(x)$ is the distribution of $x$, with pdf $f(x)$: $$F(x) = 1 - x \cdot f(x) $$
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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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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 ...
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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
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: ...
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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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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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). ...
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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, ...
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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 ...
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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 ...
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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 ...
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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} +...
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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 ...
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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 ...
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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-...
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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 ...