Description Usage Arguments Details Value Note References See Also Examples

`WavTransf1D`

computes a forward intrinsic average-interpolating (AI) wavelet transform for a
curve in the manifold of HPD matrices equipped with a metric specified by the user, such as the
affine-invariant Riemannian metric, as described in \insertCiteCvS17pdSpecEst and Chapter 3 of
\insertCiteC18pdSpecEst.

1 2 | ```
WavTransf1D(P, order = 5, jmax, periodic = FALSE,
metric = "Riemannian", ...)
``` |

`P` |
a ( |

`order` |
an odd integer larger or equal to 1 corresponding to the order of the intrinsic AI refinement scheme,
defaults to |

`jmax` |
the maximum scale up to which the wavelet coefficients are computed. If |

`periodic` |
a logical value determining whether the curve of HPD matrices can be reflected at the boundary for
improved wavelet refinement schemes near the boundaries of the domain. This is useful for spectral matrix estimation,
in which case the spectral matrix is a symmetric and periodic curve in the frequency domain. Defaults to |

`metric` |
the metric that the space of HPD matrices is equipped with. The default choice is |

`...` |
additional arguments for internal use. |

The input array `P`

corresponds to a discretized curve of *(d,d)*-dimensional HPD matrices of
dyadic length. `WavTransf1D`

then computes the intrinsic AI wavelet transform of `P`

based on
the given refinement order and the chosen metric. If the refinement order is an odd integer smaller or
equal to 9, the function computes the wavelet transform using a fast wavelet refinement scheme based on weighted
intrinsic averages with pre-determined weights as explained in \insertCiteCvS17pdSpecEst and Chapter 3 of
\insertCiteC18pdSpecEst. If the refinement order is an odd integer larger than 9, the wavelet refinement
scheme uses intrinsic polynomial prediction based on Neville's algorithm in the Riemannian manifold (via `pdNeville`

).

The function computes the intrinsic AI wavelet transform in the space of HPD matrices equipped with
one of the following metrics: (i) the affine-invariant Riemannian metric (default) as detailed in e.g., \insertCiteB09pdSpecEst[Chapter 6]
or \insertCitePFA05pdSpecEst; (ii) the log-Euclidean metric, the Euclidean inner product between matrix logarithms;
(iii) the Cholesky metric, the Euclidean inner product between Cholesky decompositions; (iv) the Euclidean metric; or
(v) the root-Euclidean metric. The default choice of metric (affine-invariant Riemannian) satisfies several useful properties
not shared by the other metrics, see \insertCiteCvS17pdSpecEst or \insertCiteC18pdSpecEst for more details. Note that this comes
at the cost of increased computation time in comparison to one of the other metrics.

The function returns a list with three components:

`D ` |
the pyramid of wavelet coefficients. This is a list of arrays, where each array contains the
( |

.

`D.white ` |
the pyramid of whitened wavelet coefficients. The structure of |

`M0 ` |
a numeric array containing the midpoint(s) at the coarsest scale |

The function does not check for positive definiteness of the input matrices, and (depending on the specified metric) may fail if matrices are close to being singular.

`InvWavTransf1D`

, `pdSpecEst1D`

, `pdNeville`

1 2 | ```
P <- rExamples1D(2^8, example = "bumps")
P.wt <- WavTransf1D(P$f, periodic = FALSE)
``` |

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.