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

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19 views

Population versions of multiple correlation coefficients and least squares estimates

I'm reading an old paper (Wold and Faxer (1957)) which considers the theoretical relation $$ y=\beta_1x_1+\cdots+\beta_hx_h+\zeta $$ where $y,x_1,\ldots,x_h,\zeta$ are (scalar) random variables ...
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1answer
19 views

rotating and exchanging x for y's in regression

I was just wondering what happens generally if i send all my x points to y's and y's to x's (i.e reflect along the y=x line) - if I change the x's and y's will my old error minimizing line still be ...
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28 views

var(AB) when A,B not independent

I need to find the variance of $\hat\beta_1 * \bar X_1 / \bar Y$ , where we have the regression equation Y= $\beta_0 + \beta_1* X_1 +…+ \beta_j* X_j$ I initially was thinking the answer is simply ...
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48 views

Fitting a equation to a spiral curve

I am completely new to this forum and also to this type of mathematical modeling. I am interested to fit the following equation to the points obtained from experimental data. I am looking for an ...
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1answer
37 views

Critical points of quadratic forms

Let $A$ be an $n\times n$ symmetric matrix, let $b$ be an $n$-vector, let $c \in \mathbb{R}$ and set $Q(x) = 1/2 x^T Ax-x^T b+c$. Prove that $x_0$, defined as a solution to $Ax_0=b$ is a critical ...
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28 views

t test vs f test

For conducting statistical tests concerning the parameter $\beta_1$ (the slope of the estimated linear regression function), why is the $t$ test more versatile than the $F$ test? This is a question ...
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10 views

Analysis of block-greedy algorithms for function approximation?

I consider the problem of selecting a final basis set $\{\phi_{c_j}\}_{c_1}^{c_n}$ approximation of function $f \in \cal{H}$ in a Hilbert space that minimizes $L_2$ error. One can use a greedy ...
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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 ...
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37 views

Searching a function for data

I have a dataset and I am trying to find the appropriate function to fit them. So far, I have fitted the data into a two variable polynomial: $$ y(t,v)=(-525.958 + 4.88502 t - 0.0149025 t^2 + ...
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1answer
14 views

For the three measurements b=0, 3, 12 at times t=0, 1, 2 find the best parabola y=C+Dt+E$t^2$

So I know how to do least squares regression using matrices to solve for Ax=b. I simply do $A^TAx=A^Tb$. However I don't really know how to account for the second power in a typical parabola ...
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1answer
25 views

What is the least squares solution given a line passes through original and following points?

So I am looking for the line y=Dt through the origin that fits the data y=4 at t=1, y=5 at t=2 and y=8 at t=3. This is what I have done so far. I know the three equations that are supposed to be ...
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0answers
21 views

Proof – OLS estimator regression [closed]

I am having trouble figuring out how I need to form and present an answer to a question. I completely understand the concepts of the math and analysis, I just don't understand how to give an answer ...
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0answers
36 views

consistency of OLS on misspecified AR(1) process

Suppose the true relationship in data is driven by AR(1) process as follows: $$X_t=\rho X_{t-1}+\epsilon_t\hbox{ , }|\rho|<1$$ and $\epsilon$ is a white noise of $(0,1)$ expectation and variance. ...
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1answer
22 views

Relation between Regularization and correlation

I was going through Chapter 3 (page 63 bottom) of Elements of Statistical Learning. While explaining regularization in ridge regression authors make the following statements. "When there are many ...
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0answers
12 views

What is the proper name of a model that takes as input the output of another model?

Thanks in advance for the help. I am writing a paper and for the life of me can't remember the proper term for a model that works as follows. rawData -> model1 -> outputModel1 -> model2 -> ...
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30 views

Finding better curved line of best fit

I have a set of hand generated data that follows somewhat closely to an exponential curve: I can come up with an exponential equation to the line that gives the values on the 3rd row, and Someone ...
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1answer
25 views

Calculating decreased cost with increasing quantity

I have a hand made table I've been using to give customers price per unit on my items, which gives a better price for the more items that they buy. My sample table right now I need to keep the ...
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1answer
54 views

Solution of overdetermined polynomial system

Some of you will find this question pretty straightforward to answer, but I desperately need some help in solving a problem involving several equations and 2 unknowns, for an engineering application. ...
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0answers
45 views

Standard deviation errors in log scale

I have a not so common issue with error bars in the log log scale. To be more precise, I have measurements of a quantity Y with an associated standard error Yer that has normal distribution and these ...
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0answers
78 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
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3answers
188 views

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
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1answer
31 views

Effects of feature scaling on weight vectors for linear regression

Given that linear regression or polynomial regression can be represented as: $\textbf{w} = (X^{T}X)^{-1}X^{T}Y$ It is standard practice in machine learning to scale each column in their training ...
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3answers
54 views

Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
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20 views

How to find the closest integer linear equation to given real linear equation

I am given a set of points in an n-dimensional plane. I want to find the closest (lowest co-variance) integer linear equation that characterizes the points. I find the real linear equation using r^2 ...
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1answer
29 views

Invertibility of $X^TX$ when sever multicollinearity in regression

I am studying about multicollinearity in regression and in the book it says, "if there is severe (but not perfect) multicollinearity, two or more predictor variables are highly correlated, so $X^TX$ ...
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37 views

Notating the components of the $\hat{\beta}$ matrix when $\hat{Y}$ is multidimensional

This is a question on The Elements of Statistical Learning. We have from the linear model $$\hat{Y} = X^{T}\hat{\beta}$$ where $\hat{Y}^{T} = \begin{bmatrix} \hat{Y}_1 & \hat{Y}_2 & \cdots ...
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1answer
20 views

How to derive this solution to this minimization problem in vector form?

