Questions tagged [polynomial]
A mathematical expression w/ >1 term containing the same variable (eg, x & x^2). Polynomials are commonly used to model curvilinear relationships.
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25 views
Is there a formal test for model comparison between polynomial regression and piecewise/segmented regression? (i.e., are these models nested?)
I am interested in comparing three models
a linear regression model: mod_linear <- lm (dv ~ iv)
a polynomial regression model: mod_polynomial <- lm (dv ~ iv + I(iv^2))
a one-breakpoint ...
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0answers
7 views
Use values of one variable to inform predicted values on a related variable
I want to use the values of one variable to inform the prediction of missing values from another variable.
For example, in the below image, the missing values of the ...
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0answers
9 views
Polynomial Kernel and similarity across dimensions
I am using SVM to classify hand grasps, and as such I am reading up on the polynomial kernel.
A section on the polynomial kernel in SKlearn reads:
Conceptually, the polynomial kernels considers not ...
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0answers
34 views
Using VIF, Interaction Effects, polynomial associations for feature selection in multiple linear regression
Is there a guide, tradition, or accepted practice on what to take into account and in what sequence between VIF, interaction effects, variable transformations and polynomial associations when ...
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6 views
Order of Magnitude on Very Flexible Model
I am taking an online class that rarely has an instructor. Upon reading the lab note of the class, I have a question.
We are doing Polynomial Regression and Splines which are very flexible models. In ...
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0answers
22 views
Regression models with special categorical variables
I need to build a regression model. I have 3 features: x1, x2, x3.
Based on those 3 features I want to predict 3 dependent variables: y1, y2, y3.
So far I created 3 separate models (Polynomial ...
1
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0answers
16 views
Approximation of a rational function [closed]
Suppose a function $f$ has a known form $ f(x) = \dfrac{P(x)}{Q(x)}$ where both $P,Q$ are polynomials of degree at most $d$. Assume $d$ is fairly low, take $d\leq 5$ for example.
What is the "...
2
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1answer
43 views
Multicollinearity: quadratic correlation between two independent variables in polynomial regression
Consider a polynomial regression of the form
$y = α*x_1^2 + β*x_1 + ɣ*x_2$
My question is: how to deal with multicollinearity between $x_1^2$ and $x2$? Or in other words, how to control for a ...
0
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1answer
27 views
Errors only in variables model, and polynomial fitting
I have a bunch of data points $(x, y)$, and I know that they fit well to a model of the form $y = a + bx + c x^2$, with $a \approx 0.01, \ b \approx 1\ \textrm{and}\ c \lesssim 0.1$. I'd like to fit ...
3
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1answer
36 views
LOOCV formula for polynomial regression
Is the LOOCV formula $$\text{CV} = \frac{1}{N}\sum_{i = 1}^N\bigg(\frac{y_i - \hat{y}_i}{1 - h_i}\bigg)^2 $$also valid for polynomial regression problems, or only for linear problems?
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17 views
D-Optimality for regression of polynomial models in one variable with missing terms
Let's say I have a model that looks as follows:
y = x + ax^3 + bx^5 + cx^7 + dx^9
Given n free choices for x as input measurements how can I determine which x's I should input to best determine a,b,c,...
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0answers
18 views
Multiple nonlinear regression model with p>n
I'd like to perform a nonlinear regression on a dataset that has more predictors than observations ($p$ =100, $n$=50), and predictors can also be multicollinear; i.e. the data is similar to a gene ...
1
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1answer
15 views
Polynomial regression with a binary dataset that does not fit logistic regression assumptions
Assumptions.
In normal logistic regressions probability has a fixed relationship with the independent variable. It either increases or stays the same as the independent variable increases OR it ...
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0answers
39 views
Singular value decomposition on a polynomial
I'm messing around with the SVD to find a best fit solution. The way I understand (never taking a stat class, only linear alg.) is that that the SVD captures the data variation by its projection onto ...
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0answers
14 views
How to compare importance of polynomial regression predictors?
I tried to make a logistic model using multiple predictors in r. Pre-analysis showed that one of the predictor (TEMP) has a bell curve effect on y variable, so I added a new variable that is a square ...
0
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0answers
23 views
Interpretation of p value in polynomial interaction
I am investigating changes in abundance of four species over time. The trends are obviously non-linear so I included a quadratic interaction term with species in my regression model. One species, &...
