Questions tagged [regression]
This tag is for questions on (linear or nonlinear) regression, which is a way of describing how one variable, the outcome, is numerically related to predictor variables. The dependent variable is also referred to as $~Y~$, dependent or response and is plotted on the vertical axis (ordinate) of a graph.
2,663
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Linear regression plot does not look linear, does it mean I can do better?
so i am using the fetch_california_housing() dataset from scikit-learn and did some simple linear regression with this. Here is a screenshot of the dataset characteristics.
I plotted the predicted ...
- 61
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21
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Line of Best-Fit Between Two Polymetric Curves
I have been using basic regression in excel to find the correlation between two variables.
I have since been able to produce two curves in an attempt to model the variable against each other.
Two ...
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37
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Function fitting curve
I have $(x_1,y_1),\dots (n_n,y_n)$ experimental data. I know that the function for fitting data is
$$y=21-A\left(1-\text{exp}\left(-(B+Cx)^D\right)\right)$$
Of course, I can not use Linear Regression. ...
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28
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How to correct the output when using LPM (Linear Probability Method)?
I have a .csv with 10 inputs and 1 output. The "problem" is that the output is binary ("0" or "1"). I am using multiple linear regression to predict the output. From ...
- 11
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15
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Conditional mean square convergence and consistency
Let $\{(Y_i,X_i)\}$ be i.i.d. random pairs that satisfies
$$Y_i=m(X_i)+e_i, \;\; \mathsf{E}(e_i|X_i)=0, \ i\in\{1,\dotsc,n\}.$$
Let $\hat m(x)$ be an estimator of $m(x)$ at point $x$ and $\boldsymbol{...
- 2,587
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24
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How to fill the temporal gaps in ESA(European Space Agency) CCI soil moisture data. [closed]
I downloaded the soil moisture data from ESA Soil Moisture Climate Change Initiative (Soil_Moisture_cci): Version 07.1 (Link:https://catalogue.ceda.ac.uk/uuid/ea3eb0714dc6402b905fe9f7ee50dbbc?jump=...
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21
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Mapping between LSQ for several samples and one single sample
I have a sysid problem of the type:
$Y = AX$,
where X is a matrix whose columns are different input vectors, and Y is the output.
Therefore, A is a general matrix that maps all the input vectors(...
- 59
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15
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Simple Linear Regression on 4 datapoints and their slope and intercept -- thought experiment
Question: Conduct simple linear regression on y=beta0+beta1*x+epsilon with the given simple data set {(1,1),(-1,-1),(1,0),(-1,0)} with the first coordinate being x and second coordinate being y. What ...
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27
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Finding an exponential model for a set of given data
I have a problem here I am working on that is asking me to "Find an equation of the line through the first and last data points. Then find an exponential model $P(x)$ for the population data. Use ...
- 2,130
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Derivation of Feasible Generalized Least Squares Process (Wooldridge 2013)
I'm using Wooldridge's Introductory Econometrics (2013) and having some trouble understanding his derivation of the feasible generalized least squares (FGLS) method. He writes, assume that the ...
- 111
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38
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Derivative Least Square Regression (Numerator vs denominator layout)
Assume I have the following expression:
$$\frac{\partial}{\partial w} \lVert X^Tw - y \rVert^2 = 0$$
which is trying to find the solution for the Least Squares Approach in Regression.
Let's assume $X \...
- 11
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26
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Mean value of linear regression with another regression
A survey is given to adults in each family. $a$ is the age of the
adult,and $b$ is the age of the oldest child - these are recorded. The
ages of the oldest child are summarized in the following box ...
- 621
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36
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Best regression model for points that follow a sigmoidal pattern
I have the following list of points :
I'm trying to find the best regression model to fit these points.
The logistic regression is not close enough to the points :
I guess I need something closer to ...
- 2,027
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29
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Simplification - Power of diagonal matrix and circulant matrix [closed]
I am solving a matrix equation for the elements of the diagonal matrix $D$. Therefore, I must find vector real vector $e$, where $D = diag(e)$.
Vector $x$ and $y$ are given and are complex vectors. V ...
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28
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Find parameters of a gaussian transformation
I have a system of equations where the relationship between input and output is derived from a pixel lattice:
\begin{equation}
x_i(k+1) = \sum_j \alpha e^{ \frac{\left(dr_{ji}^2 + dc_{ji}^2 \right)}{2 ...
- 59
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37
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Do Gaussian Processes not generalize well out of the training range?
I'm a beginner of the Gaussian Processes (GPs). I would like to know the generalizability of GPs. I trained a GP using training data generated by an objective function (the range $-5<x<5$, 10 ...
