WAVEPAL. A Python software for the frequency and wavelet analyses of irregularly sampled time series. Guillaume Lenoir

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1 WAVEPAL A Python software for the frequency and wavelet analyses of irregularly sampled time series Guillaume Lenoir Talk at the VUB 24/05/2018

3 MY RESEARCH WORLD Statistics & signal processing { Paleoclimate data State-of-the-art time series softwares Applied maths Code New softwares NB: I do not interpret the data or explain physical mechanisms.

4 THE 2 REFERENCE PAPERS

5 TIME SERIES REGULARLY SAMPLED Proxy Variable (e.g. time)

6 TIME SERIES REGULARLY SAMPLED Proxy t t t t t t Variable (e.g. time)

7 <latexit sha1_base64="4fbdfpwla4iil2nz8lbqd6tp1cu=">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</latexit> TIME SERIES IRREGULARLY SAMPLED Proxy Variable (e.g. time) t is no longer constant

8 PALEOCLIMATE TIME SERIES ODP Benthic Forams (Jian & al, 2003)

9 PALEOCLIMATE TIME SERIES ODP Benthic Forams (Jian & al, 2003)

10 PALEOCLIMATE TIME SERIES ODP Benthic Forams (Jian & al, 2003) Irregularly sampled

11 PALEOCLIMATE TIME SERIES ODP Benthic Forams (Jian & al, 2003) Global trend

12 PALEOCLIMATE TIME SERIES Goals: Detection of cycles Determine their periods and amplitudes Constraints: Irregular sampling Global trend Noisy data

13 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

14 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

15 NOTATIONS N data points <latexit sha1_base64="xy/9r6whepyiqrc3fy3z73ezpq4=">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</latexit> <latexit sha1_base64="xy/9r6whepyiqrc3fy3z73ezpq4=">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</latexit> <latexit sha1_base64="xy/9r6whepyiqrc3fy3z73ezpq4=">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</latexit> ti = 0 t 1 t 2. t N 1 C A et Xi = 0 X(t 1 ) X(t 2 ). X(t N ) 1 C A 2 R N

16 CYCLES DETECTION AT A FIXED PERIOD 0 Time series Xi 0 Test signal = sine/cosine signal period T Temps (échelle Time linéaire) Perform the analysis for many probing periods

17 <latexit sha1_base64="pnon26maldv5zf9b4hny+369icw=">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</latexit> <latexit sha1_base64="pnon26maldv5zf9b4hny+369icw=">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</latexit> <latexit sha1_base64="pnon26maldv5zf9b4hny+369icw=">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</latexit> ORTHOGONAL PROJECTION In R N Xi c! i = cos(! ti) s! i =sin(! ti) s! i c! i P sp{ c! i, s! i} Xi Lomb-Scargle Periodogram: Per(!) = P sp{ c! i, s! i} Xi 2 (Lomb, 1976) & (Scargle, 1982)

18 LOMB-SCARGLE PERIODOGRAM EXAMPLE Irregularly sampled time series Xi = sin(2 ti/40) + sin(2 ti/60) + sin(2 ti/100) Lomb-Scargle periodogram

19 AMPLITUDE OF THE CYCLES AT A FIXED PERIOD 0 Time series Minimum distance Amplitude E 0 Test signal period T Temps (échelle Time linéaire) Least squares method

20 AMPLITUDE VS PERIODOGRAM Xi Minimum distance s! i c! i P sp{ c! i, s! i} Xi be 2! 2 N Per(!) (Lenoir and Crucifix, 2018) squared amplitude periodogram

21 AMPLITUDE PERIODOGRAM EXAMPLE Irregularly sampled time series Xi = sin(2 ti/40) + sin(2 ti/60) + sin(2 ti/100) Amplitude periodogram (weighted Lomb-Scargle)

22 AMPLITUDE PERIODOGRAM EXAMPLE Irregularly sampled time series Xi = sin(2 ti/40) + sin(2 ti/60) + sin(2 ti/100) Amplitude periodogram (weighted Lomb-Scargle)

23 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

24 MOTIVATIONS State-of-the-art softwares in climate for the continuous wavelet transform only deal with regularly sampled time series. The data must then be interpolated => This may seriously affect the analysis, esp. when performing significance testing. A few algorithms which do not need interpolation exist in other fields (e.g. in astronomy) but are not suitable for climate time series Mathematically, there does not exist so far any formal extension of the Lomb-Scargle periodogram to the wavelet case

25 CYCLES DETECTION LOCALLY IN TIME AND AT A FIXED PERIOD 0 time series Xi 0 Test signal = sine/cosine weighted period T time by a Gaussian Perform the analysis for many probing times and periods Time-frequency analysis

