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

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Econometrics regression model BLUE [on hold]

Consider the regression model $Y_i=α+ϵi$ where $E(ϵi)=0$ and $Var(ϵi)=σ^2({x_i}^2)$ now how can I construct the GLS and derive the BLUE of α and compute its variance?
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1answer
29 views

Correlation and Linear Regression

I'm tasked with this question but unable to proceed on. Q: Calculate the linear product moment correlation coefficient between x and m for these samples: $$ \Sigma x=205,\\ \Sigma m=1240, \\ \Sigma ...
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1answer
42 views

Linear Regression without X? :

(Have been working in matrix algebra) Given model: $ y_i = a + e_i$ ( $y_i= α+ϵ_i$ ) That is $y$ subset $i$ and error term subset $i$ Where the expected value of each error term for each entry ...
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1answer
19 views

Deriving the identity: $\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

For some reason I am having an extremely hard time finding out how the following expression is derived $$ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} $$ Is ...
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13 views

Cox proportional hazards nested regression models - dummy variables vs. numerical variables [closed]

Background: Suppose there is a Cox proportional hazards model for estimating hazard for cancer patient at different stages of the disease: Stage 1, 2, 3, 4 The first model takes the stages as a ...
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28 views

Curve fitting on non-linear ODE data

Background The graph below was generated by a set non-linear ODEs. For those of you who might want to know: It shows the maximum distance achieved by a cylinder when fired at a specified ...
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0answers
14 views

How to use leave-one-out cross-validation scheme to compute the accuracy of a linear model fit

Using the least squares estimation I calculated the model fit for a dataset where: $$ p = \beta_{0} + \beta_{1} * t $$ How could I use leave one out cross-validation(CV) scheme to compute accuracy ...
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3answers
28 views

What is the difference between linear regression on y with x and x with y

I'm plotting the regression line of (GDP$\%$ Change, Poverty Rate$\%$)$\to (x,y)$ in Mathematica What would it mean if I were to switch the axis? (Poverty Rate $\%$, GDP change $%$) ...
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13 views

Estimating elasticity of y with respect to x in a log-log specification

The question My rudimentary workings so far is that; log(y_i/x_i) = log(y_i)-log(x_i) Factorise, so, log(y_i/x_i) = log(y_i) + upsilon_i - log(gamma_i + 1) Thus, elasticity of y to x is always >1 ...
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3answers
46 views

Question regarding Sum Notation in the least squares formula [closed]

I'm attempting to figure out the difference between Σx^2 and (Σx)^2 in this least squares regression formula http://i.imgur.com/HwxnM28.jpg. Any ideas? I figure there must be a difference.
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Logistic Regression Varimp Always Different From Other Models; Text Analytics R

I've been running logistic regression, neural networks, naive bayes, and SVM models on my tweets dataset. I'm doing a sentiment analysis, where R is predicting whether a text is positive, neutral, or ...
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2answers
35 views

The best fit for variables in a number of equations?

Let's say I have 2 variables $x$ and $y$ and 4 equations. The parameters in capital are known parameters. $$I_1=xA_1+yB_1$$ $$I_2=xA_2+yB_2$$ $$I_3=xA_3+yB_3$$ $$I_4=xA_4+yB_4$$ What's the strategy ...
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14 views

Fourier transform used for time series prediction?

For a given time series data set $(t=0,...T)$, we can use Fourier transform to data fitness $$ X(n) = \mu + \sum_{k}\left( A_k cos \frac{2\pi k n}{N} + B_k cos \frac{2\pi k n}{N} \right) +\varepsilon ...
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1answer
23 views

Can a prediction interval be interpreted as a probability?

Suppose I find a 90% prediction interval for some data distribution. This implies that if I sample large enough data from this distribution, then 90% of such data will lie inside the prediction ...
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9 views

Hat matrix and leverages in classical multiple regression

What is Hat matrix and leverages in classical multiple regression? What are their roles? And Why do use them? Please explain them or give satisfactory book/ article references to understand them. ...
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27 views

Assigning levels in factorial design.

I am sorry if the question is too basic. Actually while doing some experiment on 2-level factorial design, I assigned +1 to a low level and -1 to high level. I just need the sign of the regression ...
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1answer
22 views

If X and Z are uncorrelated and Z is normal with mean zero and constant variance, why can I assume Z is zero?

