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

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2
votes
0answers
23 views

Can I adjust linear growth of a a subpopulation to a linear decay of the general population?

I need to estimate the amount of CF patients in Poland in the next four years. I have: estimations of the Polish population for the future years a CF patients' register for the last couple of years ...
2
votes
0answers
28 views

Show $\hat{\beta}$ and $s^2$ are independent?

I have the model: $y=X{\beta}+{\epsilon}$ I know $\hat{\beta}=(X'X)^{-1}X'y$ and that it is an unbiased estimator of ${\beta}$ and that $s^2=\hat{\epsilon}'\hat{\epsilon}/(n-k)$ and is an unbiased ...
2
votes
1answer
13 views

Problem with verifying variance of residual

I am supposed to show the following: $$Var(e_{ij}) = \sigma^{2}\left(1-\frac{1}{n_i}\right)$$ Where the follwing is known: $$y_{ij} = \mu + \alpha_{i} + \varepsilon_{ij}$$ $$e_{ij} = y_{ij} - ...
2
votes
0answers
21 views

Proof – OLS estimator regression [closed]

I am having trouble figuring out how I need to form and present an answer to a question. I completely understand the concepts of the math and analysis, I just don't understand how to give an answer ...
2
votes
0answers
92 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
2
votes
1answer
76 views

Machine Learning: Linear Regression models

I'm currently in a course learning about neural networks and machine learning, and I came across these two formulas in this textbook page on linear regression: 1) $y(x) = a + bx$ and 2) $y(x) = ...
2
votes
2answers
39 views

Likelyhood function analysis

I've done some calculations on a large number of data, and created the following graph in excel representing the data: How do I go about analysing this regression in order to find the formula that ...
2
votes
1answer
39 views

Predicting the increase/decrease of number

I have these entries in my database that looks like this: ...
2
votes
1answer
22 views

Multiple regression model

I have a multiple regression equation which as four quarters (maybe called them as parameters) ...
2
votes
0answers
60 views

Predicting profit with price variation

I am currently working on a high school project that aims to predict profit from X amount of items to Y amount of profit based off a deviated sale price. For instance: I sale 10 cookies for 10 ...
2
votes
0answers
195 views

Normal equations for minimization of Frobenius norm least squares error

I'm having a hard time understanding the most efficient sequence of steps for deriving the normal equations for Frobenius norm least squares minimization. Here I want to minimize the norm of a matrix ...
2
votes
0answers
35 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
2
votes
1answer
55 views

Is there a site that will allow me to calculate a best fit for a set of data?

I have a bunch of x's and their corresponding y values, but do not have a Wolfram Pro account. Is there another site where I can input my dataset and have it spit out a best-fit regression (be it ...
2
votes
0answers
19 views

Showing Hat matrix equal specific values

Consider a one way layout model $y_{ij}$ = $\mu_i + e_{ij}$ (1 $\leq$ i $\leq$ a, 1 $\leq$ j $\leq$ $n_i$) where a = 3 and $n_1$ = 2, $n_2$ = 3, $n_3$ = 4. Show that the hat matrix for this design ...
2
votes
1answer
59 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
2
votes
0answers
101 views

Is it compulsory to make transformation to the econometric model in order to have only diagonal elements on variance-covariance matrix of errors?

I need some sharped and advanced advices for the following issue ... Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $y_{jis} = ...
2
votes
0answers
87 views

What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing ...
2
votes
1answer
410 views

Understanding Regularization parameters in Machine Learning/Statistics

Suppose I have the following $k$ degree polynomial regression model with a data set of size $n$ which includes a $k$-dimensional feature vector $x$ and an outcome denoted $t_i$ for each vector in the ...
2
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0answers
35 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
2
votes
0answers
120 views

A* vs D* vs Dijkstra [closed]

I understand the basis of A* as being a derivative of Dijkstra, however, I recently found out about D*. From wikipedia, I can understand the algorithm. What I do not understand is why I would use D* ...
2
votes
1answer
76 views

