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

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2
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1answer
46 views

How can I plot this?

Given a bunch of data $x_i$ , $y_i$, how do I plot $$f(\theta_2,\theta_2)= \frac{1}{2M} \sum_{i=1}^{M} (\theta_1\cdot x_i -\theta_2 y_i)^2$$ in matlab? I know it should be parabolic, but my code ...
2
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1answer
100 views

Fitting a simple linear regression

A professor in the School of Business in a university polled a dozen colleagues about the number of professional meetings they attended in the past five years $x$ and the number of papers they ...
2
votes
2answers
174 views

connection between PCA and linear regression

Is there a formal link between linear regression and PCA? The goal of PCA is to decompose a matrix into a linear combination of variables that contain most of the information in the matrix. Suppose ...
2
votes
1answer
1k views

Convert nonlinear regression equation to a linear regression equation

The question is: "Show how the nonlinear regression equation y=aX^B can be converted to a linear regression equation solvable by the method of least ...
2
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1answer
62 views

Choosing the right regression

I'm trying to analyze my sleep using regression analysis. Each night is rated (dependent variable). I'm trying to explain this rating with, for example, my sleep duration and each night's bed time's ...
2
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1answer
30 views

Determine which parameter has correlation with result and which is not

sorry for probably silly question, it's the first time when I need to do such work. I have large data set with regarding clicks on some element on web page. It contains some characteristics of such ...
2
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4answers
100 views

Condition for $\det(A^{T}A)=0$

Is it always true that $\det(A^{T}A)=0$, $\hspace{0.5mm}$ for $A=n \times m$ matrix with $n<m$? From some notes I am reading on Regression analysis, and from some trials, it would appear this is ...
2
votes
1answer
428 views

Arriving at the Logistic function from a Binomial Distribution and Maximum Likelihood

I've been trying to understand the origin of the Logistic function in Logistic regression: $$\Pr(Y=1|x;\theta)=\frac{1}{1+e^{-\theta x}}$$ I was lead to beilive that one could somehow arrive at this ...
2
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1answer
130 views

Find data to perform regression analysis

I'm trying to find some data (two continuous variables that I believe are correlated) online for which I can perform a regression anaylsis, my assignment sheet says: The data may be found anywhere ...
2
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1answer
4k views

Equations For Quadratic Regression

Does anyone know the specific equations for the three parameters in a least-squares quadratic regression? I'm looking for something like $\beta_1=,\beta_2=,\beta_3=$ for each of ...
2
votes
1answer
104 views

Linear relationship of a company's profit

Assume a linear relationship for a company that has several shops is not known. Let $Y_i$ be the profit the shop number $i$ makes in the coming year. Let $x_i$ be the size of the shop number ...
2
votes
4answers
89 views

Seeking a function based on its level set

I'm trying to create a function for a research project, but I fear my math knowledge is insufficient to derive it from the attached diagram I've created showing its desired behavior. I'm hoping ...
2
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3answers
471 views

How to deal with Linear Regression model with some data aggregated

Lets say I am trying to find a linear regression between Weight and Height of a person. $W=b_0+b_1 H+e$ The data I have gathered from 8 people is like this: ...
2
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1answer
4k views

How to calculate hyperbola from data points?

I have 4 data points, from which I want to calculate a hyperbola. It seems that the Excel trendline feature can't do it for me, so how do I find the relationship? The points are: (x,y) (3, 0.008) ...
2
votes
1answer
75 views

Least-squares solution to a matrix equation?

Suppose I have $n$ observations of $m$ dependent variables $y_1,\dots,y_m$, and I believe they follow some model wherein they can all be written as linear combinations of some underlying variables ...
2
votes
1answer
59 views

Outlier detection with robust multiple regression model

I have a set of features (eg, location, income, budget, education) that I use to predict a continuous variable (say, amount spent per day on the internet). I am interested in detecting outliers. I ...
2
votes
1answer
29 views

Prove a result in multiple linear regression

This arises in multiple linear regression. Given $m, n \in \mathbb{N}$ and matrices $X \in \mathbb{R}^{m \times (n+1)} (m > n + 1), H = X(X'X)^{-1}X' \in \mathbb{R}^{m\times m}, I = I_m$ and $J ...
2
votes
1answer
51 views

Is it possible to have two lines of best fit?

