Skip to main content

Questions tagged [inverse-problem]

In science an inverse problem is the problem of calculating from a set of observations the causes that produced the observations. Examples are tomography and seismic reconstruction, and many others. Use this tag for statistical methods used with inverse problems.

Filter by
Sorted by
Tagged with
1 vote
0 answers
40 views

Fitting a Gaussian function to Poisson noisy data

Let $A$, $\mu$, $\sigma$ be some positive, a priori unknown parameters. Define a Gaussian function $f$ as $$f(x) = A \mathrm{exp}\left(-\frac{1}{2} \left( \frac{x-\mu}{\sigma}\right)^2\right).$$ One ...
mathslover's user avatar
1 vote
0 answers
10 views

Issues with gradient of standard deviation in GPR using skopt.learning.gaussian_process

I'm currently working on a Gaussian Process Regression (GPR) model using the implementation provided in skopt.learning.GaussianProcessRegressor which is a wrapper for the sklearn implementation. This ...
Dave's user avatar
  • 11
0 votes
0 answers
29 views

Can estimated value plus one standard deviation be less than true value in a synthetic experiment?

I perform a synthetic calculation and then an inversion, that is, I have actual value in the synthetic calculation and I estimate it in the inverse problem stage. The inverse problem gives me the ...
Lager's user avatar
  • 11
0 votes
1 answer
105 views

Solve equation for a given set of parameters

I'm modelling a process for which the probability of the event (stop) not happening before time $t$ is $e^{-\lambda\cdot t }|\lambda>0$. When the event happens, the process stops running for a ...
Jon Nagra's user avatar
  • 313
1 vote
2 answers
312 views

Is Gaussian process functional regression a truly Bayesian method (again)?

This question has been asked before but I'd like to come back to it because I point a precise issue out. Suppose we want to estimate a function $f\left( x \right)$ from data $D = \left( {\left( {{x_1},...
Student's user avatar
  • 39
2 votes
2 answers
528 views

How to do competing risks regression after IPW?

There are 4 types of treatment in my data. To balance the covariables of different treatment groups, I have used twang::mnps function to perform inverse probability weighting and successfully got the ...
Chao Li's user avatar
  • 21
1 vote
0 answers
27 views

simultaneous parameter and variance estimation in statistical inverse problem

Background: I'm working on a geophysical inverse problem and interested in regularization parameter selection. Just by way of explanation, we're trying to estimate the friction underneath a flowing ...
Daniel Shapero's user avatar
7 votes
2 answers
1k views

In GD-optimisation, if the gradient of the error function is w.r.t to the weights, isn't the target value dropped since it's a lone constant?

Suppose we have the absolute difference as an error function: $\mathit{loss}(w) = |m_x(w) - t|$ where $m_x$ is simply some model with input $x$ and weight setting $w$, and $t$ is the target value. In ...
mesllo's user avatar
  • 679
1 vote
1 answer
93 views

How to decompose a random walk (array) into its Markov Chain transition matrix?

The algorithm, PageRank, receives a Markov Chain transition matrix (page links from one to another.) Either by random walk, or more efficiently, eigenvectors, the stationary distribution of the Markov ...
jbuddy_13's user avatar
  • 3,216
1 vote
0 answers
30 views

Does blind source separation (ICA) work if channels of mixture are observed asynchronously?

Does Independent Component Analysis (ICA - fastICA, SOBI, etc.) work reliably when applied to a multidimensional mixture (observation) $X = (X^1, \cdots, X^d)$ if the different channels $X^i$ of the ...
fsp-b's user avatar
  • 155
0 votes
0 answers
28 views

Mean and covariance estimators for indirect (transformed) observations of Gaussian

Suppose we have a multivariate Gaussian random variable $\mathbf{x}\sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and $n$ measurements $\mathbf{y}_1 = H_1 \mathbf{x}_1, \mathbf{y}_2 = H_2 \mathbf{x}...
Hypercube's user avatar
  • 116
1 vote
1 answer
35 views

Estimate the Image Using Multi Many Realizations of Its Convolution with a Known Filters Using Wiener Filter

Suppose we have a corrupted image $Y = H*X + \epsilon$ that is formed by taking an image $X$, convolving it with a point-spread function $H$, and adding gaussian noise $\epsilon$. Then we know that ...
Sunay Joshi's user avatar
2 votes
1 answer
92 views

