1. Filtering/smoothing.
Use of A(,). bandpass filtering
Suppose X(x,y)  j,k jk exp{i(j x + k y)}
Y(x,y) = A[X](x,y)
 j,k A(j,k) jk exp{i(j x + k y)}
e.g. If A(,) = 1, | ± 0|, |±0|  
= otherwise
Y(x,y) contains only these terms
Repeated xeroxing

2. Approximating an ideal low-pass filter.
Transfer function
A() = ||  
Ideal
Y(t) =  a(u) X(t-u) t,u in Z
A() =  a(u) exp{-i  u) - <   
a(u) =  exp{iu}A()d / 2
= |lamda|<Omega exp{i u}d/2
= / u=0
= sin u/u u  0

3. Bank of bandpass filters

6. Fourier series.
How close is A(n)() to A() ?

8. By substitution

10. Error

11. Convergence factors. Fejer (1900)
Replace (*) by
Fejer kernel
integrates to 1
non-negative
approximate Dirac delta

16. General class. h(u) = 0, |u|>1
 h(u/n) exp{-iu} a(u)
=  H(n)() A(-) d (**)
with
H(n)() = (2)-1  h(u/n) exp{-iu}
h(.): convergence factor, taper, data window, fader
(**) = A() + n-1 H()d A'()
+ ½n-22H()d A"() + ...

17. Lowpass filter.

18. Smoothing/smoothers.
goal: retain smooth/low frequency components of signal while reducing the more irregular/high frequency ones
difficulty: no universal definition of smooth curve
Example. running mean
avet-kst+k Y(s)

19. Kernel smoother.
S(t) =  wb(t-s)Y(s) /  wb(t-s)
wb(t) = w(t/b)
b: bandwidth
ksmooth()

20. Local polynomial.
Linear case
Obtain at , bt OLS intercept and slope of points
{(s,Y(s)): t-k  s  t+k}
S(t) = at + btt
span: (2k+1)/n
lowess(), loess(): WLS
can be made resistant

22. Running median
medt-kst+k Y(s)
Repeat til no change
Other things: parametric model, splines, ...
choice of bandwidth/binwidth

23. Finite Fourier transforms. Considered

24. Empirical Fourier analysis.
Uses.
Estimation - parameters and periods
Unification of data types
Approximation of distributions
System identification
Speeding up computations
Model assessment
...

26. Examples. 1. Constant. X(t)=1

27. Inversion.
fft()

28. Convolution.
Lemma If |X(t)M, a(0) and |ua(u)| A,
Y(t) =  a(t-u)X(u) then,
|dYT() – A() dYT() |  4MA
Application. Filtering
Add S-T zeroes

29. Periodogram. |dT ()|2

30. Chandler wobble.

31. Interpretation of frequency.

32. Some other empirical FTs.
1. Point process on the line.
{0j <T}, j=1,...,N
N(t), 0t<T
dN(t)/dt = j (t-j)
Might approximate by a 0-1 time series
Yt = point in [0,t)
= otherwise
j Yt exp{-it}

33. 2. M.p.p. (sampled time series).
{j , Mj } {Y(j )}
j Mj exp{-ij}
j Y(j ) exp{-ij}

34. 3. Measure, processes of increments
4. Discrete state-valued process
Y(t) values in N, g:NR
t g(Y(t)) exp{-it}
5. Process on circle
Y(),   < 
Y() = k k exp{ik}

35. Other processes.
process on sphere, line process, generalized process, vector-valued time, LCA group