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How many parameters best describe data in muon spectroscopy?Here we find that the Bayes factor demands the inclusion of more physically-meaningful parameters than the BIC or significance tests. Figure 1a presents some data that might reasonably be fitted with as few as three or as many as 22 physically-meaningful parameters. We find that the Bayes factor encourages the inclusion of all these parameters until the onset of over-fitting. Even though many of them have fitted values that fail significance tests (i.e. are consistent with zero), their omission distorts the fitting results severely.Figure 1Full size imageFigure 1a shows an anti-level-crossing spectrum observed in photo-excited muon-spin spectroscopy26 from an organic molecule27. The data are presented in Fig. 2a of Ref.27 and are given in the SI. These spectra are expected to be Lorentzian peaks. Theory permits optical excitation to affect the peak position, the width and the strength (photosensitivity). In the field region over which the measurements are carried out, there is a background from detection of positrons, which has been subtracted from the data presented27. Wang et al.27 did not attempt to fit the data rigorously; they did report a model-independent integration of the data, which demonstrated a change in area and position.The model that we fit hypothesises one or more Lorentzian peaks, with optional photosensitivity on each fitting parameter and with optional linear backgrounds y = a + bx underlying the peaks, described by the full equation given in the SI, equation (S3). To do a single LS fit to all the data, we extend the data to three dimensions, (x gauss, y asymmetry, z) where z = 0 for data in the dark and z = 1 for photoexcited data. Including all the data in a single LS fit in this way, rather than fitting the dark and photoexcited data separately, simplifies both setting up the fit and doing the subsequent analysis.Figure 1b shows the evolution of the SBIC and the lnBF as the number of fitting parameters in the model is increased. Starting with a single Lorentzian peak, three parameters are required, peak position P, width W and intensity A. Three photosensitivity parameters ΔLP, ΔLW and ΔLA are then introduced successively to the fit, (open and small data points for n = 3–6). The SBIC decreases and the lnMLI scarcely increases. It is only with the inclusion of one background term (n = 7) that any figure of merit shows any substantial increase. There is no evidence here for photosensitivity. The weak peak around 7050 G does not seem worth including in a fit, as it is evidenced by only two or three data points and is scarcely outside the error bars. However, a good fit with two peaks (P1 ~ 7210 G, P2 ~ 7150 G, the subscripts 1 and 2 in accordance with the site labelling of Fig. 2a of Ref.27) can be obtained with just five parameters (P1, P2, A1, A2, W). This gives substantial increases in the SBIC and lnMLI, further increased when W1 and W2 are distinguished and then when the single background term and the three photosensitivity parameters ΔLP2, ΔLW2 and ΔLA2 are successively included (solid or large data points for n = 5–10 in Fig. 1b). The SBIC reaches its maximum here, at n = 10, and then decreases substantially when the other three photosensitivity parameters and the other three background terms are included. These additional parameters fail significance tests as well as decreasing the SBIC (Fig. 1b). Conventionally, the n = 10 fit would be accepted as best. The outcome would be reported as two peaks, with significant photo-sensitivities ΔLP2, ΔLW2 and ΔLA2 for all three of the 7150 G peak parameters, but no photosensitivity for the 7210 G peak (Table 1).Table 1 Photosensitivity results of fitting the data of Fig. 1a with 10, 16 and 19 parameters. Parameter units as implied by Fig. 1a.Full size tableThe Bayes factor gives a very different outcome. From 10 to 16 parameters, the Bayes factor between any two of these seven models is close to unity (Fig. 1b). That is, they have approximately equal probability. The Bayes factor shows that what the conventional n = 10 analysis would report is false. Specifically, it is not the case that ΔLP2, reported as − 14 ± 4 G, has a roughly ({raise0.5exhbox{$scriptstyle 2$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle 3$}}) probability of lying between − 10 and − 18 G. That is not consistent with the roughly equal probability that it lies in the n = 16 range (− 24 ± 8 G). Table 1 shows that at n = 16, ΔLP2 is the only photosensitivity parameter to pass significance tests. ΔLA2, which had the highest significance level at n = 10, is now the parameter most consistent with zero. The other four are suggestively (about 1({raise0.5exhbox{$scriptstyle 1$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle 2$}})σ) different from zero.Since the Bayes factor has already radically changed the outcome by encouraging more physically-meaningful parameters, it is appropriate to try the 7050 G peak parameters in the fit. With only 28 data-points, we should be alert to over-fitting. We can include P3 and A3 (n = 18), and ΔLP3 (n = 19), but W3 and ΔLA3 do cause overfitting. Figure 1b shows substantial increases of both the SBIC and the lnMLI for n = 18 to n = 20, where the twentieth parameter is in fact ΔLA3. The symptom of over-fitting that we observe here is an increase in the logarithm of the Occam Factor (lnMLI − lnL), the values of which decrease, − 26.9, − 33.5, − 34.8, and then increase, − 33.4, for n = 16, 18, 19 and 20 respectively. Just as lnL must increase with every additional parameter, so should the Occam factor decrease, as the prior parameter volume should increase more with a new parameter than the posterior parameter volume. So we stop at n = 19. The outcome, Table 1, is that the uncertainties on the n = 16 parameters have decreased markedly. This is due to the better fit, with a substantial increase in lnL corresponding to reduced residuals on all the data. The 7210 G peak 2 now has photosensitivities on all its parameters, significant to at least the 2σ or p value ~ 0.05 level. And the photosensitivities ΔLW2 and ΔLA2, both so significant at n = 10, and already dwindling in significance at n = 16, are both now taking values quite consistent with zero. In the light of Table 1, we see that stopping the fit at