Questions tagged [logistic-regression]
For questions about logistic regressions, a regression model where the dependent variable is categorical.
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A “softmax”-like function for deciding on a partition
Softmax can be derived as follows. Say that we are given $k$ "log priors" $b_i$ that our data belongs to the $i$th category in some categorical distribution. Then we can solve for the category ...
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2answers
34 views
Using a variable in logit units in a regression (interpretation)
Suppose a dependent variable amath measures student's ability in math. The range of this variable is -5 to 5 and it is measured in logit units (it classifies the ...
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1answer
51 views
Derive $ \frac{1}{1 + exp(-(\beta_0 + \beta_1x))} $ from conditional and total probabilities
Related to: Show posterior probability takes the form of the logistic function
I basically want to derive the sigmoid function from conditional and total probabilities.
In other words, I want to ...
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25 views
Can anyone help with the inverse problem and tuning parameters
For my final year project i want to model the population of London using the Verhulst logistic model. However, to gain more marks i wish to use the inverse problem to tune the parameters to make the ...
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2answers
60 views
How to decelerate from velocity v to stop time t over distance d?
I'd be grateful for some help with this problem I am trying to solve.
Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
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19 views
How to distort a sigmoid / logistic function?
Please help. I need to move an object such that its distance / time profile resembles a sigmoid curve but in a non-symmetrical manner.
1- Let's say that I need to move 800cm in 3 seconds. How do I ...
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22 views
For a Maximum Likelihood Estimation with events that implicate each other, how should the likelihood function be constructed?
The probability of a student with a skill parameter of "s" to obtain at least a score of "k" in a certain test is defined as:
$$\frac{1}{e^{b_k-s}+1}$$
Where $b_k$ is a difficulty parameter of ...
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2answers
34 views
Motivation/derivation for Logistic Growth formula?
$$P(t) = \frac{c}{1+ae^{-bt}}$$
I see that $ae^{-bt}$ is basically compounding growth formula: $Pe^{rt}$
Not sure what the +1 does. Includes the original 100% quantity?
What about the reciprocal $\...
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1answer
26 views
Logistic Regression Explanation
I have two questions regarding logistic regression.
1) I understand that the results of a logistic regression model yield a table stating coefficients together with a p-statistic for each variable . ...
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12 views
Integral of logistic normal distribution approximation
Following paper about Glicko rating have a expression below:
Parameter estimation in large dynamic paired
comparison experiments (Equation 18, 19 on Appendix A)
$$
\int \frac{ (10^{(\theta-\theta_j)...
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1answer
203 views
How to prove the logistic loss function is strongly convex?
The logistic loss function is:
$$\mathcal{L}=\frac{1}{n}\sum_{i=1}^n\log(1+\exp(-y_ix_i^T\theta))$$
in which $y_i\in\{-1,+1\},x\in \mathbb{R}^d$. How to show that $\mathcal{L}$ is strongly convex.
My ...
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1answer
26 views
I have one question about mathematical modeling
My school does Orienteering in PE, and basically the grades are from 0 to 20, the is the best,0 is the worst and 10 is 50%, well my teacher calculates the grades this way, if you do the task in 5 ...
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72 views
Minimum sufficient statistic for logistic regression model
For the question in the link below, I am seeking the minimal sufficient statistic for $\theta$={$\beta_1$,$\beta_2$} in the linear regression model given.
I have taken the ratio of likelihoods $...
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7 views
What type of analysis with binomial dependent variable that’s repeated
I can’t for the life of me figure out what type of analysis I need to run in SPSS and I am EXTREMELY statistically delayed so any responses in the most simplest explanation would be amazing.
I had 14 ...
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19 views
Formula for ordinal complementary log-log regression
I have a model developed in R using the polr() function in the MASS package. The model is an ordinal regression with a complementary log-log link (method=cloglog). The model uses four predictors ...
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1answer
108 views
How is the log-likelihood for a multinomial logistic regression calculated?
In a multinomial logistic regression, the predicted probability $\pi$ of each outcome $j$ (in a total of $J$ possible outcomes) is given by:
$
\pi_j = \frac{e^{A_j}}{1+\sum_{g \neq j}^Je^{A_j}}
$
...
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1answer
32 views
Finding population growth rate given initial population and average pubs per female reproduction
I am new to mathematical modeling and having trouble modelling the following population growth scenario:
Starting population: $100$ individuals
Average age: $7$ years
Assumption: population equally ...
