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|   = 0 otherwise Y(x,y) contains only these terms Repeated xeroxing

Approximating an ideal low-pass filter. Transfer function A() = 1 ||   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

Bank of bandpass filters

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

By substitution

Error

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

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"() + ...

Lowpass filter.

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)

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

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

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

Finite Fourier transforms. Considered

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

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

Inversion. fft()

Convolution. Lemma 3.4.1. 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

Periodogram. |dT ()|2

Chandler wobble.

Interpretation of frequency.

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 = 1 point in [0,t) = 0 otherwise j Yt exp{-it}

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

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(), 0   <  Y() = k k exp{ik}

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