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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1answer
234 views

Books on Polynomial Regression

I have read some books on statistics, and it covers most of the regression algorithm, but does not cover up much on polynomial regression. Any suggestions on books covering up this topic, specifically ...
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1answer
5k views

R polynomial expansion error on “unique points” and degree

Why I am getting errors in R for "'degree' must be less than number of unique points" in this case? ...
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1answer
661 views

Orthogonal polynomials for regression

Is it possible to define orthogonal polynomials on the interval $[0, +\infty[$ ? Maybe using the Gram-Schmidt process from the monomial basis $(1, x, x^2, ...)$? My problem is that I have some data ...
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2answers
696 views

Rank of a matrix in Polynomial regression model?

We know, that by the assumption the rank of matrix $X$ in linear model is $k$, where $k$ is nummber of columns. What about polynomial regression model? What is the rank of matrix $X$ in this model?...
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2k views

R2 cross validation metric vs linear regression R2

I've been using sklearn for a project I'm working on. I'm following this example for analyzing my data. I believe I have an overfitting problem, but before I try to address it I want to make sure I'm ...
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0answers
182 views

What's the justification for normalizing inputs when solving using closed form?

In Machine Learning I understand that feature scaling or mean-normalizing your input is useful when trying to learn a model using gradient descent. It's useful in the sense that it speeds up training ...
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3answers
3k views

Why are there large coefficents for higher-order polynomial

In Bishop's book on machine learning, it discusses the problem of curve-fitting a polynomial function to a set of data points. Let M be the order of the polynomial fitted. It states as that We ...
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0answers
223 views

R: Fit sinusoidal curve to timeseries, and interpolate at desired times

I have ocean tide data with just high and low tides. I need to fit an appropriate curve (sinusoidal presumably, maybe just polynomial) to this, and then extract values interpolated at specific times. ...
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2answers
987 views

When to report quadratic versus linear relationships

I seem to remember from my graduate statistics course that if higher order variables (i.e., X^2, X^3, etc) are significant in a polynomial regression analysis such as our quadratic regression, then ...
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1answer
252 views

How can I fit a polynomial given some prior knowledge but fewer observations than coefficients?

I want to fit a second degree polynomial to data from a fractional factorial design in five dimensions. Because of the cost of trials, I am initially only going to do eight runs. So I have fewer ...
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0answers
766 views

Interpreting results of cubic regression [duplicate]

I have a simple question. I'm doing a regression with countries (346 countries). I have a variable that measures level of previous conflict. I rescaled this variable in a variable that goes from 0.0 (...
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1answer
95 views

Expected root of quadratic random polynomial

Suppose $A,B,C$ are i.i.d. random variables with uniform distribution on $[-1,1]$. I'm interested in the expected roots of the polynomial $Ax^2 + Bx + C$, which are complex random variables given by $$...
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434 views

units of residuals of a polynomial regression

I constructed two models: mdl <- lm(y ~ x) mdl.res <- mdl$residuals mdl1 <- lm(y ~ x + I(x^2)) mdl1.res <- mdl1$residuals Since ...
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1answer
94 views

Making the linear and quadratic terms independent in temporal data

I'm looking at the abundance of different groups of humans over time. These abundance data closely follow a quadratic model of the form $y=a_{0}+a_{1}x+a_{2}x^{2}+ \varepsilon$ for all groups of ...
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4answers
6k views

Why use regularisation in polynomial regression instead of lowering the degree?

When doing regression, for example, two hyper parameters to choose are often the capacity of the function (eg. the largest exponent of a polynomial), and the amount of regularisation. What I'm ...
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595 views

Orthogonal polynomial expansion and QR decomposition

Here is the source code of R poly function (boundary checking are removed). Why we can use QR to build polynomial expansion, ...
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2answers
1k views

How to extrapolate this simple trend line into the future for the purpose of forecasting in Matlab?

