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

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52 views

Least-squares fit of a nonlinear (polar) system

I want to determine the six unknown coefficients (uppercase letters) of the model $$x=X_c+(Au+B)\cos(Cv+D),\\y=Y_c+(Au+B)\sin(Cv+D)$$ given a set of data $(x_k,y_k,u_k,v_k)$, by least-squares. As ...
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2answers
92 views

Nonlinear LS regression

• Problem formulation I have to fit the following nonlinear model to a dataset: $$f(x)=\frac{C_1 \cdot a}{a^2 + C_2 \cdot x^2}$$ $a$: fitting parameter $C_1, C_2$: Given constants I can't apply ...
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33 views

Hypothesis testing involving regression variable

Consider the regression model $Y = \textbf{X}\beta + \epsilon$, where $\epsilon \sim N(0, \sigma^2I_n)$, $\beta = (\beta_0, \beta_1, \dots, \beta_{10})^{T}$ Construct a test with significance level ...
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1answer
36 views

For linear regression: compute $\Theta T X$

I have started learning linear regression and the equation $h(X) = \Theta T X$ has puzzled me. Let's say we have a training set of $m$ and $n$ features such that $X$ is a $m \times n$ matrix. ...
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1answer
36 views

small sample (1 point) MLE estimation

I'm trying to estimate $m$ in $y = mx+e$ from a single $(x,y)$ observation. $m,x,e$ are all $N(0,1)$ and independent. If I do a Monte-Carlo estimation on say $(1,3)$ I get about $1.5$. This seems ...
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0answers
37 views

Linear regression where explanatory variable of 0 has no meaning

I want to build a predictive model, where given a few numeric explanatory variable n1, n2, ...
1
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1answer
45 views

Intuition for least square regression line involving joint distribution

Let $X$ and $Y$ be random variables of the continuous type having the joint pdf $f(x,y) = 8xy$, with $0 \leq x \leq y \leq 1$. Determine the equation of the least square regression line. Does the ...
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34 views

Prove multiple OLS t-test follows t-distribution

I'm trying to prove the multiple regression test has a t distribution, i.e.: $\frac{\hat B_j - Bj}{se(\hat B_j)} \sim t (df=n-k-1)$ I was able to prove $\frac{\hat B_j - B_j}{sd(\hat B_j)} \sim ...
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0answers
11 views

Problems getting the covariance matrix of the ressiduals

In order to get the variance-covariance matrix of the residuals of a linear regression model, I do the following: Considering that the residual vector $e$ is: $e = Y - \hat{y} = XB+\epsilon - Xb$ ...
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0answers
20 views

Comparing an isotonic model to an additive model

Say I have a dataset in $x,y$, and say I fit a few different models to the dataset. Examples could be an isotonic regression, a smoothing spline and a simple linear regression. What are some ways I ...
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24 views

Why is $\hat y_i=\hat\beta_1-\hat\beta_2x_i$?

I'm trying to show that $\hat\sigma^2 =\frac{\sum\hat\epsilon^2}{n-2}$ is an estimator without biais and I started with: $$\hat\epsilon_i=y_i-\hat y_i$$ and my teacher suggested me to use the ...
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1answer
123 views

Expected mean squared error and MSR

In a small-scale regression study, five observations on $Y$ were obtained corresponding to $X = 1,4,10, 11$, and $14$. Assume that $\sigma=0.6,B_0=5,B_1=3$ a. What are the expected values ...
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0answers
68 views

Prove that OLS estimator of the intercept has minimum variance

Let $$y_i=B_0+B_1X_i+\epsilon_i$$ where $\epsilon_i\sim > N(0,\sigma^2)$. Find the least squares estimator of $B_0$ and show that it is unbiased and has minimum variance. I will not write in ...
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0answers
43 views

Linear Least Squares vs. Ordinary Least Squares

My understanding is that Ordinary Least Squares (Usually taught in Statistics classes) uses the vertical distance only when minimizing error/residuals (see Wikipedia for Ordinary Least Squares) with a ...
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1answer
94 views

polynomial curve fitting: higher order models' root mean square error does not decrease

I am trying to fit a curve for 15 data points. I started by creating a linear model and observing the root mean square difference, followed by quadratic, cubic and increasing the degree of polynomial ...
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0answers
18 views

What is the algorithm for training and testing a logistic regression model using Newton's method and l2 regularization?