- 153
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How to calculate sample standard deviation of the intercepts that are obtained from all possible combinations (having $n \ge 3$ points)?
Starting with an example:
Having the points $P_1(1,29),P_2(2,50),P_3(4,88)$. Then there are $4$ possible ways to choose points to find the "best" line (using ordinary least-squares). The $4$ ...
- 4,800
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Proving $\overline{(x-\overline x)(y-\overline y)}=\overline{xy}-\overline x \cdot \overline y$
In ordinary least squares, it can be shown that $\hat\beta_1 = \frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum (x_i - \bar x)^2}$. I'm trying to prove this. I arrived at $$\hat\beta_1=\frac{\overline{xy}-\...
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Understanding an Inequality In a Paper
The paper is the following:
https://arxiv.org/pdf/1608.06412.pdf
Right above (5) on page 5, the first inequality of the line where they use lemma 8, I'm not quite sure I see how they are using lemma 8....
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Model Significance linear Regression
I have a more general question. Do you know about methods to investigate model significance in a sense of testing whether $Y=X\beta+\varepsilon$ is valid, i.e $H_0:\beta=0$ -apart from the typical F-...
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24
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Multi-class least sqares with a single coefficient per intput dimension
Suppose we are given multi-class labels $\mathbf{y} \in \mathbb{R}^K$ where $K \in \mathbb{N^+}$ and data $\mathbf{X} \in \mathbb{R}^{P \times K}$ where $P \in \mathbb{N^+}$. We wish to minimise the ...
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Intrinsic Growth Rate of Beverton-Holt Model
I am reading ahead in some lecture notes that give the Beverton-Holt model with zero harvest:
$r_t=\dfrac{ar_{t-1}}{b+r_{t-1}}$.
It states that under a standard assumption of $a>b>0$, the '...
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85
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Understanding how a log-log plot models an electronic component
In a research paper (available here) that I'm reading, there is a description of the behavior of a photo resistor used in the construction of a more complex sensor, that goes like this :
Its ...
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3
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75
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Trying to fit a quasi-periodic sequence
I am trying to fit dataset with a function, but with little success, as illustrated below. The blue line is the data I wish to fit, and the red line is my attempt.
Below is the function I used:
$$ g(...
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1
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29
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Loss function for regression - vanishing cross term
[From PRML Bishop, p:48]
I do not understand how the cross term vanishes in the integration. I have tried writing this out, but it does not really make sense to me.
The same operation happens in "...
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Regression calculation of initial individual weight considering variations in weighing dates and missing data
I need to determine the initial individual weight of each animal as accurately as possible, taking into account the following challenges:
The animals may be weighed on different days.
There may be ...
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Solution of RBF constraint least squares
I have to design a matrix A that solves a linear system:
$𝑦=𝐴𝑥$, where x is a known complex vector and y is a known complex vector.
The requirement is that A is an RBF kernel, i.e., it has the ...
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$[1]$ Does good $R^2$ necessarily tells us that the experiment went right? $[2]$ Is not extrapolation dangerous when used for a far point?
In an experiment, the relation between $x$ and $y$ is linear.
$x_\text{actual} = \alpha_x x_\text{observed} + \beta_x + \gamma_x$. [Here, $\alpha$ is the slope, $\beta$ is the intercept, and $\gamma$ ...
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High values of r^2 and residual sum of squares
The table below shows the average time T it takes for a computer to sort n items
n (items) 5 10 50 100 500 1000
T (microseconds) 17.4 28.2 157.6 305.5 ...
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23
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A parametric model for inverse of quadratic equation
I have data $\left\{ q_i, p_i \right\}_{i = 1}^{N}$ from a model $p_i = a q_i^2 + b q_i + c$:
I know data is always in the positive quadrant as $q_i \geq 0, \, p_i \geq 0$ and I also know $a > 0$ (...
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How can I calculate p-values, t-statistics, and Standard Error for multiple regression by hand?
I'm attempting to calculate p-values and t-statistics for multiple regression, but all the resources I've been finding on the topic have confusing or inconsistent information. The best resource I've ...
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Need help with regression exercises
I'd an exam recently and I got 2 exercises from regression part of an exam I couldn't have solved. Can you help me. I will provide both exercises and my attempts to solving them.
Exercises:
Based on ...
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When does averaging affect a linear regression?
Given the following:
data $(x_{data},y_{data})$, where x values may be measured multiple times.
For example,
$$ x_{data} = ( 1, 2, 2, 3,3,3, 4 ) $$
$$ y_{data}=(8,10,9,1,2,1.5,−7) $$
And a fitting ...
