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

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What is the difference in how $\mathrm{R}^2$ and $\mathrm{R}$ values are interpreted?

In statistics, there is the $\mathrm{R}$ value for the product moment correlation coefficient and the $\mathrm{R}^2$ value for the coefficient of determination. In both cases they are described as a ...
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42 views

Automatic curve fitting to find order of an algorithm?

I'm a newbies in mathematics. I'm looking for an automatic best curve fitting function to find the order of an algorithm. I would like to know if it does exists a math library function that would ...
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30 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
49 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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37 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 initial ...
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16 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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16 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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12 views

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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10 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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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 $\text{...
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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) − (\hat{...
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6 views

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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14 views

Codification of matrix $X$ in $Y=XB+\epsilon$

The variables for the data below is age, group (treatment 1,2,3), Y response variable. \begin{bmatrix}60&1&5.5\\57&2&4.5\\59&3&8.1\\68&1&0.6\\67&2&1.9\\...
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25 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 ~ X+Z+...
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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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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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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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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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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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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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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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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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13 views

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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1answer
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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13 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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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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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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38 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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24 views

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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1answer
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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28 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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10 views

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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32 views

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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44 views

Kalman Filter and OLS Results Are Different

I read that Kalman Filters can be used for continuous / online linear regression and at the end of the regression its results and ordinary linear regression (OLS) results would be the same. I tried it ...
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1answer
29 views

Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify $$\...
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Linear regression using gradient descent: is the whole weight vector updated with the same number?

I'm using gradient descent with mean squared error as error function to do linear regression. Take a look at the equations first. As you can see in eq.1, the prediction is done with a bias term b ...
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22 views

Variance in the sum of batch-correlated residuals in a regression

I am looking at a regression model of the following form: $Y=intercept+\beta_{Yf.n}X_f+\beta_{Yn.f}X_n +error$ where $X_f$ and $X_n$ are predictors. A value for $Y$ will be sampled from the ...
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14 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
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Relationship/correlation between data - does it exist?

Data I refer to in this question Some analysis has been conducted for my business by an external company. The data, as it stands, only really tells part of the story and doesn't provide any real ...
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2answers
42 views

exponential regression for bacteria growth

I'm studying regression lines and curves, and I've learn the methods for working with curves of the types $ax^2+bx+c$ and $ax+b$ as well as $a\sin(x)+b\cos(x)$. Now I'm asked this: $$(0,32), (2,65),(...
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1answer
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How do you calculate the correlation between the intercept's and beta's standard error in a univariate linear regression?

I am running a regression to predict a variable Y as follows: $Y=\alpha+\beta\times x+\epsilon$ I am trying to get a distribution of the expected value of Y given standard errors in the model ...
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26 views

Using least squares regression to apply nonlinear function to time series data

If you have a nonlinear function (see example), can you use a least squares regression approach to fit it to time series data ? Is this approach also valid for n variables? How many time points are ...
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14 views

Predict the probabilty of a value belonging to a particular class

I have got two classes : BG and FG and a set of values assigned to each of these classes. Given a new value how can I find ...
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67 views

Help Determining Gradient for Equation

I am writing an OO program with geometric objects. My Plane object is capable of taking a collection of 3d points and determining the plane of best fit. I'm using this popular document on it from ...
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1answer
13 views

Statistic test for comparing two regression models

I'm having two linear regression models as follows: $y = a_1x_1 + a_2x_2 + c$ and $y = b_1x_1 + b_2x_3 + c$. I'm looking for a statistical test for proving which model is better. I've obtained the $R^...