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Questions tagged [logistic-regression]

For questions about logistic regressions, a regression model where the dependent variable is categorical.

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gradient descent in multi logistics regression with softmax

suppose we have a 2 dimensional dataset 20 x 3 matrix the third one is bias $$x= \begin{bmatrix} -0.1 & 1.4 &1 \\ -0.4 & 0.2 &1 \\ 1.3 & 0.9 &1 \\ ... &...
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link function interpretation for models

In the categorial regression using the logit link function, the estimated coefficients are used to calculate the odds ratios or the ratio between two odds, that is, the model is interpreted through ...
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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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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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Logistic regression: Prove that the cost function is convex

I'm reading this You can do a find on "convex" to see the part that relates to my question. Background: $h_\theta(X) = sigmoid(\theta^T X)$ --- hypothesis/prediction function $y \in \{0,1\}$ ...
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Loss function for logistic regression

If we are doing a binary classification using logistic regression, we often use the cross entropy function as our loss function. More specifically, suppose we have $T$ training examples of the form $(...
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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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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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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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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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1answer
87 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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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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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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Invert the softmax function

Is it possible to revert the softmax function in order to obtain the original values $x_i$? $$S_i=\frac{e^{x_i}}{\sum e^{x_i}} $$ In case of 3 input variables this problem boils down to finding $a$, ...
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1answer
114 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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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
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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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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232 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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705 views

Negative-log-likelihood dimensions in logistic regression

I'm starting to attempt to learn how regularized multi-class logistic regression classifiers work, but I'm stuck at the very beginning. The negative log-likelihood function of logistic regression for $...
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1answer
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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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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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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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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
112 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
944 views

How does one write the equation for a logit model, and then the odds ratio, with multiple explanatory variables?

I see logit equations always written with a single dependent variable, however I am running a logit model which outputs the coefficients for three explanatory variables (X1, X2, and X3) with respect ...
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1answer
58 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
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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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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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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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320 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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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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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 ...