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

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4
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0answers
195 views

How to compare the similarity between functions?

I'm designing a web service that finds the regression function of a pattern within an image. I analyzed three images and found the following three regressions: 1) $f(x) = 74.7602 + 0.2005x - ...
3
votes
0answers
49 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
3
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0answers
179 views

Can the sigmoid function approximate any function (or relation) where 0<y<1

I'm studying Machine Learning and Artificial Neural Networks. Some basic principles of Machine Learning are linear regression, multivariate linear regression, and nonlinear regression. The last of ...
3
votes
0answers
161 views

Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
3
votes
0answers
151 views

How to perform nonlinear regression with correlated errors?

I have a nonlinear least squares problem, but the errors are correlated. I could use R's nls function to do the regression if the errors were independent, but I don't know the right way to handle ...
3
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0answers
298 views

Least Square Method with Positive Parameters

this is my first post here in the Stack Exchange. A friend told me about this forum and I'm giving it a try. I searched a bit past threads, but couldn't find what I wanted, so I'm posting the problem ...
3
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0answers
992 views

Correlation and Regression Question

Two separate tests are designed to measure a students ability to solve problems. Several students are randomly selected to take both tests and the results are: $$ \begin{matrix} \text{Test A}(x) ...
2
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0answers
53 views

Predicting profit with price variation

I am currently working on a high school project that aims to predict profit from X amount of items to Y amount of profit based off a deviated sale price. For instance: I sale 10 cookies for 10 ...
2
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0answers
177 views

Normal equations for minimization of Frobenius norm least squares error

I'm having a hard time understanding the most efficient sequence of steps for deriving the normal equations for Frobenius norm least squares minimization. Here I want to minimize the norm of a matrix ...
2
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0answers
35 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
2
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0answers
19 views

Showing Hat matrix equal specific values

Consider a one way layout model $y_{ij}$ = $\mu_i + e_{ij}$ (1 $\leq$ i $\leq$ a, 1 $\leq$ j $\leq$ $n_i$) where a = 3 and $n_1$ = 2, $n_2$ = 3, $n_3$ = 4. Show that the hat matrix for this design ...
2
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0answers
100 views

Is it compulsory to make transformation to the econometric model in order to have only diagonal elements on variance-covariance matrix of errors?

I need some sharped and advanced advices for the following issue ... Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $y_{jis} = ...
2
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0answers
118 views

Expected in-sample error of linear regression with respect to a dataset D

In my textbook, there is a statement mentioned on the topic of linear regression/machine learning, and a question, which is simply quoted as, Consider a noisy target, $ y = (w^{*})^T \textbf{x} + ...
2
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0answers
85 views

What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing ...
2
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0answers
34 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
2
votes
0answers
15k views

Linear regression: degrees of freedom of SST, SSR, and RSS

I'm trying to understand the concept of degrees of freedom in the specific case of the three quantities involved in a linear regression solution, i.e. $SST=SSR+SSE, $ i.e. Total sum of squares = ...
2
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0answers
51 views

Computing evidence for least-squares fit

I'm at a loss trying to implement Bayesian model selection for standard least-squares polynomials fits. I have three polynomials of order $1$, $2$, and $3$, and a sequence of $(x,y)$ data points. ...
2
votes
0answers
572 views

What is the Moore-Penrose pseudoinverse for scaled linear regression?

The matrix equation for linear regression is: $$ \vec{y} = X\vec{\beta}+\vec{\epsilon} $$ The Least Square Error solution of this forms the normal equations: $$ ({\bf{X}}^T \bf{X}) \vec{\beta}= ...
2
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0answers
43 views

Coefficient of determination

$$ \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 = ( \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 - \displaystyle \sum^n_{i = 1} (y_i - \hat{y}_i)^2 ) + \displaystyle \sum^n_{i = 1} (y_i - ...
2
votes
0answers
150 views

Minimizing L4/ L6/ L2N norm for linear regression

OLS regression minimizes the sum of the squared errors. The normal equation for an OLS for $L_2$ minimization is as follows: $$b= (A'A)^{-1}A'y$$ What would be the equation to minimize the $L_4$ norm ...
2
votes
0answers
161 views

Orthonormal Matrix weighted regression

$Q$ is a rectangular matrix with orthonormal columns. A linear system composed of $$Qx= b$$ is really easy to solve as: $$Q'Q=I$$ hence: $$x=Q'b$$ Given that $Q$ is orthonormal can this be used to ...
2
votes
0answers
87 views

Regressing $Y$ back on the residuals

Suppose I have the linear regression model $ \hat{y_i} = a + b x_i $ for $a,b$ obtained via OLS. How does one regress $y$ back on the residuals $\hat{e}_i = y_i - \hat{y}_i$? If we write $ ...
2
votes
0answers
157 views

Effective model for calculating momentum or growth rate for a time series

I have a series of numbers tracking the performance of an entity on any given day. It's nothing but a simple integer for each date. For example, here's a series for Entity "X" ...
2
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0answers
327 views

Bare minimum of points in multiple polynomial regression

I have a question on multiple polynomial regression and the absolute minimum amount of points in the different terms. The minimum amount of points required for a second order polynomial would (in one ...
2
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0answers
92 views

