A6523 Modeling, Inference, and Mining Jim Cordes, Cornell University. False Positives in Fourier Spectra. For N = DFT length: Lecture 5 Reading

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1 A6523 Modeling, Inference, and Mining Jim Cordes, Cornell University Lecture 5 Reading Notes on web page Stochas<c processes Correla<on func<ons Genera<ng correlated variables Wave propaga<on: using the FT to propagate waves Webpage: 1 False Positives in Fourier Spectra For N = DFT length: Is it bener to have large N or small N with respect to the number of false posi<ves? 2 1

2 False Positives in Spectra with Exponentially Distributed Fluctuations Consider a complex exponential with additive complex noise xn = A e iω 0 nδt + nn, n = 0,, N 1. The noise has variance σn 2 so the signal to noise ratio in the time domain is S A ( =. N ) t σn The DFT, defined with the normalization, N 1 X k = N 1 x ne 2πink/N n=0 is N sin ( δt 2πk/N) X k = A N 1 e iϕ n 2 ω0 + N 1 sin ( δt 2πk/N) 2 ω0 k The power spectrum (periodogram) with this normalization is S k N X k 2 = S kline + S knoise + 2 { X N kline k } [Note: An alternative normalization defines the DFT without the N 1 prefactor, N 1 X k = x, ne 2πink/N n=0 and in that case the spectrum is defined as X k 2 Sk. N The two definitions of DFT yield the same power spectrum.] The ensemble average mean is S k = S kline + S knoise because the first term is deterministic and the cross term has zero expectation. If the spectral line frequency falls on an integer frequency bin, the maximum of the line is max [ S kline ] = N A The mean of the noise term is S knoise = N kn k = σn 2. Then the signal-to-noise ratio of the spectral line is S NA = 2 2 S ( ) = N. N f σn 2 ( N ) t For the null hypothesis (no signal) the PDF of spectral amplitudes is a one-sided exponential f S (S) = S 1 e S/ S The probability of exceeding a threshold S thresh is P(S > S thresh ) = e S thresh / S If we evaluate this probability at the amplitude of the signal, i.e. let S thresh = max [ S kline ] = N A 2, the false-alarm probability is P(S > N A 2 N ) = e A2 / S = e N A2/ σn 2 = e N(S/N)2 t, where we have used the time-domain signal-to-noise ratio (S/N) t = A/ σ n. What really matters is the number of false alarms (or false positives) in the spectrum. The number of spectral values expected to exceed the threshold is N false positives = NP(S > N A 2 ) = Ne N A2/ σn 2 For fixed A/σ n, the number of false alarms 0 as N. (Exponentials always win over a power!) In [37]: %matplotlib inline from numpy import * import scipy import matplotlib import matplotlib.pyplot as plt import astropy from scipy import constants as spconstants from scipy.special import gamma randn = random.randn SNRt = array([0.01, 0.05, 0.1]) Nvec = 10.**(arange(0, 5.05, 0.05)) 4 2

3 In [38]: fig=plt.figure() for s in SNRt: Nfa = Nvec * exp(-nvec*s**2) plt.plot(nvec, Nfa, '-', lw=2, label=r'$\rm (S/N)_t = %6.2f $'%(s) ) Nmax = 1./s**2 Nfa_max = Nmax / e plt.plot(nmax, Nfa_max, 'ko') plt.xscale('log') plt.yscale('log') plt.axis(ymin=0.1) plt.xlabel(r'$\rm DFT \ Length $') plt.ylabel(r'$\rm Number \ of \ False \ Alarms $') plt.title(r'$\rm False \ positives \ for \ threshold \ = \ expected \ line \ amplitude $') plt.legend(loc=2) plt.show() 5 Spectral False Alarms 2/9/17, 12:43 PM The plotted curves N until the signal-to-noise ratio of the spectral line becomes (1). For larger N the number of false positives declines exponentially. The maximum number of false positives is given by dn false positives d = [ N e N A2/ σn 2 ] = e N A2/ σn 2 (1 N A 2 / σn 2 ) = 0, dn dn or 1 N max = (S/N) 2 t This maximum occurs when the signal-to-noise ratio of the line is unity. The maximum number of false positives is N N = max false positives,max. e For the three curves N max = 100, 400 and 10 4 for (S/N) t = 0.1, 0.05, 0.01 respectively. 6 3

4 Sampling Theorem Some short comments here See longer document on web page 7 Sampling Example 8 4

5 Searching for Monochromatic Signals in Noise We derived the spectrum of a <me series containing a complex exponen<al and addi<ve noise The shape of the spectral line is a sinc func<on. For con<nuous <me and frequency this is sinc(x) = sin(πx)/πx For the discrete case it is slightly different The sinc func-on underlies many of the problems associated with spectral analyis based on the Fourier transform The sinc func<on is the response of the Fourier transform to a sinusoid. Any func<on or stochas<c process can be represented as a sum of sinusoids à its power spectrum is convolved with the appropriate sinc func<on. sinc(x) sinc(x) sidelobes main lobe Sinc func<on aligned so that its zeros fall on integer values of x. If we plot only the black dots, we get a Kronecker delta func<on Sinc func<on mis- aligned from integer x values. We no longer get a Kronecker delta x So what? Generally a monochroma<c signal will not be in integer mul<ple of the frequency resolu<on δf = 1 / T so power in sinc 2 will leak into nearby (main lobe) and distant (sidelobe) frequencies. The envelope of sidelobe amplitudes ~ 1 / f 2 5

