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Decomposition to Analyze Water Levels" +author: "Falk Mielke" +date: "2026-01-13" +format: + html: + toc: true + html-math-method: katex +--- + + +# Introduction + +Conventional analysis methods for continuous signals are designed to either reduce temporal variation to simplified quantities (e.g. "mean", "span = max - min" of a signal). +If signals oscillate regularly, Fourier methods are applicable. +These methods are well known and readily applied. + + +However, these methods occasionally fail to capture the complex characteristics of a signal. +Irregular changes, varying amplitudes, and changing frequencies can result in derived measures which are no good representation of the data. + + +A less widespread method is **Empirical Mode Decomposition**. +It is, as the name suggests, an empirical method which can help to separate meaningful components of a signal. + +::: {.callout-tip} +EMD can in principle be used to separate the following components of an oscillating signal: + +- envelope and momentary amplitude +- noise +- different oscillation modes + +::: + + +The steps involved are trivial, as I will demonstrate in this tutorial. +For most part, code is available in Python and R, though the Python implementation is somewhat more comprehensive. + + +::: {.panel-tabset group="language"} +### R + +```{r r-setup} +suppressMessages(library("dplyr")) +suppressMessages(library("ggplot2")) +suppressMessages(library("interpolators")) + +``` + + +### Python + + +```{python py-setup} + +# numeric analysis +import numpy as NP + +# data frames +import pandas as PD + +# empirical mode decomposition +import emd as EMD # (just to peek how the pro's do it) + +# signal processing +import scipy.signal as SIG +import scipy.ndimage as NDI +import scipy.interpolate as INTP + +# plotting +import matplotlib as MPP +import matplotlib.pyplot as PLT +``` + +::: + +# Example I: Canary Gape Angle + +## gape angle data + +For exploration, take this brief episode of beak gape angle of a canary cracking hemp seeds, courtesy of [Maja Mielke](https://orcid.org/0000-0001-6328-0589) ([*website*](https://maja.mielke.ws/blog/)). + + +::: {.panel-tabset group="language"} +### R + +```{r r-load-data} +data <- read.csv2("canary_gape_example.csv", sep = ",", dec = ".") +x <- data$'frame' +x <- x - min(x) +x <- x / max(x) + +y <- data$'totalGape' +``` + + +```{r r-plot-data} +#| label: fig-data-r +#| fig-cap: "The data. Grey lines show minimum, maximum and mean of the signal." + +ggplot(NULL, aes(x = x, y = y)) + + geom_line(color = "black") + + geom_hline(yintercept = min(y), color = "grey", alpha = 0.6) + + geom_hline(yintercept = max(y), color = "grey", alpha = 0.6) + + geom_hline(yintercept = mean(y), color = "grey", alpha = 0.8) + + theme_bw() + + labs(x = "time (arb. units)", y = "gape angle (deg)") + + theme(legend.position = "none") + +``` + +### Python + +```{python py-load-data} +data = PD.read_csv("canary_gape_example.csv") +x = data['frame'].values.ravel() +x -= NP.min(x) +x = NP.array(x, dtype = float) / NP.max(x) + +y = data["totalGape"].values +``` + + +```{python py-plot-data} +#| label: fig-data-py +#| fig-cap: "The data. Grey lines show minimum, maximum and mean of the signal." + +def ShowPlot(ax = None): + if ax is None: + ax = PLT.gca() + PLT.legend(loc = "lower right") + ax.set_xlabel("time (arb. units)") + ax.set_ylabel("gape angle (deg)") + ax.spines[["top", "right"]].set_visible(False) + PLT.show() + + +PLT.plot(x, y, "k-", label = "canary beak gape angle") +PLT.axhline(min(y), lw = 0.5, color = "0.4", alpha = 0.6) +PLT.axhline(max(y), lw = 0.5, color = "0.4", alpha = 0.6) +PLT.axhline(NP.mean(y), lw = 0.5, color = "0.4", alpha = 0.8) +ShowPlot() +``` + +::: + +This bird was in ["positioning" and "biting" phase](https://doi.org/10.1242/jeb.244360), placing a hemp seed in the right position for cracking with the help of their upper beak, lower beak and tongue. +On the x axis of [@fig-data-py]/[@fig-data-r], you see the time (available in video frames, or normalized). +On the y axis, gape angle indicates (approximately) the angle between the beak tips and corner[^1]. + +[^1]: "Beak corner" is not the real reference, I just used it for illustration. In fact, an anatomical coordinate reference system was used. + + +You can clearly see an initial episode with large gape angle oscillations, and a later episode with less extreme, still oscillatory movement. + + + +## Basic Algorithm + +Engineers often talk about "peak amplitude" (as in: amplitude along the peaks) and quantify a **momentary amplitude** of a noticably oscillating signal ([see here](https://www.keysight.com/used/be/en/knowledge/guides/how-to-measure-amplitude-engineers-guide)). + +This goes by the framework of ["Hilbert Transform"](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.hilbert.html#scipy.signal.hilbert) and ["Analytical Signal"](https://en.wikipedia.org/wiki/Analytic_signal). +All fascinating, yet we will here stick to the practical implementation. + + +To find the peak amplitude of the signal, one obviously has to detect the **peaks** (i.e. local maxima) of the signal. +Just to get a reference, [the `emd` toolbox in Python](https://emd.readthedocs.io/en/stable/stubs/emd.sift.interp_envelope.html) can achieve this. +(I did not find the equivalent function in