All Questions
Tagged with ordinary-differential-equations statistics
50
questions
1
vote
1
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30
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Question about Statistical measures
I have recently tried parameter estimation for nonlinear ODE using non-linear fitting techniques. I learned about Statistical measures like p-tests, t-tests, $R^2$ squared, adjusted $R^2$ square, etc. ...
- 31
-1
votes
1
answer
39
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Verify whether it's a Bregman loss function, maybe by solving a differential equation
I have a function $f(x, y;\mu) = \frac{\mu}{x}(x-y)^2$, where $\mu > 0$ is a parameter. I want to see whether it's a Bregman loss function. A Bregman loss function is define as:
$D_\phi(x,y) = \phi(...
- 31
0
votes
0
answers
57
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Time Series Analysis and Recurrence Relations/Differential Equations
I am beginning to watch a video playlist on the subject of time series analysis, and it seems pretty clear both from notation and some of the terminology (such as "characteristic equation") ...
- 1,922
6
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0
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121
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The statistical average of a continuous value: $\overline{O} = \int O(x) \rho(x) dx$, but coordinate invariant
I am trying to solve a Lagrange multiplier problem for the following equation
$$
L= - \int_{-\infty}^\infty \rho(x) \ln \frac{\rho(x)}{q(x)} dx + \alpha \left( 1- \int_{-\infty}^\infty \rho(x) dx \...
- 2,589
1
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0
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60
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Literature on Principal differential analysis
I'm currently dealing with topics in Functional Data Analysis (FDA), specifically Principal Differential Analysis (PDA). By the corresponding R package description, this is related to estimating a ...
- 61
6
votes
1
answer
110
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Optimal speed for approaching red light to maximize velocity with non-uniform probability
Problem statement
When I cross red lights, my goal is to being going as fast as possible when the light turns green.
I am at distance $D$ from a traffic light when it turns red.
Let the time length of ...
- 337
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0
answers
31
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Determining correlations of derivatives of a function given only measurements of that function
Cross-posted from statistics stackexchange:
Say we have a permanent-magnet DC motor that roughly obeys the system equation $\ddot{x}(t) = \alpha \dot{x}(t) + \beta u(t) + \gamma$, where $x(t)$ is the ...
- 1,378
1
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1
answer
47
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How to find an expression for an MGF
The MGF, $M_x(t)$ is a function of $t$. It has the property that $\lim_{t\to 0} M_x(t)=1$. It can be shown that:
$\lim_{t\to 0}\frac{d}{dt} \log[M_x(t)]=E[X^1]=E[X]$
Find an expression for
$\lim_{t\to ...
- 31
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votes
1
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23
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Calculate normal distribution from $\frac{1}{f(x)}\frac{d([f(x)])}{dx} = \frac{d-x}{a}$
Calculate normal distribution from $\frac{1}{f(x)}\frac{d([f(x)])}{dx} = \frac{d-x}{a+bx+cx^2}$ when $b=c=0$ then we have $\frac{d-x}{a}$. This is taken from Mathematical Statistics with Applications ...
- 393
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votes
1
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Why does this data not line up with the differential equation that's supposed to model it?
Sorry for the bad title, I wasn't sure how to ask this specific question.
So for a (extra credit) homework assignment, I wrote a python program for my differential equations class that should model ...
0
votes
0
answers
64
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Solving for the $k$, given the survival function for a newborn as: $S(t) = \frac{\left(121 - t\right)^{1/2}}{k},\; t\in\left(0, 121\right]$
I'm doing an assignment and I can't seem to solve the following question:
Given the survival function for a newborn as:
$$S(t) = \frac{\left(121 - t\right)^{1/2}}{k},\; t\in\left(0, 121\right].$$
What ...
- 19
1
vote
1
answer
140
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CDF as a result of a Cauchy problem: how to solve it?
I'm studying a particular class of random variables.
In order to find the CDF $F(x)$ of my variable, I should solve the following Cauchy problem:
$$
\begin{cases}
F(x)=e^{-\lambda F'(x)} \\
F(0)=0
\...
- 97
1
vote
0
answers
19
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How to obtain the parameter update for the multiclass classification (general loss and activation function)?
Consider the feature space $\mathcal{X}=\mathbb R^{d}$ and $\mathcal{Y}=\{1,...,c\}$ such that $c > 2$. We consider some activation function $\alpha: \mathbb R^{c} \to \mathbb R^{c}$ and out weight ...
- 998
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1
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46
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How to identify linear non-homogenous ODE from data?
