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Questions tagged [basis-function]

A basis function is an element of a set of functions that span a function space.

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Problems interpreting GAM output

I have been advised to run General Additive Models to be able to describe trends in my data, my data being animal harvest numbers by year. I have done so, but have a problem with interpreting the ...
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How to use a gaussian basis function

I am trying to build a linear regression model and one of my features when plotted against the output looks like a bell curve. So intuitively I figured that if I use the raw data for that feature my ...
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48 views

Smoothing splines as basis expansion

I am following the discussion on chapter 5 of Elements of statistical learning which discusses basis expansion using splines. The data set I used is the Ozone data which can be found at :http://web....
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Difference between s() and ti() terms in mgcv package when applied to one variable

I am using the mgcv package in R to fit logistic GAMs to survey data. In one of my models I use an interaction between two covariates. I am currently trying to fit ...
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1answer
74 views

Are $h_i(x)=x^{-\alpha_i}$ okay basis functions for fitting?

I have some pairs of data ${(x_1,y_1),..., (x_n,y_n)}$ genereated by some process and would like to fit it with a function so that $y_i \approx \hat{f}(x_i)$. By plotting the $(X,Y)$ on a 2D plot, ...
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1answer
106 views

Please correct my assumption on how regression trees work

I'm trying to understand how regression trees work, I've been experimenting with catboost and xgboost in python, and I'm getting results which I don't expect, can someone please clarify (and apologies ...
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37 views

Adaptive knot selection for B-spline fitting

When fitting a B-spline for regression purposes I've seen a lot of cases where knots are fixed uniformly ,but in some situations this could lead to poor estimations because the behaviour of the curve ...
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29 views

residualized covariance matrix from pca/eigenvalue decomposition

I understand that given N dimensional data you can use PCA to construct an N dimensional orthonormal basis that explains 100% of the variance of the original data. However, you can also construct ...
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1answer
479 views

Number of basis functions in natural cubic spline

According to ESL, natural cubic basic spline with $K$ knots is represented by $K$ basis function. However, the ns() function in R with ...
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94 views

How to calculate the output of ns() function in R by hand

ns() function in R can generate the natural cubic spline basis matrix. I checked the reference but still do not know how to calculate such matrix by hand. For ...
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94 views

Kernel function from polynomial basis functions

In chapter 3 & 6 of Bishop's Pattern Recognition and Machine Learning, he showed that the equivalent kernel based on eqn (3.62) $$ k(x,x') = \beta \phi(x)^T (\alpha I + \beta \Phi^T \Phi )^{-1}\...
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1answer
21 views

Generate funciton with specified bases

I am trying to simulate functional data using specified bases and a set of random coefficients. For example, I can generate a function on $[0,1]$ with Fourier basis like $$f(x) = \sum_{i=0}^k a_i sin(...
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How to measure the effect of each covariate in GLM

I have a GLM model with two variables (Z1 and Z2). I have also used some basis functions to estimate these variables. In other words, Z1 is substituted by 10 basis functions and Z2 with 12 basis ...
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3answers
120 views

Why does it matter that bases are linearly independent?

I am taking an advanced linear algebra course and am once again confused by a lot of the concepts. I understand that the definition of a basis is a set of vectors that spans the vector space and is ...
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1answer
523 views

Is spline basis orthogonal?

When we talk about the basis, we have the concept like orthogonal, unit length etc. for vectors. I think the same concept also exist in Fourier basis and Polynomial basis. But how about spline (Say ...
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1answer
618 views

Gaussian basis function in Bayesian Linear Regression

I am learning about Bayesian Linear Regression from the book "Pattern Recognition And Machine Learning" (Bishop, Christopher M.). I want to recreate graphs from illustration 3.8 and this require to ...
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1answer
239 views

In using a basis model to represent a cubic spline, what is the meaning of the truncated power basis function and its math in lay terms?

I am going over “Introduction to Statistical Learning” (James, et al). While I understand the concept of a spline (piecewise polynomial with continuous 1st and 2nd derivative at each knot), I’ve been ...
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156 views

Regression Spline basis functions

I'm reading this book - Generalized Additive Models by Simon Wood. On page 124, he defines basis for regression spline as (language is R) ...
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1answer
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bad fit - nomenclature for breeds

Question: What is it called when one uses a basis, like the pure line instead of the sigmoid/logistic, in a manner that grossly departs from the "physics" of the problem? There should be a word for ...
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1answer
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Understanding Gaussian Process Regression via infinite dimensional basis function view

It is often said that gaussian process regression corresponds (GPR) to bayesian linear regression with a (possibly) infinite amount of basis functions. I am currently trying to understand this in ...
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1answer
651 views

Splines - basis functions - clarification

I have been reading the very helpful introduction on splines on http://freakonometrics.hypotheses.org/9184 and on http://www.stats.uwo.ca/faculty/braun/ss3859/chapters/splines/splines.pdf, as well as ...
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775 views

How are radial basis functions (RBFs) networks extended to use multiple layers?

I am trying to understand the interpretation of radial basis functions (RBFs) as networks and then trying to understand the relationship it has to "normal" neural networks and how to extend them to ...
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3answers
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Understanding Gaussian Basis function parameters to be used in linear regression

I'd like to apply the Gaussian basis function into a linear regression implementation. Unfortunately I'm having a hard time understanding a couple parameters in the basis function. Specifically $\mu$...
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1answer
667 views

Spline – basis functions

I am trying to wrap my head around splines and the concept of basis functions using the Elements of Statistical Learning. I understand that the goal is to find polynomials that are continuous at first ...