Moving Average (MA)

Rolling Average

The name rolling average instead of moving average is choosen here, because by default the functions works like a rolling window that slides through the entire time series from $t = 1,...,T$, calculating a value for each datum and it is up to the user to deceide to have the boundary values discarded.

For centered MA the choosen order p has to be odd and thus can be written as $p = 2k + 1$ so that

\[ MA_t(p) = \frac{1}{p} \sum_{i=-k}^{k} x_{t+i}\]

With centered=false the last (p-1) lagging values are used instead, for this option the order p can be either even or odd.

\[ MA_t(p) = \frac{1}{p} \sum_{i=0}^{p-1} x_{t-i}\]

rollingAverage(y :: Vector, p :: Int; centered::Bool = true, discard::Bool = false, offset::Int = 0)

Both methods allow an offset for calculating the MA. With a positive offset an additional lead value is included at the expense of a lagged value and the reverse holds for negative offsets. Negativ offset are only allowed for centered MA, since for centered=false the maximum amount of lagged values are already used by default.

MA with weights

One can also specify weights $\omega_i$ for each i-th term, subject to that they sum to unity, e.g. for centered MA

\[ WMA_t(p) = \frac{1}{p} \sum_{i=-k}^{k} \omega_i x_{t+i} \qquad s.t \sum_{i=-k}^{k} \omega_{i} = 1.\]

For a MA of order p, the weights provided by the user have to be a vector of size (p x 1).

rollingAverage(y :: Vector, p :: Int, weights :: Vector;
    centered::Bool = true, discard::Bool = false, offset::Int = 0)

Calculation for boundary or edge values

When using a centered moving average of order p, for the first $\frac{p-1}{2}$ and the last $\frac{p-1}{2}$ values of the time series there are not enought data points available. To generalise about these boundary cases, the normal centered MA can also be thought as a special case of weighted MA with uniform weights, where each weight must equal $w_i = \frac{1}{p}$. For k missing values, the weights that still can be applied are normalized so that they sum to 1, so that each of the (p-k) remaining weights now equals $\frac{1}{p-k}$. As an example, the calculation of the first value of a time series y using a centered MA of order 5 becomes $MA_{1}(5) = \sum_{i=1}^{3} \frac{1}{3} y_i = \frac{1}{3} \sum_{i=1}^{3} y_i$.

For weighted MA with given weights $w_1,..,w_p$, in case that for the first k weights there are no data points available for the calculation, the remaining weights are normalized as follows $w'_i = \frac{w_i}{\sum_{j=k+1}^{p}w_j}$. Correspondingly, in case the last k weights cannot be used, the remaining $p-k$ weights are normalized: $w'_i = \frac{w_i}{\sum_{j=1}^{p-k}w_j}$

As it is common to drop data for which the full order of the specified MA cannot be computed, discard=true will return NaN^[1] for each datum in the output vector, where not enough data points are available due to boundary cases.

Seasonal Average

For each season component $S_i$ this function calculates the average for all observation that where recorded in the same type of season given $1,2,...s$ types.

Since the formal mathematical notation requires the use of sets, the interested reader can see the formula for calculating the seasonal averages in the notes down below^[2].

maSeason(y :: Vector, seasons :: Int; repeating::Bool = false)

Classical decomposition

Given the implementations of maSeason and rollingAverage, the following two decompositions using moving averages can be archived:

The additive model

\[ y_t = \tau_t + S_t + u_t\]

or the multiplicative model

\[ y_t = \tau_t * S_t * u_t,\]

where $\tau_t$ is the trend component estimated using moving averages, $S_t$ is the average season component estimated using the detrended time series and $u_t$ is simply the residual factor defined as $u_t = y_t - \tau_t - S_t$.

maDecompose(y :: Vector, seasons :: Int; combine::Bool = false, model=:add)