Could you rig a data set to have two lines of equally good (and best) fit? Or is it impossible?
2
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1answer
20 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 ...
2
votes
1answer
515 views

Analytic solution for matrix factorization using alternating least squares

The standard form for ridge regression aims to minimize the following cost function. $$ \min\ \ \sum_i(y_i-x_i^T\beta)^2 + \lambda\sum_j\beta^2_j $$ As described here, it's possible to differentiate ...
2
votes
1answer
45 views

Standard Error in OLS Regression

Assuming I have the following linear regression set-up: $y_i = \alpha + x_i * \beta + \epsilon_i$ for $i = 1,2,..., n$. When I run the regession, I get a $\beta$ and $\alpha$ estimates, along ...
2
votes
1answer
85 views

Prove that $E(\mathbf{u}|\mathbf{X})=\mathbf{0}$ implies $Cov(\mathbf{x},\mathbf{u})=\mathbf{0}$

Let \begin{equation} \mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{u} \end{equation} where $\mathbf{y}=\begin{bmatrix}y_1 \\ \vdots \\ y_n\end{bmatrix}$, $\mathbf{X}=\begin{bmatrix}X_{11} & ...
2
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1answer
175 views

Least Squares Regression Matrix for Rational Functions

So first off no, this isn't a homework problem. Second, I'm trying to understand how this works, NOT find a program that will do it for me. Okay so I've known for a while how to use Gaussian-Jordan ...
2
votes
1answer
34 views

Gaussian prior favors values closest to zero?

I am reading an article on Bayesian Logistic Regression, where they're using Logistic Regression, imposing a Gaussian prior (with mean = 0) on its parameters. They state that a Gaussian prior favors ...
2
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2answers
117 views

derivation of simple linear regression parameters

I know there are some proof in the internet, but I attempted to proove the formulas for the intercept and the slope in simple linear regression using Least squares, some algebra, and partial ...
2
votes
2answers
67 views

Find parameters for curve fitting (simple linear regression involved?)

I would like to fit data in g~t scatterplot, where ...
2
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2answers
45 views

Merging Linear Regression

If I have built two linear regression models over sets $A$ and $B$, and now want a linear regression over set $A\cup{}B$. Is there a way to reuse what I already have?
2
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2answers
39 views

Regression model when under-estimations costs us more than over-estimations

We have a factory and we are planning how many items produce in 2014. During the learning process we minimize the mean squared error. But, under-estimations costs us more than over-estimations. Let's ...
2
votes
2answers
508 views

How to fit a sinusoidal function through 2 points with known slopes?

I can define my sinusoidal function as $y(x) = A\sin(B x+c) + D$ or as $y(x) = A \sin(B x) + C \cos(B x) + D$ Now, I have two points with known slopes that I must fit this sine wave to, thus my ...
2
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1answer
61 views

Creating a lift chart for a classification tree

This is likely a simple question but I'm new to data mining techniques and am trying to compare two different predictive models. I've created a logistic regression and a classification tree and would ...
2
votes
1answer
996 views

Lasso - constraint form equivalent to penalty form

We know that there are two definitions to describe lasso. Regression with constraint definition: $$\min\limits_{\beta} \|y-X\beta\|^2, \sum\limits_{p}|\beta_p|\leq t, \exists t $$ Regression with ...
2
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2answers
1k views

Linear Regression: Expectation Proof

I found the following proof in my notes: $E(Y_i) = E[\beta_0 + \beta X_i + \varepsilon_i] =\cdots= \beta_0 + \beta X_i$. This does not seem right to me, however. Why would $E(\beta_1 X_i) = \beta_1 ...
2
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1answer
193 views

Techniques to find regression parameters for multiple datasets where a subset of parameters should be the same for all datasets

I have five sets of observations of measured y as some function of measured $x_1, x_2, x_3,\ldots$ and I want to fit five functions to these observations. They have the form $$ y = f(x_1, x_2, ...
2
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1answer
390 views

Finding uncertainty in the slope/intercept for a non-linear least squares fit

I have the following function: $$M = a(\log_{10}W-2.5)+b$$ I also have a set of data with actual measured values of $W$ and $M$ (each have individual $\pm$ errors). Here's a small sampling of the ...
2
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1answer
132 views

