All Questions
Tagged with ordinary-differential-equations statistics
50
questions
23
votes
1
answer
774
views
Kähler Geodesics
Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric
$$\begin{pmatrix} \frac{1}{1-|a|^2}&\frac{1}{1-a\bar{b}}\\\frac{1}{1-\bar{a}b}&\frac{1}{1-|b|^2}\end{...
- 3,838
6
votes
1
answer
110
views
Optimal speed for approaching red light to maximize velocity with non-uniform probability
Problem statement
When I cross red lights, my goal is to being going as fast as possible when the light turns green.
I am at distance $D$ from a traffic light when it turns red.
Let the time length of ...
- 337
6
votes
0
answers
121
views
The statistical average of a continuous value: $\overline{O} = \int O(x) \rho(x) dx$, but coordinate invariant
I am trying to solve a Lagrange multiplier problem for the following equation
$$
L= - \int_{-\infty}^\infty \rho(x) \ln \frac{\rho(x)}{q(x)} dx + \alpha \left( 1- \int_{-\infty}^\infty \rho(x) dx \...
- 2,589
5
votes
0
answers
153
views
Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} D_{\alpha}\{q(x,t),p(x,\...
- 706
3
votes
1
answer
318
views
Geodesics of Fisher-Rao metric on the open interior of the finite-dimensional simplex.
I am curious about the explicit form of the geodesics of the Fisher-Rao metric tensor on the open interior of the n-dimensional simplex.
In the 2-dimensional case (only 1 parameter on the 2-simplex), ...
- 1,322
2
votes
1
answer
763
views
Linear regression with 2 unknown intercepts
The linear equation $y=2.2+0.6(x+1.2)$ has the slope $0.6$, the given y-intercept $2.2$ and x-intercept $-1.2$. The table is
$$
\begin{array}{c|lcr}
x & y \\
\hline
1 & 3.52 \\
2 & 4.12 ...
- 131
1
vote
1
answer
418
views
writing a piecewise regression model as a linear model
lets write the following piecewise regression model
$$y= \alpha_0 + \alpha_1 x +\epsilon ;\ \ x\le x_0 $$
$$ y=\beta_0 +\beta_1 x + \epsilon \ \ x\gt x_0$$
according to the variable $x_0$ is known,...
- 6,510
1
vote
1
answer
2k
views
Just learned about the bell curve in statistics. How is calculus related to this curve?
I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
- 319
1
vote
1
answer
140
views
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
1
answer
121
views
Trick in integration with Taylor expansion
I am struggling with the expression of the LHS of the following equation.
The RHS is just the Taylor expansion of the first function around point y and the differentiation wrp to the argument y.
How ...
- 145
1
vote
1
answer
30
views
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
vote
1
answer
47
views
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
1
vote
1
answer
101
views
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
1
vote
1
answer
89
views
What's the explicit difference when looking at Differential Equations vs Stochastic Differential Equations?
I learned differential equations from mathematics text and have having some trouble with its application in statistics. Specifically applied to stochastic time series.
From my time series text reads "...
- 365
1
vote
1
answer
578
views
Solving equation involving error function and normal distribution
Let $\phi(x)$ be a normal distribution, that is $\phi=\frac{1}{\sigma}\phi(x|\mu,\sigma^2)$ and $\Phi(x)$ be the CDF of $\phi$, i.e., $\Phi(x|\mu,\sigma^2)=\frac{1}{2}[1+erf(\frac{x-\mu}{\sqrt{2}\...
- 23