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Notes
- 1.
In Section A.9 of the Appendix we give the definition of a basis for \(R^n\), which is an \(n\)-dimensional vector space. The basis function terminology refers to an extension of this idea to infinitely many dimensions: the functions \(f(x)\) on an interval \([a,b]\) that satisfy
$$ \int _a^b f(x)dx < \infty $$(here the Lebesgue integral is used) form an infinite-dimensional vector space and if the functions \(B_j(x)\) form a basis then every \(f(x)\) may be written as
$$ f(x)=\sum _{j=1}^{\infty } c_jB_j(x). $$ - 2.
Because the span of the columns of the \(X\) matrix using \(B\)-splines will be the same as the span of \(X\) matrix using the orthogonalized power basis, the resulting least-squares estimated fits \(X\hat{\beta }\) will be the same in both cases.
- 3.
One method, known as backfitting, cycles through the variables \(x_j\), using smoothing (here, spline smoothing) to fit the residuals from a regression on all other variables.
- 4.
There remain upward trends in the residual plots. This is due to the penalized fitting, which induces correlation of residuals and fitted values.
- 5.
The names Gabor and Morlet both get attached to what is perhaps more properly known as the Morlet wavelet, which has the form of a product of a normal pdf and a complex exponential, the real and imaginary parts of which are sinusoidal.
- 6.
The terminology comes from spectral analysis (see Section 18.3.3) where the width corresponds to a band of frequencies.
- 7.
A popular variation on this theme, called loess, modifies the weights so that large residuals (outliers) exert less influence on the fit. The terminology comes from the English meaning of loess, which is a silt-like sediment, and is derived from German word löss, which means “loose.”
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Kass, R.E., Eden, U.T., Brown, E.N. (2014). Nonparametric Regression. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_15
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