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61 views
logistic regression without $L_2$-regularization does not have optimum?
I have a question related to machine learning. Consider the case when in the problem of binary classification the training set is linearly separable. How to show that in this case the optimization ...
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1answer
57 views
Fisher Matrix and Hessian matrix
I know that the Fisher matrix is easily obtained from the Hessian matrix
$I\left(\hat{\beta}\right)=-H\left(\hat{\beta}\right)$
Why is the covariance variance matrix the inverse of the Fisher ...
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41 views
how log likelihood's derivative is equal to zero in maximum log likelihood.
if the log-likelihood function is strictly increasing and it has not horizontal asymptote then how it's derivative is equal to zero in maximum log likelihood. Now since it is strictly increasing every ...
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39 views
Odds Ratios with independent linear trends
I have a question regarding odds ratios for a project I'm working on. For my analysis to determine if there was a significant change in access to a service at a certain type of facility from one year ...
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1answer
83 views
Kolmogorov–Smirnov test in logistic regression
When applying KS-test (as goodness-of-fit test) on logistic regression (class: 0,1), where x-axis = probability of being classified as class 1, sorting ascendingly. Here are the 2 questions:
1. Why ...
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14 views
How to select a threshold that penalize more on Type I error?
I am trying to do a binary classification in which Type I error is more important than Type I error.
Which metric should I use to select the threshold value to penalize more on Type II error?
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24 views
How to do prediction in a binary classification with logistic regression when we care much more about type I error than type II error?
How to do prediction in a binary classification with logistic regression when we care much more about type I error than type II error?
Which criteria should I use to select the threshold value and ...
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1answer
80 views
fix point solution or approximation available? logistic regression?
please, is there a simple closed-form or approximation to the following fixed-point problem in $x$? $x$ is the value searched for. $m$, $g$ and $N$ are real parameters, all greater than 0,
\begin{...
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1answer
124 views
Logistic Regression: Asymptotic confidence interval for the lethal dose
For the logistic model:
$$\log \Big( \frac{\pi(x)}{1-\pi(x)}\Big) = b_0 +b_1x$$
I want to construct a asymptotic confidence interval for the ratio of the m.l.e's of $b_0$, $b_1$:
$LD50 = -\frac{\...
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1answer
84 views
Deriving the odds ratio of a 3-way interaction logistic regression model
Suppose a logistic regression model has three binary explanatory variables $x_1$, $x_2$ and $x_3$ used
to estimate the probability of success. This model includes all three main effects, the three $2$-...
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1answer
5 views
How to introduce different costs by class in a binary logistic regression?
What is the form of the Negative Log-Likelihood Goal function in Logistic regression if we introduce different costs per class (e.g. the cost of erring class 1 is errc1, and class 2 is errc2)?
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1answer
18 views
Generalised Linear Models: Binary data
I am currently working on GLM problem.
My response variable is binary as are some of my explanatory variable,others are categorical i.e. 1-1day, 2- 2-3days, 3-5+days and so forth.
I have coded it ...
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1answer
27 views
Affine transform of logistic map to a dynamic system
I am reading the following technic report:
https://homepages.laas.fr/henrion/papers/odds.pdf
For the section 5 Example,
It gives a logistic map: $\bar{x}_{k+1} = 4\bar{x}_k(1-\bar{x}_k)$
...
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101 views
Why Sigmoid function is so popular in every field of science?
I don't know if this is a silly question but still I couldn't resist my self from asking this.
I recently came to know about the standard sigmoid function while learning about logistic regression ...
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33 views
Loss function : finding the criterion for which a given solution is the optimal classifier
For a binary classification problem, let $\eta(x) = P[Y=1 \mid x],$ and, for a given classifier $g$, we define the asymmetric cost :
$$ L(g) = P[g(X)=0, Y=1] +\lambda P[g(X)=1,Y=0]$$
For this cost, ...
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60 views
Solving a logistic regression problem with my calculator gives “singular matrix” error. Why?
In my homework, I was given the following problem:
The following data represent the population of a country. An ecologist is interested in building a model that describes the population.
$$\...
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11 views
Is a Logistic Regression always viable for having a dichotomous response variable?
Yes, I've asked this question already on stat.stackexchange, but I'm asking here as well to see if I can get any other feedback on the subject
I have learned some about a simple logistic regression ...
