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

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Regression coefficient in simple regression

Let's say we have two random variables $Y$ and $X$ used to form regression model $$Y=\alpha+\beta X+\mu$$ It also holds that $E(\mu)=0$, $\text{Var}(\mu)=\sigma_{\mu}^2$, $\text{Var}(X)=\sigma_{X}^2$, ...
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17 views

Relationship among parameters from models with different link function and scaled response variable

Given the model, $\log(A_i) = \alpha + \beta \, covar_i$, with $i=1,\dots,1000$, $\alpha=4$, $\beta=0.2$, and covariate $covar \sim U(-1,1)$, I derived $\log(A)$ values (in $\texttt{R}$) as: ...
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Appropriate regression test apart from MLR for crime data?

thanks in advance. I'm looking to run some statistical methods to find the correlation of crime rates to crime factors. I know about MLR, which is pretty simple to run in SPSS, but what are the other ...
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15 views

curve fittin with non-gaussian noise

Fitting with the least squares method results in the ML fit assuming the given points have a gaussian distributed noise. What methods are there for non-gaussian noise distributions, especially ...
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2answers
32 views

How to start using R (or free alternatives)?

How to start using software R to make regression analysis and forecasting? Are there any other free software to work with this kind of analysis?
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36 views

Linear Regression Application

I have a linear equation as follows: $B_0*x_0 + B_1*x_1 + ... + B_8*x_8 = result$ And i have about 200 different situations that are categorized into two different groups, depending on whether ...
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1answer
28 views

Multicollinearity: Why does highly correlated columns in the design matrix lead to high variance of the regression coefficient?

I came across the term "Multicollinearity" in statistics, particularly statistics. However, I never really understand mathematically why highly correlated (almost linearly dependent) columns in the ...
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21 views

Relationship between Kriging and Gaussian process regression models

In the science of Bayesian modeling one method involves using Gaussian processes to derive regression functions on data. I notice in looking at the plots for such regressions that they resemble ...
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25 views

Expected Value of an inefficient estimator of the $\beta$ parameter of a simple linear regression

A simple linear regression is defined as follow: $$y_i=\alpha+\beta x_i+\epsilon_i \qquad i=1,...,n$$ An inefficient way of estimating $\beta$ is defined as follow: ...
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19 views

Is the regression model Identified? (is it possible to obtain a least squares estimator of the parameters?)

$y_t$ is the dependent variable, $x_t$ and $z_t$ are explanatory variables, and $α$, $β$ and $γ$ are unknown parameters. $y_t$ = $α$ + $β$$x^3_t$ + $γ$/$log$($x_t$) + $u_t$ $y_t$ = $α$ + $β$$x_t$ + ...
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27 views

Calculate $m_1, m_2 $ for $y = m_1x_1 + m_2x_2$

Given these values: $$x_1 = \left\{1, 3, 6, 8\right\}$$ $$x_2 = \left\{2, 8, 5, 10\right\}$$ $$y = \left\{8.6, 30.8, 34.1, 53.8\right\}$$ And this formula $$y = m_1x_1 + m_2x_2$$ How do I ...
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19 views

Can we see the beta coefficients in OLS as mean values?

Can we see the beta coefficients in OLS as mean values? I mean the estimator β alone. y=Xβ+ε
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1answer
16 views

Find the OLS estimator $β_1$ when a new variable is added to the regression

Suppose $y_t$ = $β$$x_t$ + $u_t$ , where t = 1, 2, ..., n. We know, in this case, the OLS estimator is $\hatβ$ = ∑$x_t$$y_t$ / ∑$x_t^2$ . Now suppose one more observation $x_{n+1}$ is added. At the ...
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15 views

Regression model, find $Var(y_i-\hat y_i)$

For the model of $y_i = \beta_0 + \beta_1x_{i1} + e_i$ for $i = 1,...,n$, where $e_i \sim N(0,\sigma^2)$ Find $E(\hat y_i)$ and $Var(\hat y_i)$. Hence or otherwise, find $E(y_i-\hat y_i)$ and ...
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2answers
56 views