the `R::emd` library, but no worries, this just serves as reference.) + + +```{python py-emd-extrema} +#| label: fig-emd-py +#| fig-cap: "Empirical mode decomposition involves finding and averaging the envelope. Here, this is done in Python with the `emd` library and default settings." + + +envelope = NP.stack( + EMD.sift.interp_envelope(y, interp_method = "pchip"), + axis = 1) +# Feel free to play with the `interp_method`! + +PLT.plot(x, y, label = "raw signal", color = "k") +PLT.plot(x, envelope[:, 0], color = "orange", label = "lower and upper envelope") +PLT.plot(x, envelope[:, 1], color = "orange", label = None) +PLT.plot(x, NP.mean(envelope, axis = 1), color = "darkblue", label = "mean of envelope") + +ShowPlot() + +``` + + + +This is the formally correct application of [Empirical Mode Decomposition](https://en.wikipedia.org/wiki/Hilbert%E2%80%93Huang_transform#Empirical_mode_decomposition). + + +::: { .callout-tip } + +To extract an empirical mode, the conventional EMD procedure involves: + +- finding the peaks (local extrema) of a signal +- interpolating values at the peaks, connecting them as the envelope +- then taking the mean of the envelope (interpolated peaks) + +::: + + +Those are relatively simple steps for which primitive functions exist. + +An aesthetic problem remains: the brief ups- and downs ("wiggles") that cause narrow envelope where it shouldn't be (e.g. around $t=0.05$). +Note that these wiggles do not matter much for the actual EMD procedure. + + +## Peak Detection: Prominence! + +While the `EMD` toolbox is quite comprehensive, we will have much more control by getting our hands on the individual steps outlined above. + +The first one is **peak detection**. +Scipy holds a default function for it: [`scipy.signal.find_peaks()`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.find_peaks.html). +Non-comprehensive equivalents in R are `signal::findpeaks()` and [`pracma::findpeaks()`](https://search.r-project.org/CRAN/refmans/pracma/html/findpeaks.html), and there is [`gsignal`](https://cran.r-project.org/web/packages/gsignal/vignettes/gsignal.html). + + +The functions only find the local *maxima*, but that is no challenge (just flip the signal to get minima). + + + +::: {.panel-tabset group="language"} +### R + +```{r r-find-peaks} +#| label: fig-peaks-r +#| fig-cap: "Peak detection using `pracma::findpeaks()` wiht a threshold of `0.4`. Note that the first and last sample were manually appended to avoid end effects." +peaks <- sort(c(pracma::findpeaks(y, threshold = 0.4, minpeakdistance = 20)[, 2], length(y))) +troughs <- sort(c(1, pracma::findpeaks(-y, threshold = 0.4, minpeakdistance = 20)[, 2])) + +ggplot(NULL, aes(x = x, y = y)) + + geom_line(color = "black") + + geom_point(aes(x = x[peaks], y = y[peaks]), color = "orange", size = 6, alpha = 0.6) + + geom_point(aes(x = x[troughs], y = y[troughs]), color = "orange", size = 6, alpha = 0.6) + + theme_bw() + + labs(x = "time (arb. units)", y = "gape angle (deg)") + + theme(legend.position = "none") + +``` + +The purpose of the `pracma` package is matlab portability. +In consequence, it is less feature-rich than `scipy`, yet still the best peak finding tool in R I found. + + +### Python + +```{python py-scipy-find-peaks} +#| label: fig-peaks-py +#| fig-cap: "Peak detection using `scipy.signal.find_peaks()` with a prominence of `0.5`. Note that the first and last sample were manually appended to avoid end effects." + +peaks = NP.append(SIG.find_peaks(y, prominence = 0.5)[0], len(x)-1) +troughs = NP.append(0,SIG.find_peaks(-y, prominence = 0.5)[0]) + +print('num peaks: ', len(peaks), ", num troughs: ", len(troughs)) +# print('peaks: ', peaks) + +PLT.plot(x, y, lw = 0.5, color = "k", label = "raw signal") +PLT.scatter(x[peaks], y[peaks], s = 10, marker = "o", facecolor = "orange", edgecolor = "k", alpha = 0.6, label = "detected peaks") +PLT.scatter(x[troughs], y[troughs], s = 10, marker = "o", facecolor = "orange", edgecolor = "k", alpha = 0.6, label = None) + +ShowPlot() +``` + + +The `scipy` function is quite elaborate and feature-rich. + +::: + + +::: { .callout-note } + +- Depending on the toolbox of choice, parameters such as `threshold`, `distance`/`minpeakdistance `, `width`, and `prominence` can be used to refine peak detection in case there are meaningful priors for either of those. +- It is recommended to double-check that every peak follows a trough (though not strictly necessary, brave and daring users may skip one or two). Generally, best plot and inspect all signals with putative peaks to detect errors. +- It remains to be demonstrated how this step performs on less oscillatory trials. + +::: + + +Make sure to do something about your `NA` and `NaN` values: `find_peaks` does not like them. +And keep an eye on those edges (i.e. start and end). + + + +## Interpolation: Choice! + +There are many interpolation methods ([splines, anyone?](https://docs.scipy.org/doc/scipy/tutorial/interpolate/1D.html)). +One which is used by the reference implementation is `pchip`, "Piecewise Cubic"[^pchip]. + +[^pchip]: Fritsch and Butland (1984): "A Method for Constructing Local Monotone Piecewise Cubic Interpolants". SIAM Journal on Scientific and Statistical Computing 5:2, 300-304. + + +::: {.panel-tabset group="language"} +### R + +In R, the `pracma` and `signal` libraries contain interpolation functions