n = 10 results in completely incorrect results—misleading fitted values, with certainly false uncertainties.Discriminating between models for the pressure dependence of the GaAs bandgapThe main purpose of this example is to show how the Bayes factor can be used to decide between two models which have equal goodness of fit to the data (equal values of lnL and BIC, as well as p values, etc.). This illustrates the distinction it makes between physically-meaningful and physically meaningless parameters. This example also shows how ML fitting can be used together with the Bayes factor to obtain better results. For details, see SI §7.Figure 2 shows two datasets for the pressure dependence of the bandgap of GaAs (data given in the SI). The original authors published quadratic fits, ({E}_{g}(P)={E}_{0}+bP+c{P}^{2}), with b = 10.8 ± 0.3 meV kbar−1 (Goñi et al.28) and 11.6 ± 0.2 meV kbar−1 (Perlin et al.29). Other reported experimental and calculated values for b ranged from 10.02 to 12.3 meV kbar−130. These discrepancies of about ± 10% were attributed to experimental errors in high-pressure experimentation. However, from a comparison of six such datasets, Frogley et al.30 were able to show that the discrepancies arose from fitting the data with the quadratic formula. The different datasets were reconciled by using the Murnaghan equation of state and supposing the band-gap to vary linearly with the density (see SI, §7, equations (S4) and (S5)30. The curvature c of the quadratic is constant, while the curvature of the density, due to the pressure dependence Bʹ of the bulk modulus B0, decreases with pressure—and the six datasets were recorded over very different pressure ranges, as in Fig. 2. So the fitted values of c, c0, were very different, and the correlation between b and c resulted in the variations in b0.Here, using the Bayes factor, we obtain the same result from a single dataset, that of Goñi et al.28 The two fits are shown in Fig. 2. They are equally good, with values of lnL and SBIC the same to 0.01. The key curvature parameters, c and ({text{B}}^{prime }), are both returned as non-zero by 13.5σ (SI, §7, Table S1), consequently both with p-values less than 10−18. However, c is a physically-meaningless parameter. The tightest constraint we have for setting its range is the values previously reported, ranging from 0 to 60 μeV kbar−2, so we use Δc = 100 μeV kbar−2. In contrast, ({text{B}}^{prime }) is known for GaAs to be 4.4931. For many other materials and from theory the range 4–5 is expected, so we use (Delta {text{B}}^{prime } = 1). The other ranges are same for both models (see SI §7). This difference gives a lnBF of 3.8 in favour of the Murnaghan model against the quadratic, which is strong evidence for it. Moreover, the value of ({text{B}}^{prime }) returned is 4.47 ± 0.33, in excellent agreement with the literature value. Had it been far out of range, the model would have to be rejected. The quadratic model is under no such constraint; indeed, a poor fit might be handled by adding cubic and higher terms ad lib. This justifies adding about 5 to lnBF (see “Background in fitting a carbon nanotube Raman spectrum” section), giving a decisive preference to the Murnaghan model, and the value of b it returns, 11.6 ± 0.3. Note the good agreement with the value from Perlin et al.29 If additionally we fix ({mathrm{B}}^{prime}) at its literature value of 4.4931, lnBF is scarcely improved, because the Occam factor against this parameter is small, but the uncertainty on the pressure coefficient, Ξ/B0, is much improved.When we fit the Perlin data, the Murnaghan fit returns ({text{B}}^{prime }) = 6.6 ± 2.4. This is outside range, and indicates that this data cannot give a reliable value—attempting it is over-fitting. However, it is good to fit this data together with the Goñi data. The Perlin data, very precise but at low pressures only, complement the Goñi data with their lower precision but large pressure range. We notice also that the Perlin data has a proportion of outlier data points. Weighted or rescaled LS fitting can handle the different precisions, but it cannot handle the outliers satisfactorily. Maximum Likelihood fitting handles both issues. We construct lnL using different pdfs P(r) for the two datasets, and with a double-Gaussian pdf for the Perlin data (see equation (S6) in the SI §7). Fixing ({text{B}}^{prime }) at 4.49, fitting with the same Ξ/B0 returns 11.42 ± 0.04 meV kbar−1. Separate Ξ/B0 parameters for the two datasets give an increase of lnL of 4.6, with values 11.28 ± 0.06 and 11.60 ± 0.04 meV kbar−1—a difference in b of 0.32 ± 0.07 meV kbar−1, which is significant at 4½σ. This difference could be due to systematic error, e.g. in pressure calibration. Or it could be real. Goñi et al.28 used absorption spectroscopy to measure the band-gap; Perlin et al.29 used photoluminescence. The increase of the electron effective mass with pressure might give rise to the difference. In any case, it is clear that high-pressure experimentation is much more accurate than previously thought, and that ML fitting exploits the information in the data much better than LS fitting.Figure 2GaAs band-gap. Data for Eg(P) in GaAs from Goñi et al.28 (
) and from Perlin et al.29 (
) are shown after subtraction of the straight line E0 + 8.5P to make the curvature more visible. The Perlin data is expanded × 10 on both axes for clarity. Two least-squares fits to the Goñi data are shown, polynomial (dashed red line) and Murnaghan (solid blue line). (Figure prepared using Mathematica 12.0, www.wolfram.com/mathematica/).Full size imageBackground in fitting a carbon nanotube Raman spectrumThis example demonstrates how the Bayes Factor provides a quantitative answer to the problem, whether we should accept a lower quality of fit to the data if the parameter set is intuitively preferable. It also provides a simple example of a case where the MLI calculated by Eq. (1) is in error and can readily be corrected (see SI §8 Fig. S3).The dataset is a Raman spectrum of the radial breathing modes of a sample of carbon nanotubes under pressure32. The whole spectrum at several pressures is shown with fits in Fig. 1 of Ref.32. The traditional fitting procedure used there was to include Lorentzian peaks for the clear peaks in the spectra, and then to add broad peaks as required to get a good fit, but without quantitative figures