26 MORLET WAVELET (Formal def. in Grosmann et Morlet, 1984) 0 0 T T Temps (échelle linéaire) Time The Gaussian envelope widens when the period increases & The number of cycles in the envelope remains unchanged

27 <latexit sha1_base64="pnon26maldv5zf9b4hny+369icw=">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</latexit> <latexit sha1_base64="pnon26maldv5zf9b4hny+369icw=">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</latexit> <latexit sha1_base64="pnon26maldv5zf9b4hny+369icw=">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</latexit> <latexit sha1_base64="azlh0touirgsumjm6iscs0ajuma=">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</latexit> SCALOGRAM Same formalism as for the periodogram: In R N Xi G,! s! i c! i = cos(! ti) s! i =sin(! ti) G,! = diagonal matrix with Gaussian weights <latexit sha1_base64="wv3zssdc4lcv5aw0z/sxf5hk42c=">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</latexit> <latexit sha1_base64="wv3zssdc4lcv5aw0z/sxf5hk42c=">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</latexit> <latexit sha1_base64="wv3zssdc4lcv5aw0z/sxf5hk42c=">aaadsnicjvlbbhmxej10gzzyavreey8wq0ogowgtcqfvvfsaj0uitfjsvd6nm1j1xmr7g1whh8mrfbf/ah/b8csrobuxr9yen5k5m8d23hjtfjz966wlt27fwd+4u3nv/oohw93tny+ubm2hhkvtanucs6emrttqa2/ucwovlhojjvlzg+a/uldw6br64c8bdvlkaaxpdce9onpu47n2m/fwts5pwym50tozd6fdnotlpmrnox+nloi4rlc7bzsmcdvuueslkariwzykyeebuz8yaocd0akyhaxzr+iknphbikohqgi9xzzfbhtrcvva6ti7qbwd3yjt0c5yasrz2kgayh/lzah9e/ecounvl1jzyfuc9tqd+q+8vet/5gutns7onwvq0nqwetqvkaxluwmdi19ueta0wii9gd/cljhzdc6ccxxrd2cr2f+diwma9kwmbelh7licmuftkrn/cuwl4a38v+h3ge4fi4a9u64yphdau9xnsd7xnci2rgouvyom97rlsixqtfbzc6+ouwycsy2cnl3ryqk/khban//llglmms9x1mtl7v9/pzen4ad3ppe9h6r7+/ejb9ateopg+vsk9ugdhdiqhav6tf/oa/iygsuykzaha52y84h+g4n5cfcxvf8=</latexit> G,!ii =exp( c! 2 (t i ) 2 ) G,! c! i P sp{g,! c! i,g,! s! i} Xi Scalogram: Scal(,!) = P sp{g,! c! i,g,! s! i} Xi 2 (Lenoir and Crucifix, 2018) Formal extension of the Lomb-Scargle periodogram to the wavelet case.

28 AMPLITUDE ESTIMATION LOCALLY IN TIME AND AT A FIXED PERIOD Periodogram: We have seen: be 2! 2 N Per(!) For the scalogram: generalization be,! 2 2tr(G2,!) Scal(,!) tr(g,! ) 2 (Lenoir and Crucifix, 2018)

29 SCALOGRAM EXAMPLE Irregularly sampled time series with beats Time <latexit sha1_base64="padgplk0/mmorsiz7w7mukovwpu=">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</latexit> NB: If regular sampling: same results as with the classical algorithms (e.g. Torrence & Compo, 1998)

30 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

31 We have seen: CYCLES AND TREND 0 Time series 0 Test signal What if the time series does not oscillate around zero?

32 CYCLES AND TREND Remove the trend? (detrending) Estimated trend True trend Alternative: Build a periodogram/scalogram which is blind to the coefficient of the trend. =) No need to remove the trend This is possible with a polynomial trend (Lenoir and Crucifix, 2018)

33 CYCLES AND TREND EXAMPLE WITH A LINEAR TREND Xi = sin(2 ti/40) + sin(2 ti/60) + sin(2 ti/100) + Trendi <latexit sha1_base64="iqen3vylpfdw46495w9zuvht+xu=">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</latexit> <latexit sha1_base64="iqen3vylpfdw46495w9zuvht+xu=">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</latexit> <latexit sha1_base64="iqen3vylpfdw46495w9zuvht+xu=">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</latexit> Exactly the same! NB: Dependence on the guess of the degree of the polynomial trend

34 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

35 CYCLES + UNCERTAINTIES Measurement errors Unknown processes =) Modeled as «noise» = stochastic process which is added to the cycles and trend (additive noise)

36 <latexit sha1_base64="+dkivi0hobbs0md7mtuvbsartu8=">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</latexit> STOCHASTIC PROCESS ALSO CALLED «BACKGROUND NOISE» In climate, the background noise is often modeled by a process with memory: X(t i )= i X(t i 1 )+ (t i ) where (t i ) is a Gaussian white noise This process is called red noise or AR-1 process. In this work, we consider a more general class of processes with memory, called ARMA processes => larger choice than with other softwares (e.g. REDFIT) When the time series is irregularly sampled, we have to work with continuous-time ARMA processes (CARMA), which are then discretized on the irregular time grid.