which are not accurate (copies of matlab). + +Then, there is the `interpolators` library, yet for some weird design choice it does not offer the choice to extrapolate beyond the data range (though splines can do that easily, see Python). + + +```{r r-interpolate-peaks} +#| label: fig-interpolate-r +#| fig-cap: "Peak interpolation using `interpolators::iprPCHIP`." + +# py <- signal::pchip(sort(as.vector(x[peaks])), y[peaks], x) +# ty <- signal::pchip(sort(as.vector(x[troughs])), y[troughs], x) +# py <- pracma::interp1(sort(as.vector(x[peaks])), y[peaks], x, method = "cubic") +# ty <- pracma::interp1(sort(as.vector(x[troughs])), y[troughs], x, method = "cubic") + +select <- x >= x[max(c(peaks[1], troughs[1]))] & x <= x[min(c(peaks[length(peaks)], troughs[length(troughs)]))] +xi <- x[select] +yi <- y[select] +py <- interpolators::evalInterpolator(interpolators::iprPCHIP(x[peaks], y[peaks]), xi) +ty <- interpolators::evalInterpolator(interpolators::iprPCHIP(x[troughs], y[troughs]), xi) + + +ggplot(NULL, aes(x = x, y = y)) + + geom_line(color = "black") + + geom_point(aes(x = x[peaks], y = y[peaks]), color = "orange", size = 6, alpha = 0.6) + + geom_point(aes(x = x[troughs], y = y[troughs]), color = "orange", size = 6, alpha = 0.6) + + geom_line(aes(x = xi, y = py), color = "orange", alpha = 0.8) + + geom_line(aes(x = xi, y = ty), color = "orange", alpha = 0.8) + + theme_bw() + + labs(x = "time (arb. units)", y = "gape angle (deg)") + + theme(legend.position = "none") + + +``` + + + +### Python + +The interpolation tools of `scipy.interpolate` work like a charm. + + +```{python py-scipy-interpolate-peaks} +#| label: fig-interpolate-py +#| fig-cap: "Peak interpolation using `scipy.interpolate.pchip_interpolate()`." + +py = INTP.pchip_interpolate(x[peaks], y[peaks], x) +ty = INTP.pchip_interpolate(x[troughs], y[troughs], x) + +PLT.plot(x, y, lw = 0.5, color = "k", label = "raw signal") +PLT.scatter(x[peaks], y[peaks], s = 10, marker = "o", facecolor = "orange", edgecolor = "k", alpha = 0.6, label = "detected peaks") +PLT.scatter(x[troughs], y[troughs], s = 10, marker = "o", facecolor = "orange", edgecolor = "k", alpha = 0.6, label = None) +PLT.plot(x, py, lw = 0.5, color = "darkorange", label = "peak interpolation") +PLT.plot(x, ty, lw = 0.5, color = "darkorange", label = None) +ShowPlot() +``` + +Keep in mind that the first- and last sample were appended to the lists of peaks and troughs above to avoid edge effects. + +::: + + +Interpolation in R is quite rudimentary: I miss the option to extrapolate, and one to fix the actual values. +I did not find a package for [RBF interpolation](https://en.wikipedia.org/wiki/Radial_basis_function_interpolation). +Python is quite accurate and versatile in terms of interpolation: [`scipy.interpolate`](https://docs.scipy.org/doc/scipy/tutorial/interpolate.html) is feature rich and can do [RBF](https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.RBFInterpolator.html#scipy.interpolate.RBFInterpolator) (not shown). + + + +## Mode Extraction and Residual + +One last step: take the upper and lower peak interpolation. +This is the **envelope** of the signal. +Its difference is the **momentary amplitude**. +The mean of upper and lower envelope line is an **empirical mode**. + +This is less a coding exercise, and more of vocabulary practice. + + + +::: {.panel-tabset group="language"} +### R + + +```{r r-calc-emd} +envelope <- cbind("peaks" = py, "troughs" = ty) +rownames(envelope) <- xi +firstmode <- rowMeans(envelope) +residual <- yi - firstmode +amplitude <- abs(py-ty)/2 +``` + + +```{r r-plot-firstmode} +#| label: fig-firstmode-r +#| fig-cap: "the extracted first empirical mode (top) and the residual (bottom)." + +ggplot(NULL, aes(x = xi, y = firstmode)) + + geom_line(color = "black") + + theme_bw() + + labs(x = "time (arb. units)", y = "gape angle (deg)") + + theme(legend.position = "none") + +``` + + +```{r r-plot-firstmode-residual} +#| label: fig-firstmode-residual-r +#| fig-cap: "The residual after extraction of the first mode." + +ggplot(NULL, aes(x = xi, y = residual)) + + geom_line(color = "black") + + geom_line(aes(x = xi, y = amplitude), color = "darkred", alpha = 0.8) + + geom_line(aes(x = xi, y = -amplitude), color = "darkred", alpha = 0.8) + + theme_bw() + + labs(x = "time (arb. units)", y = "gape angle (deg)") + + theme(legend.position = "none") + +``` + +### Python + +```{python py-plot-firstmode} +#| label: fig-firstmode-py +#| fig-cap: "The extracted first empirical mode (top) and the residual (bottom)." + +envelope = NP.stack([py, ty], axis = 1) +firstmode = NP.mean(envelope, axis = 1) +residual = y - firstmode +amplitude = NP.abs(NP.diff(envelope, axis = 1)/2) + + +fig, axes = PLT.subplots(2, 1, sharex = True) + +axes[0].plot(x, firstmode, + lw = 0.5, color = "k", label = "residual") +axes[0].set_ylabel("gape angle (deg)") +axes[0].spines[["top", "right", "bottom"]].set_visible(False) +axes[0].get_xaxis().set_visible(False) + +axes[1].plot(x, residual, + lw = 0.5, color = "k", label = "residual") + +axes[1].plot(x, amplitude, + lw = 0.5, color = "darkred", label = "peak amplitude") +axes[1].plot(x, -amplitude, + lw = 0.5, color = "darkred", label = None) +axes[1].axhline(0, zorder = 0, color = "grey", linewidth = 0.5) +ShowPlot(axes[1]) +``` + + +::: + +I do not know why the envelope touches the peaks of the