I have a data set that I have used to calculate the coefficients for a linear regression. The data set is of the form $\lbrace x_i,y_i\rbrace_{i=1}^{n} $ Let $$Y = \alpha + \beta X + Z$$ where ...
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1answer
22 views

Help with Matrix Regression

The question i have is: Consider two independent random variables $ξ_1$ and $ξ_2$, such that $ξ_1 ∼ N(0,1) $ and $ ξ_2 ∼ N(0,2)$. Let $η_1 =(ξ_1+ξ_2, ξ_2)^{T} ,η_2 =(ξ_1, ξ_1−ξ_2)^{T}$. Find the ...
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8 views

Help with Regression question for Revision.

I Have my exam coming up in a few weeks, and am not sure how to go about answering a few questions. One being: For a fixed i = 1,...,n, derive $Cov(\hat{β},Y_{i})$ and $Cov(Y − (\hat{α} − α) − ...
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Multiple Calculations of Dummy Variable effects?

If I am using dummy variables to fit a regression model, I know that I am comparing each variable to whatever the baseline that I decide is. So let's say that I have a dummy variable with 5 levels in ...
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1answer
18 views

Can regressors be considered as random variables?

In the linear regression model $$y = \beta_1 X_1 + \cdots + \beta_p X_p + \varepsilon \, ,$$ can the regressors $\{X_i\}_{i \in \{1, \ldots, p\}}$ be considered as random variables? I know that what ...
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1answer
93 views

Complexity of Gaussian Process algorithms is $\mathcal{O}(n^3)$

It is often quoted that the complexity of Gaussian Process algorithms is $\mathcal{O}(n^3)$ due to the need to invert an $n \times n$ matrix, where $n$ is the number of data points. But as far as I ...
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Codification of matrix $X$ in $Y=XB+\epsilon$

The variables for the data below is age, group (treatment 1,2,3), Y response variable. ...
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1answer
24 views

Formula for finding variables by regression

I'm trying to fit data to the following formula: $$y = a + b x + c/(Sqrt[x]+d)$$ $y=a + b x$ can be fitted easily with linear regression, but I'm lost when it comes to anything more complicated. ...
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24 views

what's the difference between the following two main functions

what's the difference between the following two main functions? Let's say if I have a response Y and predictor X and Z and Z is a factor, what's the difference between these two functions 1). Y ~ ...
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Matrix Regression help for exam revision

My regression exam is a month away and i am trying to learn Matrix regression however and struggling with the questions as a whole they are: (a) Consider two independent random variables ξ1 and ξ2, ...
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1answer
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Covariance Matrices Help

Consider two independent random variables $ξ_1$ and $ξ_2$, such that $ξ_1 ∼ N(0,1)$ and $ξ_2 ∼ N(0,2)$. Let $η_1 =(ξ_1+ξ_2, ξ_2)^{T}$, $η_2 =(ξ_1, ξ_1−ξ_2)^{T}$. Find the covariance matrix between ...
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Regression and Variance Help

If a value for $σ^{2}$ is known, explain briefly how you could use the information from the n pairs of observations to decide whether for every unit change of x there is a significant change in the ...
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1answer
16 views

how to find AIC values for both models using R software?

I'm studying survival analysis. I estimated both Cox regression model and Buckley&James regression model. In order to determine which model is better for my dataset, I used Akaike Information ...
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What would be the best method to model population growth? [migrated]

I am trying to find the best method to model the population growth in a school. In my possession I have the student count by semester in the last 16 years. Moreover the first information I consist of ...
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1answer
17 views

calculate the internally studentized residual

This is from my textbook: it says that ...an ordinary residual divided by an estimate of its standard deviation $s(e_{i})$ As we can see from the example that mean for four residuals is 0, so ...
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1answer
58 views

Geometrical Properties of a curve in 3D

I have $n$ curves in the 3d space, which I represented with a certain amount of points. (That is, for every curve $i$, there is a vector $v_i$ with $m$ points which belong to the curve) My goal is to ...
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Hierarchical model without fixed intercept

I understood it is common sense to almost always use the intercept in a model. Let's assume I have a model withouth intercept, that measure heights. The predictor are also physical measurements ...
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2answers
12 views

Can you break up a regression slope coefficient into a product of slopes? $\beta_{A,C}=\beta_{A,B}\beta_{B,C}$

Any regression slope coefficient $\beta$ is defined as: $\beta_{X,Y}=Cov(X,Y)/Var(Y)$ It seems intuitive that you can break up a regression slope coefficient like this: ...
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1answer
68 views

Non-Linear Regression for Parameter Estimation

I have a second order system, it's response to a step change can be expressed in the s-space as: $$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$ Which can be inverse ...
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35 views

How do I find the line of best fit with only the points?

