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

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93 views

How to gather useful information from a residue plot

You can usually see how good your linear regression line is by looking at the residue plot. If you see the points randomly distributed, you're good. But if you see a pattern, it means there is ...
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1answer
48 views

What is the way to determin how good a sequence will interpolate?

Say I have to sequences of numbers: $$[5, 10, 14, 21, 27, 31]$$ $$[1, 20, 21, 22, 30, 31]$$ Even though they both get to $31$ by the $6$th element, logic tells me that only the first one is a good ...
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1answer
44 views

Cointegration for Price levels Time Series

I don't understand why is the difference between price levels is a stationary process while the time series of price levels themselves is a non-stationary process. For example: ...
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2answers
255 views

Question about logistic regression

A logistic regression is meant for a binary/categorical variable. Sort of like age vs baldness. 1) So, does the "S-curve" regression equation output give the odds of having that condition for a ...
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1answer
208 views

Linear regression: b1 has the minimum variance among all unbiased linear estimators of beta1

There is a proof provided in Applied Linear Regression Models (1983) by Kutner et al. (Page 64), which is quite clear and easy to understand, except one point, namely, it assumes that $\sum k_i d_i = ...
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1answer
224 views

QR factorization for ridge regression

I am solving an overdetermined system of equations: $$Ax= b$$ Using QR factorization, we can solve this system easily by posing it as: $$Rx= Q'b$$ I would like to regularize my estimate of $x$. I ...
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1answer
315 views

A couple of questions on the NURBS basis functions

I read a little about NURBS curves (specifically from http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/), and I have a couple of questions about the motivation behind the choices made in designing ...
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2answers
129 views

Least Squares Derivation

I was reading this to review the derivation of the ordinary least squares estimator but I'm having trouble differentiating (4). Can someone please help explain why $ \dfrac{\partial ...
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2answers
482 views

Exponential Regression Model

I need to model my data ($(x,y)$ pairs) using the following exponential function: $$f(x) = \exp((x + a)/b) - c$$ So, I need to find $a, b, c$ coefficients that are the best fit for my data. What is ...
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1answer
216 views

Approximation using Legendre polynomials

my aim is to fit data points by the use of Legendre Polynomials. Has anybody experience with this task? My final aim is to do this automatically with mathematica. Thanks, rainer
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2answers
361 views

How to do a regression with only integer values and a fixed intercept?

I need to write some code for an application that takes in a series of 2D points whose values are integers, and determines a polynomial regression that passes through the origin. I know how to do this ...
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1answer
94 views

Orthogonal fitted values

I have two regression models $$Y=X\beta+\varepsilon,\quad \beta\in\mathbb{R}^k$$ $$Y=Z\alpha+u\quad \alpha\in\mathbb{R}^m$$ it is known that using OLS estimates $\hat{\beta},\hat{\alpha}$ fitted ...
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359 views

vertical vs. horizontal regression

A horizontal regression is defined as the following: $$m=\frac{\sum_{i=1}^n (x_i-\operatorname{average(x)})(y_i-\operatorname{average(y))}}{\sum_{i=1}^n (x_i-\operatorname{average(x)})^2}$$ whereas ...
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2answers
689 views

Log-likelihood gradient and Hessian

Considering a binary classification problem with data $D = \{(x_i,y_i)\}_{i=1}^n$, $x_i \in \mathbb{R}^d$ and $y_i \in \{0,1\}$. Given the following definitions: $f(x) = x^T \beta$ $p(x) = ...
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1answer
112 views

Econometrics OLS estimates

I cant seem to use the formula to calculate B1 without knowing xi and yi. Is it possible to calculate using just the variances and covariance? Please help! The classical linear regression model ...
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1answer
39 views

Getting the formula of a live counter

I'm looking to replicate this greenhouse gases counter in my website. Poking around i found the initial data for the formula. The counter use the following information: Beginnig date: 2012/03/01 ...
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2answers
350 views

Simple Least Squares Regression?

I have a vector X of 50 real numbers and a vector Y of 50 real numbers. I want to model them as y = ax + b How do I determine a and b such that it minimizes the ...
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3answers
593 views

Find parameters for exponential function fitting to datapoints

I have a set of datapoints, in this case the temperature of an object adjusting to the environment temperature over time. Because I know these kind of processes take the form of $$f(x)=Ae^{x/B}+C$$ I ...
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4answers
202 views

How to fit a curve to my data

I have a datasheet. It looks like an hyperbola. How can I fit a curve to it? And how can I plot a curve of the first derivative? ...
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1answer
339 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 ...
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1answer
82 views

Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...
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1answer
1k views

help with using the “simple regression (least squares) method” of forecasting

This problem is from an engineering management textbook (Morse & Babcock, 5th ed) : 2005 $48k 2006 $64k 2007 $67k 2008 $83k "What is the ...
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293 views

bayesian networks for regression

Would it be possible to use bayesian network for regression and/or prediction? I understand that it is a tool one can use to compute probabilities, but I haven't found much material about possible ...
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323 views

Recursive coefficient of determination (R2)

Is there a way to compute the coefficient of determination $R^2$ in a recursive way? $R^2$ is defined as following: $$R^2 \equiv 1 - \frac{SS_{\rm err} }{ SS_{\rm tot}} = 1 - \frac{\sum_i (y_i - ...
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1answer
129 views

can an artificial neural network with only one hidden layer fit all purposes/applications/functions?

