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

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How to make a covariance matrix from multiple observations of different objects?

I have $N$ objects. From each object, I sample $M$ values $(x,y)$ like so: ...
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16 views

Weighted Linear Regression

I am performing linear regression analysis on a time-series of data. Data contains some missing values, My question is, if I impute the missing values using mean of all the values and I want to ...
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28 views

Minimize the sum of distances between a sample and two “centers”

Suppose we have a set of readings $\{X_{i}\}$, each of which is a real number. What I want is to find 2 numbers, $a$ and $b$, such that minimize the sum of distances between each $X_{i}$ and ...
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27 views

coefficient of determination: absence of cross products [closed]

With regard to the coefficient of determination, why is the total variation equal to the sum of the explained variation and the unexplained variation and there are no cross-products?
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15 views

Question on optimization algorithm to train peculiar regression

I've been in my operations research course, and we have been working on optimization in particular problems within regression. We hypothesize that for variables $h,s,d,t,$ there is this set ...
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1answer
21 views

Regression of y/x on x

I have a simple question but I do not manage to be sure! I would be very grateful if you can confirm me! Do we have the possibility to estimate the following model : $$\frac{y}{x}= \alpha+\beta x+\...
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14 views

What are the differences between stochastic v.s. fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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26 views

Degrees of freedom of t-test in multiple regression .

Formula of t-test in regression is, $ t=\frac{\hat{\beta}-\beta}{se (\hat{\beta})} $ and the degrees of freedom of t-test is (n-k) because we estimate $\hat{\sigma}^2$ from RSS and the RSS has (n-k) ...
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14 views

Can the dependent variable in a multivariate linear regression be binary when the independent variables are continuous?

Can the dependent variable in a multivariate linear regression be binary when the independent variables are continuous?
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6 views

Calculating person product moment correlation coefficient on a 3 X 3 table

Usually we are given problems that only involve 2 rows (x and y), but recently saw a problem asking how to compute the correlation coefficient on a table of data that has 3 rows and am not sure how to ...
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25 views

Linear or nonlinear modell

Given those three modells and the assignment to decide whether or not those modells can be transformed into linear modells: (a) $Y_i = \beta_0 + \dfrac{\beta_1}{x_{i1}}+\beta_2x_{i2}+\beta_3x_{i1}^{...
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Coefficient Correlation r of Exponential Functions Regression

I'm writing an exponent regression calculator $Ae^{Bx}$ Sample Data Set (X,Y) is (9, 1) (7, 10) (6,11) (20, 10) (15, 1) A = 5.287 and B = -0.0232. So $F(x) = 5....
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What are the causes of overfitting in regression/classification for statistical data?

Say I have some n-dimensional data, and I want to come up with some hypothesis function which generalizes that data for future predictions in the model. "Overfitness" of my hypothesis function is a ...
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19 views

$n$th order Polynomial for $(n+1)$ points

I was reading about Polynomial Fitting and found this sentence: How can one reach this conclusion and prove it?
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19 views

What is the equivalent of $R^2$ ( coefficient of determination in linear regression) for non linear regression?

I have a dataset with two correlating variables. The relation cannot be described as: $y= a+bx$. Therefore I told my math programm to calculate a nonlinear regression line. But unlike in linear ...
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16 views

Using optim and fitdistr in R to find parameters

I am using R to fit distributions. I have been given the data and have been asked to find the optimised parameters(for lognormal, weibull, exponential and gamma functions) using: 1) fitdistr and 2) ...
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9 views

What does fixed regressor say about our linearity condition?

The linearity condition states that $y_i=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for those $\vec{...
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14 views

An example where pearson is wildy different to spearman? [duplicate]

Im looking to spearman and pearson, and from what i understand spearman is better at looking at curves. Can i see an example of a small set of data (10 or less) where this difference is large.
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An example of when pearson or regression analysis is drastically different to spearman?

Im looking into spearmans rank. I know pearson and regression struggles with curves, but does anyone have any example of when pearson or regression differs with spearman and what this means? Ideally ...
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24 views

Consequences of fitting a regression model with an intercept term when it should be through the origin

Suppose a true model is $Y_i=\beta X_i +e_i$, where $e$ is the random error. Suppose instead we fit the model (using least squares) as $Y_i=\alpha_0+\alpha_1 X_i +v_i$, where $v$ is the random error. ...
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7 views

Without homoscedasticity, is OLS still the best estimator (aka BestLinearUnbiasedEstimator…BLUE)?

Consider the Gauss Markov assumptions. Suppose we have a random sample $\lbrace x_n,y_n \rbrace_{n=1}^{N}$. Assume for a simple linear regression model $y_n = \beta_0 + \beta_1 x_n + \varepsilon_n$ we ...
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1answer
26 views

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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48 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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31 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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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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41 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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17 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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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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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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15 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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1answer
14 views

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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1answer
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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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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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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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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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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14 views

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 ...