Find $w$ as the minimizer of regularized logistic regression

Could someone point me to a reference on how to find $w$ as the minimizer of: $$ \frac{1}{2}\sum_{i=1}^{d}q_i(w_i-m_i)^2+\sum_{j=1}^{n}log(1+\exp(-y_jw^Tx_j)) $$ where $q_i$ (initialized with ...
2
votes
0answers
424 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
2
votes
0answers
320 views

Finding a model for multiple non-linear regression

I want to implement a regression analysis, but I have problems with finding a model for the given data. There are $149$ $(x,y,z)$-values. $y$ values are all positive, $x$ is between $[-10, 10]$ and ...
2
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0answers
3k views

Derivation of standard error of beta in simple linear regression

Countless web pages show the equation for the standard error of the slope in a simple linear regression. For example: ...
2
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0answers
3k views

Help with problem: Estimated Standard Deviation of Regression Equation (Simple Linear)

This is a practice problem. I've solved part (a). I have provided verified answers (from the published key) to all parts (a), (b) and & (c). I need help solving (b) and (c). Consider a simple ...
2
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0answers
272 views

Surface Function Fitting to Spherical Data

I have a set of geographic (longitude,latitude,value) data to which I would like to fit surface functions, specifically, the set of quadratic surfaces: $f(x,y)=Ax^2+Bx^2+Cxy+Dx+Ey+F$ At the moment, ...
2
votes
0answers
67 views

Accurate computation for Linear Regression case

I am writing a program that inputs a sequence of points $(x_i,y_i)$ based on the user clicking on certain pixels in an image shown. The program should then find the "best -fitting" line in the least ...
1
vote
0answers
10 views

How is genetic programming used in Symbolic regression

I am in highschool and have not taken any courses on this. Rather I am working on an outside project. I don't quite understand how Genetic Programming could be used effectively to generate a set of ...
1
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0answers
43 views

Why do the components of an equivalent kernel sum to 1?

Let $\textbf{x} = (x_1, \dots, x_n)^T \in \mathbb{R}^n$ and $k \in \mathbb{N}$. We define $$ X := \begin{pmatrix} 1 & x_1 & \cdots & x_1^k \\ \vdots & \vdots & & ...
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0answers
14 views

Approximation technique when data is missing?

I am doing some statistical studies and I would appreciate some guidance to some approximation techniques when not all data is available. I have a model that takes certain input parameters (discrete, ...
1
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0answers
24 views

Does gradient descent and normal equation give the same answer?

I tried to optimize for a linear regression model using both approaches and they gave me two completely different answers. My sample data set was: df <- data.frame(c(1,5,6),c(3,5,6),c(4,6,8)) ...
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0answers
33 views

Odd Ratio and Logistic Regression

Windy and Play Tennis = 9 Windy and Not Playing Tennis = 8 Not Windy and Play Tennis = 14 Not Windy and Not Playing Tennis = 8 I performed logistic regression in Weka and got odd ratio as 0.3448 for ...
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0answers
28 views

Increase the probability of correct prediction using multiple regression

First off let me begin by saying that I'm brand new to statistics and I would appreciate it if you could dumb down any answers for my problem. I am trying to create a general prediction of how much a ...
1
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0answers
34 views

Linear regression of time series data - moving linear regression

Situation A suitable analogy for my real-world problem would be a shop - customers arrive, spend a random amount of time in the shop and leave. The arrival behaviour of customers follows a Poisson ...
1
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0answers
29 views

Estimate b when $\hat{y}$ and x are given

Using linear algebra for solving this linear regression problem, I've got the equation $\hat{y}$ = xb. Finding the projection $\hat{y}$ of y onto the columns of the matrix X containing the ...
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0answers
26 views

Decomposition of matrix in a general linear model to find $\boldsymbol{\beta} = \mathbf{G}\mathbf{P}^{-1}$

In a simple least squares linear regression such as $y_{i} = \beta_{0} + \beta_{1}x_{i} + \epsilon_{i}$, there is a nice property that the slope of the regression line can be written $\beta_{1} = ...
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0answers
18 views

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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0answers
28 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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0answers
19 views

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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0answers
28 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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0answers
36 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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0answers
78 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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0answers
37 views

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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0answers
16 views

Regression question (details inside). Measuring the incremental impact on the dependent variable of one category over other categories

Formulate a regression equation you would use to test for the differences in ROE between firms that used tier 1 investment banks as their advisors and those that used tier 2 or tier 3 banks (note: ...
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0answers
16 views

Statistics linear modelling in R

Suppose I have a date set of the form: Test Subject/Sex/No. of mistakes made in the morning/ No. of mistakes made in the afternoon A / M / 2 / 5 B / F / 1 / 4 C / M / 3 / 5 D / F / 1 / 5 Suppose ...
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0answers
19 views

Cannot figure out autocavariance

The moving average model of order q has the form $$Y_t =β_0 +e_t +b_1e_{t−1} +b_2e_{t−2} +...+b_qe_{t−q}$$ where $e_t$ is a serially uncorrelated random variable with mean $0$ and variance $σ^2_e$. ...