Standardizes (multivariate) time series by their median, MAD and transforms the standardized time series by a \(\psi\) function.
psi(y, fun = c("HLm", "HLg", "SLm", "SLg", "HCm", "HCg", "SCm", "SCg"), k,
constant = 1.4826)Transformed numeric vector or matrix with the same number of rows as y.
vector or matrix with each column representing a time series (numeric).
character string specifying the transformation function \(\psi\) (more in Details). If fun = "none", no transformation is performed.
numeric bound used for Huber type psi-functions which determines robustness and efficiency of the test. Default for psi = "HLg" or "HCg" is sqrt(qchisq(0.8, df = m) where m are the number of time series, and otherwise it is 1.5.
scale factor of the MAD.
Sheila Görz
Let \(x = \frac{y - med(y)}{MAD(y)}\) be the standardized values of a univariate time series.
Available \(\psi\) functions are:
marginal Huber for location:
fun = "HLm".
\(\psi_{HLm}(x) = k * 1_{\{x > k\}} + x * 1_{\{-k \le x \le k\}} - k * 1_{\{x < -k\}}\).
global Huber for location:
fun = "HLg".
\(\psi_{HLg}(x) = x * 1_{\{0 < |x| \le k\}} + \frac{k x}{|x|} * 1_{\{|x| > k\}}\).
marginal sign for location:
fun = "SLm".
\(\psi_{SLm}(x_i) = sign(x_i)\).
global sign for location:
fun = "SLg".
\(\psi_{SLg}(x) = x / |x| * 1_{\{|x| > 0\}}\).
marginal Huber for covariance:
fun = "HCm".
\(\psi_{HCm}(x) = \psi_{HLm}(x) \psi_{HLm}(x)^T\).
global Huber for covariance:
fun = "HCg".
\(\psi_{HCg}(x) = \psi_{HLg}(x) \psi_{HLg}(x)^T\).
marginal sign covariance:
fun = "SCm".
\(\psi_{SCm}(x) = \psi_{SLm}(x) \psi_{SLm}(x)^T\).
gloabl sign covariance:
fun = "SCg".
\(\psi_{SCg}(x) = \psi_{SCg}(x) \psi_{SCg}(x)^T\).
Note that for all covariances only the upper diagonal is used and turned into a vector. In case of the marginal sign covariance, the main diagonal is also left out. For the global sign covariance matrix the last element of the main diagonal is left out.
psi_cumsum,
CUSUM