residual... I honestly find this a beautiful mystery. + + + +# What is EMD good for? + +Separating the **first empirical mode** and the **residual** ([@fig-firstmode-py]/[@fig-firstmode-residual-r]) illustrates what you can achieve with the presented procedure. + + +::: { .callout-tip } +The *first mode*: + +- captures a rather long-term change of beak opening, by extracting something close to the mean of the angular range in which the canary beak move +- is smooth, compared to the raw signal +- yet still shows jump-like and wiggly behavior. + + +The *residual*: + +- is far more regular than the raw signal +- is centered +- can be subject of further analysis (e.g. Fourier Analysis). + +The mode extraction can be repeated iteratively. + +::: + + +You might find this too canary-centric. +I will now attempt to generailze the procedure by applying it to (i) random walks and (ii) real groundwater level data. + + +But, first things first: for the purpose of conceptual generalization, a more general function will be useful. + + +::: {.panel-tabset group="language"} +### R + +```{r r-firstmode-function} +#' extracts from a signal (with an optional timeline vector) +#' +#' extra arguments will propagate to `findpeaks` +#' +#' @returns a list with +#' - first empirical mode: mean of envelope +#' - envelope: the upper and lower peak amplitude envelope +#' - select: boolean vector of time subset +#' - extrema: peaks and troughs +#' +#' +#' usage: +#' > emd <- extract_firstmode(y, t = x, threshold = 0.4, minpeakdistance = 20) +extract_firstmode <- function (signal, t = NULL, ...) { + + # general purpose time vector + if (is.null(t)) { + t <- seq(0., 1., length.out = length(signal)) + } + + ## 1. find peaks + peaks <- sort(pracma::findpeaks(signal, ...)[, 2]) + troughs <- sort(pracma::findpeaks(-signal, ...)[, 2]) + extrema <- sort(c(peaks, troughs)) + + if (peaks[1] < troughs[1]) { + peaks <- c(peaks, length(signal)) + troughs = c(1, troughs) + } else if (peaks[1] > troughs[1]) { + peaks = c(1, peaks) + troughs = c(troughs, length(signal)) + } + + ## 2. interpolate -> envelope + select <- t >= t[max(c(peaks[1], troughs[1]))] & t <= t[min(c(peaks[length(peaks)], troughs[length(troughs)]))] + ti <- as.numeric(t[select]) + yi <- as.numeric(signal[select]) + py <- interpolators::evalInterpolator(interpolators::iprPCHIP(t[peaks], as.numeric(signal[peaks])), ti) + ty <- interpolators::evalInterpolator(interpolators::iprPCHIP(t[troughs], as.numeric(signal[troughs])), ti) + + ## 3. empirical mode + envelope <- cbind("peaks" = py, "troughs" = ty) + rownames(envelope) <- ti + firstmode <- rowMeans(envelope) + # residual <- yi - firstmode + # amplitude <- abs(py-ty)/2 + + return(list("firstmode" = firstmode, "envelope" = envelope, "select" = select, "extrema" = extrema)) +} + + + +``` + +### Python + +```{python py-firstmode-function} + +def ExtractFirstmode(signal, t = None, **peak_kwargs): + # extracts from a signal (with an optional timeline vector) + # the following components: + # - first empirical mode + # - envelope + # - extrema + # + # usage: + # > md1, env, _ = ExtractFirstmode(y, x, prominence = 0.5) + + # general purpose time vector + if t is None: + t = NP.linspace(0., 1., len(signal), endpoint = True) + + ## 1. find peaks + peaks = SIG.find_peaks(signal, **peak_kwargs)[0] + troughs = SIG.find_peaks(-signal, **peak_kwargs)[0] + extrema = NP.array(sorted([*peaks, *troughs]), dtype = int) + + if peaks[0] < troughs[0]: + peaks = NP.append(peaks, len(signal)-1) + troughs = NP.append(0, troughs) + elif peaks[0] > troughs[0]: + peaks = NP.append(0, peaks) + troughs = NP.append(troughs, len(signal)-1) + + ## 2. interpolate -> envelope + py = INTP.pchip_interpolate(t[peaks], signal[peaks], t) + ty = INTP.pchip_interpolate(t[troughs], signal[troughs], t) + + ## 3. empirical mode + envelope = NP.stack([py, ty], axis = 1) + firstmode = NP.mean(envelope, axis = 1) + + # residual = signal - firstmode + # amplitude = NP.abs(NP.diff(envelope, axis = 1)/2) + + return firstmode, envelope, extrema + + +``` + +::: + + +These functions can help a lot to play around with the peak finding parameters and inspect the outcome. + + + +# Example II: Random Walk on Water +## anachronistic measures + +Ultimately, I would like to apply these functions to quasi-continuous water level measurements from [`watina`](https://www.vlaanderen.be/inbo/datasets/watina/). +My goal is to slightly improve water level analysis, for the following reason. + + +Conventional water level analysis is subject to a few anachronisms. +In *the good old days*™, long before the time of GPS, automatic data loggers, or computers, people went to the field in spring to gather bi-weekly measures of water level from observation wells. + +What they were really interested in were approximate measures of highest and lowest ground water, so-called `xG3` values. +"Approximate", because measurement frequency was limited: a bi-weekly rhythm was about as good as we could get with actual humans putting yardsticks into holes in the ground. + + +There are `LG3` for a lower- and `HG3` for an upper reference of water levels [^ritzema2012]. +They are defined as follows: + +> Gemiddelde van de drie laagste|hoogste grondwaterstanden in een hydrologisch jaar (1 april t/m 31 maart) bij een meetfrequentie van tweemaal per maand (rond de 14e en 28e). + +[^ritzema2012]: Ritzema *et al.