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

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SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
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18 views

Properties of Fitted Regression Line

Suppose the following regression model $$Y_i=BX_i+\epsilon_i$$ where $\epsilon_i\sim N(0,\sigma^2)$independents is the random error. Verify which properties of the estimated regression line are ...
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24 views

Optimal place to measure for simple linear regression/fitting

Suppose I have a linear model $y_i=a\cdot x_i+b+\epsilon_i$, where $y_i,\epsilon_i,a,b\in\mathbb{R}$, $x_i\in[-1,1]$. I can take n measurements of $y_i$ at $x_i$, where $n\in\mathbb{N}$. $\epsilon_i$ ...
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1answer
29 views

Multiplicative version of Maclaurin or Talyor series

Is there a multiplicative version of Maclaurin or Talyor series? May be in the format $\ln y = b_0 + b_1 \ln x + b_2 (\ln x)^2 + \cdots $ I want to use that as an approximation in a regression ...
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9 views

regression relationship

with data of two variable(say X and Y ),My math teacher gave me two formulas -one for finding Y for given value of X $$ Y- \bar Y= B(X-\bar X)$$ -another for finding X for given value of Y $$ X- ...
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41 views

General formula for leverage in a simple regression model using a single binary covariate

Need some help to start derive the general formula for leverage in a simple regression model using a single binary covariate? Consider a data set where the variable X takes only 0 or 1 values (e.g., ...
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1answer
36 views

Do 'X' and "y' have 'zero' correlation , or can be anything between '-1' and '+1'?

let , we have bi-variate data on X and Y . Now corresponding to the value $x_0$ , y can take any value.but for all other values of x , y takes a constant value. what will be the correlation ...
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17 views

What are some possible reasons for a large condition number?

For this question, please assume that I am talking about the condition number with respect to the spectral norm. That is, $\kappa_2(A) = \|A\|_2\|A^{-1}\|_2 = \frac{\sigma_{max}(A)}{\sigma_{min}(A)}$. ...
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20 views

Why is $\beta$ a linear combination of $\epsilon$

I have a multiple linear regression question. Why is $\beta$ a linear combination of $\epsilon$? I don't know how one comes to this conclusion. Any help will be greatly appreciated.
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18 views

Dummy variables

In the following regression, i want to predict if a new fund depends on their previous old fund returns, y= a + bTYPE + 0.24 Size + 0.2 Leverage + epsilon where TYPE is a dummy variable, 1 for a new ...
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17 views

How to compare experimental data with teorethical prediction

I would like to know, what is the method to approximate experimental data to teorethical one. I have heard about polynomial regression. After calculating particular matrices and solving set of ...
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1answer
15 views

unequal sample size and linear regression

Can i use simple linear regression when there is unequal sample sizes for the dependent and independent variable? Will this be a problem? That is; y=a+bx+epsilon, where y and x have unequal sample ...
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1answer
30 views

Sum of random error in regression

If you know that $\sum_{i=1}^n e_i=0$.What can you say about $\sum_{i=1}^n\epsilon_i=0$? Where $e_i=Y_i-\hat{Y_i}$ and $\epsilon_i=Y_i-E[Y_i]$. I know that $$Y_i=B_0+B_1X_i+\epsilon_i$$ and ...
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1answer
19 views

Polynomial Fitting of Circular Data Object

This is a very odd question. I have a one dimensional data set that is graphed on a histogram. I am trying to curve fit this data set (using the class midpoints as the x values, and the frequencies as ...
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65 views

I would like to Reverse Engineer a formula, given a set of coordinates.

I'm trying to find the original function without trial and error, also I know that every single function can be represented as a polynomial function but I am not looking for a best fit as much as an ...
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18 views

Comparison of performance

i am testing for a sample of fund performance against a benchmark sample of funds performance (i.e. a 'peer' group).each benchmark is calculated one for one that corresponds to each fund strategy and ...
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20 views

How to derive one matrix algebra equation from another

I have one matrix equation: s = A' X^-1 A (where s is scalar, A is a vector, A' is its transpose, and X^-1 is the inverse of a square, symmetric matrix) Which can be transformed into another ...
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2answers
31 views

Prove that $E(|(X+Y)(X-Y)|) \leq 2\sqrt{1-\rho²}$, where $\rho$ is correlation.

For two random variables $X,Y$ with mean $0$ and variance $1$, their correlation is $\rho$. We have to prove that $$E(|(X+Y)(X-Y)|) \leq 2\sqrt{1-\rho^2}.$$ But, I can't understand how the $\rho$ ...
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13 views

Universal polynomial approximation algorythm

I would like to ask, is there any universal algorythm to fill this matrix for any n value? $\textbf{A} = \matrix{n & \sum x_i & \sum x_i^2 & \cdots & \sum x_i^n \cr \sum ...
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5 views

Error in Joint Modelling in R

I have a dataset which has the following variables: sex (indicator), risk (indicator), and age. I've done cox models with these variables no problem and even basic joint models with them no problem. ...
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1answer
18 views

Can a dummy variable help me in a linear regression where my slope changes based on that variable

Sorry if that wasn't a particularly helpful title, let me explain the situation. If I have a scatter chart of y ~ x1 and I notice that if I filter the data by tiers of a different variable, my slope ...
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1answer
17 views

How does measurement error affect the standard error of regression coefficients?

How does inclusion of the measurement error in the model, as $$Y_i + \varDelta_i = bX_i + \varepsilon_i$$ affect the standard error of least square estimators $\hat{b}$ of coefficients $b$? If I ...
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1answer
34 views

How to propagate uncertainties in the dependent variable when doing linear regression?