How to invert/infer a parameter in nonlinear conditional expectation function

I wouldn't be surprised if this question has already been asked, as it sounds like a standard bookwork result. However, I'm not sure I know the language to describe it, and when I type in the the ...
Marses's user avatar
  • 333
0 votes
0 answers
111 views

Inverse Neural Networks

Suppose there is a series of transformation applied to the random variable $z_0$ such that $$ z_M = f_{\theta_{M}} \circ f_{\theta_{M-1}} \circ \ldots \circ f_{\theta_{1}}(z_0) =: f_{\theta}(z_0). $$ ...
Josh Pilipovsky's user avatar
0 votes
1 answer
111 views

Validation of inverse problem solution based on Bayesian method

Recently, I read a paper about the inverse problem and parameter estimation. The main approach of the paper is based on the Bayesian method. The answer in this method is a posterior probability ...
VOLCANIC_9's user avatar
1 vote
0 answers
120 views

Is it possible to recover original normal distributions from observations of sums of normally distributed variables?

I have been trying to solve the following problem for several weeks. I would greatly appreciate it if you help me solve the following problem. Problem description Assume that there are seven iid ...
mossan's user avatar
  • 11
2 votes
0 answers
87 views

Inverse problem with normal distributions and max

Consider $n$ independent normally distributed random variables $$X_i\sim N(\mu_i,\sigma_i^2)$$ and denote $Y = \max\limits_{1\leq i\leq n}\{X_i\}$. We can define the probabilities, for each $1\leq i\...
Dennis's user avatar
  • 71
1 vote
2 answers
205 views

Distribution of population size $n$ given binomial sampled quantity $k$ and selection probability $\pi$

Given a drawn (without replacement) sample size $k$ from a binomial distribution with known probability parameter $\pi$, is there a function which gives distribution of likely population size $n$ from ...
phdmba7of12's user avatar
1 vote
1 answer
249 views

Existence of least squares and maximum likelihood estimators?

In statistical parameter estimation where there is a deterministic and stochastic component to the observation-generating model, do least squares and maximum likelihood estimators always exist? ...
hatmatrix's user avatar
  • 859
1 vote
0 answers
67 views

What is the inference behind the momentum variable and the Kinetic energy for a weakly non-linear inverse problems in the HMC method?

We generate an auxiliary momentum variable in the HMC method to provide gradient for the propagation of trajectory (m, p) (model or position, momentum) in the phase space. If we look into Newton's ...
Nirmit's user avatar
  • 11
1 vote
1 answer
116 views

Probability of Crossing a Threshold as a Function of Time (Forward Model)

I know the state of some object at a given time. Let's assume that the state is temperature. At each time, I also know the mean and variance of that temperature. I would like to obtain a probability ...
js16's user avatar
  • 49
2 votes
0 answers
76 views

Can we get the input from a multilayer perceptron based on the output of one of its hidden layers?

I was reading a relatively new paper that proposed to split a nerual networks layers into groups and sending each group to different nodes to train them in a distributed manner. In order to not send ...
sgaseretto's user avatar
3 votes
0 answers
175 views

Neural Network Inversion and its consequences

I am currently looking at Federated Learning. Here is a good example from google. The idea is that training happens on multiple devices. This means on one hand that training data never leaves a user (...
Mr.Sh4nnon's user avatar
1 vote
0 answers
100 views

Population Monte Carlo Algorithm using L2 Distance Measure/ Likelihood Distribution

I am currently struggling with some concepts of the Population Monte Carlo Framework. Initially, I came across this set of algorithms as I am currently trying to infer parameters from a 7D ...
NewKidAround's user avatar
1 vote
0 answers
77 views

How to properly solve for the inverse problem of OLS? [duplicate]

In textbook ordinary least squares we want to find a vector of coefficients $b_{k+1\times n}$ such that the sum of the squared deviations of what's observed ($y_{n\times 1}$) from what's assumed to be ...
yosimitsu kodanuri's user avatar
1 vote
1 answer
144 views

What is the error of my regression? [closed]

I'm conducting a polynomial of a third degree upon a diode measurement where Amplification was measured against Voltage. It's a very exponential behavior. However, I used the ...
Ben's user avatar
  • 3,473
1 vote
0 answers
728 views

Is the covariance matrix a diagonal matrix with variances on the diagonals?