0
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1answer
26 views
What is the purpose of the first (1s) element in sklearn.preprocessing.PolynomialFeatures?
I got confused when I used 10 degrees and got 11 outputs.
I checked the https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.PolynomialFeatures.html and there seems to be (1) column ...
4
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2answers
71 views
Polynomial Regression: Can you tell what type of non-linear relationship there is by difference in statistics when there is a better fit?
Can you tell just from the statistics from polynomial regression specifically what type of relationship there is?
Ive run the two following linear regression models:
y= a+ bx_1+ bx_2 +bx_3 + bx_4
y= a+...
2
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0answers
38 views
Fitting a function to ECDF [closed]
I have some ECDFs. I would like to summaries the ECDFs with functional approximations. I was thinking that a polynomial, spline, or other line fitting procedure would generate a nice parsimonious ...
0
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1answer
42 views
How do I predict x values with confidence intervals from non-linear polynomial fit in R?
I have the following data.frame:
...
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0answers
17 views
How to get the Breier function to have a positive skew instead of a negative skew?
I'm trying to use the classic Briere function that typically gives a negative skew.
Brier: y = aa * Tdata * (Tdata - Tmin) * ((Tmax - Tdata )^(1/2))
I was trying ...
0
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1answer
19 views
Curve fit in SPSS repeated measures
I have cognitive data collected at 8 timepoints, which are 18 months apart. I would like to know what curve cognitive performance fits over time. I.e. is cognitive performance best explained by a ...
3
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0answers
113 views
Shape of MSE in polynomial regression
In polynomial regression, an increase in the degree of freedom can cause high variance and low bias. So model overfits on the training set and loses its generalization. On the other hand with a low ...
1
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1answer
44 views
How do we come up with the SVM Kernel giving $n+d\choose d$ feature space?
I was going through the CS229 notes on SVM and Kernel tricks and I came across the following line.
More generally the kernel $K(x,z)=(xTz+c)^d$ corresponds to a feature
mapping to an $n+d\choose d$ ...
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0answers
31 views
Interpreting orthogonal polynomial interaction terms with continuous predictors?
I'm struggling with the interpretation of orthogonal polynomial interactions when both predictors are continuous and would like to make sure my interpretation is correct. Thank you in advance for your ...
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0answers
30 views
Why does the bias decrease in this polynomial regression setup
I have questions regarding the bias variance tradeoff (more about bias actually):
Suppose $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3$, and that $y=f(x) + \epsilon $ where $\epsilon $ is normally distributed ...
3
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1answer
323 views
Given a truncated power basis function show that it represents a cubic spline for one knot
Given the truncated power basis function
$$h_1(x)=1, h_2(x)=x, h_3(x)=x^2, h_4(x)=x^3, h_5(x)=(x-\epsilon)^3_+$$
Show that a function of the form $f(x)=\beta_0+\beta_1x+\beta_2x^2+\beta_3x^3+\beta(x-\...
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2answers
384 views
How to test for overfit in lm() regression?
For a variable x, I find that coefficients of x, x^2, x^3 are all significant in lm(). But I ...
4
votes
2answers
252 views
“Spline fitting” in a piece-wise regression sense
I'm looking to better understand how a Octave built in function splinefit works. That itself is a wrapper around something on the MATLAB file exchange.
As I ...
0
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2answers
40 views
Relationship between variables in a proposed model using linear regression?
I am new to linear regression and I am currently working on a linear regression problem - I have 8 features and one output.
The features I am using seem unrelated to each other and I found an article (...
2
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0answers
82 views
Quantifying the universal approximation theorem
Let $m\geq 1$ be an integer and $F\in \mathbb{R}[x_1, \dots, x_m]$ be a polynomial. I want to approximate $F$ on the unit hypercube $[0, 1]^m$ by a (possibly multilayer) feedforward neural network. ...
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0answers
5 views
Response surface methodology - can I use them when there's no discrepancy?
I'm trying to use the response surface analyses and I'm wondering if I can use it when my data are skewed, so to say.
For example, I'm looking at provided support, received support, and well-being (...
