Additional Functions

Useful or Helper Functions

Here are functions that have a usecase in other areas that fall not strictlly into any offical categories for trend estimation.

TrendDecomposition.greatestConvexMinorant — Function

greatestConvexMinorant(y :: Vector)

Computes greatest convex minorant of series y using the pool-adjecent-violators algorithm

Returns the coordinates of the knots as a tuple (x, y)

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TrendDecomposition.leastConcaveMajorant — Method

leastConcaveMajorant(y :: Vector)

Computes least concave majorant (lcm) of series y using the pool-adjecent-violators algorithm

Returns the coordinates of the knots as a tuple (x, y)

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Autoregression AR(p) Estimation

TrendDecomposition.arBurg — Function

arBurg(y :: Vector, p :: Int; intercept::Bool = false)

Fits an autoregressive model of order p to time series y using Burg's method.

Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.

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TrendDecomposition.arOLS — Function

arOLS(y :: Vector, p :: Int; intercept::Bool = false)

Fits an autoregressive model of order p to time series y using ordinary least square (OLS).

Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.

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TrendDecomposition.arYuleWalker — Function

arYuleWalker(y :: Vector, p :: Int)

Fits an autoregressive model of order p to time series y using the Yule-Walker estimators.

Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.

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TrendDecomposition.arDurbinLevinson — Function

arDurbinLevinson(y :: Vector, p :: Int;
                      intercept::Bool = false, pacf::Bool = false)

Fits an autoregressive model of order p to time series y using the Durbin-Levinson recursion.

Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.

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TrendDecomposition.autoCovariance — Function

autoCovariance(X :: Vector, T::Int;
             maxLag::Int=T-1, method::Symbol=:classic, covar::Bool=false, demean::Bool=false)

Calculates the autocovariance of time series X with length T, by default it is assumed that the data is already centered (demean=false). For large series the method :fourier can be choosen, which uses the fast fourier transform.

Returns a (maxLag x 1) vector of the autocovariance. For the method :fourier with covar=true a tulpe containing the covariance sclar and autocovariance vector is returned.

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Experimental

Here are functions that are only experimental in use but could be useful for some specific purposes.

TrendDecomposition.trendL1Filter — Function

trendL1Filter(y :: Vector, λ :: Real; m = 2, max_iter=20, method = :ADMM)

Placeholder for trendL1Filter extension, when using TrendDecomposition together with other Julia packages like Convex.jl and SCS.jl.

This function provides the generic use of serveral optimization methods to compute a numerical solution. Following methods are implmented: :ADMM -> alternating direction method of multipliers :ConvexSCS -> SCS solver with Convex.jl. Prerequisite! Import necessary modules with: using Convex, SCS

The function returns the estimated trend component

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