For s.e. of flux multiply cv by mean flux over time period Damage: penetration depends on size

sagtu17.pdf Ascona12.pdf

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 Filtering/smoothing.

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  2 H(  )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 ave t-k  s  t+k Y(s)

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

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

Running median med t-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 If |X(t)  M, a(0) and  |ua(u)|  A, Y(t) =  a(t-u)X(u) then, |d Y T ( ) – A( ) d Y T ( ) |  4MA Application. Filtering Add S-T zeroes

Periodogram. |d T ( )| 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 Y t = 1 point in [0,t) = 0 otherwise  j Y t exp{-i t}

2. M.p.p. (sampled time series). {  j, M j } {Y(  j )}  j M j 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