Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

ISSN1842-6298 Volume1(2006), 1 – 12

MODELING SEASONAL TIME SERIES

Alexandra Colojoar¼a

Abstract. The paper studies the seasonal time series as elements of a (…nite dimensional) Hilbert space and proves that it is always better to consider a trend together with a seasonal component even the time series seams not to has one. We give a formula that determines the seasonal component in function of the considered trend that permits to compare the di¤erent kind of trends.

1

Preliminary notions

In the following we will considerSTn , thevector space of n-time series (the space

Rn_{endowed with the canonical structure of}R_{-vectorial space) ; an element ofSTn}

will be denoted by

X_=fxig

n=fx1;:::;xng;

and then

X₊Y _{= fxig+fyig:=fxi+yig;} X _{= fxig:=f xig:}

Particularly, each real number kde…ne a constant time series:

K_{=fkgn=fk; :::; kg}

De…nition 1. a) If X _and Y _{are two time series, we de…ne the product by:} X Y_=fxig

nfyign:=fxiyign; and their inner product by:

hX_jY_i:= 1 n

n

X

i=1

xiyi:

The spaceSTn of a n-time series endowed with this scalar product is a Hilbert space .

2000 Mathematics Subject Classi…cation: 62M10 Keywords: seasonal time series, model, regressor.

b) We associate to a given time seriesX_=fxig

(we observe that scalar product between two time series is the average of their prod-uct),

-its canonical norm (de…ned by the scalar product)

kX_k=p_hX_jX_i:=pX2₌ 1

-its centred time series:

nX_:=X X_;

-its variance:

V arX_:=knX_k2 _=kX_k2 X 2 _=kX_k2 _hX_j1_i2_:

c) Let X _and Y _{be are two time series, we de…ne their covariance by:}

Cov(X_;Y_{) :=}X Y X Y_=hX_jY_{i h}X_j1_{i h}Y_j1_i=hX_jY_{i h}X_{j h}Y_j1_ii=hX_jnY_i;

We observe that the covariance is a bilinear form and veri…es the following conditions:

Cov(X_;Y_{) = h}nX_jY_i=nX Y_; Cov(X_;nY_{) = Cov(}X_;Y_);

Cov(X_;X_{) = V ar}X_;

V ar(X Y_{) = V ar}X_{+V ar}Y _2Cov(X_;Y_):

De…nition 2. Ifn=ks; we will callsthe periodandkthe numbers of periods.The time series X _{will be called a periodical time series if xi = xs+i = x2s+i = ::: =}

x(k 1)s+i for everyi= 1; :::; s;that means that a periodical time series is of the form:

C_{=fc1; :::; csg}

n:=fc1; :::; cs; c1; :::; cs; ::: ; c1; :::; csg: The mean and the norm of such a series are:

C₌ 1

De…nition 3. Suppose that n = ks: At at a given time series X _{2 STn we will}

associate three time series :

a) its periodical time series:

cX_:=fc1(X_{); :::; cs(}X_)g

n=fc1(X); :::; cs(X); :::; c1(X); :::; cs(X)g 2 STn;

where

ci(X) :=

1

k xi+xs+i+x2s+i+:::+x(k 1)s+i ;

We observe that, generally,c₍X_{)is not a seasonal time series ( because}c₍X_{) =} X_),

b) its seasonal time series: the normed time series n₍cX_{) of its periodical time}

series,

c) its deseasoned time series:

sX_:=X cX_:

Proposition 4. Suppose thatn=ks; and consider a time series X _{2 STn: Then:}

a) its periodical time series cX_{has the following properties:}

c₍X_{) =}X_; c₍nX_{) =}n₍cX_);

kcnX_k2 _{=V ar(}cX_{) =k}cX_k2 X 2_; c₍C_{) =}C_;

X cY₌cX Y₌cX cY_; X cX₌cX cX_=kcX_k2_;

