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
98 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
votes
1answer
63 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
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
162 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
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2answers
484 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
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1answer
92 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
341 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
63 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
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1answer
232 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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2answers
192 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
135 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
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2answers
559 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
votes
1answer
46 views

Is there a site that will allow me to calculate a best fit for a set of data?

I have a bunch of x's and their corresponding y values, but do not have a Wolfram Pro account. Is there another site where I can input my dataset and have it spit out a best-fit regression (be it ...
2
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0answers
11 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
votes
1answer
24 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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1answer
48 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
2
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0answers
96 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
53 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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1answer
133 views

Understanding Regularization parameters in Machine Learning/Statistics

Suppose I have the following $k$ degree polynomial regression model with a data set of size $n$ which includes a $k$-dimensional feature vector $x$ and an outcome denoted $t_i$ for each vector in the ...
2
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0answers
31 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
1answer
49 views

How to calculate $\sum(X_i-\bar{X})^2$ in R

I'm trying to figure out how to calculate $\sum(X_i-\bar{X})^2$ in R, specifically identifying it in either the aov function or $\operatorname{lm}(y\sim x)$ function. I am trying to use it to ...
2
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1answer
24 views

Aproximate data with this equation (or linearize the equation)

I have found an equation that describes the behaviour of a phisical system: $$ y=a_1e^{-a_2x} + a_3 + a_4x + a_5e^{{-a_6} / {(1-x)}}$$ Now I have data of that physical system and I want to ...
2
votes
1answer
51 views

Basic Multilinear regression question for finding examples or counterexamples.

Hello Wise mathematicians! I have few quenstions about Multi linear regresstion. I've been asked from my friend, but I have very weak knowledge background from that field. It seems my friend is in ...
2
votes
1answer
81 views

Why does the regression line of $x$ on $y$ and $y$ on $x$ meet at $\bar{x}$ and $\bar{y}$?

Why does the least squares regression line of $x$ on $y$ and $y$ on $x$ intersect at $\bar{x}$ and $\bar{y}$? Also, why are the form of regression lines as they are? For the general form ...
2
votes
1answer
41 views

Probit model question (regression)

I'm reading a thesis and I need your help to understand the equation below. $$\Pr(\text{failure}=1 \mid X_1,X_3,X_3,X_4)=\int_{-\infty}^z \varphi(k) \, dk\tag{1}$$ $\varphi(k)$ is the standard ...
2
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0answers
44 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
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0answers
362 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
votes
1answer
484 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
votes
2answers
28 views

Linear regression. Lowering response maintaining equal independent variable.

I have put some data together and modelled the behaviour of the response ($y$) as function of three independent variables $x_1$, $x_2$ and $x_3$. A simple multi-linear regression. The model looks ...
2
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0answers
37 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
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0answers
124 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
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0answers
130 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
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0answers
82 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
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0answers
132 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
251 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
85 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
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1answer
179 views

Projection Pursuit Regression

This is with reference to projection pursuit regressions. I kind of get the idea behind approximating a continuous function using weighted sums of ridge functions. I am not sure why ridge functions ...
2
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0answers
341 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
2answers
571 views

Least squares estimator of mu

The question is: Assuming that $y_i = \mu + \epsilon_i $,$i = 1,\ldots,n$ with independent and identically distributed errors $\epsilon_i$ such that $E[\epsilon_i] = 0$ and $Var[\epsilon_i] = ...
2
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0answers
291 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
2k 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
votes
1answer
287 views

Multiple linear regression with interaction

I'm doing a multiple linear regression with interacting variables. I'll give you an example: y=value a1=material a2=weight a3=color a1 and a2 are interacting variables but a3 is not. Right now I'm ...
2
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0answers
85 views

Polynomial and exponential regression [duplicate]

Possible Duplicate: Determining computational complexity of stochastic processes I have some points $(x_i,y_i)$ generated by a program. These values are not exact, but are random ...
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0answers
2k 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
247 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
64 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 ...
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5answers
518 views

Find square root approximation function (tool)

first I have to apologize for any uncorrect naming or categorisation of my question, as I am an electrical engineer rather than a mathematican. I try to find a simple solution for my problem: I have ...
1
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4answers
284 views

Parabola from 4 approximate points

I have calculated four approximate points from a sensors to get information. I would like to deduce the closest parabola to my points. The problem is that I can't solve it to get an appropriate ...
1
vote
3answers
528 views

Simple Least-Squares Regression Question

Given a set of 5 points (i.e. (1, 3), (2, 8) etc...), how can I get just the slope of the best fit line? I've been looking up least squares regression, but I'm rather statistics ignorant and don't ...
1
vote
1answer
321 views

Why can we assume that the expected value of the error term is zero? [closed]

Why can we assume that the expected value of the error term in a linear regression model is zero? This is with regard to a simple linear regression.