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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Interpreting Results of Logistic Regression when both x, y variables are nominal

I've been trying to analyze the result from my experiment. But since I'm new to the field of statistics, I'm struggling in every step, including the interpretation of results. I have 4 groups of ...
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Proof that a necessary condition for characteristic roots to lie inside unit circle is $\sum\limits_{i=1}^{n} a_{1} < 1$

I have been trying to show that given $$P_{n}(\alpha) = \alpha^{n} - a_{1}\alpha_{n-1} - a_{2}\alpha^{n-2}... - a_{n} = 0,$$ the $\alpha$'s that solve this equation (real-valued or complex) lie in the ...
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Confidence Interval for a 3rd Degree Polynomial Distribution

I've tried looking for an answer online (including in this website), but I can't seem to find one. I have sample x and y values (in the thousands), which I estimated as a 3rd degree polynomial (by ...
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R poly what family of polynomials? [duplicate]

The R poly function documentation says that it gives orthogonal polynomials, but it's not clear what orthogonal polynomials it gives. Are they Legendre? Are they ...
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Can I use zscore to calculate pvalue of coefficients in polynomial regression model?

Recently I am trying to reproduce results from this paper. The author performed microassay to detect gene expression in mutiple overlapping time window. Take time window 1 for example, let's say there ...
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Help Understanding Polynomial/Least Squares Regression

I have a dataset of 2 variables (called x with shape n x 2 values of x1 and x2) and 1 output (called y). I am having trouble understanding how to calculate predicted output values from the polynomial ...
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How to do data imputation and normalization when using polynomial regression?

The question is about the practical use of polynomial regression. Let's say there is a dataset with columns A, B, T where T is a dependent variable, A and B are independent variables. A and B contain ...
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rank elements to minimize the median of the response at a certain rank

I have the following data. ...
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Fit a polynomial model column of time series data

I'm asked to fit a polynomial model on the FTSE column from the EuStockMarkets dataset available in R. How is it possible to fit on just one column in this case? Is there another column of sorts in ...
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How to interpret polynomial regression models that involve categorical variables?

I want to ask the way to interpret polynomial regression models that involve categorical variables. So far, I know that when there is no polynomial, it is straightforward to interpret estimates as the ...
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Coefficients for higher dimensional NIPC (non intrusive polynomial chaos expansion) heavily biased towards 0 degree function

Currently, I am trying to apply the PCE to the thermal fin problem as solved using finite element. Although I cannot attach the code I use, suffice to say I solve a linear system $AU=F$ where $F$ is ...
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Log-Binomial Polynomial Terms Interpretation

I have a log binomial model of tested (yes/no) by number of years since 2011. The quadratic term is also significant. I am trying to find the probability of being tested at each year. I thought I ...
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Is it appropriate to fit a linear model to my data?

I have a bunch of outcome/exposure relationships I am trying to fit models to: From these graphs, I am not sure if a simple lm is appropriate. Some of them look ...
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How can we explain the "bad reputation" of higher-order polynomials?

We all must have heard it by now - when we start learning about statistical models overfitting data, the first example we are often given is about "polynomial functions" (e.g., see the ...
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Why adding $\log(x)$ for repeated powers in Fractional Polynomial Models?

I try to understand Fractional Polynomial Models like they are defined in Royston and Altman (1994). According to the paper, a Fractional Polynomial Model of degree 2 with a single covariate $X$ is ...
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When adding polynomial features, the issue of multicollinearity doesn't hold. Why?

In the regression model, sometimes to capture the non-linear relationship between dependent and independent variables, we use polynomial features. But in regression, if the two or more features are ...
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Interpreting Quadratic Variables in Negative Binomial Regression

I want to include a quadratic term for age in my negative binomial model as past work has suggested it may be curvilinear. I know how to interpret a quadratic coefficient in OLS, but am unsure with ...
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How to interpret polynomial regression bootstrap output?

I have the following data: ...
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Polynomial regression and multicollinearity [duplicate]

to use linear regression we should have the predictors/features be independent from one another. However, in polynomial regression, x^2 is perfectly correlated to x ( knowing x value gives us the ...
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Why are odd-degreed polynomial kernels slower than those with even degrees for SVM?

I have been using one-class support vector classifiers to extract features for multinomial classification. I noticed that fitting time is much longer when the degree of the polynomial kernel is odd. ...
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Backward elimination in multiple polynomial regression?

I'm working on multiple regression models (with R), and the use of polynomials can increase my adj r-squared. However, I'm a bit confused because if I add polynomials, the adj r-squared decreases, but ...
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Finding Inverse CDF of a Polynomially Distributed RV

Let's say there exists f(x) = 3(x-1)^2 0 < x < 1, 0 otherwise Then the CDF would surely be F(x) = 0 if x < 0, 3x^3 -9x^2 + 9x for 0 < x < 1 and 1 otherwise. How would I go about ...
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How can I improve my model's prediction based on test set accuracy socres?

Perhaps that's a bad way to phrase this question, but I'm much more of a coder than a statistician. I would really appreciate help on the following: This pretty picture shows three predictions by my ...
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Why not always use Polynomial Regression to solve classification problems? [closed]

Consider this simple classification problem: You can solve it using Logistic Regression. But there's another way. As @whuber noted in this answer, in hypothesis $h(x) = \frac{1}{1 + e^{-P(x)}}$, ...
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Difference in simple polynomial regression coefficients using `poly()` function in R vs doing manually [duplicate]

I'm not sure whether to ask this here or on Stack OverFlow, but I think here is better because the question is based more around the theoretical aspects than specific code. I'm currently working ...
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how to configure nlmer/nmle for polynomial logistic regression?

I previously modeled my data with glmer and it is OK: ...
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How can I use the link(g, lam) function from "psyphy" package to adjuste asymptotes?