Cov(X_;cX_{) =} X cX X cX _{= k}cX_k2 cX 2 _{=V ar(}cX_):

b) its deseasoned time series sX_{has the following properties:}

sX_{= 0;} n₍sX_{) =}sX_;

ksX_k2 _{= V ar}X Var₍cX_{) = k}X_k2 _kcX_k2_;

sC₌0_; s₍nX_{) =}n₍sX_{) =}sX_;

sX sY₌X Y cX cY _{= Cov(}X_;Y_{) Cov(}cX_;cY_):

Lemma 5. Suppose thatn=ks: A time seriesX _{2 STn is a periodical (seasonal)}

2

The problem

In this paragraph we will considerF =fF_{1; :::;}F_{kga …nite system of time series and}

the constant time series 1 _{as elements of the Hilbert spaceSTn, then a time series}

from vF the subspace generated byf1_;F

1; :::;Fkgwill be of the form 0+ 1F1+

:::+ kFk ( i2R):

At each time series X_{and each …nite system of vectors F we will associate:}

FX , the a¢ne space generated by nF = fnF₁; :::;nFkg and the constant series

X_:

FX=fX+ ₁nF₁+:::+ _knFkj _i 2Rg:

, the colon matrix de…ned by the coe¢cients of a time series fromFX:

= ( 1::: k)t

Cov(F;F);the matrix of which elements are Cov(F_i;F_{j);and Cov(}X_{;F) the}

colon matrix of which elements areCov(X_;F_i).

Lemma 6. Let X _{be a time series, F =f}F_{1; :::;}F_{kg a …nite system of time}

se-ries and Y_{= 0+ 1}F_{1 +:::+ k}F_{k a time series from the subspace generated by}

f1_;F_{1; :::;}F_kg.

a) The following assertions are equivalent:

a1) the time seriesX Y _{is orthogonal on} 1_,

a2) 0 =X Pk₁ iFi,

a3) Y _{is of the form:} Y₌X_{+ 1}nF_{1+:::+ k}nF_{k ( i.e.} Y _{is an element of}

the a¢ne space FX),

b) ([1]) The Time series X Y _{is orthogonal on the system F if and only if} = ( 1::: k)t veri…es the system:

~ Cov(X_{;F) =Cov(F;F) :}

Proof. a) From the orthogonality of X Y _on 1_:

0 =hX Y_j1_i=X Y ₌X ₀

k

X

1 iFi;

it results that

0 =X

k

X

and thenY _{is of the form:}

Y₌X₊

k

X

1

inFi:

Obviously, ifY _{is of the last form we have}

hX Y_j1_i= *

nX₊

k X

1

inFij1 +

= 0.

Corollary 7. Xb_{F, the projection of the time series} X _{onto the subspace generated}

byf1_;F_{1; :::;}F_{kg;coincides with the projection of} X_{onto F}X and it is of the form b

XF =X+ 1nF1+:::+ knFk;

where = ( 1::: k)t is the solution of the system~ (Lemma 6.b).

Proof. The time seriesX Xb_{F is orthogonal on}1_;F_{1; :::;}F_{k([4], V1 A), then where}

i are solutions of the system ~:

De…nition 8. a) FX (the a¢ne space generated by nF and the constant series X ) will be called a class of models forX: Their elements

F₌X_{+ 1}nF_{1+:::+ k}nF_{k= 0+ 1}F_{1+:::+ k}F_{k ; 0=}X

k

X

1 iFi;

will called a models ofX;and the time series"F =X Fwill called the error of the model F_{; the squared of his norm}

k"Fk2=

1

n(xi fi)

2 _=d2_(X;F);

being the mean squared errors for the modelF_{and will measure the accuracy}

of the model.

b) XbF;the projection ofXontoFX, (or the the regressor ofXontoFX, [2] ),will be called best modelof X_{from the class of modelsF}X, and its coe¢cients will be called the best coe¢cients of the decomposition:

