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

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71 views

Is it possible to fit any regression line to a set of data points?

If you have a set of data points (x1,y1), (x2,y2),...,(xn,yn) And you know it fits a trend y=f(x) where ...
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3answers
76 views

How to fit logarithmic curve to data, in the least squares sense?

How to fit logarithmic curve to data, in the least squares sense? I have simple data of the type $(x,y)$, that is 2D. I need to fit curve of the type: $y = c_1 + c_2\ln(x)$. So I have the $x$'s and ...
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233 views

How to find a line of best fit of the form $y=ax$?

We have the following points: $$ (0,0)(1,51.8)(1.9,101.3)(2.8,148.4)(3.7,201.5)(4.7,251.1)(5.6,302.3)(6.6,350.9)(7.5,397.1)(8.5,452.5)(9.3,496.3)$$ How can we find the best fitting line $y=ax$ ...
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51 views

How come least square can have many solutions?

I know there always exists a least-square solution $\hat{x}$, regardless of the properties of the matrix $A$. However, I keep finding online that least-square can have infinitely many solutions, if ...
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62 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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46 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
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39 views

Would it be any concern if we find correlation between intercept and other regression coefficients?

During a multiple linear regression analysis, I found correlation between intercept (beta-0) and two of the other regression coefficients. Is there any problem or concern in this case? If no, please ...
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142 views

Polynomial regression - correctness and accuracy

I have just finished a code that performs polynomial regression, doing $(X'X)^{-1}X'y$ (where $X'$ is the transpose) to estimate the vector of coefficients. Now I'd like to add some check procedures ...
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1k views

Exponential extrapolation

Given a set of points on 2D surface $(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)$ and a function $f(x)=k+ab^x$, the task is to find values of $k,a$ and $b$ that minimize the following sum: $$\sum_{i=1}^n ...
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351 views

Linear trend has to pass through a point

I need to interpolate a linear trend surface through a number of points but with the condition that the surface has to pass exactly through one of them. Can somebody give me any advice?
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147 views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
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1answer
47 views

Linear Regression with independent but non-identical noise

If I have this linear regression equation: $$y=X\beta+\epsilon $$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ...
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1answer
98 views

Modeling non-linear data using least squares best fit

I have some data for liquid viscosity as a function of pressure and temperature. I would like to learn how come up with a single equation that would determine this fluid's viscosity with pressure and ...
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60 views

Linear regression throug the origin versus mean?

Assume that I have data that can be described by: $y_i = \beta x_i + \epsilon_i, \epsilon_i \sim (0,\sigma_{\epsilon})$, then the least squares estimator is given by $\hat{\beta_1} = ...
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1answer
156 views

Optimizing Independent Variables to Maximize Dependent Variable

I looked around online and couldn't find anything that was answering my question so I thought I would take to the stack! I'm interested in knowing if there is a statistical or mathematical way of ...
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1answer
164 views

Extremely poor polyfit, what am I doing wrong?

I have a dataset with me: http://pastebin.com/YZArky1j, which I am trying to polyfit. This is what I used to perform the polyfit: ...
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90 views

Is there a way to fit an even function using only odd functions?

I was wondering if there is a way to make an infinite series of odd functions equal to an even function. For example, I would like to know if the next equation is valid ...
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1answer
58 views

How to derive the equations in 3:19-3:30 provide in a MIT opencourse ware lecture about the least square method?

How to derive the equations in 3:19-3:30 provide in a MIT opencourse ware lecture about the least square method? Link: http://www.youtube.com/watch?v=YwZYSTQs-Hk Thanks in advance!
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593 views

degrees of freedom for linear regression

If you have a reduced model with $H_0:\beta_1 = 1$, $H_a: \beta_1 \neq 1$, then the reduced model is: $$Y_i = 1X_i + \beta_0 + \epsilon_i$$ Are the degrees of freedom for the error term SSE $n-1$?
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1k views

Finding the change point in data from a piecewise linear function

Greetings, I'm performing research that will help determine the size of observed space and the time elapsed since the big bang. Hopefully you can help! I have data conforming to a piecewise linear ...
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1answer
10k views

Find equation of curve fit programmatically in Matlab?

In MATLAB, when you plot something, there's a tool available which is called "curve fitting". And if you have a set of data points and a linear correlation, this tool will easily come up with an ...
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50 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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37 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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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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1answer
28 views

Multiple Linear Regression in Matrix Form

I am currently studying for my exams and came across the following question: State the multiple linear regression equation in matrix form. Write down the order of each matrix and explain what the ...
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1answer
39 views

finding column vectors - linear transformations

$L:\mathbb{R}^3\rightarrow \mathbb{R}^2$ with bases $\mathcal{S}=\left\{\left(-1,1,0\right),\left(0,1,1\right),\left(1,0,0\right)\right\} \: \text{for} \:\mathbb{R}^3 \:\text{and} \\ ...
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1answer
37 views

Multicollinearity: Why does highly correlated columns in the design matrix lead to high variance of the regression coefficient?