We want to minimize the mean squared error $$ \sum_{t=1}^n (y_t - \theta^T x_t - \theta_0)^2. $$ Letting $X = [x_t, 1]$, we can rewrite the above problem in vector form as $$ \sum_{t=1}^n (y_t - ...
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1answer
19 views

Correlated explanatory variables in linear regression

Is it any reason to assume that if two strongly correlated explanatory variables have impact on response that regression coefficients for these variables have the same signs ? Could such assumption be ...
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32 views

Least squares: Calculus to find residual minimizers?

Reading a section on simple regression in "An Introduction to Statistical Learning with Applications in R" I got a question on residual sum of squares minimization. Quoting from the book: [quote] ... ...
2
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1answer
56 views

Machine Learning: Linear Regression models

I'm currently in a course learning about neural networks and machine learning, and I came across these two formulas in this textbook page on linear regression: 1) $y(x) = a + bx$ and 2) $y(x) = ...
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21 views

Mutiple Regression, calculating R-squared

If I have two regressors in multiple regression equation y=b0 + b1*X1 + b2*X2, how can I find R-squared for the model?I need to know the written formula(not in excel) for two independent variables as ...
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53 views

Fitting an ellipse such that the ratio of its radii is in a range

I need to fit an ellipse to a group of points. However, I have an issue and I appreciate if anyone can help me. The issue is that I need to have the fitted ellipse such that the ratio of its radii is ...
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1answer
205 views

Example of a real-world situation where multivariate analysis is applicable.

I have searched a lot of site to understand the situation where multivariate analysis is applicable. But not got any easily understandable example. Would you please give me a real-world example where ...
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27 views

Application of Multivariate Analysis

The following situation is proven valuable where multivariate analysis can be applied. This example is taken from the book Applied Multivariate Statistical Analysis ...
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1answer
35 views

Linear regression with constrained weights

I have a set of $n$ linear combinations, each with $m$ parameters and desired value $b$. I want to find the set of weights $w$ which minimizes the total equations distances (e.g. the sum of distances ...
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16 views

Chi sqaured table for degrees of freedom 616?

In order to check heteroskedasticity, we use the White's test. I tried to follow this method below, however, could not find a table with df=2016 and 95,5% confidence. I don't understand how we get ...
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1answer
29 views

estimate coefficients of $y = \alpha x + \beta y + \gamma z + \epsilon$

I know how to find $m$ and $b$ for $y= mx +b$, which is : $m= \frac{\bar{x}\bar{y}- \bar{xy}}{(\bar{x})^2 - \bar{x^2}}$ and $b= \bar{y} - m\bar{x}$ How can we estimate $\alpha, \beta, \gamma$ and ...
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2answers
38 views

Likelyhood function analysis

I've done some calculations on a large number of data, and created the following graph in excel representing the data: How do I go about analysing this regression in order to find the formula that ...
1
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1answer
70 views

How to apply non-linear regression to Logistic (sigmoid) curve

I've been looking at a useful way to represent Doppler shift from a satellite passing over a ground station. I've calculated the Doppler shift frequency values at 1-second interval for the duration of ...
1
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2answers
76 views

Maple: How do I type “solve” with an arrow under?

I am trying to learn using Maple 18 (Mac). I have defined a function with a list of X and Y values. f := x->LinReg(X, Y, x) Now I would like to output the unknown "x" value that correlates with ...
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1answer
65 views

Rates of convergence of an OLS estimator

I have a linear regression model $$ y_t=x_t\beta+e_t,\quad t=1,\ldots,N. $$ Here $x_t$ is non-random and given by $(1,\delta_t t)$ where $\delta_t$ is 1 for odd $t$ and $0$ otherwise. Moreover, ...
2
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1answer
37 views

Predicting the increase/decrease of number

I have these entries in my database that looks like this: ...
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2answers
61 views

Using inverse of matrix A as approximate inverse of matrix that is very close to A

Say we have two matrices, $A$ and $A'$ so that $A\approx A'$, and we have the inverse of $A$, $B$, where $AB=I$, and the inverse of $A'$ where $A'B'=I$. If we have some guarantee about how big any ...
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1answer
17 views

Develop relation between dependent and independent Using Tobit model

Depenent variable (Y): Range (0 to 10) (Not less than 0 and not more than 10) (range which i collected from field survey) Independent Variables: X1 - Time (in sec) X2 - Distance (in meter) X3 - ...
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1answer
287 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 ...
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61 views

standard deviation and adjusted R-squared for simultaneous regressions

I am conducting a study that requires two steps of statistical estimation. First, I run a regular OLS regression, from which I gather three outputs that I need: coefficient values standard ...
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1answer
371 views

MATLAB curve fitting - least squares method - wrong “fit” using high degrees

Anyone here that could help me with the following problem? The following code calculates the best polynomial fit to a given data-set, that is; a polynomial of a specified degree. Unfortunately, ...
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0answers
16 views

Multivariate regression with nonindependent variables

I'm trying to run a multivariate regression in which not all variables are independent, and an not sure if this is possible. The reason is as follows: Let's say we have a large number of contracts, ...
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1answer
46 views

adjusted R squared with multiple dependent varialbles

A question about regression in statistics. What is the formula for adjusted R squared if there are multiple dependent variables
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39 views

How to proof that least square estimator $\hat{B}$ doesnt exist when $x$ is linearly dependent?

For the linear regression model $Y=xB+e$, prove that if the columns of $X$ are linearly dependent, the least square estimator $\hat{B}$ does not exist I know that since $\hat{B}$ is an unbiased ...