Questions tagged [regression-analysis]

This tag is for questions about regression analysis. In statistical modeling, regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (or 'predictors').

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7 views

Let the population regression function be $Yt=B1+B2Xt+Ut$ .

$$ Yt=B1+B2Xt+Ut $$ i) Consider the estimator B2 , $b2= {x̅ /y̅}$. is it unbiased ii) Derive an expression for the variance of b2. I am not able to start solving the question . Any help would be ...
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15 views

Two datasets need to be of comparable size [closed]

It is so that there is two datasets, one of unit kBq/cc and the other in unit of Bq, the first set is measuered each half second, while the other is measured each whole second. The first set contains ...
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11 views

Can we make confidence interval to the mean with the confidence interval of the parameters in a linear regression

https://i.stack.imgur.com/VkJO0.png As we can see above, i plotted some blue points that its y position is normally distributed bettwen a mean $(2\cdot x + 3)$ such that x is its x position and an ...
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1answer
86 views

Fitting Data to Model Equation

I am attempting to fit an equation which models how fast a chemical reaction progresses. Given the following equation: $$\ln(k)= \ln\biggl(\frac{d\,k_0\,M}{1+P_r}\biggl)+\ln\biggl(\frac{a\exp(-b/T)+\...
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1answer
36 views

Fitting data that follows specific equation

Given the following equation: $y = \frac{80-x}{100-x}$ If I had real world data that followed this exact equation, how would I back out an equation to always predict the correct $y$ value given an $x$ ...
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11 views

Regression of a polynomial model in one variable with missing terms (D-optimality)

Let's say we are given a model that looks as follows: $y = x + ax^3 + bx^5 + cx^7 + dx^9 + \mathcal{N}(0,\sigma_1)$ Given $n$ free choices for the input variable $x$ how can we determine which $x$'s ...
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15 views

please explain this feature of isotonic regression

i posted this yesterday on a very old blog entry but it doesn't seem to be getting looked at, so i thought i'd repost here: i have an application where i know that ground truth is a monotonic function ...
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1answer
34 views

Intuition behind optimzation problem of ridge regression?

In one of the texts that I am reading it is given that regularization parameter restricts the choice of functions in case data given is not sufficient for processing of signal. It is given that lambda ...
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1answer
20 views

Linear regression vector calculus confusion

For linear regression in the form $$|y\rangle = X|\beta\rangle +|\epsilon\rangle $$ Where, $$|y\rangle = \begin{bmatrix} y_1 \\\vdots \\ y_n\end{bmatrix}, \ X=\begin{bmatrix} 1 & x_1 \\ \vdots &...
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8 views

How to solve Weighted Least Square estimator for Linear model with dummy variables?

Suppose we collected data $(X_i,Y_i,Z_i = 1), i = 1,\ldots,n, $ and $(X_i,Y_i,Z_i = 2), i = n+1,\ldots,2n$, and wanted to estimate the mean function $E(Y \mid X = x,Z = z)$. If we assume that $Var(Y \...
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21 views

How to find a vector of constant $P$ such that $\Lambda' \beta = P' X \beta$?

There is an example in a textbook (without any explanation) on finding $\lambda$, and I am struggling to know how to explicitly find $\rho$ and $\lambda$ such that $\lambda' \beta = \rho'X\beta$, i.e.,...
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11 views

Test the hypothesis that the rapists tend to receive sentences different from those of armed robbers.

It is known that armed robbers receive sentences normally distributed with a mean of 7 and standard deviation of 7. A criminologist is interested in whether rapists tend to receive sentences different ...
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9 views

chi square GLM inference

Suppose at $m$ different positions on a line $a_1,....,a_m$, we sample from a i.i.d normal distribution $N(\mu_i,\sigma_i^2)$, $n_i$ times for each of the $1\le i\le m$ different points. Here of ...
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59 views

For Total Least Squares, why do we minimize Frobenius instead of spectral norm?

Say we have a linear model $y + e = X\theta + F$, with input matrix $X$ and output vector $y$, so one-dimensional outputs for simplicity. In ordinary least squares (OLS), we would then have $F=0$ and ...
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38 views

Properties of ridge regression hat matrix and ridge residuals

I'm referencing https://arxiv.org/pdf/1509.09169.pdf on ridge regression. On page 34 question 1.5 we need to prove : Ridge fit $\widehat{Y}(\lambda)=X(X^{\top}X+\lambda I_p)^{-1}X^{\top}Y$ is not ...
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1answer
31 views

What is the distribution of residual in simple linear regression?