of merit and without any attempt to explain the origin of the broad peaks, and therefore with no constraints on their position, widths or intensities. The key issue in the fitting was to get the intensities of the peaks as accurately as possible, to help understand their evolution with pressure. Here, we take a part of the spectrum recorded at 0.23 GPa (the data is given in the SI.) and we monitor the quality of fit and the Bayes factor while parameters are added in four models. This part of the spectrum has seven sharp pseudo-Voigt peaks (Fig. 3a; the two strong peaks are clearly doublets). With seven peak positions Pi, peak widths Wi and peak intensities Ai, and a factor describing the Gaussian content in the pseudo-Voigt peak shape, there are already 22 parameters (for details, see SI §8). This gives a visibly very poor fit, with lnL = − 440, SBIC = − 510 and lnMLI = − 546. The ranges chosen for these parameters for calculating the MLI (see SI §8) are not important because they are used in all the subsequent models, and so they cancel out in the Bayes factors between the models.Figure 3Carbon nanotube Raman spectrum. In (a), the carbon nanotube Raman spectrum is plotted (black datapoints) with a fit (cyan solid line) using the Fourier model. The residuals for four good fits are shown, × 10 and displaced successively downwards (Fourier, Polynomial, Peaks and Tails; all at lnL about − 60, see text). The backgrounds are shown, × 8 (long dashed, chain-dotted, short dashed and solid, respectively. In (b), the evolution of the MLIs is shown against the number of parameters for these four models. (Figure prepared using Mathematica 12.0, www.wolfram.com/mathematica/).Full size imageTo improve the fit, in the Fourier model we add a Fourier background (y=sum {c}_{i}mathrm{cos}ix+{s}_{i}mathrm{sin}ix) (i = 0,..) and in the Polynomial model, we add (y=sum {a}_{i}{x}^{i}) (i = 0,..) for the background. In both, the variable x is centred (x = 0) at the centre of the fitted spectrum and scaled to be ± π or ± 1 at the ends. In the Peaks model we add extra broad peaks as background, invoking extra parameter triplets (Pi, Wi, Ai). These three models all gave good fits; at the stage shown in Fig. 3a they gave lnL values of − 65, − 54 and − 51 and BIC values of − 156, − 153 and − 148 respectively. Thus there is not much to choose between the three models, but it is noteworthy that they give quite different values for the intensities of the weaker peaks, with the peak at 265 cm−1 at 20.5 ± 1.1, 25.5 ± 1.3 and 27 ± 1.7 respectively (this is related to the curvature of the background function under the peak). So it is important to choose wisely.A fourth model was motivated by the observation that the three backgrounds look as if they are related to the sharp peaks, rather like heavily broadened replicas (see Fig. 3a). Accordingly, in the fourth model, we use no background apart from the zeroth term c0 or a0 to account for dark current). Instead, the peak shape is modified, giving it stronger, fatter tails than the pseudo-Voigt peaks (Tails model). This was done by adding to the Lorentzian peak function a smooth function approximating to exponential tails on both sides of the peak position (for details, see SI §8) with widths and amplitudes as fitting parameters. What is added may be considered as background and is shown in Fig. 3a. This model, at the stage of Fig. 3a, returned lnL = − 62, BIC = − 146, and yet another, much smaller value of 15.5 ± 1.0 for the intensity of the 265 cm−1 peak.The Tails model is intuitively preferable to the other three because it does not span the data space—e.g. if there was really were broad peaks at the positions identified by the Peaks model, or elsewhere, the Tails model could not fit them well. That it does fit the data is intuitively strong evidence for its correctness. The Bayes factor confirms this intuition quantitatively. At the stage of Fig. 3a, the lnMLI values are − 251, − 237 and − 223 for the Fourier, Poly and Peaks models, and − 211 for the Tails model. This gives a lnBF value of 12 for the Tails model over the Peaks model—decisive—and still larger lnBF values for these models over the Fourier and Poly models.All models can be taken further, with more fitting parameters. More Fourier or polynomial terms or more peaks can be added, and for the Tails model more parameters distinguishing the tails attached to each of the seven Lorentizian peaks. In this way, the three background models can improve to a lnL ~ − 20; the Tails model does not improve above lnL ~ − 50. However, as seen in Fig. 3b, the MLIs get worse with too many parameters, except when over-fitting occurs, as seen for the Poly model at 35 parameters. The Tails model retains its positive lnBF  > 10 over the other models.The other models can have an indefinite number of additional parameters—more coefficients or more peaks, to fit any data set. It is in this sense that they span the data space. The actual number used is therefore itself a fitting parameter, with an uncertainty perhaps of the order of ± 1, and a range from 0 to perhaps a quarter or a half of the number of data points m. We may therefore penalise their lnMLIs by ~ ln 4 m−1 or about − 5 for a few hundred data points. This takes Tails to a lnBF  > 15 over the other models—overwhelmingly decisive. This quantifies the intuition that a model that is not guaranteed to fit the data, but which does, is preferable to a model that certainly can fit the data because it spans the data space. It quantifies the question, how much worse a quality of fit should we accept for a model that is intuitively more satisfying. Here we accept a loss of − 30 on lnL for a greater gain of + 45 in the Occam factor. It quantifies the argument that the Tails model is the most worthy of further investigation because the fat tails probably have a physical interpretation worth seeking. In this context, it is interesting that in Fig. 3a fat tails have been added only to the 250, 265 and 299 cm−1 peaks; adding fat tails to the others did not improve the fit; however, a full analysis and interpretation is outside the scope of this paper. In the Peaks model it is not probable (though possible) that the extra peaks would have physical meaning. In the other two models it is certainly not the case that their Fourier or polynomial coefficients will have physical meaning. More