X₌F_+"F; F2FX:

The corresponding error will be denoted byb"F:

Remark 9. The norm of_b"F is the distance from X to the a¢ne space FX (in the Hilbert space of the n-time series).

k_b"Fk= min F2F_Xk"

Fk= min F2F_Xk

X F_k=d(X_;FX)

Proposition 10. Let X _{be a time series,F}X a class of models for X, and let Fbe a model of this class:

X₌F_+"F:

Then:

a) For every F_2FX, the norm of the error (the mean squared errors) of the decom-position

X₌F_+"F:

veri…es:

k"Fk2 =M SE=

1 n

n

X

i=1

"2i =V arX+V arF 2Cov(X;F);

b) The minimum of all these norms (minimum mean squared errors) veri…es: k_b"Fk2 = X XbF

2

= minfk"Fk2jF2 FXg= minM SE =V arX V arXbF:

Proof. a) BecauseX₌F_:

k"Fk2=kX Fk2 =V ar(X F) =V arX+V arF 2Cov(X;F):

b) Because X XbF is orthogonal on F ([4], [3]), particularly on XbF, and then

0 = X Xb_F Xb_{F =}XXb_F Xb_F 2_{, it results that Cov(}X_;Xb_{F) =V ar}Xb_{F, and}

then

k_b"F(X)k2 =V arX V arXbF:

3

The three cases

In the following we will studies the class of the models that contain only a seasonal components C _{, that contain only a trend} nU _{and that contain a trend and a}

seasonal componentC₊bnU_:

Using9 we will …nd Xb_{F , its coe¢cients, and the norm of the associated errors for}

all the three models considered above .

In all the cases the idea is the same: we have to …nd the minimum of the function k"Fk2 constrained by the fact thatC= 0 ; that is to consider the function :

(b_;C_{; ) =k"}Fk2 C;

and solve the associate system

F

Theorem 11. The best coe¢cients, the regressors and the minimum squared errors of the three following decompositions are:

Proof. a)Because"=X X₊C ₌Y C_(whereY₌nX₌X X_{), the function}

and the systemF becomes

F

From the …rst equation we obtain

b

b) Because Y₌nX₌bnU_{+"; it results that the function to be considered is}

=kY bnU_k2_=kY_k2₊b2_{V ar}U 2b_Cov(U_;Y_);

and the systemF has a single equation

F 2b_{V ar}U 2_Cov(U_;Y_{) =0:}

U _{being non-constant, V ar}U_{6=0;and then}

b

b_U=Cov(U;Y) V arU =

Cov(U_;X₎ V arU :

Evidently, the regressor of X _is

b

and then

From the …rst equation we obtain

We used the fact thatU_{is nonseasonal and thenV ar(}sU_{) =V ar(}sU_{) V ar(}sU_):

Evidently, the regressor of X _is

b

X_U+C=X₊bb_U+CnU₊Cb_U+C=bb_U+CsU₊cX_;

and then

k_b"U+Ck2 = V arX V arcX bbU+C

2

V arsU₌

= V arX bb_U+C 2_{V ar}sU_{=V ar}sX (Cov(sU;sX))

2

Var₍sU₎ :

4

Comparisons

In this section we will determine the di¤erence between:

- the regressions of a seasonal time series, if we consider or not a trend, - the regression of a seasonal time series with a trend and the model obtained by the Cesus decomposition.

Theorem 12. For a given time series X_{; a regression to a model that considers a}

trend U _{and a seasonal component} C _{is better that the one that considers only a}

seasonal component; that means that the error given by the decomposition

X₌X₊C_+";

is always greater than that given by

X₌X₊bnU₊C_+":

The two errors are equal if and only if

Cov(sU_;sX_{) =0;}

or equivalent, if and only if Cov(U_;X_{) =Cov(}cU_;cX_{), or} UX₌cUcX_.