Suppose that $$Y_i=\beta_0+\beta_1x_i+\epsilon_i,$$ where $\epsilon_1,\ldots,\epsilon_n$ are independent random variables with identical normal distribution $N(0,\sigma^2)$. Let $$\hat{Y}_i=B_0+B_1x_i$...
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78 views

Semisupervised vs supervised learning

I am trying to understand the mathematical properties of supervised learning and semi-supervised learning. Let us consider the case for the mean $\mu$. Then the supervised learning estimator can just ...
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30 views

estimated standard deviation of coefficients of a polynomial regression in python

I know how to find the coefficients of a polynomial regression using 'numpy'. The problem I have is carrying out confidence intervals for the coefficients. To carry out the confidence intervals, I ...
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11 views

Hello everyone, I'm trying to come up with a modelling technique that accounts for unexplained variation. I'd like to get some input/ suggestions.

Aim: To find a correlation b/w two function f(X1,X2) and g (X1,X2), which are dependent on X1 and X2 and some other unknown parameters (Xi, where i != 1,2). X1 and X2 are continuous variables, f and g ...
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Line through tops of time series stock price data

I am trying to automate a feature of technical analysis of stock prices basically a ‘Top joining Trendline’. I have identified Tops and bottoms in time series data through local maxima and minima. I ...
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How to write up a model with composite variables

The original model, I wished to estimate was: $$y_{t}= \alpha+\sum^{14}_{k=-1}\beta_{k}N_{t-k}+\varepsilon_{t} \quad(1)$$ , where $t$ is the time variable sampled at every five minutes; $N_t$ is the ...
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1answer
73 views

Determine if estimator is unbiased

I need to find out if the following estimator for a regression with no intercept (ie. $Y_i = \beta X_i + \epsilon_i$) is unbiased. $$\hat{\beta} = \frac{\sum x_i^2y_i}{\sum x_i^2}$$ I'm given $\bar{x},...
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28 views

One way analysis of variance in Regression Model perspective

Recently, I have some trouble with One-way ANOVA. Before I raise my doubts, I think it is necessary to make some notation. Suppose that the form of One-way analysis of variance is as follows: \begin{...
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30 views

Interaction between dummy variables in Linear Regression

I know as a fact that dummy variables always reflect the deviation from the reference category of the original variable after controlling for relevant other variables. Unfortunately I don't understand ...
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28 views

Linear regression with error depending on the x

Consider the following model: $y_i = \beta x_i + \epsilon_i x_i$ where $y_i$, i = 1,2,3...n are observed; $x_i$, i = 1,2...n are known positive constants and $\beta$ is an unknown parameter. There ...
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1answer
55 views

MS Excel polynomial regression curve vs. Calculated polynomial regreesion curve results on different equations

I've been doing some research based on polynomial regression curves: $ax^2 + bx + c = 0$ I did all the research on MS Excel, and built some notes based on that. Then, I created a C program that ...
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16 views

Regression based on function recurrence

There is a hypothetical machine which takes an integer $x$ and returns an integer $y$ such that $y=F(x)+\varepsilon$ where $\varepsilon$ is an integer. It is known that the function is of the form $F(\...
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1answer
32 views

Regression/forecast with an added linear constraint

I am not sure if I am asking on the right place. But given a set of independent variables $X_i$ and the dependent variable $Y_i = f(X_i, b) +c$, how can I estimate the regression equation given a set ...
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43 views

Are the general and affine fundamental matrices nested models?

Given two models with the parameters in matrix form as $\mathbf{F}_\text{G} = \begin{pmatrix} f_{1} & f_{2} & a \\ f_{3} & f_{4} & b \\ c & d & e \\ \end{pmatrix} \hspace{...
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57 views

What happens when we minimize the sum of errors instead of the sum of errors squared?

Basically title says it all. Say we run a regression where instead of OLS we try to minimize the errors alone. Is this even possible. When I try to differentiate with respect to $\beta_0$ for the FOC ...
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22 views

NFL Stadium Hedonic Pricing Model

Long time lurker, so first time posting on here, but I think I have something a lot of you may find interesting. I am very interested in the real estate field and thought a fun project to take on ...
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69 views

Closed form solution for Restricted Weighted Least Squares

From Greene, we know that the closed-form solution of a restricted least squares is: $\beta_{Constrained} = \beta_{Uncon} - (X'X)^{-1}R'[R(X'X)^{-1}R']^{-1}(R\beta_{Uncon}-r)$. Is there any similar ...
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1answer
27 views

Intersection point of two curves with errors - covariance matrix

I have measured the parameters for a hyperbola and an ellipse, let us call them $$ \begin{cases} a^2x^2 + b^2y^2 = 1 \\ c^2x^2 + d^2y^2 = 1 \end{cases} $$ and I have errors associated with each ...
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17 views

On the difference between the main effect in a one-factor and a two-factor regression

This question was asked on Cross Validated where it received little attention and no comments or answers, but as it is purely mathematically oriented it may well be more suitable here. Consider a ...
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1answer
26 views

How can we interpret residual plot in case we have many variables?

