Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [differential-equations]

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

A differential equation is an equation that contains at least one derivative. A derivative is the rate of change in a function with respect to changes in a variable. For example, Newton's second law states that force equals mass times acceleration ($F = ma$), but acceleration is the rate of change in velocity over time ($a = dv/dt$), thus:
$$ F = m\frac{dv}{dt} $$

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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 NeuralODE, which is a method that fuses concepts of…
Firebug
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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.…
vonjd
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How does Hamiltonian Monte Carlo work?

I made the below graphic to explain how I currently understand the HMC algorithm. I'd like verification from a subject matter expert if this understanding is or isn't correct. The text in the below slide is copied below for ease of…
mjake
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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: $$E = -\sum_jt_j\ln o_j$$ Where $t_j$ is the target output for the Neuron $j$,…
Geno Racklin Asher
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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 function being proportional to the distance from…
Bravo
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Where does the logistic function come from?

I first learned the logistic function in machine learning, where it is just a function that map a real number to 0 to 1. We can use calculus to get the derivative and use it for some optimization eventually to have some binary classifier. Later, I…
Haitao Du
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What's up with Neural Stochastic Differential Equations from a practical standpoint?

I've spent a few days reading some of the new papers about Neural SDEs. For example, here is one from Tzen and Raginsky and here is one that came out simultaneously by Peluchetti and Favaro. There are others which I plan to read next. This work all…
jeffery_the_wind
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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 infected, and $R$ is the recovered. I'm trying…
Poisson Fish
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On solving ode/pde with Neural Networks

Recently, I watched this video on YouTube on the solution of ode/pde with neural network and it motivated me to write a short code in Keras. Also, I believe the video is referencing this paper found here. I selected an example…
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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 wanted to try something different. I have really…
Islands
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Why are most epidemic models continuous-time?

Most classical epidemic models such as SIR and variants are formulated as differential equations. However, to me discrete-time models feel more natural to measure the evolution of a disease on a day-by-day basis: human activity come in daily…
Federico Poloni
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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 increments are normally distributed. I played around a…
RJ Nowling
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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 using deSolve and fitting using nloptr. One of…
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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 satisfying the following differential…
orrillo
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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 perhaps are driven by shared hidden variables. I…
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