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Comparison of regression coefficients between simple and multiple linear regression models
Let's say we want to predict weight ($y$) of a person based on age ($x_1$) using linear regression.
Assume a simple linear regression
$y = m*x_1 + c_1 $
Now we want to include height ($x_2$) of a ...
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How do I fit "diamond" ellipsis curve?
I have bunch of x,y data that at first seem to be fittable with a logarithmic function. My current plot looks like this:
As you can see, the logarithmic fit using ...
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How is the regression coefficient derived?
Searched quite a bit in the forum but cant find this. In Wikipedia the derivation of the regression coefficient for simple linear regression is skipped and points to a book that isn't freely available....
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Geometry of linear regression
In this image, a linear regression fit is made to the data points $X\in \mathbb R^{N\times 2}$. As you can see, the fitted values $\hat y$ is inside a plane/hyperplane.
However, in linear regression, ...
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What is the difference between a coefficient in an OLS regression model and the mean difference in a Tukey's HSD test?
I'm wondering if someone could explain in high-school-level language what the difference is between these concepts?
I'm looking at the results of an OLS regression for a single categorical variable (...
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Finding the form of $z$ in terms of $x$ and $y$, letting $z(x,0)$ to be $0$ forcefully.
In a practical experiment, we have two independent variables, $x>0$ and $y \ge 0$, and one dependent variable, $z \ge 0$.
Theoretically, we know that at any fixed $x$, we have the relation between $...
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Finding best 2D curve passing by waypoints
For a robotic application, I am looking for a mathematical tool for finding the best curve passing by or near (up to a certain radius) from waypoints as shown below. The pink circles are obstacles ...
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10
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Consistency and unbiasedness of the estimator
Let ($\epsilon_t - \alpha$)~i.i.d Bernoulli [(1-$\alpha) -\alpha$)].
Define the regression of ($\epsilon_t - \alpha$) on $r^2_{t-1}$ without a constant, and where $r^2_{t-1}$ is a return series.
Given ...
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Should the data matrix associated with a random vector be transposed?
Given a $p$-dimensional zero mean random (column) vector $X \in \mathbb{R}^{p}$, the auto-covariance is given by $$\Sigma = Cov(X) = \mathbb{E}[XX^T].$$
Now given $n$ observations of this column ...
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How to interpret covariance(A,B) divided by percentile rank of std dev(A)?
Is there a way to make meaning of it? I was thinking that maybe because the covariance is divided by the percentile rank of std dev of A, the magnitude of A is to a degree bounded. But how does it ...
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Linear quantile regression v.s. two-step OLS regression
Assume we have a linear model $y = 10 + 0.5x + \epsilon$ where the $\epsilon$ is a random noise. We have $n$ samples $(y_1, x_1),\cdots,(y_n, x_n)$ and want to calculate the 90th percentile y ...
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Instantaneous rate of change equation for exponential and data
So, I recently became interested in the number $e$ and wanted to get an expression that when applied to an investment gave the "equivalent instantaneous rate". The equation is:
$$EIR = f_{y}...
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2
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71
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Metric to compare function similarity
I have 3 functions (built by x,y values: CSV file link).
The red one is a baseline function. The green and blue ones are approximated versions of the red ones. It is clear that the green one is much ...
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2
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275
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How do you find the best fit line for circular data
I have $n$ data points all of which are measurements of angles which I know to be increasing linearly with equal spacing. I am trying to find the best fit line for this data. Researching it, it seems ...
0
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1
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57
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Power curve fitting with 4 components [closed]
I'm trying to fit a set of data to a power curve of the form f(x)=a*(x+b)^c+d, is there a formula to determine the a/b/c/d components?
For example, if I have the following data:
X
Y
11
740
12
420
...
- 3
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26
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Effects of Scaling and Parallel Shifting of Data on Least Squares Estimators in Multiple Regression Analysis
Model
$$
y =
\begin{pmatrix}
y_1\\ \vdots \\ y_n
\end{pmatrix} \in \mathbb{R}^n,\
X =
\begin{pmatrix}
1 & x_{11} & \cdots & x_{1p} \\
1 & x_{21} & \cdots & x_{2p} \\
\vdots &...
- 347
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0
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26
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Least Angle Regression for Complex Rational Models
Background
I am doing a project and need to fit the following rational model, where the terms $\Psi$s are real but $y$, $p$, and $q$ are complex:
$$
y_{i} = \frac{\sum_{j=1}^{n_{p}}\Psi_{ij}p_{j}}{1+\...
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