I have a spam dataset with 57 features and 3065 data points. I also have test data with around 1500 data points. The classes are spam/non-spam. I have to fit a logistic regression model on MatLab ...
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0answers
25 views

A property of Piece-wise continuous simple linear regression model

My fellow members I attempted to model the growth of capital of a small business person who does business with the aim of just raising his or her capital, as follows: Assumptions A fixed capital ...
1
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1answer
80 views

Statistical property proof

Consider the multiple linear regression model $y = Xβ + ε$, where $X$ is the $n × p$ design matrix, $y$ is the $n × 1$ dimensional vector of response and $ε ∼ N(0,σ^2I)$. The vector consisting of the ...
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0answers
20 views

Piecewise linear regression with a given cost function

If we're given a cluster of $(x, y)$ values that appear non-linear [1](example image), we wish to partition the set of points into $r$ sets of continuous points [2] and then find regression lines on ...
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2answers
27 views

How to understand this multiple regression question without having an example in the textbook?

How to understand this multiple regression question without having an example in the textbook? Please briefly show how to do the final question and give the answer to the final question.
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16 views

Proof reviewing: If z¹ and z² two random variables get by two $x_i$ centered and reducted, what may be $ρ_z¹_z²$ in terms of $ρ_x¹_x²$.

Let be z¹ and z² two random variables get by x¹ and x² centered and scaled, give the expression of $ρ_z¹_z²$ in terms of $ρ_x¹_x²$. what I know: the Matrix $R$ of correlation is $$\left\{ ...
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1answer
49 views

Difference between $\sigma^2\{\text{pred}\}$ and $\sigma^2\{\hat Y_h\}$?

Can someone explain this to me? I've read the relevant section of the text about a million times, and it was even explained in class, but I can't seem to wrap my head around it. The Statement of the ...
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0answers
21 views

Comparing ridge regressions

I am applying ridge regression to biological problem and using the ridge coefficients as measure of explanatory power of each predictor. I have 2 sets of reponse variables with 2 matching sets of ...
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0answers
21 views

Explanatory power of a multiple regression function

What is the explanatory power of a multiple regression model? How is the explanatory power measured?
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12 views

Is division by $\sum x_i-\bar{x}$ actually null?

I'm trying to find out what are $\hat{β_1}, \hat{β_2}$ $ \left \{ \begin{array}{c @{=} c} \frac{∂S( \hat{β_1}, \hat{β_2})}{∂S \hat{β_1}} =-2\sum(yi − \hat{β_1} − \hat{β_2}xi) = 0, \\ ...
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0answers
22 views

Showing that a linear regression by method of leasts squares exists for any set of $n \geq 2$ points

As part of a project for school I am conducting a statistical study that involves the use of linear regressions on many sets of points. Before being able to apply my results using a computer programme ...
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0answers
28 views

what does is mean by 'overfitting' of data?

I have the following equation that is to be estimated: y = a + bA + cB + dC + eF + dG + e and i got 2 other additional variables, fH and gI, that i do not wish to add in.. can i reason this out by ...
2
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2answers
55 views

Fitting two parallel lines to a set of points

In two dimension I have a set of points X = $\{x_1,..., x_N\}$. I want to fit two parallel lines to these points like $l_1$ and $l_2$ $$l_1 = p_1 + \lambda n^\perp$$ $$l_2 = p_2 + \lambda n^\perp$$ ...
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19 views

distribution of linear regression predicted values

Let $\tilde{y}_*$ the prediction for a new observation at level $X = x_*$. Assume that $\sigma^2$ and $\tilde{y}_*$ are independent. Show that $$\frac{\tilde{y}_* - y_*}{se(\textrm{pred})(\tilde{y}_* ...
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23 views

3 variables 'linked' to one outcome (for lack of better words)

Paper route salary formula (reverse engineer...again for lack of a better word;) Ok suppose I have these variables: Pieces of mail : 302 Number of stops: 177 TotalWeight* : 272 Hours allocated: ...
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0answers
40 views

how to compute joint pdf of (x,y) where y is correlated with x

I want to know how to compute joint pdf of (x,y) where x follows exponential distribution with parameter $\lambda$ and $y=x+\epsilon $, $\epsilon$ follows standard normal distribution. x and ...
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1answer
40 views

Corollary of the Frisch-Waugh Theorem

Consider the following linear regression model: $$y=X \beta + \epsilon = X_1 \beta_1 + X_2 \beta_2 + \epsilon $$ Where we have $n$ obsevations and $k$ variables, and hence $X$ is a matrix $nk$, and ...
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34 views

single variable is significant but overall test is not

I do a multiple regression with 3 independent variables $X_1$, $X_2$ and $X_3$. The correlation between $Y$ and $X_1$, $Y$ and $X_2$, and $Y$ and $X_3$, are each large and statistically significant. ...
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1answer
22 views

Regression Model with (Y,X) non-random?