I have heard that only a single layer is needed for an ANN to fit any possible function (input to output). Is this true and where is this investigated/state/found? Then what is the advantage of having ...
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1answer
30 views

Derivative of logistic loss function

I am using logistic in classification task. The task equivalents with find $\omega, b$ to minimize loss function: That means we will take derivative of L with respect to $\omega$ and $b$ (assume y ...
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16 views

How to represent the parameters in logistic function

I want to find the parameters in logistic function. I read the guide at here. It very clear to explain. But it did not has final solution that I need. Now, we will consider a basis logistic function ...
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1answer
17 views

Model selection in regression: Estimated parameters seem to be “non-significant”

I have conducted an experiment which manipulated three factors (Factor 1: 3 levels, Factor 2: 2 levels, Factor 3: 2 levels). The response variable is binomially distributed (1 = correct or 0 = not ...
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1answer
15 views

How to perform a monotonic function fitting of data points?

I'm seeking suggestions for general purpose function fitting of a set of data points, where, based on physical intuition, the relationship is expected to be "monotonic", i.e. the function should be ...
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2answers
36 views

Autocorrelation and var-cov matrix

$$Y_t=\beta_1+\beta_2 X_{t2}+\dots +\beta_k X_{tk}+\epsilon_t \qquad (t=1,\dots,T)$$ $$\epsilon_t=\rho \epsilon_{t-1}+v_t, \qquad v_t \sim \mathrm{i.i.d.}(0,\sigma^2_v)$$ GLS estimation under AR(1) ...
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0answers
25 views

Properties of best linear predictor?

Conside two scalar random variables, $Y,X$. The best linear predictor of $Y\mid X$ under square loss function is $\theta_0=\operatorname{argmin}_{\theta} ...
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22 views

How does Linear Regression classification work?

I am currently trying to understand the following: Logistic regression is a probabilistic, linear classifier. It is parametrized by a weight matrix $W$ and a bias vector $b$. Classification is ...
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1answer
35 views

Does more data give you a better forecast?

Say I have a large set of data. Each data point corresponds to a particular day in the year, so for 1 year I will have 365 points. Say I have collected this sort of data for 5 years. Now, I want to ...
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36 views
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24 views

Simple linear regression prove variables are uncorrelated:

I am working on the following problem: In a problem of simple linear regression, $$Y = \hat\beta_0 + \hat\beta_1 x(bar),$$ show that the random variables $\hat\beta_1$ and $Y$ are un-correlated (All ...
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0answers
11 views

Proofs on regression analysis

How can I prove: 1) estimating population variance $\hat\sigma^2={1 \over n-2}[S_{YY}-{S^2_{XY} \over S_{XX}}]$. 2)expected value of error mean square=$E(EMS)=\sigma^2$ To prove (2): I showed that ...
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1answer
26 views

properties of least square estimators in regression

$Y_i=\beta_0+\beta_1 X_i+\epsilon_i$ where $\epsilon_i$ is normally distributed with mean $0$ and variance $\sigma^2$ . The least square estimators of this model are $\hat\beta_0$ and $\hat\beta_1$. ...
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19 views

Problem on Linear Regression

Consider the following 2-variable linear regression where the error $e_i$ 's are independently and identically distributed with mean 0 and variance 1; $y_i = α + β(x_i − \bar x) + e_i , i = ...
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1answer
8 views

Linear regression or ANOVA with unordered independent variable

I have a set of data, let's say describing a group of people. Let's say we know their income and color of hair: ...
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25 views

Multinomial Logistic Regression

(1) $$P(y^{(i)} =1\mid X,W) = \frac{\exp(W^{(i)^T}X)}{\sum_{j=1}^m \exp(W^{(j)^T}X)}$$ $W$ and $y$ are vectors where the superscript is an index. And there are $m$ classes (that is, there are $m$ ...
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14 views

General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
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17 views

Predicting y from a log-linear regression

I was wondering if someone could explain to me the very last step on the right hand slide. Why is do we have a sum rather than a product. Thank you very much.
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Logistic regression eye treacting data (need model)

I have two sets of time course data, they are for an eye-tracking study. The data is 20 100ms chunks, one category being percent fixations for canonical sentences, and the other being percent looks ...
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2answers
73 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 ...
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14 views

Regression factors and covariance matrix

I am trying to follow some notes. They have two matrices. One is called comfact (company factors). This is a $580 \times 5$ matrix. The $580$ rows represent $580$ different companies. The $5$ ...
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1answer
17 views

Regression analysis question

I do writing that involves correlation studies, but I am not a mathematician. I am considering extending my research to golf, but wonder whether the nature of golf scoring makes that unfeasible. In ...
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36 views

Linear Regression Question (Linear Algebra) Help!!

Hey guys, I have a quick question. I am trying to prove that the squared sample correlation between fitted and observed values is equal to $R^2$ (coefficient of determination). I am having a lot of ...
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2answers
55 views

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

I would like to fit data in g~t scatterplot, where ...
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23 views

How to find a function that can approximate another blackbox function programmaticly?

This question has been posted on http://stackoverflow.com/questions/21758016/how-to-find-a-function-that-can-approximate-another-blackbox-function-programmat I was suggested to post it here. I ...
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1answer
45 views

Calculating variance of estimated intercept parameter, $\hat\beta_0$

I have the following sample : $$ \begin{array}{c|lr} X&80&100&120&140&160&180&200&220&240&260\\ \hline Y & 70 ...