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Questions tagged [regression]

Questions on (linear or nonlinear) regression, the fitting of functions that best approximate empirical data.

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Computing SSREG and SSRES (Regression)

SSREG is calculated with $\sum_{i=1}^{n} (\hat Y_i - \bar{Y})^2$ which is the regression sum of squares SSRES is calculated with $\sum_{i=1}^{n} ( Y_i - \hat Y_i)^2$ which is the residual sum of ...
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For every $x$ $\%$ change in variable $y$, there is an implied $z$ $\%$ change in variable $N$ - statistics

I had a general maths question. I am trying to find the right topic to study for my problem. I would appreciate if someone could point me in the correct direction to study. The general problem is: ...
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1answer
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Stochastic process vs regression for modelling data

why one would model something (anything?) with a stochastic process, such as Ornstein–Uhlenbeck (OU), rather than a regression, like $y(t) = \beta_0 + \beta_1e^{( - rt)}$? With regression, you can use ...
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How do we solve design system matrix when there is no line that passes through all points

When solving for Ordinary Least Squares regression line I was taught that you solve the system of equations $$ y = X\beta \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$$ where $X$ is the design matrix. The way ...
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Fitting a quadratic model using simple linear regression

Suppose that the true model is $Y_i = \beta_0 + \beta_1X_i + \beta_2X_i^2 + \epsilon_i, 1 \leq i \leq n,$ where $ \{\epsilon_i\}$ are i.i.d normal random variables with mean zero and variance $ \...
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Calculating Ego-Motion from planar keypoints

I'm working on a module to estimate the ego motion from a robot with the help of a camera. In order to simplify that system I decided to track keypoints on the floor below the robot instead of solving ...
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econometrics: weighting frequency over time

I want to measure a model like: Score = arunning_sum_of_all_exercises + brunning_sum_of_identical_exercise + c*exercise_density_value_in_period_x --other variables -- Lets say I want to test ...
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Regression with incomplete basis

Suppose that $f(x) = \frac{1}{2}\text{max}(x+10,0) + \frac{1}{3}\text{max}(x+20,0) + \frac{1}{4}\text{max}(x+110,0) + \frac{1}{5}\text{max}(x+120,0)$ If I randomly simulate values for $x$ such that $$...
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Regression convergence

By simulation we create a vector $Y = (y_1,y_2,...,y_n)$, where each $y_i \in R$ is independently drawn from a given non-degenerate distribution. Next we create by simulation vector $\xi = (\xi_1,\...
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estimator lasso

I would like to know if you could either give me a hint to answering the following problem or a reference (article, book, etc) to have a better idea on how to address it the prove for LASSO estimator ...
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What will be the value of sub gradient at $0$ for function $|x|$

I am learning about Lasso Regression and came across taking gradient with respect to $0$. I came to know about subgradient but could not understand what will be its value at $0$. In lasso regression, ...
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What is Kalman filter?

I am not an economics grad. I am an engineer. I was working on beta value in market model regression.This is how I proceed. Firstly, I calculated daily returns from adjusted closing price of all ...
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How does scale Hyperbolic Distance if the radius change?

I trained a neural network which makes a regression to points in Poincarè Disk Model with radius $r = 1$. I want to optimize using the hyperbolic distance $$ \operatorname{arcosh} \left( 1 + \frac{...
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1answer
62 views

Converting tabulated data to equation of multiple variables

I've come across a mathematical problem while working on my school project. I've got tables for airplane takeoff distance which is dependent on airport altitude, temperature and mass of an airplane. ...
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Calculate derivative of partial effect in probit with respect to the parameter

In the standard probit regression we have that $ Pr(y>0| X) = \Phi(\frac{X\beta}{\sigma})$. With $\Phi$ Cumulative Density Function of the Normal Distribution. The marginal effect of the ...
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Linear regression model

Assume the usual linear regression model with $Y = X \beta + \epsilon$, where $X$ is fixed and known and $E(\epsilon) = 0$ and $\operatorname{var}(\epsilon) = \sigma^2I$. Let $a$ be an $n$-...
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Regression: width of confidence interval vs t-statistic

My question is about confidence intervals of the slopes estimated in a multivariate regression. I would like to clear up something that is probably a fault somewhere in my understanding. As I ...
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Sum of residuals in instrumental variable when constant included

Do residuals from an IV regression sum to 0? i.e Let e = y- Xb* where e is the residual and b* is the IV estimator using Z as an instrument for X. (Assume that dimension of Z = dimension of X and both ...
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Where does the identity matrix come from in the formula for ridge regression coefficients?