We have the following data points in variable data pertaining to a problem that we are solving: ...
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0answers
791 views

Selecting polynomial terms in regression

I'm developing a nonlinear response correction for a sensor (to transform "raw.peak" to "target"). I don't care about interpretability. I do care about future accuracy. One might first just throw it ...
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1answer
1k views

Comparing two models through ANOVA with different types of functions

See this file here: Decay.TXT. I first tried to fit the logarithmic model first ...
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1answer
1k views

How to use predict() function to average over polynomial terms in a linear model, so that it does not over- or under-estimate the data?

I have been having trouble with the predict function underestimating (or overestimating) the predictions from an lmer model with ...
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1answer
87 views

What is the name of figure where the fitted curve becomes the straight center line (figure in description)?

There's a figure attached to this post. The left hand side of the figure shows a graph with a polynomial fitting the data. The right hand side shows the same data plotted on a different scale but ...
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1answer
1k views

Kernel feature mapping: Derivation of polynomial kernel

The question is related to the derivation shown in section 3.1 Examples in the following lecture: http://people.eecs.berkeley.edu/~jordan/courses/281B-spring04/lectures/lec4.pdf I am confused about ...
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1answer
2k views

Linear regression polynomial slope constraint in R

My problem is how to find the best decreasing 3rd degree polynomial regression in R. I have data, lets say ...
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2answers
157 views

Is there a black box that can extract polynomial (quadratic) relationships?

I have an equation of the form $f(x)=\alpha x + \beta x^2$ that I want to match to experimental data. My current approach is to plot the data on a log-log scale and find the slope unity region (by '...
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1answer
282 views

How to translate orthogonal polynomial parameters back to the original metric

I am trying to work out how the parameters from a lmer model using orthogonal polynomials can be translated back to their original metric. Chapter 5.3.3 in Hedeker, Donald, and Robert D. Gibbons. ...
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1answer
2k views

Nonlinear Regression with linear method from Python's scikit-learn/ sklearn using a polynom

I am trying to do a regression analysis for some data, say 20 variables $\left( {{x_1},{x_2},{x_3},...} \right)$ where the underlying probability distribution is known (e. g. ${x_1} \in {\rm N}({\mu ...
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40 views

OLS, phenomenon { alpha = - mean(beta_2*x_orig)} : coincidence?

as suggested in the title, when with some data I perform this model: y ~ alpha + beta_1 * x_1 + beta_2 * (x_1)^2 + error term with OLS I SOMETIMES fall into the ...
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4answers
410 views

Something wrong with my implementation of the bias/variance diagnostic in polynomial regression

I'm trying to diagnosing bias/variance so I have the below Octavecode: ...
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1answer
4k views

Intuition behind the characteristic equation of an AR or MA process

Ok, so I've just started learning Time Series Analysis. We can write an MA(q) process as Yt = θ(L) ϵt and an AR(p) process as ϵt = φ(L) Yt in terms of the lag operator. Then, with no explanation (...
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1answer
1k views

Distribution of a second degree polynomial of a Gaussian random variable

I would like to compute $$P(Y=aX^2+bX+c<0)$$ where $X \sim N(0,\sigma)$. I can do it quite easily using Monte Carlo. However, I've been asked to find the analytical pdf $f_Y(y)$ of $Y$ and then ...
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0answers
66 views

Stationarity in the Almon lag model

I have a quick question regarding the Almon approach (Shirley Almon) as presented in chapter 17 of Gujarati's Basic Econometrics. In an example given in the textbook, they use non-stationary data ...
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0answers
32 views

how to get polynomial equation for given data in regression analysis? [duplicate]

I have following data and I want apply regression analysis for this data.Based on the data I want derive equation of 2nd or 3rd or 4th order polynomial equation. How can I do that by using math ...
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2answers
7k views

Quadratic terms in logistic regression

I am looking at the results of a logistic regression model (i dont have the data) and the person who has developed the model has included quadratic terms in the model. I understand the use of such ...
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0answers
28 views

Interpolate from curve data

I have these curves, From this curve I can determine the life of a prop shaft due to gyroscopic forces at different yaw angles and certain speeds. I performed curve fitting on data points to get ...
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1answer
346 views