Study area and datasetThe study area covers all Estonia located between 57.5(^circ ) N, 21.5(^circ ) E and 59.8(^circ ) N, 28.2(^circ ) E. The study area is relatively flat with no steep slopes and altitudes ranging between 0 and 200m above the sea level. Data about events were collected directly from field books that contained information about the mowing activity’s start and end date and the covered area. Considering the main agricultural areas of the country, we consider 2000 fields in which events are geographically evenly distributed across all Estonia, as shown in Fig. 1. In total, data about 1800 mowing and 200 non-mown events were collected in 2018, based on manual labelling. During manual labelling, the specific mowing days were labelled based on the following: a) information recorded by farmers in field books regarding mowing days, b) domain experts knowledge about the most probable days for mowing based on the climate, weather, and field conditions, c) rapid decrease in the Normalized Difference Vegetation Index (NDVI) and rapid increase in the coherence compared to past measurements. The average field size is 6.0ha, and around 95% of the fields were mown during the year. 90% of the fields are in the range of (0.5-10)ha. The greatest density of the fields is located in Lääne-Viru, Tartu and Jõgeva countries. Grassland parcels vector layer is provided by Estonian Agricultural Registers and Information Board (ARIB)50. The satellite imagery used in the study is from Copernicus program that provides free open Earth observation data to help service providers, public authorities, and international organizations improve European citizens’ quality of life.Figure 1Geographic distribution of events used in this study (This map was created by QGIS version 3.16, which can be accessed on https://qgis.org/en/site/).Full size imageSentinel-1 and Sentinel-2 dataFor Sentinel-1 data, in total, 400 S1A/BSLCIW products acquired between 1st of May 20017 and 30th of October 2018, were processed. 87 products were from relative orbit number (RON)160, 62 from RON131, 84 from RON87, 93 from RON58, and 60 from RON29. These were organised into S1A/S1B 6-day pairs. Sentinel-2 provides high spatial resolution optical imagery to perform terrestrial observations with global coverage of the Earth’s land surface. Sentinel-2 data is provided by the European Space Agency (ESA) together with a cloud mask, which can filter clouds on the image with moderately good accuracy. 400 Sentinel-2A and -2B L2A products acquired between 1 May 2017 and 30 October 2018 were processed. Each Sentinel-2 image is a maximum of three days off from the closest Sentinel-1 image. Only the NDVI was derived from Sentinel-2. NDVI has been widely used in the classification of grassland24,51 and that is mainly due to its ability in limiting spectral noise. The spatial resolution of the derived Sentinel-2 NDVI feature is 10 m.MethodsThe goal of the analysis is to detect mowing events from Sentinel-1 (S-1) and Sentinel-2 (S-2) data. For this, coherence time series were calculated about every field in the database about the event. Average coherence of a field, imaging geometry parameters, imaging time and average NDVI were stored in a database. The database formation process involved preprocessing many satellite images where average coherence and NDVI value was calculated for every parcel for every available date (constrained by image availability and cloud cover). The overall scheme of the proposed methodology is illustrated in Fig. 2. First, the time-series data from S-1 and S-2 images are preprocessed. Then, the most important features are used in a deep neural network to predict mowing events. The model has a reject region option that enables the model to abstain from the prediction in case of uncertainty, which increases trust in the model.We used the Sentinel Application Platform (SNAP) toolbox for processing S-1 data. More specifically, we followed the same following pre-processing steps in16: apply orbit file, back-geocoding (using Shuttle Radar Topography Mission (SRTM) data), coherence calculation, deburst, terrain correction, and reprojection to the local projection (EPSG:3301). Lastly, we resampled the data to 4m resolution to preserve the maximum spatial resolution and square-shaped pixels. Because the study areas’ terrain is relatively flat, there are few topographic distortions in the SAR data. Each swath’s coherence was calculated independently. Only pixels totally inside the parcel boundaries (including the average window used for coherence computation) were utilized to calculate results, and any interference beyond the parcel limits was discarded. Pair-wise coherence was calculated with 6-day time step. The data was stored into a database using a forward-looking convention: coherence regarding date X refers to the coherence between S-1 images over the period between date X and X + 6 days. For preprocessing S-2 data, L1C and L2A Sentinel-2 products were obtained through Copernicus Open Access Hub6. Next, a rule-based cloud mask solution was applied52. Finally, the fourth and eighth bands were extracted to compute NDVI values.Figure 2Flowchart of the proposed approach to detect mowing events.Full size imageFeature extraction from Sentinel-1 