In regression, we assume that $(X,Y)$ are random variables following some certain distribution. How would the problem change if we do not assume $(X,Y)$ are randoms. Why can we just have ...
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1answer
62 views

Calculate power regression

I've searched for quite a while, but google has very little on what I actually need. I have a series of points that form a power function. How do I calculate the what the power function is? I can ...
0
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1answer
32 views

Doubt about proving if number of parameters equals to sample size, the error sum of squares is zero

Consider the linear model $Y=X\beta+\varepsilon$, where $Y$ is an $n$ by $1$ vector, $X$ is a known $n$ by $p$ matrix, and $\varepsilon$ is an $n$ by $1$ vector of random errors following normal ...
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0answers
34 views

Can a supervised learning technique perfectly learn a function?

Obviously the goal of a supervised learning algorithm is to learn some function, $f : x \mapsto y$. In essence, we want $f$ to be a be a good predictor of $y$ when given $x$. I'm curious if a ...
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2answers
50 views

Calculus derivation of OLS regression formula

Edited to make the question more clear. In deriving the OLS formula from calculus, that is, solving $$\min_{\beta}(Y-X\beta)^T(Y-X\beta).$$ How could be guarantee the solution is actually the ...
5
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1answer
349 views

How to prove SSE and SSR are independent

Consider $Y=X\beta+\varepsilon$, where $X$ is n by p, $\beta$ is p by 1 and $\varepsilon$ is n by 1 with covariance matrix = var($\varepsilon$)=$\sigma^2 I$. Give expression for the regression and ...
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0answers
59 views

Deriving the slope variance of linear regression line through the origin

I have derived the value of the slope of a linear regression line (through the origin) to be to be: I am having difficulty deriving the variance of the slope! Any help is appreciated!
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1answer
547 views

Linear Regression Model, linearity in parameters/ variables

I am confusing with the wording here. I was reading a book on linear regression. "The primary concern for linear models is that they display linearity in the parameters. Therefore, when we refer to a ...
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1answer
208 views

Is the sum of residuals in the weighted least squares equal to zero?

So I know that in OLS, the sum of the residuals is equal to zero. This makes sense. I also know that given any slope parameter its possible to rescale the intercept to where the sum of the u will be ...
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0answers
55 views

LMS Update Rule

I'm starting to study machine learning using Andrew Ng's class notes. I understand conceptually how linear regression works, but am having trouble with this equation: $$ \theta_j := \theta_j + ...
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20 views

What to do about Missing values for Multivariate regression analysis

I am required to perform multivariate analysis on all countries in the World bank database regarding digital divide. I am confused becasue when I look at factors such as School enrollment, there are ...
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0answers
17 views

Problems with Exchange Procedure in Remez Algorithm

So first off: *** This code is not being used in production software. It is a personal project of mine, trying to understand approximation theory and advanced curve fitting. In other words, I'm ...
2
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1answer
149 views

Solving for a 3D point in a 5D graph given 3 pairs of 2D points.

I am attempting to solve the values $C$, $D$, and $S$, given three pairs of $[M,R]$. $$R = \frac {M}{C - MDC + DC\left(MS\right)^2}$$ I have been able to solve for a related equation (or rather, ...
5
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1answer
60 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
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0answers
48 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
2
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1answer
43 views

Optimal place to measure for simple linear regression/fitting

Suppose I have a linear model $y_i=a\cdot x_i+b+\epsilon_i$, where $y_i,\epsilon_i,a,b\in\mathbb{R}$, $x_i\in[-1,1]$. I can take n measurements of $y_i$ at $x_i$, where $n\in\mathbb{N}$. $\epsilon_i$ ...
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1answer
31 views

Multiplicative version of Maclaurin or Talyor series

Is there a multiplicative version of Maclaurin or Talyor series? May be in the format $\ln y = b_0 + b_1 \ln x + b_2 (\ln x)^2 + \cdots $ I want to use that as an approximation in a regression ...