Factoring a multivariate linear polynomial

I'm a computer programmer trying to solve a particular toy problem, and my understanding of linear algebra is far too lacking to solve it! I have a data set that can be modeled using this function: ...
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1answer
28 views

Multilinear fit vs Polynomial fit

I have a program that generates some physics data in 1D and 2D functions. In this program, the user defines a number of models that are used to compute a 2D function. That 2D function, and it's ...
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27 views

Predictive models

Given a set of temperatures of different cities for a month, which prediction model should I use for a two day look ahead prediction? Regression models or Time series?
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97 views

The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$

We have $n$ sensors $X_i$ which estimate the scalar value $\alpha$ with different relative accuracies $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
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23 views

Show that $E[g(X)u]=0$ in a standard regression model where $u = \hat{\beta}-E(\hat{\beta})$

Consider the standard regression model $y = X\beta + \epsilon$ where $y$ and $\epsilon$ are $(n \times 1)$ vectors and $X$ a $(n \times K)$ matrix. Let $\beta$ be any estimator of $\beta$. Let $u = ...
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Population versions of multiple correlation coefficients and least squares estimates

I'm reading an old paper (Wold and Faxer (1957)) which considers the theoretical relation $$ y=\beta_1x_1+\cdots+\beta_hx_h+\zeta $$ where $y,x_1,\ldots,x_h,\zeta$ are (scalar) random variables ...
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1answer
19 views

rotating and exchanging x for y's in regression

I was just wondering what happens generally if i send all my x points to y's and y's to x's (i.e reflect along the y=x line) - if I change the x's and y's will my old error minimizing line still be ...
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22 views

var(AB) when A,B not independent

I need to find the variance of $\hat\beta_1 * \bar X_1 / \bar Y$ , where we have the regression equation Y= $\beta_0 + \beta_1* X_1 +…+ \beta_j* X_j$ I initially was thinking the answer is simply ...
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44 views

Fitting a equation to a spiral curve

I am completely new to this forum and also to this type of mathematical modeling. I am interested to fit the following equation to the points obtained from experimental data. I am looking for an ...
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1answer
28 views

Critical points of quadratic forms

Let $A$ be an $n\times n$ symmetric matrix, let $b$ be an $n$-vector, let $c \in \mathbb{R}$ and set $Q(x) = 1/2 x^T Ax-x^T b+c$. Prove that $x_0$, defined as a solution to $Ax_0=b$ is a critical ...
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25 views

t test vs f test

For conducting statistical tests concerning the parameter $\beta_1$ (the slope of the estimated linear regression function), why is the $t$ test more versatile than the $F$ test? This is a question ...
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10 views

Analysis of block-greedy algorithms for function approximation?

I consider the problem of selecting a final basis set $\{\phi_{c_j}\}_{c_1}^{c_n}$ approximation of function $f \in \cal{H}$ in a Hilbert space that minimizes $L_2$ error. One can use a greedy ...
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1answer
18 views

Three-Perpendicular Theorem for linear regressions

For a random vector $X=(X_1,\ldots,X_p)'$, we define $$ \mathcal{L}(X)=\{b_0+b_1X_1+\cdots+b_pX_p,b_0,\ldots,b_p\in\mathbb{R}\}. $$ The linear regression of the $q$-dimensional random vector ...
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37 views

Searching a function for data

I have a dataset and I am trying to find the appropriate function to fit them. So far, I have fitted the data into a two variable polynomial: $$ y(t,v)=(-525.958 + 4.88502 t - 0.0149025 t^2 + ...
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For the three measurements b=0, 3, 12 at times t=0, 1, 2 find the best parabola y=C+Dt+E$t^2$

So I know how to do least squares regression using matrices to solve for Ax=b. I simply do $A^TAx=A^Tb$. However I don't really know how to account for the second power in a typical parabola ...
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25 views

What is the least squares solution given a line passes through original and following points?

