Proximities between functional data (semi-metrics)
Computes semi-metric distances of functional data based on Ferraty F and Vieu, P. (2006).
semimetric.deriv(
fdata1,
fdata2 = fdata1,
nderiv = 1,
nknot = ifelse(floor(ncol(DATA1)/3) > floor((ncol(DATA1) - nderiv - 4)/2),
floor((ncol(DATA1) - nderiv - 4)/2), floor(ncol(DATA1)/3)),
...
)
semimetric.fourier(
fdata1,
fdata2 = fdata1,
nderiv = 0,
nbasis = ifelse(floor(ncol(DATA1)/3) > floor((ncol(DATA1) - nderiv - 4)/2),
floor((ncol(DATA1) - nderiv - 4)/2), floor(ncol(DATA1)/3)),
period = NULL,
...
)
semimetric.hshift(fdata1, fdata2 = fdata1, t = 1:ncol(DATA1), ...)
semimetric.mplsr(fdata1, fdata2 = fdata1, q = 2, class1, ...)
semimetric.pca(fdata1, fdata2 = fdata1, q = 1, ...)fdata1 |
Functional data 1 or curve 1. |
fdata2 |
Functional data 2 or curve 2. |
nderiv |
Order of derivation, used in |
nknot |
semimetric.deriv argument: number of interior knots (needed for defining the B-spline basis). |
... |
Further arguments passed to or from other methods. |
nbasis |
|
period |
|
t |
|
q |
If |
class1 |
|
semimetric.deriv: approximates L_2 metric
between derivatives of the curves based on ther B-spline representation. The
derivatives set with the argument nderiv.semimetric.fourier: approximates L_2 metric between the curves
based on ther B-spline representation. The derivatives set with the argument
nderiv.semimetric.hshift: computes distance between curves
taking into account an horizontal shift effect.semimetric.mplsr:
computes distance between curves based on the partial least squares
method.semimetric.pca: computes distance between curves based on
the functional principal components analysis method.
In the next semi-metric functions the functional data X is
approximated by k_n elements of the Fourier, B–spline, PC or PLS basis
using, \hat{X_i} =∑_{k=1}^{k_n}ν_{k,i}ξ_k, where ν_k
are the coefficient of the expansion on the basis function
≤ft\{ξ_k\right\}_{k=1}^{∞}.
The distances between the q-order derivatives of two curves X_{1} and
X_2 is,
d_{2}^{(q)}≤ft(X_1,X_2\right)_{k_n}=√{\frac{1}{T}\int_{T}≤ft(X_{1}^{(q)}(t)-X_{2}^{(q)}(t)\right)^2 dt}
where X_{i}^{(q)}≤ft(t\right) denot the q derivative of X_i.
semimetric.deriv and semimetric.fourier function use a
B-spline and Fourier approximation respectively for each curve and the
derivatives are directly computed by differentiating several times their
analytic form, by default q=1 and q=0 respectively.
semimetric.pca and semimetric.mprls function compute
proximities between curves based on the functional principal components
analysis (FPCA) and the functional partial least square analysis (FPLS),
respectively. The FPC and FPLS reduce the functional data in a reduced
dimensional space (q components). semimetric.mprls function requires
a scalar response.
d_{2}^{(q)}≤ft(X_1,X_2\right)_{k_n}\approx√{∑_{k=1}^{k_n}≤ft(ν_{k,1}-ν_{k,2}\right)^2≤ft\|ξ_k^{(q)}\right\|dt}
semimetric.hshift computes proximities between curves taking into
account an horizontal shift effect.
d_{hshift}≤ft(X_1,X_2\right)=\min_{h\in≤ft[-mh,mh\right]}d_2(X_1(t),X_2(t+h))
where mh is the maximum horizontal shifted allowed.
Returns a proximities matrix between two functional datasets.
Ferraty, F. and Vieu, P. (2006). Nonparametric functional data analysis. Springer Series in Statistics, New York.
Ferraty, F. and Vieu, P. (2006). NPFDA in practice. Free access on line at http://www.lsp.ups-tlse.fr/staph/npfda/
See also metric.lp and semimetric.basis
## Not run: # INFERENCE PHONDAT data(phoneme) ind=1:100 # 2 groups mlearn<-phoneme$learn[ind,] mtest<-phoneme$test[ind,] n=nrow(mlearn[["data"]]) np=ncol(mlearn[["data"]]) mdist1=semimetric.pca(mlearn,mtest) mdist2=semimetric.pca(mlearn,mtest,q=2) mdist3=semimetric.deriv(mlearn,mtest,nderiv=0) mdist4=semimetric.fourier(mlearn,mtest,nderiv=2,nbasis=21) #uses hshift function #mdist5=semimetric.hshift(mlearn,mtest) #takes a lot glearn<-phoneme$classlearn[ind] #uses mplsr function mdist6=semimetric.mplsr(mlearn,mtest,5,glearn) mdist0=metric.lp(mlearn,mtest) b=as.dist(mdist6) c2=hclust(b) plot(c2) memb <- cutree(c2, k = 2) table(memb,phoneme$classlearn[ind]) ## End(Not run)
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