dataCoherence is a normalized measure of similarity between two consecutive (same relative orbit) S-1 images. Interferometric 6 day repeat pass coherence in VV polarization (cohvv), and coherence in VH polarization (cohvh) are chosen features as they are shown to be sensitive to changes in vegetation and agricultural events25. The shorter the time interval after the mowing event and the first interferometric acquisition, the higher the coherence value. Generally, up to 24 to 36 days after a mowing event, coherence stays relatively high. Precipitation caused the coherence to drop, which disturbs the detection of a mowing event. The spatial resolution of the S-1 6-day repeat pass interferometric coherence is 70 m. Given two S-1 images (s_{1}) and (s_{2}), coherence is calculated as follows:$$begin{aligned} wp =frac{|langle s_{1}s*_{2}rangle |}{sqrt{langle s_{1}s*_{1}rangle | langle s_{2}s*_{2}rangle |}} end{aligned}$$
(1)
where (|langle s_{1}s*_{2}rangle |) is the absolute value of the spatial average of the complex conjugate product.Coherence between two S-1 images (s_1) and (s_2) reaches its maximum value of 1 when both images have the same position and physical characteristics of the scatters. In contrast, the coherence value declines when the position or properties of the scatters change.Feature extraction from Sentinel-2 dataNDVI is related to the amount of live green vegetation. Generally, NDVI increases and decreases over the season, indicating the natural growth decay of vegetation, while the significant drops in the NDVI indicate an agricultural event such as mowing. NDVI is derived from S2 images and is calculated as follows:$$begin{aligned} NDVI=frac{band8 – band4}{band8 + band4} end{aligned}$$
(2)
Figure 3Typical signature of NDVI and coherence in VV and VH polarisation for non mown field during the year.Full size imageFigure 4Field with single mowing event during the year.Full size imageFigure 5NDVI measurement for a field example with a single mowing event during the season.Full size imageFigures 3, 4 and 5 show different samples of mown and non mown fields. NDVI measurements are green, cohvh and cohvv are blue and black, respectively. For non mown field, the typical signature of NDVI during the year is shown in Fig. 3. For non mown field, the typical signature of NDVI during the season is a half-oval curve; coherence is not stable but remains at almost the same level without apparent trend changes, as shown in Fig. 3. An example of a field with a single mowing event during the season is shown in Fig. 4. A mowing event is characterized by a rapid increase in both cohvh and cohvv and a sharp decrease in NDVI, as observed at day 150 (See Fig. 4). Forty days later, a similar signature is probably not due to a mowing event but likely caused by drought during summer.Notably, NDVI measurements are irregular and relatively sparse. Around 75% of total NDVI measurements are invalid in Estonia, and the percentage is slightly lower in Southern Sweden and Denmark due to cloud cover. The Cloud mask indicates the percentage of cloud coverage and allows the cloudy and cloud-free pixels to be identified. Using the standard cloud mask technique by the European Space Agency (ESA) leads to outliers noticed in the sudden decrease in the NDVI. Figure 5 shows an extreme value of NDVI that is supposed to be an outlier due to high differences to the precedent and subsequent values. The outlier is marked with a yellow dot (NDVI=0.38), nearest previous (NDVI=0.75), and next (NDVI=0.78) measurements are marked with a blue colour.Sentinel-1 and Sentinel-2 data preprocessingTo detect NDVI outliers effectively, a good understanding of the data is needed. NDVI outliers due to cloud mask errors rarely co-occur together, and hence, they can be treated as independent events53. NDVI outliers are usually identified with a sudden drop to almost zero and do not form a sequence. It is enough to look at neighbouring measurements (one before and one after) to detect individual outliers. If the difference between the adjacent measurements is high, this is an outlier signature. Hence, outliers can be handled by iterating through every three consecutive NDVI measurements for a given field and checking the difference between the first and second values and between third and second values. Figure 6 shows the scatter plot of all three consecutive NDVI measurements. The Y-axis shows the difference between third and second NDVI values in a triplet, while X-axis represents the difference between second and first NDVI values in a triplet. Triplets with up to 7 days difference are shown in blue, and triplets from 7 to 14 days are shown in green. The points structure forms a rhombus shape with a small cloud of possible outliers in the upper left corner. To filter outliers from the list of actual mowing events, we only consider triplets within up to 10 days interval (as the mowing event signature can recover in 10 days). Knowing rhombus equation (the centre is approximately in (0, 0), and the side length is around 0.6), the filtering rule can be easily applied as follows:$$begin{aligned} ndvi_3 – 2 cdot ndvi_2 + ndvi_1 ge 0.6 end{aligned}$$
(3)
