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

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3
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44 views

Determine whether ARMA(p,q) is stationary and/or invertible?

Determine whether an ARMA(p,q) process is stationary and invertible such that $y_t = \sum_{i=1}^{p} \phi_i y_{t-i} + \sum_{i=1}^{p} \theta_{i} \epsilon_{t-i}$ with the restriction that ...
3
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1answer
32 views

Logarithmic Functions algebra question

This is my first post and I honestly just want a second opinion on my answer to a question I got incorrect on an exam before I go arguing over it with my professor. Basically, is this mathematically ...
3
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0answers
96 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
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0answers
134 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
255 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
818 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) ...
3
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1answer
134 views

Bound of linear regression's object function

Randomly uniformly select $n$ numbers from a set $\{1,2,...,U\}$ with/without replacement, $y_i$ is the $i$th number selected, and $x_i$ is the rank of $y_i$ in the $n$ numbers. The rank is the order ...
2
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1answer
553 views

Polynomial fitting - how to fit and what is _polynomial fitting_

I don't understand what is polynomial fitting. Can anyone explain me how to fit a curve to given points?
2
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3answers
172 views

least squares regression in 3space

robjohn is giving me a hand with this, but in case anybody else knows... I need to do a least-squares regression for linearity on a set of coordinates in 3space. If the dataset is linear, I need to ...
2
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4answers
168 views

Fitting curve for exponential: $y = A - B\mathrm{e}^{-t/\tau}$

I have some data that follows a saturation or charging profile such as $y = A - B\mathrm{e}^{-t/\tau}$. To begin with, is there a proper name for this function? I have seen it many times, including: ...
2
votes
2answers
166 views

Closed form for coefficients in Multiple Regression model

I want to find $\hat{\beta}$ in ordinary least squares s.t. $\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X_1 + \cdots + \hat{\beta}_n X_n $. I know the way to do this is through the normal equation using ...
2
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2answers
236 views

design matrices

Given a linear model $Y = X\beta + \epsilon$ with three treatments and six subjects where $X$ is the design matrix, suppose $X = \begin{matrix}1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 ...
2
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1answer
527 views

Linear Regression with 3x3 Matrices

Here's my Homework Problem: We can generalize the least squares method to other polynomial curves. To find the quadratic equation $y=a x^2+b x+c$ that best fits the points $(-1, −3)$, $(0, 0)$, ...
2
votes
1answer
251 views

How do I do a least squares fit of $a x + b y = 1$?

How do I do a least squares fit of the line equation $a x + b y = 1$, so that the points are as close to the line as possible? (Not just vertically close) If I use the matrices $$X = ...
2
votes
2answers
96 views

Linear Models - Regression Analysis

As a student learning Applied Regression Analysis, I come from a background with very little information about this topic. I understand that given $y = \beta_0 + \beta_1x_1 + \epsilon$ $E(y\mid x) = ...
2
votes
1answer
404 views

What is $Cov(\hat{Y},Y)$?

If $\hat{Y}$ is the OLS linear regression model for $Y$, what can I say about $Cov(\hat{Y},Y)$? Is this value 0?
2
votes
1answer
760 views

Hat Matrix Identities in Regression

I need to show that $\bar h= \sum{h_{ii}/n} = \operatorname{Tr}[H]/n = (p+1)/n$ Using the fact that $\operatorname{Tr}[AB]=\operatorname{Tr}[BA]$ and $H=X(X^TX)^{-1}X^T$. But I have no idea how to ...
2
votes
3answers
356 views

Residuals of regression model

Let's suppose I do a regression between earnings and age (and suppose I do not know the distribution of earnings). Would it be possible for the residuals to be normally distributed? I am thinking it ...
2
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2answers
399 views

Calculate Line Of Best Fit Using Exponential Weighting?

I know how to calculate a line of best fit with a set of data. I want to be able to exponentially weight the data that is more recent so that the more recent data has a greater effect on the line. ...
2
votes
1answer
173 views

How to handle constant term in Least Squares Regression?

In the well known matrix form of a least squares regression where I am trying to solve for B in Y = B1X1 + B2X2 + B3 I might be given X and Y sample data as something like $X$ = $\begin{bmatrix} ...
2
votes
1answer
40 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
51 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
82 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
51 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
votes
1answer
28 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
votes
4answers
95 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
302 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
votes
1answer
73 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
votes
1answer
87 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
82 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
votes
3answers
405 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
2k 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
156 views

What is the difference between Curve Fitting and Regression(Machine Learning)?

I know that Machine Learning regression algorithms try to find the function of the data. That is, if we have 1000 data points (x,y), to find a general continuous function that follows the trends of ...
2
votes
2answers
36 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
35 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
294 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
34 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
2answers
471 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
votes
1answer
133 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
votes
1answer
245 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
votes
1answer
114 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
votes
2answers
194 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
59 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
97 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
62 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
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1answer
2k 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
150 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
2answers
443 views

Binary Logistic Regression Model Processing

Thanks for showing interest and wanting to help out. My aim is to develop a model that - as accurately as possible - predicts how entities in a population will either cooperate or defect, as a % of ...
2
votes
1answer
91 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
334 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 ...