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.

Filter by
Sorted by
Tagged with
4
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
1
vote
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
vote
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
votes
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
votes
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
votes
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?
1
vote
0answers
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,...
0
votes
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
vote
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 ...
0
votes
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 ...
0
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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: ...
0
votes
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
votes
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
votes
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
vote
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 $n+d\choose d$ ...
0
votes
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 ...
0
votes
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
votes
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-\...
1
vote
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
votes
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
votes
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. ...
0
votes
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
votes
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 ...
0
votes
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. ...
1
vote
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. ...
0
votes
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
votes
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
votes
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 $y$-...
1
vote
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
votes
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
votes
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?
1
vote
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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?
0
votes
0answers
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,...

1
2 3 4 5
7