Least Squares Regression To Half of a Parabola

I have a set of points in two dimensional space, and I know a priori that they approximate half of a parabola. I want to find the coefficients for a quadratic function where all of the points fall on ...
2
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2answers
260 views

maximize log determinant subject to a linear constraint

Does anyone know any efficient method to solve the following problem? $ (\alpha,\beta) = \text{argmax} \log \det (\alpha K_1 + \beta K_2)$ s.t. $c_1 \alpha + c_2 \beta = c_3, \alpha\geq0, \beta\geq ...
2
votes
1answer
63 views

Possibility of Unboundedness in Least Squares Minimization

Suppose we have the quadratic minimization problem \begin{equation} \min_x \frac{1}{2} x^TPx + q^Tx +r \end{equation} We know that when $P$ is symmetric positive semi-definite, but the optimality ...
2
votes
1answer
101 views

What am I reinventing? RE: Linear regression modeling for frequency of discrete events

I'm looking to model the frequency of events to quantify how much that frequency is increasing or decreasing. For the sake of concreteness think of the events as web page hits for several low traffic ...
2
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1answer
67 views

Biased linear regression

I have a set $S$ of coordinates $(x,y)$, and am estimating $f(x) = ax + b$ where $a > 0$. I also happen to know that $\forall x,y((x,y) \in S \implies y < f(x))$. The question is how I can ...
2
votes
1answer
206 views

How to find a parametric equation?

I want to find an equation for a race track, so I could get the position of a point with respect to time. Let's say I have this track and here are a few points on it: Could it be possible to model ...
2
votes
1answer
100 views

Nonlinear regression with correlated errors

it's my first post here and I'm a newbie in statistics, so please forgive me if I'm doing something wrong or explaining myself badly. Anyway, I have a problem similar to this: How to perform ...
2
votes
1answer
437 views

variance of multiple regression coefficients

If I consider universal kriging (or multiple spatial regression) in matrix form as: ${\bf{V = XA + R }}$ where $\bf{R}$ is the residual and $\bf{A}$ are the trend coefficients, then the estimate of ...
2
votes
1answer
67 views

Probabilistic regression on outliers

I have a given data set $D = \{ x_i, y_i \}_{i=1}^n$ for a regression problem. When I plot the data, it looks like there is an underlying parabola (2nd order linear model) and some outliers. I want ...
2
votes
1answer
348 views

An intuitive explanation for neural networks as function approximators ?

We use normal linear regression for modelling functions on datasets . But Can someone explain how neural networks help in approximating more complex ,especially non-linear functions ? intuitively , ...
2
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1answer
444 views

Curve fitting with upper and lower bounds for derivatives

I compute (at a great cost) upper and lower bounds $f_u(x)$ and $f_l(x)$ of an unknown function $f(x)$ at points $x$ in $[0,1]$. Now I am interested in an estimation of the derivative $f'(x)$. I ...
2
votes
2answers
212 views

How to fit an equation to a curve with disturbances

For example, I have the following data: Y = 366 measured values X = 366 measured values t = [ 1 : 366 ], representing the days of the year (index) So at each t (day), we have value of Y and ...
2
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2answers
144 views

Choosing set of best estimators for linear least squares

I have a measured experimental dataset which is well approximated by the sum of several basis functions in linear combinations. Linear least squares of course gives me the optimal weight for each ...
2
votes
2answers
779 views

Fitting data to a portion of an ellipse or conic section

Is there a straightforward algorithm for fitting data to an ellipse or other conic section? The data generally only approximately fits a portion of the ellipse. I am looking for something that doesn't ...
2
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0answers
19 views

combining multiple regression outputs

Suppose I have multiple regressions, along with their r-squares, standard-errors, etc.: $y(t) = \alpha_1 + \beta_1 x(t) + e_1$, where $t \in (\tau_0, \tau_1)$ $y(t) = \alpha_2 + \beta_2 x(t) + ...
2
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0answers
22 views

GLM for Poisson Regression for Soccer Ratings Not Converging

I have been trying to formulate a model of soccer matches to help me predict the outcomes. The model I'm trying to formulate involves using Poisson regression to assign attack and defence ratings to ...