Proof. Using Theorem11 we obtain:

k_b"U₊C(X)k2 =V arsX

(Cov₍sU_;sX₎₎2

Var₍sU₎ V arsX=kb"C(X)k

2_;

and evidentlyk_b"U+C(X)k=kb"C(X)kif and only if Cov(sU;sX) =Cov(U;X) Cov(cU;cX) =0:

or equivalent their elements are on the unique form: X₊bnU₊C_{) : becauseC}X is a a¢ne subspace ofUX+CX;the distance of XtoCX is greater than the distance of

X_toUX+CX;and then:

k_b"U₊C(X)k=d(X;UX+CX) d(X;CX) =kb"C(X)k:

Remark 13. Let’s now consider a time seriesX_{: Its seasonal component}C0

(as is used in Census decomposition) is exactly the best seasonal components (as it results from 3.1.a)):

C0₌ncX₌CbC(X);

and his deseasoned time series will be Z₌X C0

=X ncX₌sX₊ X_:The

re-gressor ZbU = Z+bU(Z)U of Z to the subspace UZ gives us a model for X from UX+CX:

X0 ₌Zb_U+Cb_C=Z₊b_U(Z₎nU₊ncX₌X₊b_U(Z₎nU₊ncX_{2 U}X+CX. Evidently, then the norm of the is greater than the norm of _b"U+C =X XbU+C (the

error associate at the best model):

k_b"U₊C(X)k= min F2U+C

k"Fk k"X0k:

In the following proposition we will calculate the norm obtained by the Cen-sus decomposition (the mean squared error obtained by the CenCen-sus method) and determine the case when it coincide with the norm of the error obtained by the regression.

Proposition 14. In the conditions of 13 - the mean of the squared error of the model obtained by the Census model is

k"X0k2 =V arsX (Cov(

sU_;sX₎₎2

V ar(U₎ ,

it is evidently greater that the error of the regressor ofX _{to the a¢ne space U}X+CX obtained in the Proposition 3.1.c). The norm of these errors are equals if and only if the periodical time series cU _{associate at the trend} U _{is a constant time series.}

Proof. First, we observe that Z₌X C0₌X ncX₌sX ₊ X _{and then} Z₌X_:

The coe¢cient of the regressor of Z_{to the a¢ne spaceU}X becomes:

bU(Z) =

Cov(U_;Z₎ V ar(U₎ =

Cov(U_;sX₊ X₎ V ar(U₎ =

The model

X0 ₌ZbU+C0=X+bU(Z)nU+ncX2 UX+CX.

being an element of the subspaceU +Cthe norm of his error"0=X X0_{is evidently}

greater than that of the regressor ofX _{to the subspaceU +C (Proposition 3.1.c):}

k_b"U+Ck2 = X XbU+C

2

=V arsX (Cov(sU;sX))

2

V ar(sU₎ :

Evidently the two errors coincide if and only ifV ar(U_{) =V ar(}sU_{);that means if}

and only if V ar(cU_{) = 0:}

References

[1] G.E.P. Box, G.C. Jenkins and G.M.Reinsel, Time Series Analysis. Forecast-ing and Control. Third edition, Prentice Hall, Inc., Englewood Cli¤s, NJ, 1994. MR1312604(95m:62191).Zbl 0858.62072.

[2] P.J. Brockwell and R.A. Davis, Introduction to time series and forecasting. Second edition, Springer Texts in Statistics, Springer-Verlag, New York, 2002. MR1894099(2002m:62002).Zbl 0994.62085.

[3] C. Chat…eld, The Analysis of Time Series. An Introduction. Fifth edi-tion, Texts in Statistical Science Series. Chapman & Hall, London, 1996. MR1410749(97e:62117).Zbl 0870.62068.

[4] C. Gourieux et A. Monfort,Series Temporelles et modeles Dynamiques, Econom-ica, Paris, 1990.

University of Bucharest,

Bd. Regina Elisabeta, Nr. 4-12, Bucharest, Romania.