where ndvi_1, ndvi_2, and ndvi_3 are consecutive NDVI measurements within 10 days interval.All outliers are removed, which represent around 0.1% of NDVI measurements.Figure 6Scatter plot of NDVI triplets.Full size imageSmoothing is an essential pre-processing step for noisy features. In this work, cohvh and cohvv features are smoothed using different techniques, including exponential moving average (EMA), moving average54, and Kalman filter55. Smoothing using moving average is done by taking the averages of raw data sequences. The length of the sequence over which we take the average is called the filter width. Table 1 shows the performance of moving average smoothing technique using different values for the filter width. The results show that the best AUC-ROC of 0.9671 is achieved at a filter size of 7. The Kalman filter produces estimates of the current state variables and their uncertainties. Once the outcome of the subsequent measurement is observed, these estimates are updated using a weighted average, giving more weight to estimates with higher certainty. The AUC-ROC achieved using Kalman filter is 0.962. The EMA is done by taking averages of sequences of data, in addition to assigning weights to every data point. More specifically, as values get older, they are given exponentially decreasing weights. The smoothed cohvh and cohvv EMA for cohvh and cohvv are calculated using a recursive definition (i.e., from its previous value) as follows:$$begin{aligned}&cohvh_sm(cohvh_n, alpha ) = alpha cdot (cohvh_n) + (1 – alpha ) cdot cohvh_sm(cohvh_{n-1}, alpha ) end{aligned}$$
(4)
$$begin{aligned}&cohvv_sm(cohvv_n, alpha ) = alpha cdot (cohvv_n) + (1 – alpha ) cdot cohvv_sm(cohvv_{n-1}, alpha ) end{aligned}$$
(5)
where (cohvh_sm(cohvh_{n-1}, alpha )): exponential moving average for end of (cohvh_{n-1}). (cohvv_sm(cohvv_{n-1}, alpha )): exponential moving average for end of (cohvv_{n-1}). (alpha ): a smoothing parameter.The higher the smoothing parameter, the more it reacts to fluctuations in the original signal. The lower the smoothing parameter, the more the signal is smoothed. Experimentally, we found that the best value for (alpha ) to achieve the best AUC-ROC of 0.968 is (frac{1}{3}) as shown in Table 2. The different smoothing techniques achieve comparable performance. EMA technique was selected as it achieves slightly higher performance.Table 1 Performance of moving average smoothing using different filter width.Full size tableTable 2 Performance of EMA smoothing using different values of (alpha ).Full size tableDerived featuresNew derived features from S-1 and S-2 are extracted to improve the performance of the machine learning model. The features were derived based on the following knowledge about mowing events: coherence tends to increase. In contrast, ndvi tends to decrease after mowing events and, many farmers perform mowing during the same time of the year due to the good weather conditions. Such knowledge was elaborated with the derived features. In the following, we will go through the list of derived features considered in this study. Mixed coherence is derived from S-1 features to capture the overall coherence trend. Mixed coherence is a non-linear combination of cohvh and cohvv and is calculated as follows:$$begin{aligned} Mixed_coh = sqrt{cohvh cdot cohvv} end{aligned}$$
(6)
The date is an important feature for the model to adapt, as it is more likely to have mowing events in the summer rather than in early spring, especially in Estonia. The normalized day of the year is calculated as normalization improves the training process of the neural network. Some methods normalize features during the training process, such as Batch Normalization used in this study56. However, neighbouring batches could have entirely different normalization variables (batch mean and variance). At the same time, DOY is a feature susceptible to small changes, e.g., mowing prediction on day 108 or 109 could have drastically different meaning (weekend or working day, day with sunny weather or day with heavy rain). It implies that unified normalization of the DOY feature before training could help avoid the unwanted impact of Batch normalization and possible gradient computation issues. The normalized day of the year is calculated as follows:$$begin{aligned} t = frac{day_of_year}{365} end{aligned}$$
(7)
where (day_of_year) is the year’s day, which is a number between 1 and 365, January 1st is day 1.In addition, we use another time feature dt to capture the gaps in time series. dt is defined to be the normalized difference in days between the current measurement and the previous one. Normalization was performed with min-max scaling. dt is calculated as follows:$$begin{aligned} dt = frac{diff – min_diff}{max_diff – min_diff} end{aligned}$$
(8)
where (min_diff): the minimum difference in days between two previous consecutive measurements obtained from training data. (max_diff): the maximum difference in days between two previous consecutive measurements obtained from training data.Since mowing is characterized by an increase in the coherence and decline in the NDVI, it is important to capture the difference in the values of features and/or slopes of the features’ curves. In the following, we summarize the list of original and derived features extracted from Sentinel-1 and Sentinel-2 included in this study.