How to calculate $\sum(X_i-\bar{X})^2$ in R

I'm trying to figure out how to calculate $\sum(X_i-\bar{X})^2$ in R, specifically identifying it in either the aov function or $\operatorname{lm}(y\sim x)$ function. I am trying to use it to ...
2
votes
1answer
29 views

Aproximate data with this equation (or linearize the equation)

I have found an equation that describes the behaviour of a phisical system: $$ y=a_1e^{-a_2x} + a_3 + a_4x + a_5e^{{-a_6} / {(1-x)}}$$ Now I have data of that physical system and I want to ...
2
votes
0answers
19k views

Linear regression: degrees of freedom of SST, SSR, and RSS

I'm trying to understand the concept of degrees of freedom in the specific case of the three quantities involved in a linear regression solution, i.e. $SST=SSR+SSE, $ i.e. Total sum of squares = ...
2
votes
1answer
75 views

Best fit line using geometric distance (not vertical distance)

There must be a theory of finding the best fit line to a bunch of points in the plane, where "best fit" is defined by the geometric distance, not vertical distance. In other words, we are trying to ...
2
votes
1answer
55 views

Basic Multilinear regression question for finding examples or counterexamples.

Hello Wise mathematicians! I have few quenstions about Multi linear regresstion. I've been asked from my friend, but I have very weak knowledge background from that field. It seems my friend is in ...
2
votes
1answer
104 views

Why does the regression line of $x$ on $y$ and $y$ on $x$ meet at $\bar{x}$ and $\bar{y}$?

Why does the least squares regression line of $x$ on $y$ and $y$ on $x$ intersect at $\bar{x}$ and $\bar{y}$? Also, why are the form of regression lines as they are? For the general form ...
2
votes
1answer
51 views

Probit model question (regression)

I'm reading a thesis and I need your help to understand the equation below. $$\Pr(\text{failure}=1 \mid X_1,X_3,X_3,X_4)=\int_{-\infty}^z \varphi(k) \, dk\tag{1}$$ $\varphi(k)$ is the standard ...
2
votes
0answers
853 views

Least Squares “analytic expression” for fitting a 2D quadratic function to measurements

I have n scattered elevation measurements: $ \{x_i,y_i,z_i\}_{i=1..n} $ that I want to fit a quadratic function to: $ z = ax^2 + by^2 + cxy + dx + ey + f$. The problem can be written as a vector ...
2
votes
0answers
53 views

Computing evidence for least-squares fit

I'm at a loss trying to implement Bayesian model selection for standard least-squares polynomials fits. I have three polynomials of order $1$, $2$, and $3$, and a sequence of $(x,y)$ data points. ...
2
votes
0answers
628 views

What is the Moore-Penrose pseudoinverse for scaled linear regression?

The matrix equation for linear regression is: $$ \vec{y} = X\vec{\beta}+\vec{\epsilon} $$ The Least Square Error solution of this forms the normal equations: $$ ({\bf{X}}^T \bf{X}) \vec{\beta}= ...
2
votes
2answers
31 views

Linear regression. Lowering response maintaining equal independent variable.

I have put some data together and modelled the behaviour of the response ($y$) as function of three independent variables $x_1$, $x_2$ and $x_3$. A simple multi-linear regression. The model looks ...
2
votes
0answers
44 views

Coefficient of determination

$$ \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 = ( \displaystyle \sum^n_{i = 1} (y_i - \bar{y})^2 - \displaystyle \sum^n_{i = 1} (y_i - \hat{y}_i)^2 ) + \displaystyle \sum^n_{i = 1} (y_i - ...
2
votes
0answers
162 views

Orthonormal Matrix weighted regression

$Q$ is a rectangular matrix with orthonormal columns. A linear system composed of $$Qx= b$$ is really easy to solve as: $$Q'Q=I$$ hence: $$x=Q'b$$ Given that $Q$ is orthonormal can this be used to ...
2
votes
0answers
88 views

