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The Discrete Fourier Transform
Leigh J. Halliwell, FCAS, MAAA
[email protected]
Brian Fannin, ACAS, MAAA
[email protected]
CASE Spring Meeting
Middle Tennessee State University
March 22, 2016
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Outline
I.
Complex Numbers
II. Roots-of-Unity Random Variables
III. Collective-Risk Model
IV. Examples
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I. Complex Numbers
Basic Arithmetic
ii  1
z  x  iy
z  x  iy
zz   x  iy  x  iy 
 xx  xiy  iyx  iyiy
 xx  yy
Complex Plane versus Real Line
Wessel(1979)-Argand(1806) diagram [Kramer, 73]
No trichotomy between complex numbers
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I. Complex Numbers (Cont’d)
Completeness of Complex Numbers
1 i
jj  i  j  
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Fundamental Theorem of Algebra [Kramer, 71]
Actuaries should examine the meaning of their formulas with
complex numbers. Or are they just “imaginary” numbers?
Conjugate Theorems
uv uv
u v  u v
f  z    ak z k  f  z    ak z k
k


k
ak   ak  ak  f  z    ak z k  f  z 
k
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I. Complex Numbers (Cont’d)
The exponential function

ez  
k 0
zk
k!

ez  
k 0
zk
 ez
k!
Absolutely convergent, since k! outgrows zk.
k
l
k l






u
v
u
v
u v




e e  
 




k!l!
 k 0 k!  l 0 l! 

n

u k v nk
1 n
n!
 
 
u k v nk
n  0 k  0 k!n  k !
n  0 n! k  0 k!n  k !
n

n
  k nk

1
u  v
    u v

 eu v
n!
n  0 n! k  0  k 
n 0

n
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I. Complex Numbers (Cont’d)
The real exponential
e :    is 1-to-1 (increasing)
every unit is e more than before. “Real” exponential growth
The wheat and chess problem
What is “imaginary” exponential growth?
ez 
ez ez 
ezez 
e zz 
e x x 
e xe x  e x
The distance of ez from O is invariant to the imaginary part of z
e iy  e 0iy  e 0  1
We’re almost to Euler’s famous formula (see next)
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I. Complex Numbers (Cont’d)
f t   e it  f t   i  e it  i  f t 
Velocity here is proportional to position
But multiplication by i is counterclockwise rotation by 90°
So f(t) is circular motion; must repeat every 2π radians
“Will it Go Round in Circles?” Billy Preston 1972
Euler’s Formula:
e i  2  1
e i  cos   i sin 
“Imaginary” growth is circular, with period 2πi
The basis of polar coordinates:
x  iy  rei
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I. Complex Numbers (Cont’d)
Never again need you forget your trig identities!
cos     i sin      e i     e i e i
 cos   i sin  cos   i sin  
 cos  cos   sin  sin  
 i cos  sin   sin  cos  
Physical or Actuarial Application??
As
Varθ   , e iθ becomes uniform circular
Infinite variance can be “domesticated” exponentially
Integral powers of
e i1 are dense but unordered on Κ – chaos
I. Complex Numbers (Cont’d)
What is legitimate exponentiation, or zs? Use logs:
z  0s  e s ln z
 e s ln z i2 
 e s ln z e i2s
  z  0   e i2s
s
Uniquely defined only for integral values of s.
like a double-slit experiment that recombines before we can see it
real exponential allows for well defined (x>0)y, but not extendable to C.
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I. Complex Numbers (Cont’d)
The Circle Group (multiplicative):
Κ  z  C : z  zz  1
closed under multiplication
a) associative
b) identity element
c) inverse:
z 1  z
n
The “nth Roots of Unity” Group Ωn  z  C : z  1  Κ
primitive root ω
n distinct numbers
1   0 ,  1  e
i
2
n
, ,  n 1 ,  n  1
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I. Complex Numbers (Cont’d)
Excel Graphs of Ωn
Findings
nth roots of unity sum to zero for n > 1
The kth powers of Ωn equal Ωm , where m = n/GCF(k, n)
e.g., squares of tenth roots are the fifth roots of unity
e.g., outer join of second and third roots = sixth roots
e.g., outer join of second and fourth roots = fourth roots
e.g., if n is prime, kth powers of Ωn equal Ωn for 0 < k < n
Ωn groups are useful in number theory (prime factorization)
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II. Roots-of-Unity Random Variables
The Ωn RV:


Prob Z   k  pk

k  0,1,  , n  1 n  1

Because Prob Z n  1  1, only n distinct moments:
1
1  1 1 
 Z  1    k

 
  

E j   
j
jk
Z
1



 
  

 n 1  
n 1
 n 1k
 Z  1 
m   nn  p


  p0 
 n 1   p1 
  

j  n 1  

  pk 
  

 n 1 n 1  

  pn 1 
1
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II. Roots-of-Unity Random Variables (Cont’d)
Ω is the (nxn) discrete Fourier transform
jk
mod jk , n 
symmetric:
invertible:
jk    
1
  n
     
n 1
1
jl
jk
l 0
lk

n 1
n  1 n    jl lk
l 0
n 1
 1 n     j  k l   jk  I n  jk
l 0
One-to-one matching between probabilities
moments. Ω1 is the inverse Fourier transform
and
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II. Roots-of-Unity Random Variables (Cont’d)
If Z1 and Z2 are independent Ωn RVs, then:

 
    
E Z1Z 2   E Z1 Z 2  E Z1  E Z 2
j
j
j
j
j
DFT Z1Z 2   DFT Z1  DFT Z 2 
pZ1Z 2  pZ1  pZ 2

pZ1Z 2   1 pZ1  pZ 2

Isomorphism between <[Z]n, +> and < Ωn, ×>:
j  k  l mod n   j k   l
Circles fundamental to lines, for lines are circles of infinite radius.
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III. Collective-Risk Model
Severities X and claim count N are all independent:
S  X1  X 2    X N

DFT S    DFT X   ProbN  k 
k
k 0

 E DFT X 
If N ~ Poisson(λ):
 
N



k 0
k 0
E z N   z k e  k k!  e    z  k!  e  e z  e   z 1
DFT S   e   DFT  X 1
k
Exact probabilities under arithmetic modulo n
What is the mean of an exponential RV wrapped around limit l?
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IV. Examples
1. Binomial probabilities to overflow
2. Klugman example
3. Non-probabilistic DFT uses
a. Binomial coefficients
b. Combining AQ pattern in AY pattern
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References
•
Clark, Allan, Elements of Abstract Algebra, New York: Dover, 1984
- Introduction: theory of equations as the start of modern algebra
- Chapter 2: group theory, circle group, cosets, homo- and isomorphisms
•
•
Halliwell, L., “The Discrete Fourier Transform and Cyclical Overflow,”
Variance, 8:1, 2014, 73-79, www.variancejournal.org/issues/08-01/73.pdf
“
, “Complex Random Variables,” CAS E-Form, Fall 2015,
www.casact.org/pubs/forum/15fforum/Halliwell_Complex.pdf
•
Kramer, Edna E., The Nature and Growth of Modern Mathematics,
Princeton University Press, 1981.
- Ch 4 (70-82): complex numbers, Hamilton, quaternions
- Ch 28: Grassman, Noether, twentieth-century algebra and physics
•
Klugman, S. A., H. H. Panjer, and G. E. Willmot, Loss Models: From
Data to Decisions, New York: Wiley, 1998.
- §4.7.1: fast Fourier transform
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