So I am looking for the line y=Dt through the origin that fits the data y=4 at t=1, y=5 at t=2 and y=8 at t=3. This is what I have done so far. I know the three equations that are supposed to be ...
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Proof – OLS estimator regression [closed]

I am having trouble figuring out how I need to form and present an answer to a question. I completely understand the concepts of the math and analysis, I just don't understand how to give an answer ...
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28 views

consistency of OLS on misspecified AR(1) process

Suppose the true relationship in data is driven by AR(1) process as follows: $$X_t=\rho X_{t-1}+\epsilon_t\hbox{ , }|\rho|<1$$ and $\epsilon$ is a white noise of $(0,1)$ expectation and variance. ...
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1answer
20 views

Relation between Regularization and correlation

I was going through Chapter 3 (page 63 bottom) of Elements of Statistical Learning. While explaining regularization in ridge regression authors make the following statements. "When there are many ...
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12 views

What is the proper name of a model that takes as input the output of another model?

Thanks in advance for the help. I am writing a paper and for the life of me can't remember the proper term for a model that works as follows. rawData -> model1 -> outputModel1 -> model2 -> ...
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29 views

Finding better curved line of best fit

I have a set of hand generated data that follows somewhat closely to an exponential curve: I can come up with an exponential equation to the line that gives the values on the 3rd row, and Someone ...
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1answer
20 views

Calculating decreased cost with increasing quantity

I have a hand made table I've been using to give customers price per unit on my items, which gives a better price for the more items that they buy. My sample table right now I need to keep the ...
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1answer
46 views

Solution of overdetermined polynomial system

Some of you will find this question pretty straightforward to answer, but I desperately need some help in solving a problem involving several equations and 2 unknowns, for an engineering application. ...
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27 views

Standard deviation errors in log scale

I have a not so common issue with error bars in the log log scale. To be more precise, I have measurements of a quantity Y with an associated standard error Yer that has normal distribution and these ...
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77 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
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3answers
179 views

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
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1answer
28 views

Effects of feature scaling on weight vectors for linear regression

Given that linear regression or polynomial regression can be represented as: $\textbf{w} = (X^{T}X)^{-1}X^{T}Y$ It is standard practice in machine learning to scale each column in their training ...
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51 views

Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
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How to find the closest integer linear equation to given real linear equation

I am given a set of points in an n-dimensional plane. I want to find the closest (lowest co-variance) integer linear equation that characterizes the points. I find the real linear equation using r^2 ...
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1answer
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Invertibility of $X^TX$ when sever multicollinearity in regression

I am studying about multicollinearity in regression and in the book it says, "if there is severe (but not perfect) multicollinearity, two or more predictor variables are highly correlated, so $X^TX$ ...
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Notating the components of the $\hat{\beta}$ matrix when $\hat{Y}$ is multidimensional

This is a question on The Elements of Statistical Learning. We have from the linear model $$\hat{Y} = X^{T}\hat{\beta}$$ where $\hat{Y}^{T} = \begin{bmatrix} \hat{Y}_1 & \hat{Y}_2 & \cdots ...
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20 views

How to derive this solution to this minimization problem in vector form?

We want to minimize the mean squared error $$ \sum_{t=1}^n (y_t - \theta^T x_t - \theta_0)^2. $$ Letting $X = [x_t, 1]$, we can rewrite the above problem in vector form as $$ \sum_{t=1}^n (y_t - ...
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1answer
19 views

Correlated explanatory variables in linear regression

Is it any reason to assume that if two strongly correlated explanatory variables have impact on response that regression coefficients for these variables have the same signs ? Could such assumption be ...
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28 views

Least squares: Calculus to find residual minimizers?

Reading a section on simple regression in "An Introduction to Statistical Learning with Applications in R" I got a question on residual sum of squares minimization. Quoting from the book: [quote] ... ...
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1answer
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Machine Learning: Linear Regression models

I'm currently in a course learning about neural networks and machine learning, and I came across these two formulas in this textbook page on linear regression: 1) $y(x) = a + bx$ and 2) $y(x) = ...
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21 views

Mutiple Regression, calculating R-squared

If I have two regressors in multiple regression equation y=b0 + b1*X1 + b2*X2, how can I find R-squared for the model?I need to know the written formula(not in excel) for two independent variables as ...