TimeSeriesBook.pdf

Setting r 1 and θ π gives the following famous

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Setting r = 1 and θ = π , gives the following famous formula: e ıπ + 1 = (cos π + ı sin π ) + 1 = - 1 + 1 = 0 .
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369 This formula relates the most famous numbers in mathematics. From the definition of complex numbers in polar coordinates, we imme- diately derive the following implications: cos θ = e ıθ + e - ıθ 2 = a r , sin θ = e ıθ - e - ıθ 2 ı = b r . Further implications are de Moivre’s formula and Pythagoras’ theorem (see Figure A.1): de Moivre’s formula ( re ıθ ) n = r n e ınθ = r n (cos + ı sin ) Pythagoras’ theorem 1 = e ıθ e - ıθ = (cos θ + ı sin θ )(cos θ - ı sin θ ) = cos 2 θ + sin 2 θ From Pythagoras’ theorem it follows that r 2 = a 2 + b 2 . The representation in polar coordinates allows to derive many trigonometric formulas. Consider the polynomial Φ( z ) = φ 0 - φ 1 z - φ 2 z 2 - . . . - φ p z p of order p 1 with φ 0 = 1. 3 The fundamental theorem of algebra then states that every polynomial of order p 1 has exactly p roots in the field of complex numbers. Thus, the field of complex numbers is algebraically complete. Denote these roots by λ 1 , . . . , λ p , allowing that some roots may appear several times. The polynomial can then be factorized as Φ( z ) = ( 1 - λ - 1 1 z ) ( 1 - λ - 1 2 z ) . . . ( 1 - λ - 1 p z ) . This expression is well-defined because the assumption of a nonzero constant ( φ 0 = 1 6 = 0) excludes the possibility of roots equal to zero. If we assume that the coefficients φ j , j = 0 , . . . , p , are real numbers, the complex roots appear in conjugate pairs. Thus if z = a + ıb , b 6 = 0, is a root then ¯ z = a - ıb is also a root. 3 The notation with “ - φ j z j ” instead of “ φ j z j ” was chosen to conform to the notation of AR-models.
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370 APPENDIX A. COMPLEX NUMBERS
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Appendix B Linear Difference Equations Linear difference equations play an important role in time series analysis. We therefore summarize the most important results. 1 Consider the following linear difference equation of order p with constant coefficients. This equation is defined by the recursion: X t = φ 1 X t - 1 + . . . + φ p X t - p , φ p 6 = 0 , t Z . Thereby { X t } represents a sequence of real numbers and φ 1 , . . . , φ p are p constant coefficients. The above difference equation is called homogeneous because it involves no other variable than X t . A solution to this equation is a function F : Z R such that its values F ( t ) or F t reduce the difference equation to an identity. It is easy to see that if { X (1) t } and { X (2) t } are two solutions than { c 1 X (1) t + c 2 X (2) t } , for any c 1 , c 2 R , is also a solution. The set of solutions is therefore a linear space (vector space). Definition B.1. A set of solutions {{ X (1) t } , . . . , { X ( m ) t }} , m p , is called linearly independent if c 1 X (1) t + . . . + c m X ( m ) t = 0 , for t = 0 , 1 , . . . , p - 1 implies that c 1 = . . . = c m = 0 . Otherwise we call the set linearly dependent . Given arbitrary starting values x 0 , . . . , x p - 1 for X 0 , . . . , X p - 1 , the differ- ence equation determines all further through the recursion: X t = φ 1 X t - 1 + . . . + φ p X t - p t = p, p + 1 , . . . .
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