ndvi Normalized difference vegetation index, obtained from Sentinel-2.

cohvv Coherence in VV polarization, Sentinel-1 feature.

cohvh Coherence in VH polarization, Sentinel-1 feature.

t Normalized day of the year when the measurement is obtained.

dt Normalized difference in days between current and previous measurement. The data was interpolated with a daily grid, this feature differentiated between interpolated data and real data by capturing the difference between valid (not interpolated) measurements.

cohvv_sm Smoothed cohvv with exponential mowing average (with parameter (frac{1}{3})).

cohvh_sm Smoothed cohvh with exponential moving average (with parameter (frac{1}{3})).

mixed_coh Harmonic mean of cohvv and cohvh. The harmonic mean is chosen as one of the simplest options of non-linear combination.

ndvi_diff Difference between current and previous NDVI measurements. This feature captures the decrease in the ndvi, which is highly related to mowing detection.

cohvv_sm_diff difference between current and previous (cohvv_sm) measurements. This feature captures the increase in the (cohvv_sm), which is highly related to mowing detection.

cohvh_sm_diff difference between current and previous (cohvh_sm) measurements. This feature captures the increase in the (cohvh_sm), which is highly related to mowing detection.

ndvi_der The slope of the line between previous and current NDVI values.

cohvh_sm_der The slope of the line between previous and current (cohvh_sm) values. This feature captures the change in the smoothed cohvh.

cohvv_sm_der The slope of the line between previous and current (cohvv_sm) values. This feature captures the change in the smoothed cohvv.