In Residual plots, we try to visualize & interpret whether linearity is valid or not in the linear regression model. One way to do this is to plot error term wrt to the independent variable(say x)....
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1answer
34 views

Finding $z(x,y)$ from $z(x,\text{constant})$ and $z(\text{constant},y)$

I asked this question earlier, but I got no answer, maybe due to my bad English and bad explanation. Now, trying to ask in a different way, hope you can help me: In a surface, the form of $z(x,y=\text{...
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25 views

Dealing with multicollinearity in linear regression

I'm reading a stats book and using a bit of code to understand and see practical examples. I'm doing a simple regression and I'd like to understand and get a feel for multicollinearity. I'll use VIF (...
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44 views

How to predict a function with a neural network

Suppose we have the following problem that needs to be solved with a convolutional neural network. We are given an arbirary greyscale image $I(x,y)$ as an input. We want to integrate the image along ...
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1answer
70 views

Linear Regression: Finding a quadratic function that approximates a given set of points

Using linear regression, find a quadratic function that is best-fit for the following points: $(1, 1), (1, 5), (-1, -2)$. This might be a trivial question, but I'm new to this topic. I get stuck at ...
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24 views

How to Reconstruct a curves from coordinate points? How to detect curves?

I have extracted the coordinate points of a diagram using edge detection in opencv. With these coordinates I wanted to reconstruct the diagram in question, with lines and curves in separate canvas i.e ...
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59 views

Bi variate Normal Distribution

Let $(X,Y)\sim N_2(\mu,\Sigma)$ with $ \mu =(\mu_x,\mu_y)$. Also,let $\sigma_x^2,\sigma_y^2$ and $\sigma_{xy} $ denote the variance of $X$, variance of $Y$ and covariance between $X$ and $Y$ , ...
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37 views

Mathematical Proof of the Classical Assumptions.

I am studying in class regression analysis and we just went through the classical assumptions. I was wondering, is there a mathematical way to prove that $V(Y_i)=σ^2$ and $Cov(Y_i,Y_j)=0$? Since they ...
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12 views

Are there any assumptions missing from the linear regression model?

Let's assume a linear regression model: $$\\ Y^{(i)} = \theta X^{(i)} + E^{(i)},$$ where (i) is the number of observation and $\theta$ is a vector. I have found such an equation: $$\\ p(y^{(i)}|x^{(i)}...
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45 views

Why can we treat Cox's partial likelihood as a full likelihood?

I am doing some self study on Cox regression, and am trying to figure out how we can derive the partial likelihood for the Cox model from the full likelihood. Generally, I know that to get a partial ...
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1answer
40 views

Is it possible to calculate the regression sine function given three points?

Take the three points $(10,52)$, $(20,38)$ and $(50,-53)$ How would you calculate the sine regression line of the form: $$f(x)=A\ \sin{\frac{x+B}{C}}$$ In other words how would you calculate the ...
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39 views

Does high standard error and high r-square imply spurious regression?

Does a regression passed on time series data with one independent variable and one dependent variable which yields parameters with very high standard errors (t-values) and also a high r-squared imply ...
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19 views

Least absolute deviations problem minimization

Least absolute deviations problem minimization $\min_{β∈R}|y_j − x_{j1}β_1|$, for j = 1,....N Here y is the dependent variable and x is the independent variable. What happens to the case when $x_{...
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14 views

Which covariance matrix should I use for treating heteroskedasticity in my panel data?

I have a data set with panel structure (panel data) with 78 individuals observed over 5 three-year periods. I have 10 dependent variables an 1 independent variable. I applied logarithmic ...
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1answer
110 views

Taylor series approximation of Gaussian Q function , $Q\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int_x^\infty {{e^{ - \frac{{{v^2}}}{2}}}dv} $

I am trying to find a Taylor series expansion for Gaussian Q function. I have seen that error function $Erfc(x)$ is an approximation of $Q(x)$ (Is my assumption correct?). $Erfc(x)$ has Taylor ...
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1answer
81 views

What are the “moment conditions” in the GMM method? Also: GMM vs IV vs 2-stage least squares?

GMM = generalized method of moments IV = instrumental variables 2SLS = Two stage least squares OLS = ordinary least squares I keep seeing talk of 'moment conditions' or 'moment equations', but don'...

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