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1answer
18 views
Using LDA decision boundary inequality to classify an observation X
I have a single regressor $X$ and response $y$, where $y=n_k/n$ if $X$ is of class $k$, and $k=1,2$. Let $n_k$ denote the number of observations in class $k$, and $n$ the total number of observations. ...
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24 views
Can I view maximum likelihood as finding the closest probability mass function (pmf) to the pmf with all its mass on the observed value
I took a deep dive into logistic regression recently. I was bothered by the fact that the likelihood formula often explicitly incorporates the values of the coding (0 or 1) into the calculation. In ...
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33 views
How should I calculate cost function based on SSE and Sigmoid values?
Given the following Samples: x1 : X= 80, y =1, x2: X=20, y =1 , and x3: X = 120 y = 0
How should i go about these questions ?
1) Calculate the probability that y = 1 for each 𝑥0 of the data set (h𝜽...
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1answer
318 views
Hessian of negative log-likelihood of logistic regression is positive definite?
I'm trying to show that the Hessian of the negative of the log likelihood with two parameters is positive definite, but I'm not sure how to go about it once I compute the Hessian.
The function is:
$-...
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0answers
57 views
Equivalent Neural Network in Keras to Multinomial Logistic Regression Using VGG16 Bottleneck features
In the Kaggle Dog Breed Identification kernel:
https://www.kaggle.com/gaborfodor/dog-breed-pretrained-keras-models-lb-0-3
The bottleneck features from the VGG16 model (using Keras) are feed into a ...
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1answer
28 views
Deriving max. likelihood estimate of β for a logistic model of two classes with a single binary regressor
I have the log-likelihood function:
$$l(\overrightarrow\beta)=\sum_{i=1}^n [y_i log(p(\overrightarrow x_i;\overrightarrow\beta))+(1-y_i)log(1-p(\overrightarrow x_i;\overrightarrow\beta)] $$
where $...
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7 views
Which bivariate analysis to use for determining significance in predictors between two groups?
I'm conducting a "typical" statistical analysis in my research, but I have a few questions regarding appropriateness of the tests I'm using and whether or not I'm doing things the correct way. I don't ...
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1answer
45 views
Logistic Equation through three points
I am trying to solve a logistic equation $\frac{dP}{dt}=rP\left ( 1-\frac{P}{K} \right )$ for the values of $r$ and $K$ through the points $(0,100),(10,130),(20,160)$. I have derived a general ...
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1answer
113 views
Monotonic transformation to smooth the probabilities
I am studying some event for a set of objects that can be plotted on a square $[0, 100] ^ 2$. I have used logistic regression to calculate probabilities that event occur for different objects and the ...
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13 views
Logistical Curve Tansformations
Mostly I am having a brain fart.
Working on a game and wanted to use logistical curve for the experiance gain of skill levels. Idea is to create a system the reflects natural learning; early on you ...
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10 views
softmax function in logistic regression: confusion about two different forms
I read that, in multiclass logistic regression, we have a pivot class $K$ and $K-1$ set of $\vec{w}$ weights, then, for the pivot class:
\begin{eqnarray}
P( C_K | \vec{x} ) &= 1- \sum_\limits{t=...
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1answer
566 views
How to show that log likelihood function in logistic regression is concave? [closed]
I know of a proof for this which involves finding matrix of second derivatives (Hessian) for the given expression and proving that it is negative semi definite. But that is quite sophisticated for my ...
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1answer
385 views
Partial derivative of sigmoid function with sum (sigma) input
Definition of logistic sigmoid function is:
$$σ(t)=\frac{1}{1 + e^{-t}}$$
Differentiating this is very simple, It just requires simplification with exponent rule and differentiation with quotient ...
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100 views
How to get the closed form of integration of sigmoid function multiply normal distribution density function
As the title describes, I want to get the closed form of the following equation
$$
f(\sigma,\mu) = \int_{-\infty}^{\infty} \frac{1}{1+e^{-(\beta X + \alpha)}}\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(X-\...
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1answer
59 views
Bayesian Logistic Regression, conditional probability integration
In Andrew NG's Lectures (CS229), the Bayesian Logistic Regression section contained a formula;
$$P(Y|X,S)=\int_\theta P(Y|X,\theta)P(\theta|S)d\theta$$
Here, $\theta$ is treated as a random variable....
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1answer
79 views
Harvesting in logistic equation
So the question is:
Analyze the equation depending on c :
$$y'=y(a-by)-c.$$
When does the population die out in time?
My attemp for solution
So to draw phase portrait we need to find first ...