Let's say I have an independent variable $\vec x$ and a dependent variable $\vec y$ and measurement errors on my dependent variable that I know to be $\delta y$. For the sake of simplicity, let's say ...
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3answers
42 views

Is it possible to find initial parameters when fitting triple exponential term function to data?

I'm trying to fit $f(x) = A \exp(Bx) + C \exp(Dx) + E \exp(F x) $ to data. I can finish off the fitting using Levenberg-Marquardt, but I'd like to find a quick way to calculate initial parameters. ...
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1answer
15 views

Why or how can we say errors(residuals) are independent and they follow the normal probabilties in regression analysis?

While I am studying linear regression analysis and I have encountered a sentence salying "errors are independent and follow normal probabilities". I can only guess what it says but I can't trust my ...
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1answer
22 views

Simple Regression

The question asks for the slope and intercept, but I don't have a correlation coefficient or the raw data (just sample size, mean, and standard deviation).
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1answer
20 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
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29 views

Functon fitting goes wrong

Let's say I got a some function (let's say it named $B_w$) and I make a curve deped on some parameters. As example ...
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2answers
36 views

Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
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6 views

Is there a standard formula for calculating a project portfolio elasticity?

My project portfolio has a lot of variable numbers in it which could drastically change the cost, schedule and resource requirements of the portfolio. I was wondering if there is already some kind of ...
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21 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) + ...
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Averaging between different formula of same kind

There is a dataset of 8 books. We examined the similarity of each book, using a technique, with other 19 books (8*19 number of distinct books). We stored the data and then, we used ...
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14 views

tree models - adaptive basis functions

i am self-studying adaptive basis functions - and came across the following text on classification and regression trees. I am wondering: as the final region in a tree might be result of multiple ...
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1answer
22 views

Fit Quantized Piecewise Constant Function to Another Piecewise Constant Function

I have a situation where I have a function $$f(x) : [r_1,r_2]\in\mathbb{R} \rightarrow [r_3,r_4]\in\mathbb{R}$$ and I need to fit a function $$g(x) : [r_1,r_2]\in\mathbb{R} \rightarrow ...
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41 views

On a modified least square.

Given a vector $y \in \mathbb R^n$ and real constants $x_{ij}$ ($i=1,\dots,n$, $j=1,\dots,p$), we consider a vector $\beta = (\beta_0,\dots,\beta_p)$ which minimize ...
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15 views

Is total least square solution only valid for isotropic error

Let $\mathbf{y} = \mathbf{Ax}$ represent a system of equation where, $\mathbf{y}\in\mathbb{R}^n, \mathbf{A}\in\mathbb{R}^{n\times m}$. However due to error in sensor, what we observe is the following ...
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2answers
53 views

Minimizing a summation?

I have absolutely no idea how to approach this problem. I've been looking through notes, and I think I missed this when my professor discussed this in class. $$ \text{Consider the data}\\ i\: x_i\: ...
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2answers
38 views

Expected value and the standard simple regression model

Given the standard simple regression model: $y_i = β_0 + β_1 x_i + u_i$ What is the expected value of the estimator $\hat\beta_1$in terms of $x_i, \beta_0$ and $\beta_1$ when $\hat\beta_1=\sum x_i ...
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2answers
46 views

Approximate/Find Function

I have got values $x_{i}$ and targets $z_{i}$. Now I want to find a function $f(x)=z$, which approximates the mapping of my value $x_{i}$ to its targetvalue $z_{i}$ as good as possible for every ...
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48 views

Simple curve fitting smoothing algorithm

Is there any conditionally simple algorithm to obtain a best fit (equation) for points set? I have a few points for which I want to have a "smooth" function is constructed that approximately fits the ...
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1answer
31 views

Detrending sine waves accurately

I am doing some data analysis where I look at electricity demand over the course of a day, but need to separate the intra-day (constant and periodic) components from daily changes (assumed linear). At ...
3
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1answer
25 views

Quantile Regression - Linear Loss Minimization

I'm currently reading Quantile Regression by Roger Koenker, and for some reason I'm having a lot of trouble deriving one of his equation (sect. 1.3, p. 5-6). He goes on to demonstrate that $\hat{x}$ ...
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0answers
23 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 ...
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36 views

How to find a perfect regression fit in R?

I have a set of points, which I know can be described with some equation. How can I find this equation? The scatter plot for this set looks like this: I look at the plot and assume that I can use a ...
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56 views

How to calculate relation between Beats (as in BPM) and variance of the energies?

I'm working on an implementation of a Beat Detection algorithm and I can't find a relation between the statistical variance of the energies in 1 second of audio decoded as PCM and the threshold value ...
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36 views

Understanding Piecewise Logistic Function

I am trying to understand the use and application of the piecewise logistic function/regression and have recently been shown how to apply it after getting some help at another forum. However, the ...
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1answer
77 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 ...
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19 views

Regression with many discrete and continuous predictors and few rows

I want to do regression on a dataset. It has one continuous dependent variable that I want to predict. It has many categorical and some continuous predictors. It only has a few rows. A simplified ...
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3answers
53 views

Transposing matrix when differentiating it

Hi so I am trying to understand the solution of linear regression with matrices (found at the following link) and an confused about how on page 10 he says the derivative of $2Y'XB$ with respect to $B$ ...
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15 views

Softmax Regression Gradient Derivation

I'm implementing softmax regression and am deriving the max-log-likelihood update for gradient descent by hand first. Coming from the Stanford UFLDL site, they show the gradient of the cost function ...