0
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2answers
77 views
The log of a predictor and polynomial regression
I’m working with primate brain data as a predictor in regression models. In the primate brain literature it is custom to log brain data, but it is unclear to me why. It has been argued that since one ...
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0answers
18 views
Force a polynomial fit to pass through zero intercept [duplicate]
I have to perform a polynomial fit but I have to force it to pass through intercept zero. How can I do it?
Here the standard code for poly fit. If you have the answer in pure math it’s fine.
...
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1answer
500 views
How to plot quadratic model? [closed]
I have fit a polynomial glm in R with x and x^2 as the predictor of interest.
...
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0answers
39 views
Scipy.optimzie can't get the right fit for data with errors (polynomial function 3rd order)
so I have a problem with fitting a polynomial function 3rd order to my measured data points. The fit I get looks good and finds the minimum quite well, but the parameters and their errors are very big ...
0
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0answers
429 views
How to measure correlation in polynomial regression?
I have two variables that have a quadratic relationship. I can fit such an equation and get the R-squared, but how can I measure the degree to which the two variables are associated? Does a ...
2
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1answer
154 views
Polynom fit with big errors and wrong parameters
I have a question about polynomial fitting with python and I think its a more statistical question.
When I generate code for a polynomial function 3rd order with a not constant offset/error in the $y$-...
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0answers
29 views
Bivariate basis functions with span invariant to rotation about $z$-axis
Consider the following functions defined over $x,y\in\mathbb{R}$:
$f_0(x,y)=1$
$f_1(x,y)=x$
$f_2(x,y)=y$
These functions form a basis with three-dimensional span (the set of all non-vertical planes)
...
0
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1answer
24 views
How can we adjust the loss to incorporate beliefs about the test set?
Let's say we've some training data points (not a huge number), collectively referred to as $X$, and we're trying to fit a curve (a polynomial) to these points. Later, we check training and test set ...
3
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1answer
550 views
PCA with polynomial kernel vs single layer autoencoder?
What is the relationship between PCA with polynomial kernel and a single layer autoencoder ?
What if it is a deep autoencoder?
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0answers
11 views
Necessary conditions for Lagrange polynomial positive on interval
Given a Lagrange polynomial of degree $k$ in second form of the barycentric interpolation formula
$p(x) = \frac{\sum_{j=0}^k \frac{w_j}{x - x_j} y_j}{\sum_{j=0}^k \frac{w_j}{x - x_j}}$ with $x_j = \...
2
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0answers
99 views
Prove variance of locally weighted regression increases with degree
I am interested in proving the following fact for locally weighted polynomial regression from The Elements of Statistical Learning by Hastie et. al.
It can be shown that $||l(x_0)||$ increases with ...
2
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2answers
925 views
What are the pros and cons to fit data with simple polynomial regression vs. complicated ODE model?
Suppose in a disease outbreak scenario and we want to estimate number of infected people based infections over time.
Why we cannot simply fit the data with some polynomials (or some MLP neural ...
0
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1answer
155 views
How can I plot a linear and quadratic predictors from a regression model, while also controlling for other variables? [closed]
Here is some data and a model. It consists of a linear and quadratic predictor (a and a2) and a linear control variable (b).
...
0
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1answer
222 views
How to interpret a third-order regression?
I read some questions about this subject, but I couldn't find an answer.
I'm having trouble interpreting the practical effect of the polynomial predictor variable on the response variable.
My model ...
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0answers
59 views
Expected variance of quadratic regression coefficients with random x-values
I am thinking about a problem where I need to estimate the maximum of a quadratic regression, that is, fit a model $y = ax^2 + bx + c + \epsilon$ and estimate the maximum as $-\hat b/(2\hat a)$. My ...
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0answers
23 views
Multiple Regression Analysis Beginner
Background: I am using an instrument that measures two physical properties, X1~Temperture and X2~ Velocity. When gathering the data to make the curve a set of predetermined concentrations are chosen ...
0
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1answer
252 views
How do you determine what degree of polynomial to fit to data?
Say you have to fit a polynomial to data that is generated by another polynomial, for example. What is the process of determining what degree polynomial to use to fit that data?
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36 views
Squeeze a time series to fit in a range while maintaining shape
I have the following time series with intermediate highs and lows marked by the vertical lines:
I want to transform/squeeze the series so that the resulting series would fit in a range, let's say [0,...