I came across the term "Multicollinearity" in statistics, particularly statistics. However, I never really understand mathematically why highly correlated (almost linearly dependent) columns in the ...
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1answer
92 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 ...
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83 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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90 views

Low Leverage in Residuals, Logistic Regression

I am doing an interpretation of logistic regression and I have an observation withh high residuals but low leverage. I thought that means that it is an outlier(bad prediction) but not ...
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160 views

How do I find equation of this curve?

I need to find equation of the curve as shown below, for which, I need to find equation for upper part. lower part is half circle. upper part is a constant distance from circle with line passing ...
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2answers
2k views

Prove $SST=SSE+SSR$

Prove $$SST=SSE+SSR$$ I start with $$SST= \Sigma (y_i-\bar{y})^2=...=SSE+SSR+ \Sigma 2( y_i-y_i^*)(y_i^*-\bar{y} )$$ and I don't know how to prove that $\Sigma 2( y_i-y_i^*)(y_i^*-\bar{y} )=0$ a ...
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1answer
85 views

Robust Standard Errors

For OLS, my professor said that you should always test for heteroscedasticity first, rather than going straight to the adoption of robust standard errors. I didn't quite follow this and no ...
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53 views

Polynomial best fit line for very large values

not only are the x values large, the difference between them and the y values is huge. My data points: 22353120,720 24448725,671.427053270323 26544330,634.312274868634 28639935,566.291966792026 ...
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70 views

Multiple linear regression

For a homework we have to determine the effect of a predictor variable on an outcome variable using simple linear regression. We have lots of data (about 300 variables) and we may include some other ...
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1answer
238 views

Proof that a and b in linear regression are random variables

Does anyone know how to prove that the variables $a$ and $b$ that are used in linear regression are random variables? For me the assumption would be that these are dependent on the values of $x$ and ...
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115 views

Linear Regression to quadratic function

What is the optimal linear regression (w and w/o y-intercept) for a quadratic curve w.r.t. mean square error. Mathematically speaking: Given, $$y = x^2$$ for $$x = [-a,a]$$. What is the best ...
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1answer
420 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
664 views

What is the difference between a polynomial regression and a generalized linear model?

I have seen that a polynomial linear regression can have this form: $y = c_0 + c_1 x_1 + c_2 x_2 + \dots + c_k x_k $ but I have read that the general lineal model which is a form of the multiple ...
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97 views

What is the equation that fits this curve?

I have a curve that looks like this (it's cyclical): Curve I can get a partial fit by fitting a 3rd degree polynomial, but I have a feeling there must be a better fit (something that involves sin ...
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425 views

Normal Distribution from Standard Deviation?

So I have a data set $(x_{1},y_{1}), (x_{2},y_{2}),\dots,(x_{n},y_{n})$ and from it I have the values of $\sum x$, $\sum x^{2}$, $\sum y$, $\sum y^{2}$, $\sum xy$. My question is, how do I find a ...
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1answer
95 views

$(X^tX)^{-1}$ when $p>>n$

For the $n\times p$ matrix $\mathbf{X}$, is there any use in approximating $(\mathbf{X}'\mathbf{X})^{-1}$ when $p>>n$? If so, what information might this tell us? I understand when $p<n$, ...
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134 views

Middle line between points

How can we calculate LINE that most fit the points T1(1,0) T2(2,0) T3(-1,1) T4(0,1) $x= (1,2,-1,0) $ $y= (0, 0, 1, 1) $ $1= (1, 1, 1, 1)$
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5k views

Can someone explain what plim is?

In my Introductory Econometrics class we discussed a concept of "plim" or "probability limit. I'm not sure what this means though and my professor doesn't explain it well at all. Can someone tell me ...
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543 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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127 views

Calculating the regression equations

I have four data points $(1,2), (2,4), (3,5), (5,7)$ and Im looking for the least squares regression line that best fits them. I use the normal equation $A^tAx=A^tb$ in this form - ...
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919 views

Simple non linear fitting question(Least Squares Fitting--Exponential) [duplicate]

Possible Duplicate: easy to implement method to fit a power function (regression) I have the following simple function: $h = cV^n$ h and V being the variables and $c$ and $n$ are ...
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1answer
89 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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134 views

problem constructing an equation that connects variables

need a suggestion/advice/inputs from the mathematicians here. I have collected data of number of instructions/sec executed by a processor every second, number of loads/stores performed every second ...
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50 views

How to estimate $\alpha$ in $y=(1-\exp(-\alpha x))/(1+\exp(-\alpha x))$?

I have a function $$y=\dfrac{1-\exp(-\alpha x)}{1+\exp(-\alpha x)}$$ where $y$ is not binary. The range of this function is $[-1,1)$. So this does not fit into either logit or probit models. How ...