3.5.4.  Two simple averaging models for decomposition.
x     = [xi, . . . , x„]        . . .     uniform sample vector of the time series
x = {xt\tez}
e     = [ei, . . . , e„]        . . .     random errors
t      G {1,2,..., n} We are assuming both models:
-  additive:              Xt     =    Trt + Szt + Ct + Et     (AM)
or
-  multiplicative:    Xt     =    TrtSztCtEt                (MM).
Let d be the period length of Szt, and n = rad.   We introduce a matrix m x d
X :
X\             ...        X]^             ...        X d
Xd+1        ■ ■ ■        Xd+k        ■ ■ ■        X2d
lxJ,k\
where xhu = xy_i^d+k (e-g- j =yeari &=month). Let us denote as vec the operator rearranging columns of a matrix downwards into one column vector.   Then we can write x = vec(XT). MATLAB: X = reshape(a;, d, m).'
The j-row X(j,:) stands for samples from j-th cycle of Szt including Trt, C t and Et.
The k-th column X(:,k) gathers samples which are at k-th position within each cycle, again including all components TVt,Ct and Et.
Let E = [ehk\ be the matrix of errors defined alike.
The seasonal component Szt is fully determined by its values in one cycle s := (si, . . . , Sd) '■= (Szi, . . . , Szd) with mean š
si + - + sd d
satisfying:
T   n               ,7      A A/T
(3.5.2)
0	v	AM
1	v 16	MM
We are going to find estimates sk fulfilling (3.5.2).
3.5.4.1. The small trend method [MATLAB: szsmtr]
Assumptions:
Trt + Ct, or   TrtCt should be approximately constant within each
cycle of Szť-
AM:     xj,k - Sk - eJtk     =    Trt + Ct     ~     ..«j    ,   .      ...
v  in j-th pe-
MM:     xJik/(skeJik)         =    TrtCt
riod, t = (j — l)d + k.
Algorithm:
(1)   ŕhj = i X)fc=i Xj,k ■ ■ ■ estimate of rrij for j' = 1
(2)   for k = 1, . . . , d we compute:
AM:       sk     =     ^Y.J=i{x3,k-m3)
MM:
sk
sk + ej,k
Em                / ^
, = 1 x],k/m]
1      V-VÍTI
It is easy to see that these estimates fulfil (3.5.2 AM:     f     =     iEti(iEľ=iK*-^)
1      V-VÍTl
i                  i
MM:
i      /
1    1
— /   ■   i t;------/    Xi k = — ra = 1.
To be continued with the steps of 3.5.4.3.
17
3.5.4.2. Moving average method [MATLAB: szma]
We introduce vector w = (w-q, . . . , wq)    of equal weights for the common moving average filter:
w     =     1(1,1,..., 1)T                 for odd       d = 2q + l,
(2q+l)x
w     =     i(l/2,l,...,l,l/2)T     for even      d = 2q.
-------------------v------------------'
(2q+l)x
We start with the moving average operation:
rat = <    —'—i
YJqr=-qwr^t+r     for     g + l<r<n-g,
Xt                            for     1 < t < g or n - g + 1 < ŕ < n.
By the construction of weights, exactly the whole seasonal cycle is being averaged at each position of vector w. In addition to suppressing the noise component et, this averaging eliminates the seasonal component Szt as well due to its zero/unit mean in the AM/MM layout.
That is why the the following assumptions are justified: Assumptions:
AM:     xt — Szt — et     =    Trt + Ct     x     "*'lf       ^.t <-ř <~ MM:     xtI{Szt et)         =    TrtCt         w     mt   }        q+    ~    ~
n — q.
Algorithm:
(1) After having the vector of values smoothed by the moving average operation we rearrange it into a matrix of size mxd with cycles row-by-row, following the analogy with x: m = (rôi, . . . , ífi„) =: vec(M ), where M =: [ŕnj,t]. Then X - M in AM, or X./M = [xj,k/ŕn,j,k] in MM, have the structure shown below, where there are q zeros, or ones at the beginning of the first, and at the end of the last cycle.
For odd d:
X -M
X./M For even d: X -M
X./M
0      •     •
•      •     0
1      •     •
•      •     1
0      •     ..
•     o     ..
1      •     ..
•      1     ..
(2)  For k = 1, . . . , d is computed:
1 ■EľliOEj,*-™j,*)i
in AM:       sk
in MM:      šk
m-Sk   Ĺ^J =
1        [YV^7™
sk + ej,k
m-Sk lAZ-íj
[(Eľll Xj,k/mjlk) -6k].

where Sk
0     for k = ^±i (with odd d and jfc = g + 1)
1     otherwise
(3)   s=  lYfk=lSk-
(4)   We have to center the estimates to assure the validity of (3.5.2):
Sk = ~šk — ~š in AM, or   s"*; = Ik/Is in MM for k = 1, . . . , d. Again we continue with the steps of 3.5.4.3.
3.5.4.3.  The separation of Trt and Szt
(5)   Periodization: Szu+jd = Sk for k = I,..., d and j = 0, 1, . . . , ra — 1.
(6)   Separating the seasonal component Szt'.
yt := xt — Szt « Trt + Ct + et in AM, or yt := xt/Szt « Trt Ct et in MM for t = 1, . . . , n.
(7)   In yt the component Trt + Ct, or TrtCt of yt (possibly after logarithmic transformation in MM) is estimated using appropriate method (e.g. that of section 3.5.1 or 3.5.2). Just only trend may be estimated putting q = 0 in 3.5.1 or using smoothing techniques described in next sections, and then individually separate the cyclic component Ct'.
(8)   Separating the cyclic component Ct'.
Ct + et fa yt - Trt in AM, or Ct et fa yt/frt in MM.
(9)   Analyzing periodic components of Ct + et
or log-transformed Ct et in MM, using periodogram and periodicity tests of 3.4.3. Now these tests have chance to be sensitive to harmonics of the weak cyclic component, which is no more dominated by the presence the other components. Step (7) may be then repeated to obtain final decomposition of Trt and C t with improved model for Ct.
3.5.4.4.  Eliminating Szt by differencing at lag d (see 3.5.3(1) ).
Assumptions:
Xt = Szt + rat + et, where nit = Trt + Ct, and the period length of
C t is significantly longer that that of Szt-
Algorithm:
(1)   yt := xt - xt-d = Szt - Szt-d + mt — mt-d + et - et-d,
t = d + 1, . . . , n.
(2)   We estimate rat as of 3.5.4.3(7)-(9) where, of course, rat = nit — nit-d are to be interpreted like seasonal fluctuations.