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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Denormalizing Data

I am applying Polynomial Regression to my data, however the parameters theta were always =0, i noticed that my y data or output is too large in the order of 100000 so i normalized y, i got very good ...
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170 views

How to determine exact point of tangency?

I have fitted my stress-strain data with $y=ax^3+bx^2+cx+d$ and also added tangent lines as shown in figure below. I am interested to see the deviation of the fit from the linearity. I am not aware of ...
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Using inverse of cube In linear model

What's the formula for a equation that can produce the continuum from the red to green lines in this graph below? I can easily get anywhere from the green line to the blue with $$y = B_0 + B_1x + B_2x^...
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Best practices for extrapolating data

I have a set of variables that parameterize a logistic equation bacterial growth model. The parameters change based on temperature (e.g., growth speeds up at higher temperatures) and so it is ...
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Why is polynomial regression considered a special case of multiple linear regression?

If polynomial regression models nonlinear relationships, how can it be considered a special case of multiple linear regression? Wikipedia notes that "Although polynomial regression fits a nonlinear ...
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7k views

Polynomial term in logistic regression

I've made a logistic regression model that includes a polynomial term to degree 2. I'm aware that logistic regression models the response variable as a non-linear function of the predictors. Does it ...
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513 views

Is it possible to center orthogonal polynomials in multiple regression

I have a regression model that looks like the one below. $$ Y = \beta_0 + \beta_1T + \beta_2T^2 + \beta_3T^3 + \beta_4D + \beta_5D*T + \beta_6D*T^2 + \beta_7D*T^3 $$ Where ...
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195 views

Optimal Rotation Before Curve Fitting

I am fitting a 1-dimensional polynomial to a set of data points $\{(x_1,y_1,z_1),...,(x_N,y_N,z_N)\} \subset \mathbb{R}^3$ using least squares regression. My first instinct was to choose $x$ as an ...
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Number of trials needed to increment a counter variable

Truthfully I'm not even sure what the proper name for this sort of trial is, so I'll just describe the whole setup. Suppose I have an integer variable n, which starts at zero and gets incremented ...
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256 views

Overcoming small dataset anomalies in genetic algorithm

So I am currently making my 6th version of a model designed to predict the likelihood of of a particular medical condition based on a multifactorial genetic markers and I really would appreciate some ...
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1answer
417 views

How to algorithmically determine the best order of fit?

I am doing a least squares polynomial interpolation for 10,000 data sets that look mostly like one period of a sine curve, but whose values are not evenly spaced in the time domain, and can sometimes ...
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Optimal orthogonal polynomial chaos basis functions for log-normally distributed random variables

I hope this is the appropriate venue for this type of question. If not, please feel free to migrate! :) I'm trying to solve a stochastic partial differential equation of the form $$\alpha(\omega)\...
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Estimation of a power function in regression $y = ax^k$

I'm performing a case of polynomial regression. I use a power $k$ for the regressors (e.g. marketing spend), which helps me determine the nature of the response curve. I also need to estimate the ...
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Getting a second-order polynomial trend line from a set of data

Alright, so I have about a thousand datapoints that I'm plotting on a chart (scatter plot). Here's a few of the records: ...
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6k views

Python creating a polynomial model with two input variables

I have data for two input variables x_1 and x_2 and one output variable y. The two input variables seem to have a nonlinear relationship with y. Here are the plots So I am trying to fit a ...
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602 views

How to select the degree of polynomial multiple regression?

I have around 50 dependent quantities (regressor variables). I want to find the best relation between the response variable data and regressor variable data. I tried multiple linear regression with ...
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Computation of polynomial contrast variables

Please give me idea how to efficiently recode a categorical variable (factor) into the set of orthogonal polynomial contrast variables. For many types of contrast variables (e.g. deviation, simple, ...
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Polynomial regression using scikit-learn

I am trying to use scikit-learn for polynomial regression. From what I read polynomial regression is a special case of linear regression. I was hopping that maybe one of scikit's generalized linear ...
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Can the OLS residual variance suggest a polynomial relationship?

I am trying to figure out whether from the following graph of the OLS residuals that the linear relationship does not hold, and that probably a cubic relationship would do better? Since both in the ...
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Unit root test for ARIMA models

I have a slight confusion regarding seasonal models and which polynomial to use for conducting unit root tests. Given a model: $\phi(B)\Phi(B^s)\Delta^d\Delta^D_S X_t = \theta(B)\Theta(B^s)\epsilon ...
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What happens when I include a squared variable in my regression?

I start with my OLS regression: $$ y = \beta _0 + \beta_1x_1+\beta_2 D + \varepsilon $$ where D is a dummy variable, the estimates become different from zero with a low p-value. I then preform a ...
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316 views

Orthogonality of Hermite Polynomials

As we know probabilistic Hermite polynomials are orthogonal with respect to the weight function $\frac{1}{\sqrt{2 \pi}} e^{-x^2/2}$ (density of standard normal). I have a distribution which is a ...
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How would you report (in publication) the results of a linear model fit using the poly function in R?