37 <latexit sha1_base64="k/bk1i9l59dzlyyglra4ar7rnpm=">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</latexit> MODEL FOR THE TIME SERIES Xi = Trendi +E (t) sin( ti + )+ Noisei Polynomial (fixed degree) (C)ARMA process with fixed order Periodic signal OR Locally periodic signal

38 <latexit sha1_base64="nzl/c6e3f7jpjqflqym0g4fhfgi=">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</latexit> <latexit sha1_base64="k/bk1i9l59dzlyyglra4ar7rnpm=">aaad7hicjvjdaxnbfl3p+lhrv6qpviwuhyrk2brbfrckrfdjvmhsofvczjpddt2dxwzmlzlkz/gkvvop/dhiq4l+c8/ctebb/jhlz+6ce8+998zmuc6ksxh8ubusxlp85erqtbxrn27eut1ev/pavi3oxccrikopx6krhvriykutxldwii3hhtgyn+44/8eboy2s1l49q8vrmezkhssstybg7d3zmngpygvxdjbomtzxqk08svl8no3y7jwxunws3vlk6cwuful9irubzhlzssm8pmphcs/mev40+t6iyi+9ar21qwlnqkkmgipjkcilu6cudl5d6lnmnbajmgltsct7bc1pddwguqirkdbtzdl2hx5v2luchtkzqht4nzghbybtiu7ddtvc9jec2af/yj3lnk63m6xjn6seauke6l94y8j/5tktlo7pmwuq0fqz4trlpkvdp+i6d39rzzghbubscfwadsbm5tmhzdgs3z1tyv7vholqt898bem/fjckyfs+rzl7v8g+bv7dp0e/w+g0zkrgb85venoanmj9omvgc4rytyzcexnp7nwdlwjumkcy5f4nd+lylrse1dnxxuk/u2yad/9yfw6zmbc4aull9s+/04vgykv3pbe/ehhtp/npejxu0x000qdhte0vai8gspijvtbx+hzuwbvgffbhebrs8py79nsipv4exfpvza==</latexit> Null hypothesis: SIGNIFICANCE TESTING Xi = Trendi + Noisei = HYPOTHESIS TESTING Alternative hypothesis: Xi = Trendi +E (t) sin( ti + )+ Noisei Test with the periodogram/scalogram Under the null hypothesis, the distribution of the periodogram/scalogram can be computed analytically or numerically (MCMC) (Lenoir and Crucifix, 2018) Significance level, at a given percentage

39 SIGNIFICANCE TESTING EXAMPLE WITH THE PERIODOGRAM Irregularly sampled noisy time series Xi = sin(2 ti/40) + sin(2 ti/60) + sin(2 ti/100) + Red noisei <latexit sha1_base64="i2+m7pcrkmkizqsu4ym3z5qmvwu=">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</latexit> <latexit sha1_base64="i2+m7pcrkmkizqsu4ym3z5qmvwu=">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</latexit> <latexit sha1_base64="i2+m7pcrkmkizqsu4ym3z5qmvwu=">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</latexit> Amplitude periodogram and assume the background noise is red

40 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

41 SMOOTHING THE PERIODOGRAM AND THE SCALOGRAM Pros: Reduce the variance: periodogram/scalogram less noisy Improve the quality of significance tests: less false positive detections (Lenoir et al., in prep.) Cons: Computing time! Decrease of the frequency resolution Methods: Periodogram: WOSA Compute the periodogram on each tapered segment and take the average. Scalogram: moving average in time

42 SMOOTHING EXAMPLE WITH THE PERIODOGRAM Same noisy time series No smoothing With smoothing: Noise damping per frequency With smoothing

43 OVERVIEW Frequency analysis Wavelet analysis Cycles and trend Significance testing Smoothing Example of analysis with WAVEPAL

44 WAVEPAL Available on Github: Available for Mac and Linux. Possibly also for Windows in a next version. Easy to install (should be ): it also installs Python and its dependencies. Written in Python 2 => Some knowledge of Python is required. 1 small tutorial for the moment. More to come. Recommended coding practice: with the Jupyter notebook. Tutorial:

45 BACKGROUND NOISE ANALYSIS

Continuous Wavelet Transform Reconstruction Factors for Selected Wavelets

Continuous Wavelet Transform Reconstruction Factors for Selected Wavelets General Background This report expands on certain aspects of the analytical strategy for the Continuous Wavelet Transform (CWT)