Regression: find the best degree of polynomial with the best regularization parameter

When trying to predict data using linear regression or classify with logistic regression, with a polynomial, I know how to find the best degree of a polynomial to fits given data when the ...
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1answer
93 views

How to explain simply that the set of runs for Non Intrusive Polynomial Chaos cannot be used as a Monte Carlo sample

I had quite an annoying problem at work, a few days ago. I was doing a forward Uncertainty Quantification analysis using Non Intrusive Polynomial Chaos (NISP) (see for example here). Basically, you ...
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2answers
720 views

is linear regression/polynomial regression sensitive to irrelevant features/noise

is linear regression/polynomial regression sensitive to irrelevant features/noise will their respective weights/coefficients be automatically be tuned down? or is it a straight nail in the coffin? ...
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1answer
30 views

Justify choice of polynomia based on statistically significant result?

I am using an OLS. The variable of interest is Nth day to the end of the year (discrete variable). I would like to represent the relationship between y and Nth day to the end of the year with a ...
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0answers
377 views

Can a SVM with polynomial kernel lead to overfitting?

I'm currently searching for the best set of parameters for a Polynomial Kernel with a grid search. I would like to know if using a high value for Polynomial order (10, 100 or 1000) can lead to ...
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0answers
69 views

Predictive validity question using cubic linear models R

I have a data set with 2 months worth of observations from one of our clients, 1 observation per day. I am trying to predict number of sales based on two expenditure methods. My independent variables ...
2
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1answer
334 views

Centering variables in regression leads to the same model of original variables, why still doing that?

The regression model y= b0+ b1 x + b2 x^2 + b3 x^3 and the second regression model y = b0 +b1 (x-u) + b2 (x-u)^2 + b3 (x-u)^3 where u is the mean of x These two models lead to the same curves, or ...
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3answers
976 views

Best basis set for polynomial expansion

I want to do a regression of x onto y: $$f(y)=c_{1}x+c_{2}x^{2}+c_{3}x^{3}\cdots$$ Obviously a plain Taylor expansion as above is suboptimal since the coefficients will not be orthogonal/...
2
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1answer
3k views

Logistic Regression - Adding a polynomial basis to my input matrix make sense?

When I tried to run logistic regression on a 1500 X 35 input matrix, I obtained an error of 0.23 with 0 -1 loss. Then, I tried to add a polynomial basis of degree 2 or 3 to my matrix, which can be ...
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1answer
50 views

polynomial regression: what do large values of Y mean?

I have a time series of x:libor and y:money rates. using the following polynomial y=b0+b1(x)+b2(x)^2, i get values of y that exceed (or are sometimes negative) the coveriance/variance for large ...
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1answer
617 views

Regression plot and function for: heavy-tailed probability distribution

I've got data points from a simulation as coordinates in a text-files like so: ...
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1answer
828 views

Linear Ridge not correct prediction/coefficients- Scikit learn

I am using a similar code to this ridge example. The code proposed is simple. X and Y points inside [-1,1] range and predict the radius creating polynomial features and ridge linear regression. As ...
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1answer
241 views

Polynomial regression providing odd p-values

I have the data: ...
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2answers
3k views

Response Surface Methodology (RSM) for A Mathematical Model

I would like to create a second order polynomial model using Response Surface Methodology (RSM) for a non-polynomial mathematical model. For example, I would like to represent $f(x)=x_1 + \sin(x_1x_2) ...
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0answers
171 views

Problem with R-Squared value

I have a problem to determine my R-Squared value. I do a polynomial regression: fit3 <- lm(value ~ date + I(date^2)+ I(date^3),data=training) I have a R-Squared value (0.9416) when I do ...
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0answers
139 views

Orthogonal polynomials contrasts for regression (unbalanced)

We can obtain the sum of squares of a contrast for a regression of degree $j$ by: $$ SSR_j=\frac{\left(\displaystyle\sum_{i=1}^{I} C_{ji}T_i\right)^2}{rK_j}, $$ where $I$ is the number of levels of ...