I want to fit a model
$$\overset{\cdot} x = Ax + v(t)Bx + Cu(t)$$
to data, where $u(t),v(t)$ are known inputs and $A,B,C$ should be fitted. The data are assumed to be drawn from the above model but ...
- 1,504
1
vote
1
answer
101
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Use Bayes method to solve ODE system with random noise
For ODE system $\frac{du}{dt} = \beta u$, $t>0$, $u(0)=1$, where $\beta$ is unknown. But the solution to the system at t=1 up to some noise is known: $h :=u(1) + \zeta$, where $\zeta$ is a random ...
- 113
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0
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71
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Estimating parameters of SIR model and problem with real-life data
I tried to make an SIR model based on real-world data. But, I ran into a snag when I'm trying to estimate the parameters of $\beta$ and $\gamma$. With equations:
$$
\begin{cases}
\frac{dS(t)}...
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votes
0
answers
34
views
How to distinguish chaos from inaccurate?
If you have a system of ODE that gives rise to a chaotic system, you can easily find that the solutions either explicit or implicit are drastically different from real-world results. Yet, this does ...
1
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0
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84
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SIR epidemic model with vital dynamics
I am reading the Wikipedia article on the SIR model with vital dynamics.
I am wondering about the birth and death rate. The birth rate seems to be constant, ie, it seems like the population in all 3 ...
- 697
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1
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312
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Write down the backward equations for $P_{12}$ and $P_{21}$ and use the symmetry of Q to solve these equations.
Hint: Whenever confronted with an ordinary differential equation of the form x′(t) = ax(t)+b(t), it might be beneficial to consider the function y(t) = $e^{−at}x(t)$.
$$Q = \left[ \begin{matrix}
...
- 253
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1
answer
22
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Standard Theory of Linear Difference Equations, Power Function
I'm reading this paper and have a question about the math done on page 4.
We go from having
$$\lambda^{T_0} = p \lambda^{T_{0 + 1}} + q\lambda^{T_{0 - 1}}$$ to
$$p \lambda^2 - \lambda + q = 0$$
...
- 37
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votes
0
answers
94
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Validity of Coronavirus Curves - Are we using the correct baseline?
For most analysis (models), are we inaccurately assuming that the newly reported cases are the number of actual new cases? Could this exponentially growing number just be a function of the way testing ...
- 17
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votes
1
answer
41
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Logit regression change in $x$
I've been reviewing the logit regression equation to predict the probability of a response variable given $X=x$ as per below and I can't understand how a per unit change in the predictor is equal to $...
- 478
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votes
0
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117
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Understanding the stochastic SIR model
I am learning about what it means for a model to be Stochastic. To do this, I am examining the stochastic SIR model found here: https://en.wikipedia.org/wiki/Gillespie_algorithm (scroll down to the ...
1
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0
answers
46
views
How to minimize the difference between datasets
How do I go about matching a real-world dataset to a differential equation that describes it? In this case I have a real, tracked set of pendulum angles over time and self-made python script to ...
1
vote
0
answers
56
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Pearson Type III probability distribution in an old math paper
The paper I'm getting this from can be found here. It's William Gosset's original derivation of the t-distribution.
I'm interested in the author's use on page 4 of the Pearson Type III distribution ...
- 177
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0
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72
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Manifold assumption when visualizing high dimensional dynamical systems
I come from a background in statistics, where we often visualize high dimensional data sets by projecting them onto lower dimensional subspaces. The most common example of this approach is Principal ...
- 2,531
1
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0
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62
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Is there a variational problem that can provide the following class of variational derivative?
Suppose I have the variational problem
$$
E(y) = \frac{1}{2}\int_{a}^{b} y^2 + \alpha y'^2dx
$$
Variational derivative will provide
$$
\frac{\delta E}{ \delta y} = y -\alpha y'',
$$
Is there a ...
- 5,317
1
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0
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79
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Epidemic threshold on activity driven network
I am trying to understand the equations used in a paper
(https://www.nature.com/articles/srep00469.pdf)
Mainly I'm trying to understand how the epidemic thershold was calculated using the ...
- 111
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0
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119
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Branching annihilating random walks and mean field theory
I am attempting a project on modelling branching morphogenesis, but am getting very confused looking at the literature.
On the one hand, the structure formation itself is clearly best described by a ...
- 3,197
2
votes
1
answer
763
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Linear regression with 2 unknown intercepts
The linear equation $y=2.2+0.6(x+1.2)$ has the slope $0.6$, the given y-intercept $2.2$ and x-intercept $-1.2$. The table is
$$
\begin{array}{c|lcr}
x & y \\
\hline
1 & 3.52 \\
2 & 4.12 ...
- 131