6 Searching for Monochromatic Signals in Noise In the no- signal limit: The PDF of the spectrum is exponen<al (one sided) The false- alarm probability is e - η for a threshold for detec<on of η x spectral mean The spectral mean = spectral rms for an exponen<al PDF If we find a spectral line that exceeds the threshold, we would say that the line is real at 100e - η % confidence. A Puzzle about Power Spectra We found that for a noise- only spectrum (or of a very weak signal + noise) that the PDF of the power spectrum is a one- sided exponen<al. A feature of this spectrum is that the errors are 100%: rms / mean = 1. This statement is independent of the data set length, N = number of samples. But don t larger data sets mean bener sta<s<cs? What gives? 6

7 12 N = 1024 P = 2.6 samples (S/N) t = Spectrum / FFT Frequency Index Counts 13 Detection Probability The exponen<al PDF applies to the no- signal case But for the frequency bin in the spectrum that has a signal the PDF is different: What is the relevant PDF? Need to consider the PDF of phasor + noise From the PDF we can calculate the probability of detec<on (true posi<ve) and false nega<ves. 7

8 PDF of Phasor Magnitude s = 0, 3, 5, 10 sigma_n = 1 8

9 PDF of Intensity s = 0, 1, 3, 5 sigma_n = 1 Detec<on probability Z 1 P det (I min )= I min di f I (I) ROC Curves Receiver operating characteristic Relative operating characteristic In a so- called detec<on problem, we try to establish whether a signal of some assumed type is present in data that include noise This is a universal problem that applies to many laboratory and observa<onal contexts. In astronomy, ROC curves apply to finding sources/ signals in images, spectra, <me series, etc. An ROC curve = P d vs P fa (detec<on vs false- alarm probability) Binary classifier used in physics, biometrics, machine learning, data mining, 9

10 hnp://en.wikipedia.org/wiki/receiver_opera<ng_characteris<c 10

11 Fourier Comments Fourier series/transforms involve exponen<al basis func<ons that are orthogonal over a relevant interval (e.g. [0, N- 1] for the DFT) If there are unknown values of the func<on in either domain, then orthonormality is broken gaps nonuniform sampling mises<mated values before FT The three FT forms (FT, FS, DFT) are similar but not always interchangeable sampling and transforming do not commute Issues: aliasing periodicity and convolu<on 21 Vector form of DFT Foreshadowing matrix algebra and modeling: The DFT is iden<cally equal to the least squares fipng of a set of sinusoids to data A <me series can be wrinen as a data vector. So can its Fourier transform. An N- point FT can be wrinen as the product of a matrix and the data vector. What does this matrix look like? 22 11

12 Stochastic Processes on One Page 1. Stochastikos = proceeding by guesswork, literally skillful in aiming 2. X(y, ) = sequence of random variables ordered by y and associated with an ensemble { }. (y continuous or discrete) 3. Ergodic Strictly stationary Wide sense stationary (WSS) Stationary increments Nonstationary 4. WSS = stationarity up to second moments 5. Sample averages 6= ensemble averages (general) Ergodic: equivalence as sample!1 6. Second order moments (general) t can be time, spatial, wavelength, frequency,... (a) Autocorrelation function: R X (t 1,t 2 ) hx(t 1 )X (t 2 )i Autocovariance function: X(t 1,2 )! X(t 1,2 )=X(t 1,2 ) hx(t 1,2 )i (b) Crosscorrelation function: R XY (t 1,t 2 ) hx(t 1 )Y (t 2 )i Crosscovariance function: C XY (t 1,t 2 ) h X(t 1 ) Y (t 2 )i i. Orthogonal: R XY (t 1,t 2 )=0for all t 1,t 2 ii. Uncorrelated: C XY (t 1,t 2 )=0for all t 1,t 2 (c) Structure function: D X (t 1,t 2 )=h[x(t 1 ) X (t 2 )] 2 i 7. WSS processes: (a) ACF, ACV, CCF, CCV, SF: depend only on argument differences, = t 2 t 1 ( time lag ) (b) Wiener-Kinchin theorem: Power spectrum = Fourier transform of the ACF: S(!) = R dt e i! R X ( ) 8. Stationary increments: SF: D X (t 1,t 2 )! D X ( ) even if ACF, etc. do not. 9. Second-order quantities ubiquitous in modeling, simulations, mining, and inference. 10. Third moment and bispectrum: X(t 1,t 2,t 3 )=hx(t 1 )X(t 2 )X(t 3 )i 3rd order stationary! 1 =t 2 t 1, 2 =t 3 t 1 ) ( 1, 2 ) Bispectrum: S(! 1,! 2 )= R d 1 e i! 1 1 R d 2 e i! 2 2 X( 1, 2 ) 12