I need to customize the asymptotes of the model, and I am trying with psyphy package which provides parameters for adjusting asymptotes in its ...
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9 votes
2 answers
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Can any data be learned using polynomial logistic regression

We know that a Taylor polynomial can approximate any continuous function. As @DemetriPananos noticed, Logistic regression seeks to estimate the coefficients for a model and any cut off is imposed post ...
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1st order(linear?) polynomial logistic regression, is it a reasonable build? and have only one inflection?

I have been working on a third order polynomial logistic regression ...
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How to find a turning point of quadratic term for different categories?

I have built a logistic regression with polynomial relationship with variable age, which I include in the model as an interaction with variable ...
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24 votes
5 answers
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Why is the use of high order polynomials for regression discouraged?

I've read many times on this site that high order polynomials (generally more than third) shouldn't be used in linear regression, unless there is a substantial justification to do so. I understand the ...
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Which if these two models works better?

I have this time series I want to perform polynomial regression on, to estimate the trend. To start, I tried using only a second order polynomial, these are the results (AIC=30.37105) We can see how ...
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5 votes
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Why do you need non-linear regression if you can use a linear one to fit any kind of curvature to your data?

Polynomial regression fits a non-linear model to the data. But as a statistical estimation problem it's still linear in the sense that the regression function $h\left(\Theta, X\right)$ is linear in ...
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Properly set-up indepenent variables for polynomial logistic regression

I would like to conduct an analysis to identify factors that affect whether or not a first purchase was made. In this context, there are three outcomes I want to predict: Did not buy anything Did buy ...
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Error propagation for cubic relationship

I have the cubic relationship between two variables, x and y, and I need to find the error in x. y = ax^3 + bx^2 + cx + d I have the values for the coefficients and their respective uncertainties. I ...
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2 votes
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How many samples does one need to perform polynomial regression of degree $m$?

Suppose $(X_i, Y_i)$, $i = 1,\dots, n$ are random variables such that $$X_i\sim N(0,1)$$ $$Y_i = f(X_i) + \epsilon_i$$ where the $\epsilon_i$ are i.i.d. standard Gaussian and $f(x)=\sum_{k = 0}^\infty ...
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Polynomials that converges pointwise to a simple function on (-1,1) and bounded by $e^{|x|}$?

I am trying to prove a theorem related to the moment generating function. I will need a sequence of polynomial that converges to a simple function $K_{(-1,1)}(x)$ pointwise on the real line while ...
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1 answer
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Was Kernel Regression Invented to Address the Problems with Higher Order Polynomial Regressions?

I had this thought today : We all know that higher order polynomials (e.g. polynomial regression models) have a tendency of overfitting the data and performing poorly (i.e. generalizing) to new data. ...
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Polynomial regression interaction features

In polynomial regression input vector $[a,b,c]$ is transformed into $[1, a, b, c, a^2, b^2, c^2, ab, ac, bc]$ before linear regression is applied. At least this is how ...
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How to partially fit a piecewise polynomial with segment boundaries unknown?

Suppose that I have the some data like this. ...
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How to invert data that was detrended with polynomials?

I am using VAR to model some time series data. To make the data stationary I detrended them with regression with n = 2. I used the function below... ...
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Computationally + Statistically Efficient Unbiased Estimation of Chebyshev Polynomials of Expectations

Let $T_n$ denote the $n^\text{th}$ Chebyshev polynomial, defined by the recursion \begin{align} T_0(x) &= 1,\\ T_1(x) &= x,\\ T_n(x) &= 2x \cdot T_{n-1} (x) - T_{n-2} (x). \end{align} Now, ...
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Squaring Negative Explanatory Variables [closed]

Let us say I am interested in analyzing an event, which takes place at year 0. My variables are the years since the event: years before the event are coded as "negative" years (so the year ...
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Finding coefficients in nonlinear regression [duplicate]

$Curve Model f(x) = ax^2 + bx + c$ $Data(x,y) = {(x_1,y_1),(x_2,y_2)...(x_n,y_n)}$ $Square Error = {E_1,E_2...E_n}$ SSE: sum of SE $Initial values = {a=1 ,b=1, c=1}$ I'm new to statistics. In ...
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Orthogonal contrasts and the poly function in R: Interpreting the coefficients as "absolute" or "incremental" effects

in a growth curve like model, I try to test whether a trajectory (over time) is better described by a linear or quadratic trend. I fitted a linear mixed model (time nested in participants) and called ...
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Interpreting lower-order interactions when higher order interaction is NOT significant

I have a multilevel model in which I estimated a cubic effect of age (model 1). I also tested if gender moderate the linear, quadratic, and cubic effect of age (model 2). It turned out that the latter ...
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2 votes
1 answer
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Coef and Polynomial equation

I have a problem about the coef of polynomial, from machine learning. ...
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1 vote
1 answer
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How to avoid overfitting in multiple linear regression (lm) in R, as the polynomial degree increases so do the fit stats

I am fitting multiple linear regressions to a data set in which the fitted plane has to approximate the values equal to 99; reddish-orangeish dots in the figure below. I am testing fits with a ...
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Do I need to generate orthogonal polynomial in scikit learn for logistic regression classifier?

I tried to make a logistic regression classifier to predict the class of unseen data. In this data cross terms play important role to predict the multi imbalance class, with this information I cannot ...
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Is polynomial regression doomed to only work for data set with tiny amount of features?

Consider curve fitting for a dataset $D = \{(x_i, y_i)\}$, $i = 1, \ldots, N$, $x_i \in \mathbb{R}^f$, $y_i \in \mathbb{R}$. Here, $f \in \mathbb{N}$ is the number of features associated with the data....
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