The formula for the ridge regression coefficients is $$ \beta = {X^{\top}Y}({X^{\top}X+\lambda I})^{-1}$$ I have tried to derive it as follows: The loss is (I am omitting the sum before the square ...
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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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1answer
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Logistic Regression with a Time Factor for Forecasting

I have a general question with regards to modelling the chance that domestic flights within the U.S. are cancelled or not using logistic regression (which I'm relatively new to). I have used 'R' to ...
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How to predict next year monthly data when there is only one year monthly data? [closed]

I'm fairly new to statistics and time series and I'm facing a work problem. I want to predict the next year monthly revenue but I only have my last year monthly revenue data. To my knowledge, the ...
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1answer
19 views

Normal Equation Derivation Step Help

I was interested in seeing how the normal equations for least squares for linear regression are derived, and found this page: https://eli.thegreenplace.net/2014/derivation-of-the-normal-equation-for-...
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minimum significant regression coefficient value based on sample size

I have a dataset with 35 observations for five variables, namely ET, sw, ws, vpd,Ta. I am doing a simple linear regression between ET and each of the other four variables (i.e ET with sw; ET with ws; ...
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Sketching residual plots

A set of data has been analysed by fitting the model $Y_i = x_i^{'}\beta + \epsilon_i$, where $\epsilon_i$ follows a normal distribution with mean 0 and variance $\sigma^2$. Sketch a residual plot for ...
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How do you create the polynomial basis function given data?

According to Wikipedia: However, i saw that basis functions applied before linear regression looks like: Is this saying that, Each observation x has D dimensions. then applying the wikipedia ...
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Angle between regression line and SD line when plots flipped

The question I am working on is: The scatterplot y* vs x* has a regression line that makes an angle with SD line. Then the corresponding angle for the scatterplot of x* vs y* is: I have generated ...
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Error Term, Regression

Assume I have $Y_i=A+BX_i + \epsilon_i$ For a multiple linear regression I have, $Y_i=A+B_1X_1+B_2X_2+....+B_NX_N+\epsilon$ The question is, what is the distribution of $\epsilon$, knowing that $\...
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3answers
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What method does a calculator use to calculate a linear regression line?

Take three coordinates $(1,1)$, $(3,2)$ and $(4,3)$. My calculator returns the linear regression line: $$y=0.6429x+0.2857$$ of the form $$y = ax +b$$ correct to four significant figures for constants ...
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Time-series modeling

I'm wondering what methods can be used to predict a future value using past values. I looked into linear regression modeling, but this doesn't allow for a time value. As an example, say I have an ...
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Variance of a fitted model

Show that for any linear model, $\sum_{i=1}^{n}\frac{\text{Var}(\widehat{Y_{i}})}{n} = \frac{p\sigma^2}{n}$. Wasn't too sure where to start here. I know that Bias($\widehat{\sigma^2}$)=$-\frac{p\...
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Writing $\beta$ and X for a qualitative model

Express the following model in matrix form, ie: specify $\beta$ and $X$ so that the model can be written as $Y = X \beta + \epsilon$. The model $Y_{ij} = \mu_i + \epsilon_i$ where $Y_{ij}$ represents ...
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The steps to get $\beta$ in Least Square Estimation, why $x_i$ was removed?

