Estimates the derivatives of a time series using generalized local linear approximation (GLLA). GLLA is a filtering method for estimating derivatives from data that uses time delay embedding and a variant of Savitzky-Golay filtering to accomplish the task.

## Arguments

- x
Numeric vector. An observed time series

- n.embed
Numeric (length = 1). Number of embedded dimensions (the number of observations to be used in the

`Embed`

function)- tau
Numeric (length = 1). Number of observations to offset successive embeddings in the

`Embed`

function. A`tau`

of one uses adjacent observations. Default is`1`

- delta
Numeric (length = 1). The time between successive observations in the time series. Default is

`1`

- order
Numeric (length = 1). The maximum order of the derivative to be estimated. For example,

`"order = 2"`

will return a matrix with three columns with the estimates of the observed scores and the first and second derivative for each row of the embedded matrix (i.e. the reorganization of the time series implemented via the`Embed`

function)

## Value

Returns a matrix containing *n* columns in which *n*

is one plus the maximum order of the derivatives to be estimated via generalized local linear approximation

## References

**GLLA implementation**

Boker, S. M., Deboeck, P. R., Edler, C., & Keel, P. K. (2010)
Generalized local linear approximation of derivatives from time series. In S.-M. Chow, E. Ferrer, & F. Hsieh (Eds.),
*The Notre Dame series on quantitative methodology. Statistical methods for modeling human dynamics: An interdisciplinary dialogue*,
(p. 161-178). *Routledge/Taylor & Francis Group*.

Deboeck, P. R., Montpetit, M. A., Bergeman, C. S., & Boker, S. M. (2009)
Using derivative estimates to describe intraindividual variability at multiple time scales.
*Psychological Methods*, *14(4)*, 367-386.

**Filtering procedure**

Savitzky, A., & Golay, M. J. (1964).
Smoothing and differentiation of data by simplified least squares procedures.
*Analytical Chemistry*, *36(8)*, 1627-1639.