How do I calculate the line of best fit with the smallest chi-square value for multiple degrees? In other words, given some points, how do I calculate $ax^5+bx^4...fx^0 = y$ WITHOUT USING EXCEL OR A ...
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14 views

Regression on multiple output values

I am trapped into a tricky problem for a while, and due to my poor maths background, I am still unable to figure out a constructive solution.. Here is the context: I have hundreds of data samples, ...
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2answers
28 views

Is this a linear estimator?

I would like to prove if $$\hat{\beta_1} = \frac{1}{n}\sum_{i=0}^n \frac{y_i-\bar{y}}{x_i-\bar{x}}$$ where $y = {\beta_0} + {\beta_1}x+ u$ and $\Bbb E(u\mid x) = 0$, is a linear estimator or not. ...
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Book search on statistics

I am searching a book that Analysis of Failure and Survival Data (Chapman & Hall/CRC Texts in Statistical Science) by Peter Smith. Its link is here. I tried to buy it from Amazon, but it is out ...
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1answer
13 views

when fitting to linear model or non-linear model

What is the residual standard deviation? Can I see whether the model I used is accurate or not by looking at this measure? In fact, I try to understand whether my data set is fitting to linear ...
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1answer
17 views

Alternatives to Shephard interpolation?

I am a chemist, so I have little experience in the field of math. My program is that I have a set of points (approx. 20000) in some larger dimensional space (like 10-20 dimensions), and I want to be ...
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Determining best regression equation in minitab

Using minitab 16, for a regression model, how would I determine the ‘best’ regression equation by using an all-possible subset regression approach with suitable selection criteria (and how does one ...
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1answer
31 views

Non linear regression with one parameter

I need to solve the following exercise where I'm asked to find the coefficient $\beta $ of the following model $ y_i = \beta x_{i}^2 + \epsilon_{i}$ knowing only that $ E[x_{i}^2\epsilon_{i}] = 0$ and ...
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If the conditional expectation of the random variable does follow a linear function, can we show the probability a particular data set happens?

Suppose that $\mathbb{E}[Y\mid X=x]=\beta_0+\beta_1x$ where $X, Y$ are random varibles. Given a set of observations consisting pairs of $X,Y$, is it possible to attach it as probabiltiy density ...
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1answer
42 views

How many data points are “enough” for linear regression?

I have data points $(x_t,y_t)$ generated from $y_t = a + b x_t + \epsilon$ where $\epsilon$ is gaussian error term with zero mean and unknown variance. I want to estimate coefficients $a$ and $b$ but ...
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1answer
18 views

Categorical Variable and Metric Variable

If $Y$ is my dependent variable having more than $2$ categories (so $Y$ is a non-metric/Categorical variable) and $X_1, X_2, X_3\cdots X_n$ are my independent variables which are metric in nature. ...
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1answer
18 views

3rd order polynomial with a covariate?

I'm a physiologist by profession. I've conducted a research study that examines the effect of altitude on sustainable power output (let's call this variable CP) in cycling. A 3rd order polynomial ...
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9 views

Regression of Sequence f Product

So, I have an equation like below: $k_{A}*A*k_{B}*B*k_{C}*C=f$ (eq1) The only value I have is A,B, C, and f but I have several equation so that I think it is possible to use regression technique. ...
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Guess the number of eggs in a jar?

If I give you a jar filled with Easter eggs, is there a way to predict the number of eggs in it through some machine learning / regression technique ?
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1answer
26 views

Proof that the least squares regression line through the origin . . .

Prove that for the fitted least squares regression line through the origin, $\sum_i x_ie_i=0$. I am having trouble understanding how to prove this. Would someone be able to explain how to go ...