Please see the following steps Get $\beta$ in Least Square Estimation:
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setup time optimization using statistics

together I work at a manufacturer with numerous products. The diversity if variants is very high. For almost every product the machines have to be converted. There are essentially 3 characteristics, ...
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Fitting a logarithmic trendline on already logged values

This is the situation. I am running trials with a population simulator, which produces various outputs, with the variance of these outputs being dependent on the number of clones (recursions) the ...
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Probability of being late to bus

Bob takes a bus everyday to get to his school. The bus takes off exactly at 7.00 everyday. Here is when bob arrives to the bus stop: day one: 6.58 day two: 6.55 day three: 6.58 day four: 7.02 day ...
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Response surface methodology

How Can I Calculate optimal value from equation if I don't know the interaction effect value for those 2 factors? enter image description here
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Unique solution to arbitrary regression problem

I have a the following regression problem, where I have data points $(x_i, y_i)$ and $f_{\theta}$ is the regression function parametrized by $\theta$. For example, in linear regression, $f$ is just a ...
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Logistic Regression: Are thresholds and non-negative constraints equivalent?

For interpretability, I want to restrict the weights $\beta_i$ in a logisitic regression $$p(x) = \frac{1}{1 + e^{- \beta^Tx}} $$ to be non-negative, i.e. $\beta_i \geq 0$. Since the negative ...
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Is it appropriate to use clustering to partition the dependent variable into separate datasets for a home price prediction model?

I'm struggling to decide how to deal with a heteroskedasticity problem in a home price prediction model I'm developing. The training set residuals are normally distributed around zero, but they have ...
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Empirical risk minimization vs. convergence in conditional expectation

Consider the following equations: $$ \widehat{R}(f) = \frac{1}{n} \sum_{i=1}^n (Y_i - f(X_i))^2$$ $$ R(f) = \mathbb{E} (Y - f(X))^2$$ Suppose that we have $\min_{f \in \mathcal{F}} \widehat{R}(f) \...
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Rank of sub-matrix of projection matrix

Consider the projection matrix in Linear Regression $P=X(X^TX)^{-1}X^T$. If we have $n$ points, $P$ is an $n$ x $n$ matrix. We also know it satisfies a number of properties, including that it's ...
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Regression task on desirable subsets with limited supervision

Suppose I have a set of n elements. I want to have a model for how "desirable" a subset of those elements is when evaluated by a person. The training/testing data is a number of given subsets, each ...
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In a logarithmic regression, does it matter which letter is assigned to which coefficient?

I have seen a logarithmic function written as: $$y = a + b \ln(x).$$ For example, here. But also, I've seen it written as: $$ y = a \ln(x) + b .$$ For example, here. Is there a reason to assign a ...
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How to interpret the vectors and design matrix in a linear model

In regression, linear models are of the form: $$y_i = \pmb z_i^T \pmb\beta_i + \epsilon_i$$ Or we can write this in a more general form with vectors and a design matrix: $$\pmb y = \pmb Z \pmb \beta ...
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Calculating the regression coefficient of a time-based series

If i have a sample (lets say house price in millions over time) where x=1,2,...,14 samples y-values are shown in the image below, with one sample estimated per month. How do i calculate the estimated ...
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Why is the likelihood for bayesian liner regression in that form?

Let's say we have a linear model $Y = bX + e$ where $b$ represents the parameter. Normally, the posterior is written like this for data $X_n$ and $Y_n: P(B|Y_n, X_n) = P(Y_n|X_n,b)P(b|X_n)/P(Y_n|...
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Intercept and dummy variables regression significance

I am trying to find the joint significance of the alpha (the intercept) and the dummy variables, $\gamma_1$ and $\gamma_2$. I know the significance of $\alpha$ from the regression intercept t-stat ...
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what are $SSE$ and $S_{y,x}$?

I'm a BC student who is trying to solve some statistic quizzes. there is a multiple choice question that is this: In a simple regression model $y=a+bx+e$ our given data is this: $$ \bar{x} = 2, \...
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Given data matrix and error matrix, find value of $x,y,z$

In an exercise about multiple regression, I'm given the following data matrix $X$ and error vector $\epsilon$: $$ X = \begin{bmatrix} 1 & 4 & x \\ 1 & -2 & 1\\ 1 & 1 &...