I am a geophysicist learning about geophysical inverse problems. In many papers, the authors discuss the "covariance matrix" as it applies to the inverse problem. In most geophysical applications, ...
Darcy's user avatar
  • 915
4 votes
1 answer
161 views

Inferring a Markov chain from its invariant measure

Given a probability measure $p$ on $\{1,\dots,n\}$ assumed to be the invariant measure of some irreducible ergodic Markov chain with unknown transition matrix $P$, i.e., $p = pP$, what (if any) ...
S Huntsman's user avatar
0 votes
0 answers
37 views

Approximation of fractional function that has real-power numerator

I have the function $f(x)=\frac{(1+x)^k}{1+ax}$, where $x>0, 0<a<k<1$. The function has only one maximum at $x_0=\frac{a-k}{a(k-1)}$, increases on the left of $x_0$ and decreases on the ...
Tri Nguyen's user avatar
2 votes
0 answers
372 views

How to select the regularization parameter between two losses?

In deep learning, the total loss commonly consists of a task-specific loss and a weight regularized loss: loss = loss_specific + lambda * reg_loss In my case (...
mining's user avatar
  • 1,019
2 votes
0 answers
115 views

Machine learning/Deep learning to solve inverse tomographic problem

The typical simplifiled representation of a tomographic system is $y = Ax$, where $y$ is the collected data (sinogram in CT), $A$ is the projection matrix, and $x$ is the unknown image. The ...
Nick X Tsui's user avatar
0 votes
1 answer
63 views

Marginal Posterior Likelihood-Solving inverse Problem

For a university project, we were required to code our own Parallel Tempering Algorithm and use it to solve an Inverse Problem with 4 Parameters. Unfortunately, I'm not sure if I'm too stupid or have ...
NewKidAround's user avatar
1 vote
0 answers
358 views

Solving an inverse problem with machine learning

I am running up against a very tough inverse problem that I suspect might be solvable using machine learning. Here is the problem. Overview I am studying an object $X$ which, internally, is ...
rhombidodecahedron's user avatar
2 votes
0 answers
266 views

Connection between MCMC and Optimization for Inverse/Parameter-Estmation Problems

I've been considering two approaches to solving inverse/parameter-estimation problems, and I'm curious to the connection and/or difference between the two approaches. Set up: Say we have a forward ...
kjd's user avatar
  • 21
1 vote
0 answers
225 views

Accuracy of exponential decay parameter estimates comparing different models

I am wondering what approach is more accurate for estimating parameters between two different models of exponentially decaying signal data. The signal decays very rapidly and only a few samples can ...
esd100's user avatar
  • 111
1 vote
0 answers
392 views

How to compare posterior distributions for different observed data? KL-divergence?

So I'm solving an inverse problem with the Bayesian approach $p(u | y) \propto p(y| u )p(u)$. Assuming I have two datasets $y_1$ and $y_2$, what can be said about the difference in the posteriors $p(...
jenna's user avatar
  • 65
2 votes
1 answer
2k views

L-curve method for regularization parameter selection

I work on PDE inverse problems and I'm interested in how these can be viewed as problems of statistical inference. I'm looking for some model parameters $m$ which minimize the misfit with some data $d$...
Daniel Shapero's user avatar
5 votes
0 answers
225 views

Making sense of standard deviation after sampling using Cholesky

I have an inverse problem with over 65,000 degrees of freedom. I am using Bayesian formulation to solve this problem. After using the optimization algorithm to obtain MAP solution, I want to calculate ...
0b1100001's user avatar
  • 151
1 vote
1 answer
1k views

Least square regression with L1 regularization and non-negativity constraint

There are two functions associated by the model $a(x) = \int_{k_1}^{k_2} b(k)\exp(-kx)dk$ where $a(x)$ is the experimental data I have, and $b(k)$ is the information I want to get. Or I can write ...
shva's user avatar
  • 131
2 votes
0 answers
144 views

Confusion related to inverse problems in statistics [closed]

I am getting started with inverse problems in statistics. However, I didn't something related to it. I was reading this paper http://math.uni-heidelberg.de/studinfo/reiss/CavalierInvProb.pdf. It ...
user34790's user avatar
  • 6,807
3 votes
1 answer
423 views

Regression with an unknown dependent variable

I want to know if there is any literature about the following regression problem: $$ Y=X\beta +\epsilon$$ where $Y$ is unknown. But, i know $X$ and the OLS estimator of $\beta$ $$ \hat{\beta}=(X^\top ...
david's user avatar
  • 31