Regressing $Y$ back on the residuals

Suppose I have the linear regression model $ \hat{y_i} = a + b x_i $ for $a,b$ obtained via OLS. How does one regress $y$ back on the residuals $\hat{e}_i = y_i - \hat{y}_i$? If we write $ ...
2
votes
0answers
165 views

Effective model for calculating momentum or growth rate for a time series

I have a series of numbers tracking the performance of an entity on any given day. It's nothing but a simple integer for each date. For example, here's a series for Entity "X" ...
2
votes
0answers
344 views

Bare minimum of points in multiple polynomial regression

I have a question on multiple polynomial regression and the absolute minimum amount of points in the different terms. The minimum amount of points required for a second order polynomial would (in one ...
2
votes
0answers
95 views

Find $w$ as the minimizer of regularized logistic regression

Could someone point me to a reference on how to find $w$ as the minimizer of: $$ \frac{1}{2}\sum_{i=1}^{d}q_i(w_i-m_i)^2+\sum_{j=1}^{n}log(1+\exp(-y_jw^Tx_j)) $$ where $q_i$ (initialized with ...
2
votes
0answers
439 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
2
votes
2answers
885 views

Least squares estimator of mu

The question is: Assuming that $y_i = \mu + \epsilon_i $,$i = 1,\ldots,n$ with independent and identically distributed errors $\epsilon_i$ such that $E[\epsilon_i] = 0$ and $Var[\epsilon_i] = ...
2
votes
0answers
325 views

Finding a model for multiple non-linear regression

I want to implement a regression analysis, but I have problems with finding a model for the given data. There are $149$ $(x,y,z)$-values. $y$ values are all positive, $x$ is between $[-10, 10]$ and ...
2
votes
0answers
3k views

Derivation of standard error of beta in simple linear regression

Countless web pages show the equation for the standard error of the slope in a simple linear regression. For example: ...
2
votes
1answer
311 views

Multiple linear regression with interaction

I'm doing a multiple linear regression with interacting variables. I'll give you an example: $y$=value, $x_1$=material, $x_2$=weight, $x_3$=color $x_1$ and $x_2$ are interacting variables but $x_3$ ...
2
votes
0answers
85 views

Polynomial and exponential regression [duplicate]

Possible Duplicate: Determining computational complexity of stochastic processes I have some points $(x_i,y_i)$ generated by a program. These values are not exact, but are random ...
2
votes
0answers
3k views

Help with problem: Estimated Standard Deviation of Regression Equation (Simple Linear)

This is a practice problem. I've solved part (a). I have provided verified answers (from the published key) to all parts (a), (b) and & (c). I need help solving (b) and (c). Consider a simple ...
2
votes
0answers
285 views

Surface Function Fitting to Spherical Data

I have a set of geographic (longitude,latitude,value) data to which I would like to fit surface functions, specifically, the set of quadratic surfaces: $f(x,y)=Ax^2+Bx^2+Cxy+Dx+Ey+F$ At the moment, ...
2
votes
0answers
67 views

Accurate computation for Linear Regression case

I am writing a program that inputs a sequence of points $(x_i,y_i)$ based on the user clicking on certain pixels in an image shown. The program should then find the "best -fitting" line in the least ...
1
vote
5answers
595 views

Find square root approximation function (tool)

first I have to apologize for any uncorrect naming or categorisation of my question, as I am an electrical engineer rather than a mathematican. I try to find a simple solution for my problem: I have ...
1
vote
4answers
335 views

Parabola from 4 approximate points

I have calculated four approximate points from a sensors to get information. I would like to deduce the closest parabola to my points. The problem is that I can't solve it to get an appropriate ...
1
vote
3answers
615 views

Simple Least-Squares Regression Question

Given a set of 5 points (i.e. (1, 3), (2, 8) etc...), how can I get just the slope of the best fit line? I've been looking up least squares regression, but I'm rather statistics ignorant and don't ...
1
vote
2answers
67 views

Fit exponential with constant

I have data whic would fit to an exponential function with a constant. So y=aexp(bt) + c Now I can solve an exponential without a constant using least square by taking log of y and making the ...