Description Usage Arguments Details Value References Examples

The spectrum of a multivariate time series is a matrix-valued function of the
frequency *λ \in [-π, π]*, which is symmetric/Hermitian around
*λ = 0*.

`mvspectrum`

estimates it and returns a 3D array of dimension
*num.freqs \times K \times K*. Since the spectrum is symmetric/Hermitian around
*λ = 0* it is sufficient to store only positive frequencies.
In the implementation in this package we thus usually
consider only positive frequencies (omitting *0*); `num.freqs`

refers
to the number of positive frequencies only.

`normalize_mvspectrum`

normalizes the spectrum such that
it adds up to *0.5* over all positive frequencies (by symmetry it will
add up to 1 over the whole range – thus the name *normalize*).

For a *K*-dimensional time series it adds
up to a Hermitian *K \times K* matrix with 0.5 in the diagonal and
imaginary elements (real parts equal to *0*) in the off-diagonal.
Since it is Hermitian the mvspectrum will add up to the identity matrix
over the whole range of frequencies, since the off-diagonal elements
are purely imaginary (real part equals 0) and thus add up to 0.

`check_mvspectrum_normalized`

checks if the spectrum is normalized
(see `normalize_mvspectrum`

for the requirements).

`mvpgram`

computes the multivariate periodogram estimate using
bare-bone multivariate fft (`mvfft`

). Please use
`mvspectrum(..., method = 'pgram')`

instead of `mvpgram`

directly.

This function is merely included to have one method that does not require the astsa nor the sapa R packages. However, it is strongly encouraged to install either one of them to get (much) better estimates. See Details.

`get_spectrum_from_mvspectrum`

extracts the spectrum of one time series from an
`"mvspectrum"`

object by taking the i-th diagonal entry for each frequency.

`spectrum_of_linear_combination`

computes the spectrum of the linear
combination *\mathbf{y}_t = \mathbf{X}_t \boldsymbol β* of *K*
time series *\mathbf{X}_t*. This can be efficiently computed by the
quadratic form

*
f_{y}(λ) = \boldsymbol β' f_{\mathbf{X}}(λ) \boldsymbol β ≥q 0,
*

for each *λ*. This holds for any *\boldsymbol β*
(even *\boldsymbol β = \boldsymbol 0* – not only for
*||\boldsymbol β ||_2 = 1*.
For *\boldsymbol β = \boldsymbol e_i* (the i-th basis vector)
this is equivalent to `get_spectrum_from_mvspectrum(..., which = i)`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
mvspectrum(series, method = c("pgram", "multitaper", "direct", "wosa",
"mvspec", "ar"), normalize = FALSE, smoothing = FALSE, ...)
normalize_mvspectrum(mvspectrum.output)
check_mvspectrum_normalized(f.U, check.attribute.only = TRUE)
mvpgram(series)
get_spectrum_from_mvspectrum(mvspectrum.output,
which = seq_len(dim(mvspectrum.output)[2]))
spectrum_of_linear_combination(mvspectrum.output, beta)
``` |

`series` |
a |

`method` |
string; method for spectrum estimation; see |

`normalize` |
logical; if |

`smoothing` |
logical; if |

`mvspectrum.output` |
an object of class |

`f.U` |
multivariate spectrum of class |

`check.attribute.only` |
logical; if |

`which` |
integer(s); the spectrum of which series whould be extracted. By default, it returns all univariate spectra as a matrix (frequencies in rows). |

`beta` |
numeric; vector |

`...` |
additional arguments passed to |

For an orthonormal time series *\mathbf{U}_t* the raw periodogram adds up
to *I_K*
over all (negative and positive) frequencies. Since we only consider
positive frequencies, the normalized multivariate spectrum should add up to
*0.5 \cdot I_K* plus a Hermitian imaginary matrix (which will add up to zero
when combined with its symmetric counterpart.)
As we often use non-parametric smoothing for less variance, the spectrum estimates
do not satisfy this identity exactly. `normalize_mvspectrum`

thus adjust the
estimates so they satisfy it again exactly.

`mvpgram`

has no options for improving spectrum estimation whatsoever.
It thus yields very noisy (in fact, inconsistent) estimates of the
multivariate spectrum *f_{\mathbf{X}}(λ)*.
If you want to obtain better estimates then please use other `method`

s in
`mvspectrum`

(this is highly recommended to obtain more
reasonable/stable estimates).

`mvspectrum`

returns a 3D array of dimension *num.freqs \times K \times K*, where

num.freqs is the number of frequencies

K is the number of series (columns in

`series`

).

Note that it also has an attribute `"normalized"`

which is
`FALSE`

if `normalize = FALSE`

; otherwise `TRUE`

.
See `normalize_mvspectrum`

for details.

`normalize_mvspectrum`

returns a normalized spectrum over
positive frequencies, which:

- univariate:
adds up to

*0.5*,- multivariate:
adds up to Hermitian

*K \times K*matrix with 0.5 in the diagonal and purely imaginary elements in the off-diagonal.

`check_mvspectrum_normalized`

throws an error if spectrum is not
normalized correctly.

`get_spectrum_from_mvspectrum`

returns either a matrix of all univariate spectra,
or one single column (if `which`

is specified.)

`spectrum_of_linear_combination`

returns a vector with length equal to
the number of rows of `mvspectrum.output`

.

See References in `spectrum`

, `SDF`

,
`mvspec`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | ```
set.seed(1)
XX <- cbind(rnorm(100), arima.sim(n = 100, list(ar = 0.9)))
ss3d <- mvspectrum(XX)
dim(ss3d)
ss3d[2,,] # at omega_1; in general complex-valued, but Hermitian
identical(ss3d[2,,], Conj(t(ss3d[2,,]))) # is Hermitian
ss <- mvspectrum(XX[, 1], smoothing = TRUE)
## Not run:
mvspectrum(XX, normalize = TRUE)
## End(Not run)
ss <- mvspectrum(whiten(XX)$U, normalize = TRUE)
xx <- scale(rnorm(100), center = TRUE, scale = FALSE)
var(xx)
sum(mvspectrum(xx, normalize = FALSE, method = "direct")) * 2
sum(mvspectrum(xx, normalize = FALSE, method = "wosa")) * 2
xx <- scale(rnorm(100), center = TRUE, scale = FALSE)
ss <- mvspectrum(xx)
ss.n <- normalize_mvspectrum(ss)
sum(ss.n)
# multivariate
UU <- whiten(matrix(rnorm(40), ncol = 2))$U
S.U <- mvspectrum(UU, method = "wosa")
mvspectrum2wcov(normalize_mvspectrum(S.U))
XX <- matrix(rnorm(1000), ncol = 2)
SS <- mvspectrum(XX, "direct")
ss1 <- mvspectrum(XX[, 1], "direct")
SS.1 <- get_spectrum_from_mvspectrum(SS, 1)
plot.default(ss1, SS.1)
abline(0, 1, lty = 2, col = "blue")
XX <- matrix(arima.sim(n = 1000, list(ar = 0.9)), ncol = 4)
beta.tmp <- rbind(1, -1, 2, 0)
yy <- XX %*% beta.tmp
SS <- mvspectrum(XX, "wosa")
ss.yy.comb <- spectrum_of_linear_combination(SS, beta.tmp)
ss.yy <- mvspectrum(yy, "wosa")
plot(ss.yy, log = TRUE) # using plot.mvspectrum()
lines(ss.yy.comb, col = "red", lty = 1, lwd = 2)
``` |

ForeCA documentation built on May 29, 2017, 9:09 a.m.

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