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

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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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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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22 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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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 $s(e_{...
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63 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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18 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: $\beta_{A,C}=\beta_{A,B}\...
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69 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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37 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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16 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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30 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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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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19 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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32 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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24 views

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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46 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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24 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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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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10 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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45 views

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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42 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 about ...
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39 views

Application of weighted least squares to a log linear equation

I am trying to fit a curve to a set of data using a weighted least squares approach. The reason I am using the weighted approach is to bias my solution to my more reliable data. I am however having a ...
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16 views

variance of residual in use of uncertainty in gradient and intercept

For a real and differentiable function $z\left ( v_{j=1},\cdot \cdot \cdot ,v_{j=N} \right )$ the estimated uncertainty in z is $\sigma^{2}_{z}=\sum_{i=1}^{N}\left [ \left ( \frac{\partial z}{\...
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68 views

Non-Linear Model Transformation

I want to transform this Non-Linear Model $y= 8-ae^{bx}$ to Linear.And my issue is in this step $lny=ln(8-ae^{bx})$ how can simplify it to reach in a linear model which is like this $y*=b0+b1x$ ...
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Likelihood of an autoregressive model

I have the following autoregressive model: $Y_I=\lambda_t + \alpha_t(Y_t-\lambda_{t-1}) + \epsilon_t$ where $\lambda_t=\beta_1+\beta_2cos(\pi t/6)+\beta_3sin(\pi t/6)$ and $\epsilon$ has a ...
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27 views

How to compute Simple Linear Regression equation not using Least Squares Estimators?

I know how to compute the Simple Linear Regression (SLR) equation using Least Squares Estimators, $b_0$ and $b_1$. But I was given the following table: $$ \begin{array}{c|lcr} & \text{mean} &...
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15 views

Plotting data with error bars with gaussian distributed error

I want to fit a straight line model of the form $y_i = a+ b\ x_i$ to the list of $(x, y)$ pairs given below. How can I plot the data with error bars in both coordinates? ...
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53 views

Formula for the error terms squared of a linear regression model

I need to prove that the sum of the error terms squared $\sum\epsilon^2$ of a linear model is equal to a given formula: $\sum_{i=1}^{n}(Y_i-\alpha-\beta(x_i-\bar{x}))^2=n(\hat{\alpha}-\alpha)^2+(\hat{...
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94 views

linear regression-slope

Your friend in the U.S. gives you a simple regression fit for predicting house prices from square feet. The estimated intercept is -44850 and the estimated slope is 280.76. You believe that your ...
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30 views

R Squared in polynomial multipleregrssion

In a linear regression, we can use R-Squared to check if a model fits But what if I have a polynomial regression with to variable $var_1$ and $var_2$ and a model that goes like $$y=x_0+ x_1\cdot ...
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20 views

Retrieve inputs for second pass (cross section) regression from Time series linear regression

I will explain my question through simple example best demonstrating the issue. Basically I work with Matlab, so if there is already implemented lib functions for this - That's can be the answer. And ...
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47 views

Regression: Service time based on number of copiers

For a random sample of 10 service calls, both the number of copiers and the total service time were recorded. Number of Copiers (x) : 4, 2, 5, 7, 1, 3, 4, 5, 2, 6 ...
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Prove information matrix equality for binary Logit model

I need to prove the information matrix equality for a binary logit model. I know that in general the information matrix equality means the following: "The Information Matrix Equality (IME), which is ...
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39 views

Calculated product of two projection matrices

how do i show that $$P_MP_m=P_mP_M=P_m$$ where $$P_M=X_M(X'_MX_M)^{-1}X'_M$$ $$P_M=X_m(X'_mX_m)^{-1}X'_m$$ and $X_m$ is embedded into the larger matrix with the same number of columns $X_M$.
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given a set of points and vectors, expand the set

I have a set of $n$ points: $(Sx, Sy, Sz)$ and $n$ vectors: $(Vx,Vy,Vz)$, each uniquely mapped to one another. The set also exists inside of an $X*Y*Z$ size rectangular space. I want to ensure a ...
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Parameterizing conditional expectations in terms of regression coefficients (gaussian case)

Consider three jointly normally distributed random variables $X,Y$ and $Z$. I know that in the Gaussian case $$E[Z\mid X, Y]=\beta_{ZX;Y}X +\beta_{ZY;X} Y$$ where $\beta_{ZX;Y}$ notes the ...
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using simple regression to capture a “normal range” for a dependent variable against a factor?