@John recently pointed out to me that R's poly function produces less correlated values (more orthogonal) to fit polynomial predictors, i.e. the transformed ...
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Perform linear regression, but force solution to go through some particular data points

I know how to perform a linear regression on a set of points. That is, I know how to fit a polynomial of my choice, to a given data set, (in the LSE sense). However, what I do not know, is how to ...
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2k views

nearest neighbors degrees of freedom

For polynomial fitting with a polynomial of degree $n$, we have $n$ degrees of freedom. Is there a similar concept for $k$ nearest neighbors? Is there any way to compare the degrees in general? I come ...
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735 views

Model reduction - Backward elimination

The rule I use for reducing covariates (trimming) is as following: Look at the highest order interaction. If that is not significant, drop it. If it is significant, stop. If drop the highest order ...
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123 views

Sample point locations and multiple linear 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 ...
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comparing predictive model with hold out set

In Rapid Miner, I created a predictive model (SVM) with Kernel type = polynomial, c= 10, and obtained 80.77% accuracy using cross validation. When compared to hold out set my accuracy on the test ...
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Calculating Adjusted $R^2$ in Polynomial Linear Regression with Single Variable

When calculating Adjusted $R^2$ the formula is $1-(1-R^2)\frac{n-1}{n-k-1}$ with $k$ being how many predictors you have. If I am using a model with a single variable but that variable has been put ...
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Can I interpret the inclusion of a quadratic term in logistic regression as indicating a turning point?

In a Logistic Regression with linear and quadratic terms only, if I have a linear coefficient $\beta_1$ and quadratic coefficient $\beta_2$, can I say that that there is turning point of the ...
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756 views

Two negative beta's in a curvilinear regression when mean centered or using standardized values

The problem I encounter is the following: Imagine a (perfect) inverted U-shaped relation between an independent variable and a dependent variable. When you look at the curve estimation there is ...
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2answers
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Improvement of regression model

I am just learning R. I have developed a regression model with six predictor variables. While developing it, I found the relationships are not very linear. So, maybe because of this the predictions of ...
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Polynomial in linear regression

I am new to Regression and R. I know that polynomial functions are used when a regression model does not fit data (underfitting), but I want to know which degree of polynomial should be used? I also ...
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1answer
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What is the role of a categorical predictor in polynomial regression?

I understand that there is a function in R called poly() that can generate orthogonal polynomials--useful for applying on input variables before running a ...
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1answer
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Recovering raw coefficients and variances from orthogonal polynomial regression

It seems that if I have a regression model such as $y_i \sim \beta_0 + \beta_1 x_i+\beta_2 x_i^2 +\beta_3 x_i^3$ I can either fit a raw polynomial and get unreliable results or fit an orthogonal ...
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5k views

Why do I get wildly different results for poly(raw=T) vs. poly()?

I want to model two different time variables, some of which are heavily collinear in my data (age + cohort = period). Doing this I ran into some trouble with lmer ...
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889 views

How to produce a polynomial trend line equation that takes three arrays as parameters?

Does anyone know of any programming code for producing a polynomial trend line equation that takes three arrays as parameters, ie X, Y and Weight? Or even if you could explain in English how such a ...
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Does it make sense to add a quadratic term but not the linear term to a model?

I have a (mixed) model in which one of my predictors should a priori only be quadratically related to the predictor (due to the experimental manipulation). Hence, I would like to add only the ...
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1answer
131 views

Monomial distribution of $X^a \cdot Y^b$

What is the distribution of the following monomial? $$X^a \cdot Y^b$$ where $X$ and $Y$ are normal random variables and $a$ and $b$ are natural numbers. For example, when $X \sim N(0,1)$, $a=2$, and $...
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286 views

Determining polynomial model coefficients forcing slope = 1 and intercept =0

I have two observational variables, $\text{C}_{\text{obs}}$ and $R$, both subject to measurement error. I believe that a model of the form $$\text{C}_{\text{mod}} = a_0 + a_1R + a_2R^2 + a_3R^3$$ ...
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Dynamic consistency and multilevel models using lmer

I've been using nlme and more recently lmer to fit multi-level models of time course data using orthogonal polynomials. My ...
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1answer
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Using Fractional Polynomials for Logistic Regression Modelling in R

I am learning logistic regression modeling using the book "Applied Logistic Regression" by Hosmer. In chpaters, he suggested using Fractional Polynomials for fitting continuous variable which does ...
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993 views

Does the p-value in the incremental F-test determine how many trials I expect to get correct?

I've implemented an incremental F-test program that evaluates the fit of an unrestricted model $M_{UR}$ against the restricted model $M_R$ using the F statistic $\frac{SSE_{R} - SSE_{UR}}{SSE_{UR}}\...
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Is there a fast algorithm to check for AR(p) stationarity?

It is well-known that an AR(p) process $$ x_t=\sum_{i=1}^p \varrho_i x_{t-i} + \epsilon_t \,, $$ is causal and stationary if and only if the roots of the polynomial $$ \mathcal{P}(u) = 1 - \sum_{i=1}^...
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Why would one use age-squared as a covariate in a genetic association study?

Why would one use age and age-squared as covariates in a genetic association study? I can understand the use of age if it has been identified as a significant covariate, but I am at a loss as to the ...

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