* (2012): "Meten en interpreteren van grondwaterstanden: analyse van methodieken en nauwkeurigheid". Alterra-rapport, Wageningen University & Research. + + +This is literally how they are calculated: +the mean of the three lowest/highest ground water levels measured in bi-weekly measurement interval, over a period from April to next March. + + +The anachronism is that we still calculate `xG3` like this, despite the availability of high frequency sampled water levels. +We artificially pick bi-weekly values, at 8:00 AM or at noon. +We disregard the continuous, temporally periodic nature of the phenomenon. +We choose an arbitrary sampling cadence. +We ignore measurement uncertainty. + +Knowing what was demonstrated above with the momentary amplitude, we could try to do better! + + +Let us first get an intuition of what `xG3` values look like, by looking at random walk data. + + +## random walk data + +Example code in R for generating random walk traces [can be found online](https://mattgrobis.blogspot.com/2016/04/introduction-to-r-3-for-loops-and.html). +I wrapped into a function for repeated execution, integrating `LG3` and `HG3` extraction. + + +::: {.panel-tabset group="language"} +### R + +```{r r-define-randomwalk} + +#' A random walk with regular subsamples +#' +#' @param start <- 0 # The starting point of the random walk +#' @param mean.move <- 0 # For each step, is there a bias to move in a certain direction? +#' A positive value means the random walk will tend to move +#' upwards, a negative value will make it tend downward, and +#' zero is unbiased. +#' @param sd.move <- 1 # The standard deviation of the step size. The higher this is, the more the random walk will bounce around. +#' @param t <- 365 # The number of steps you want the walk to take overall +#' @param sampling_interval <- 14 # the xG3 sampling interval (bi-weekly) +#' +walk_randomly <- function(start = 0, + mean.move = 0, + sd.move = 1, + t = 365, + sampling_interval = 14 + ) { + # Begin the walk at the starting point + walk <- start + samp <- start # bi-weekly samples + + # Run the random walk + for(i in 2:t){ + # Start at 2 because we already have our starting point + walk[i] <- walk[i - 1] + # Take the previous position... + rnorm(1, mean.move, sd.move) # ... and add a random value + walk[i] <- walk[i] * 0.98 # pull towards zero + + # take a sample every two weeks + if (i %% sampling_interval == 0) { + samp[as.integer(i/sampling_interval)] <- walk[i] + } + } + + # aggregate the results + result <- list() + result$walk <- walk # the walk itself + + result$samples <- samp # bi-weekly samples + + # calculating xG3 values + result$hg3 <- mean(sort(samp, decreasing = TRUE)[1:3]) + result$lg3 <- mean(sort(samp, decreasing = FALSE)[1:3]) + + return(result) +} + +``` + + +This is what the walks look like: + +```{r r-do-the-walk} +#| label: fig-randomwalkemds-r +#| fig-cap: "Random walks as simulated water level measurements. Colored lines are the walks, circles indicate bi-weekly samples, horizontal lines mark LG3 and HG3." + +sampling_interval <- 14 +n = 5 +dat <- lapply(1:n, walk_randomly) +# print(dat[[1]]) + +par(mfrow = c(1, 1)) +skip <- 32 +plot(NA, + xlim = c(0, 365), ylim = c(0, (n+1)*skip), + xlab = "days", ylab = "measurements", + yaxt = "n" + ) + +mypal <- colorRampPalette( c( "red", "green", "blue" ) )( n ) + + +for(i in 1:n){ + lines(skip*i + dat[[i]]$walk, type = "l", col = mypal[i], , lwd=1.5) #thickness set to be higher + points(seq(sampling_interval, 365, by = sampling_interval), skip*i + dat[[i]]$samples) + abline(h = skip*i + dat[[i]]$hg3) + abline(h = skip*i + dat[[i]]$lg3) +} + + +``` + + +### Python + +Translating the R functions above to a Python object. + +```{python py-define-randomwalk} + +class RandomWalk(object): + # This is how I imagine dancing in the eighties. + + def __init__(self, \ + start: float = 0, \ + direction: float = 0, \ + spread: float = 1, \ + t: int = 365, \ + interval: int = 14 \ + ): + + # a bit of overhead. + self.measurements = [start] + self.samples = {} + self.direction = direction + self.spread = spread + self.t = t + self.interval = interval + + # start immediately. + self.Walk() + + def Sample(self): + # just for convenience. + x = len(self.measurements) + self.samples[x] = self.measurements[x-1] + + def Walk(self): + # here, we go! + + for i in range(self.t): + + # compute next step + prev = self.measurements[-1] + nxt = prev + NP.random.normal(self.direction, self.spread, 1)[0] + nxt *= 0.98 # decay/pull towards zero + + self.measurements.append(nxt) + + # occasionally sample + if (i % self.interval) == 0: + self.Sample() + + def GetSamples(self): + return [v for v in self.samples.values()] + + def GetLG3(self): + return NP.mean(sorted(self.GetSamples())[:3]) + + def GetHG3(self): + return NP.mean(sorted(self.GetSamples())[-3:]) + + + +``` + + +This is what the walks look like: + +```{python py-do-the-walk} +#| label: fig-randomwalks-py +#| fig-cap: "Random walks as simulated water level measurements. Colored lines are the walks, circles indicate bi-weekly samples, horizontal lines mark LG3 and HG3." + +dat = [RandomWalk() for _ in range(5)] + + +fig, ax = PLT.subplots(1, 1) + +skip = 32 # shifting up the y axis + +for i, wlk in enumerate(dat): + ax.plot(skip*i + NP.array(wlk.measurements), lw = 1.5, zorder = 0) + smp = NP.stack([[k, v] for k, v in wlk.samples.items()], axis = 0) + ax.scatter(smp[:, 0], skip*i+smp[:, 1], s = 6, facecolor = "none", edgecolor = "k", alpha = 0.8, zorder = 20) + ax.axhline(skip*i + wlk.GetLG3(), lw = 0.5, ls = ":", zorder = 10) + ax.axhline(skip*i + wlk.GetHG3(), lw = 0.5, ls = ":", zorder = 10) + +ax.spines[["left", "top", "right"]].set_visible(False) +ax.set_yticks([]); +ax.set_xlabel("days"); +ax.set_ylabel("measurements"); +PLT.show() + +``` + +::: + + +Before seeing these plots, I was skeptical if `xG3` values had any use. +Due to the bi-weekly sampling, we have everything in here: missed peaks, exaggeration of brief excourses, lots of ignored values. +The "mean of three lowest/highest" is a charmingly simple calculation. + + +Nevertheless, I was astonished to see what a good representation of value range the band between `LG3` and `HG3` are! +They always come out at visually meaningful values, usually covering approximately the 90% highest density interval. + +Quite reassuring. +But maybe there is room for improvement, anyways. + + +## EMD of Random Data + +Let us see what the empirical mode of a random walk looks like, and how it might add to the water level story. + + +::: {.panel-tabset group="language"} +### R + +```{r r-random-emd} +#| label: fig-randomwalks-r +#| fig-cap: "EMD of random walks: effectively smoothing the data." + +par(mfrow = c(1, 1)) +skip <- 32 +n <- length(dat) +plot(NA, + xlim = c(0, 365), ylim = c(0, (n+1)*skip), + xlab = "days", ylab = "measurements", + yaxt = "n" + ) + +for(i in 1:n){ + y <- dat[[i]]$walk + t <- seq(0., 365., length.out = length(y)) + emd <- extract_firstmode(y, t = t) #, minpeakdistance = 1) + pts <- emd$extrema + + lines(t, skip*i + y, type = "l", col = "gray", lwd=0.5) + points(t[pts], skip*i + y[pts], cex = 0.5, col = alpha("black", 0.3)) + lines(t[emd$select], skip*i + emd$firstmode, type = "l", col = "darkgreen", lwd=1.0) +} + +``` + +### Python + +```{python py-random-emd} +#| label: fig-randomwalkemds-py +#| fig-cap: "EMD of random walks: effectively smoothing the data." + +fig, ax = PLT.subplots(1, 1) +skip = 32 + +for i, wlk in enumerate(dat): + y = NP.array(wlk.measurements) + mode, env, peaks = ExtractFirstmode(y) # , width = 1) + res = y - mode + amp = NP.abs(NP.diff(env, axis = 1)/2) + + + ax.plot(skip*i + y, lw = 0.5, color = 'k', zorder = 0, alpha = 0.4) + ax.plot(skip*i + mode, lw = 1.0, color = 'darkgreen', zorder = 20) + ax.scatter(peaks, skip*i+y[peaks], s = 8, facecolor = "none", edgecolor = "k", alpha = 0.3, zorder = 10) + +ax.spines[["left", "top", "right"]].set_visible(False) +ax.set_yticks([]); +ax.set_xlabel("days"); +ax.set_ylabel("measurements"); +PLT.show() + +``` + +::: + + +Note that I did not apply any restrictions (e.g. width, prominence) to peak finding. +These would quickly lead to over-smoothing of the traces. + + +::: {.callout-note} +On random walk data, EMD seems to achieve little more than smoothing. +The reason is that there is no regular oscillation in the data. + +The *kind* of smoothing is interesting: EMD naturally captures small oscillations, which in many cases are considered "white noise". + + +Note taken: EMD is especially useful if the data contains more-or-less regular oscillations. +::: + + +Next example: real data. + + + +# Example III: Water Levels + + +## Water Levels + +Our institute assembles data from various observation wells, storing them in [a database](https://watina.inbo.be). + +Two example water level traces shall serve as a test case for EMD. + + +::: {.panel-tabset group="language"} +### R + +In R, repeated application of the EMD function above does not work. + +However, the code here provides a pointer at how to use it. + + +```{r r-load-water} +#| label: fig-waterlevel-r +#| fig-cap: "A water level measurement." + +wata <- read.csv2("water_level_example_1.csv", sep = ",", dec = ".") +t <- as.Date(wata$'t') +w <- wata$'w' + +plot(t, w, type = 'o') +``` + + +```{r r-emd-wata} +#| eval: true +#| label: fig-waterlevel-emd-r +#| fig-cap: "EMD of a water level measurement." + +wemd <- extract_firstmode(as.numeric(w)) + +fm_t <- t[wemd$select] +fm_w <- as.numeric(wemd$firstmode) +residual <- w[wemd$select] - fm_w +fm_e <- wemd$extrema + +ggplot(NULL, aes(x = t, y = w)) + + geom_line(color = "darkgray", alpha = 0.6, lwd = 0.5) + + geom_line(aes(x = fm_t, y = fm_w), color = "black") + + geom_point(aes(x = t[fm_e], y = w[fm_e]), color = "orange", size = 3, alpha = 0.4) + + theme_bw() + + +``` + +R does not find all peaks reliably (e.g. see the first minimum), and there are consecutive peaks/troughs, which is a pity. +This means that relevant parts of the signal are captured on the residual. +Normally, the residual of this first EMD iteration would be "noise", and one could proceed to EMD on the first mode. + +::: {.callout-note} +The available `Python` tools are more versatile, and I recommend switching to "Python" at this point. +Even if you are not experienced in that language, the remainder of this tutorial are more conceptual considerations, and will be understandable. +::: + + +### Python + +```{python py-load-water} +#| label: fig-waterlevel-py +#| fig-cap: "A water level measurement. The mean water level is indicated by the dashed horizontal line; note the asymmetry of the signal." + +data = PD.read_csv("water_level_example_1.csv") +data["t"] = PD.to_datetime(data['t']) +t = data["t"].values.ravel() +w = data["w"].values + +PLT.plot(t, w, lw = 0.5, color = "k"); +PLT.axhline(NP.mean(w), zorder = 0, color = "grey", linewidth = 0.5, linestyle = "--") +``` + +DRY = don't repeat yourself... +Make another function for the EMD step! + +```{python py-emd-wata} +#| label: fig-waterlevel-emd-py +#| fig-cap: "EMD of a water level measurement." + +def EMDStep(t, w, **peak_kwargs): + mode, env, peaks = ExtractFirstmode(w, **peak_kwargs) + res = w - mode + amp = NP.abs(NP.diff(env, axis = 1)/2) + + fig, axes = PLT.subplots(2, 1) + ax = axes[0] + ax.plot(t, w, lw = 0.5, color = 'k', zorder = 0, alpha = 0.4) + ax.plot(t, mode, lw = 1.0, color = 'darkgreen', zorder = 20) + ax.scatter(t[peaks], w[peaks], s = 8, facecolor = "none", edgecolor = "k", alpha = 0.3, zorder = 10) + ax.axhline(NP.mean(w), zorder = 0, color = "grey", linewidth = 0.5, linestyle = "--") + ax.spines[["left", "top", "right"]].set_visible(False) + ax.set_ylabel("water level"); + ax.get_xaxis().set_visible(False) + + ax = axes[1] + ax.plot(t, res, lw = 0.5, color = 'k', zorder = 0, alpha = 0.4) + ax.plot(t, amp, + lw = 0.5, color = "darkred", label = "peak amplitude", alpha = 0.3) + ax.plot(t, -amp, + lw = 0.5, color = "darkred", label = None, alpha = 0.3) + ax.axhline(0, zorder = 0, color = "grey", linewidth = 0.5) + ax.spines[["left", "top", "right"]].set_visible(False) + ax.set_xlabel("date"); + ax.set_ylabel("residual"); + + return(fig, mode) + +fig, mode = EMDStep(t, w); +PLT.show(); + +``` + + +This is repeatable, both on the mode itself, as well as on the residual: + +```{python py-emd-wata-mode} +#| eval: true +#| label: fig-waterlevel-emd-mode-py +#| fig-cap: "Water level measurement, EMD of the first mode." + +fig, _ = EMDStep(t, mode); +PLT.show(); + +``` + +```{python py-emd-wata-residual} +#| eval: true +#| label: fig-waterlevel-emd2-py +#| fig-cap: "Water level measurement, the second mode is the EMD of the residual." + +EMDStep(t, w - mode); +PLT.show(); + +``` + +::: + +As with the random walks above, unguided peak detection *should* find each tiny local peak, therefore initially extracting white noise where it is present. + + +## Guided Mode Search (*advanced*) + +Experimenting with the peak detection parameters is worth a try. +(Because this works much better in Python, I will omit the R code here.) + + +First, search the long-term oscillations, by setting `prominence`, `width`, or `distance`. +In this special case, `width` will exclude the local minima: the lower peaks seem to be rather narrow on this water hole. +If this is an issue, remember that you can control peak finding for maxima and minima separately. +However, a combination of distance (more than half a year; remember that peaks and troughs are found separately) and prominence (*meaningful* peak) did the job to find the yearly baseline and straighten out the data (which might not be what we want). +This **effectively eliminates drift**. + + +```{python py-emd-wata-topdown1} +#| eval: true +#| label: fig-waterlevel-emd-topdown1 +#| fig-cap: "Exemplifying 'bottom-up' emd, first smoothing the obvious yearly oscillations" + +# try: prominence, distance, width +fig, mode = EMDStep(t, w, distance = 200, prominence = 0.20); +residual = w - mode +PLT.show(); + +``` + + +Another obvious component is the fine-grained noise; again, we get it by setting no restrictions on the peak detection. + +```{python py-emd-wata-topdown2} +#| eval: true +#| label: fig-waterlevel-emd-topdown2 +#| fig-cap: "Another possible step: narrowest peaks." + +fig, mode2 = EMDStep(t, residual); +PLT.show(); + +``` + +The remainder after these two hand-selected EMD iterations is a straightened, de-noised signal which could be used for more standardized analysis. + + +```{python py-emd-wata-topdown3} +#| eval: true +#| label: fig-waterlevel-emd-topdown3 +#| fig-cap: "The residual, after two different modes were extracted." + +residual2 = residual - mode2 +PLT.plot(t, w - NP.mean(w), lw = 0.5, color = 'k', zorder = 0, alpha = 0.4, + label = "raw signal"); +PLT.plot(t, mode2 - NP.mean(mode2), lw = 1.0, color = 'darkgreen', zorder = 20, + label = "residual after 2 EMD steps"); +PLT.axhline(0, color = "k", lw = 0.5, zorder = -1); +PLT.gca().legend(loc = "best"); +PLT.show(); + +``` + +There are some obvious problems in this: + +- edge effects +- wet summer ~2015: pronounced "dry" minimum was lacking; water levels around that year are dragged towards the zero +- peak chopping: some peaks, e.g. mid-2010 and 2016, are lost by smoothing + +Note that, in this particular example, drainage might be the cause of the asymmetrical (capped) distribution of water level values; EMD still handles it relatively well. + + +There is another valuable outcome of EMD: we get the lower and upper **envelope**! + +```{python py-wateremd-envelope} +#| label: fig-water-envelope-py +#| fig-cap: "The envelope of a water level measurement can be used as a continuous measure of minimum and maximum water levels; envelope average marks a continuous middle of water level range." +mode, env, peaks = ExtractFirstmode(w, distance = 200, prominence = 0.2) +PLT.plot(t, w, lw = 0.5, color = 'k', zorder = 0, alpha = 1.0); +PLT.plot(t, env[:,0], + lw = 0.5, color = "darkred", label = "envelope", alpha = 0.6); +PLT.plot(t, env[:,1], + lw = 0.5, color = "darkred", label = None, alpha = 0.6); +PLT.plot(t, NP.mean(env, axis = 1), + lw = 1.0, color = "darkgreen", label = "first mode", alpha = 0.6); +PLT.axhline(NP.mean(w), color = "k", lw = 0.5, zorder = -1); +PLT.show(); + +``` + +This envelope, if averaged over time, is somewhat related to the GxG's. + + + +There is no guarantee that the same guiding parameters will work on every observation in your data set. +In my experience, a "guided" (top-down or bottom-up) EMD approach requires a lot of fiddling with the parameters, possibly even case distinction. + + +```{python py-load-water2} +#| label: fig-waterlevel2-py +#| fig-cap: "Another water level measurement, lacking the obvious yearly oscillations, will not work with the previous, year-focused peak detection parameters. You could smooth it, though, with default EMD settings (not shown)." + +data2 = PD.read_csv("water_level_example_2.csv") +data2["t"] = PD.to_datetime(data2['t']) +t2 = data2["t"].values.ravel() +w2 = data2["w"].values + + +_, _ = EMDStep(t2, w2, distance = 200, prominence = 0.20); +PLT.show(); + +``` + +The peak detection controls are unavailable in R (at least I did not find them), so we might better stick with the defaults anyways (which usually work). + + +## Summary: Water Level EMD + +::: {.callout-note} +To "wrap up" (envelope-pun intended), some observations: + +- "guided EMD": yearly oscillations are found first by selecting for peak characteristics +- noise can be extracted either before or after; EMD is one way of naturally smoothing a signal +- EMD has some caveats on water level measurements, where oscillation are not necessarily regular or symmetric + +And the most important take-home message: + +- EMD extracts the **envelope**, **first mode**, and **residual**, all of which can occasionally be useful for further analysis + +::: + + +One goal of my application of EMD was to find a better way to extract a yearly range of water levels, to replace the anachronistic `LG3` and `HG3` calculations (which, nevertheless, worked surprisingly well on random walks). +With plain application of EMD (i.e. no pre-processing of the data), the envelope seems to be a promising aspect for further inspection. + +The cases in which water levels turned out to be non-ideal for EMD application is when they lack regular oscillations. +They are not symmetric (e.g. winter wet plateau due to drainage, summer dry dip, drift, ...), and not necessarily regular (e.g. "wet summer"). +Water level measurements such as the first example above might be more usefully approached with [wavelets](https://docs.scipy.org/doc/scipy-1.12.0/reference/signal.html#wavelets) to find the summer minima. +This is just my [almost-uneducated guess](http://mielke-bio.info/falk/posts/27.cycle_extraction/#orgac5a9c6), based on the visual form of the curves. +CWT (Continuous Wavelet Transform, [e.g. in R](https://www.rdocumentation.org/packages/Rwave/versions/2.6-5/topics/cwt)) is a great subject for another tutorial. + + +As a general reminder, there are many tools to consider for signal analysis. +I hope this tutorial could help to bring EMD to your personal repertoire. + + +Thank you for reading! +As always, feedback and suggestions are welcome. + diff --git a/content/tutorials/empirical_mode_decomposition/notes_qmd.txt b/content/tutorials/empirical_mode_decomposition/notes_qmd.txt new file mode 100644 index 000000000..52c1d28cb --- /dev/null +++ b/content/tutorials/empirical_mode_decomposition/notes_qmd.txt @@ -0,0 +1,61 @@ + +steps to get a qmd to hugo markdown: + ++ export hugo-md: + quarto render .qmd --to hugo-md + + ++ preview procedure: + rm tutorials -rf + unzip + python -m http.server 8887 + + + +// should be handled by markdown export ++ include yaml + preserve_yaml: true + adjust date, tags, categories and authors + + +// should be handled by markdown export ++ callouts: https://rossabaker.com/configs/website/shortcodes/callout/ +```{=markdown} +{{% callout note %}} +``` +Give a call out to your colleagues! +```{=markdown} +{{% /callout %}} +``` + + +// should be handled by markdown export ++ section crosslinks: +``{=markdown} + +## Section + + +// manually added as {=markdown} export +!! careful with code ticks in markdown export ++ figure captions + + + +
Figure 1: Caption text.

+ +`
This is the ``max(y)``.

`{=markdown} + + +// should be handled by yaml export options ++ equations + in yaml header: + params: + math: true + replace $s$ -> \\(s\\), $$\ldots$$ -> \\[\ldots\\] + eqn with \\(\\) and \\[\\] + cf. math https://gohugo.io/content-management/mathematics/ + and https://github.com/quarto-dev/quarto-cli/discussions/12272 diff --git a/content/tutorials/empirical_mode_decomposition/requirements.txt b/content/tutorials/empirical_mode_decomposition/requirements.txt new file mode 100644 index 000000000..dbc84d6ec --- /dev/null +++ b/content/tutorials/empirical_mode_decomposition/requirements.txt @@ -0,0 +1,126 @@ +anyio==4.7.0 +argon2-cffi==23.1.0 +argon2-cffi-bindings==21.2.0 +arrow==1.3.0 +asttokens==3.0.0 +async-lru==2.0.4 +attrs==24.3.0 +audioread==3.0.1 +babel==2.16.0 +beautifulsoup4==4.12.3 +bleach==6.2.0 +certifi==2024.12.14 +cffi==1.17.1 +charset-normalizer==3.4.0 +comm==0.2.2 +contourpy==1.3.1 +cycler==0.12.1 +dcor==0.6 +debugpy==1.8.11 +decorator==5.1.1 +defusedxml==0.7.1 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