The question reads as follows My JMP output of the simple regression of the data is as follows: Is this all I need? The question phrasing is somewhat throwing me for a loop. Is there something ...
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234 views

Simple Regression ~ House Price Prediction

I am stuck with this question. "You have a data set consisting of the sales prices of houses in your neighborhood, with each sale time-stamped by the month and year in which the house sold. You ...
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What does it mean to regress out current features?

First of all, I'd like to say that this is the intro to a homework problem. Please do not post any answers, I am only looking for clarification on some terminology in the setup. I am trying to ...
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How to regress certain non-linear data

How can I perform a regression onto data of that follows this shape: \begin{equation} U(x):=\sum_{i=1}^N\, a_ix^ie^{-b_ix} \end{equation} where the $a_i\in \mathbb{R}$ and the $b_i \in (0,\infty)$ ...
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1answer
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Inverse function for non-linear regression purpouses

The Setting: I want to perform a regression onto data of that follows this shape: \begin{equation} U(x):=\sum_{i=1}^N\, a_ix^ie^{-b_ix} \end{equation} where the $a_i\in \mathbb{R}$ and the $b_i \in (...
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1answer
50 views

How do I calculate regression line using a data set with repeated values indicated as frequencies?

I have a data set that comprises of Independent Variable $(X)$ and Dependent Variable $(Y)$ values with a certain frequency $(F)$. I know that I have to find $x^2$ and $xy$ but how do I factor in the ...
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41 views

How to find equation of a curve from points

I have got a curved line that I would like to find the equation of using Microsoft Excel. The curve seems to be either a polynomial or some sort of a trig graph. I've done some research and looked at ...
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22 views

How to interpret parameter in proportional hazard model?

How can I show that $b$ in this proportional hazard model $y=1-x^{e^{bz}}$ is the percent change in $y$ with a unit change in $z$? For brevity, I have neglected to show the functional form of $x$, ...
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Imposing non-negativity constraint on a linear regression function

Suppose I am interested in estimating the linear regression model $$ Y_i = g(X_i)^T\beta + \epsilon_i $$ where $Y_i$ is a scalar outcome of interest, $X_i$ is a scalar covariate with support on the ...
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29 views

How to interpret b in $y=x^{e^{bz}}$ in nonlinear regression?

What is the correct way to interpret b in this nonlinear equation $y=x^{e^{bz}}$? I've estimated the model and b seems to be the percent change in y with a unit change in z, but I am unsure how to ...
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Hosmer–Lemeshow test - Best Model

I have 3 different models and I do the Hosmer–Lemeshow test. I have a p-value and a Chi2 value. How can I know which model fits the best my data? Khi-2 || Pr > Khi-2 12.04 || 0.19 7....
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Why can I plug the roots of a partial derivative of a linear optimization objective E into E without changing it?

As an example, to fit a line to 2D data $\boldsymbol x_i$ with the parameters $\theta = (a\;\;b\;\;c)^T$ with the normal equation $\langle \boldsymbol x, \left(\begin{smallmatrix}a\\b\end{smallmatrix}\...
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How to fit sum of products of sine waves?

System Model: \begin{align} Y(t_1, t_2, t_3) = A \bigg[ & 2+ k_1\cos(w_1t_1+\phi_1) +k_2\cos\left(w_2t_2+\phi_2\right)+ \\[2ex] &4k_3\cos\left(\dfrac{w_1t_1+\phi_1}{2}\right)\cos\left(\dfrac{...
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Compute linear regression slope over $[x_i, y_i]$ when $x_i$ samples over $x$ are regularly distributed.

Context: To have an idea of the trend-line of a set of samples $[x_i, y_i]$, I usually compute the slope $a$ in the linear regression ($y = ax + b$) with a spreadsheet software and the formula: $$ a ...
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60 views

Power Regression $y=Ax^B+C$

I have to do reproduce a power regression but I don't have any experience in procedures like that. I read a little bit about power fit/power regression and that a formula like $y = ax^b$ is used for ...