Feature selectionThe permutation feature importance measurement was introduced by Breiman57. The importance of a particular feature is measured by the increase in the model’s prediction error after we permuted the values of this feature, which breaks the relationship between the feature and the outcome. A feature is important if shuffling its values increases the model error and is less important otherwise. The importance of features considered in this study is ranked in Table 3. It is notable from Table 3 that the ordinal features are significantly more important than the derived ones. We used backwards elimination to select the optimal subset of features to be used by the machine learning model. More specifically, we start with all the features and then remove the least significant feature at each iteration, which improves the model’s overall performance. We repeat this until no improvement is observed on the removal of features. Figures 7 and 8 show that the end of season accuracy(EOS) and event accuracy, respectively, for training using a different subsets of the most important features. We refer to (F_{x}-F_{y}) to be the set of important features from feature x to feature y in Table 3. Figure 7 shows that using only ndvi and (mixed_{coh}) achieves EOS of 93%. Increasing the number of the most important features to 3 achieves a comparable performance to the best one, as shown in Fig. 7. The results show that using the ndvi and (mixed_{coh}) achieve around 73% event accuracy while increasing the number of features, the performance declines as shown in Fig. 8. As an outcome of the feature selection process, the developed machine learning model used all the 14 features, shown in Table 3, that achieve the highest combined performance.Table 3 Ranking features based on their performance.Full size tableFigure 7End of season accuracy for different number of features.Full size image
Figure 8Event-based accuracy for different number of features.Full size image
Machine learning modelEach record in our dataset represents specific features about a field during one season at a particular time, in addition to the target variable (mown or non mown). In this work, we use a neural network to predict mowing events. We are interested only in observations during the vegetative season, so winter measurements are not included. More specifically, we only include the data in the vegetative season, which is almost the same across all Estonia from April till October (215 days). The dataset is partitioned into 64% for training, 20% for testing and 16% for validation. All training and testing were performed using TensorFlow58 deep learning framework with default parameters. The architecture of the neural network used is shown in Fig. 9. To guarantee a fixed time interval of 1-day, all the missing values in S-1 and S-2 features are interpolated, as shown in Fig. 10. The data is processed in batches of size (64 times 215) (times )14, where 64 is the number of fields considered per patch, 215 is the number of days in the vegetation season in Estonia, 14 is the number of selected features.Figure 9Architecture of the proposed model.Full size imageThe network’s output is a vector of size 215, representing the probability of a mowing event on each day in the vegetation season. The network consists of three one dimension convolution layers. The first and second convolution layers are followed by the Softmax activation function and batch normalization layer, while the third convolution is followed by Sigmoid activation function. The NN hyperparameters required to achieve the model learning process can significantly affect model performance. These hyperparameters include the following56:

Number of epochs represents how many times you want your algorithm to train on your whole dataset.

Loss function represents the prediction error of Neural Network.

Optimizer represents algorithm or method used to change the attributes of the neural network such as weights and learning rate to reduce the loss.

Activation function is the function through which we pass our weighted suown to have a significant output, namely as a vector of probability or a 0–1 output.

Learning rate refers to the step of backpropagation, when parameters are updated according to an optimization function.

Figure 10Time series mowing events before and after linear interpolation.Full size imageA good model uses the optimal combination of these hyperparameters and achieves good generalization capability. The training was performed with the conjugate gradient descent method and the binary cross-entropy loss function. The neural network was trained during 300 epochs; an early stopping was used59. The optimizer used in our model is Nadam optimizer60 with the following parameters: (beta_1=0.9), (beta_2=0.999), (epsilon=None), (schedule_{decay}=0.004), and learning (rate=0.001). Different activation functions such as ReLU, Sigmoid, Linear, and Tanh have been experimentally evaluated on the testing dataset as shown in Fig. 11. The results show that the Softmax activation function achieves the highest combined performance (event accuracy of 72.6% and EOS of 94.5%), as shown in Fig. 11.Figure 11Performance of different activation functions.Full size imageUsing 1D convolution layer acts as a filter that slides on the time dimension allowing the model to predict future mowing events from past events. However, this approach is not suitable for real-time detection of mowing events, but we use it to predict mowing events within a fixed time frame (window). Such a time frame should be greater than half the (1-D) convolution window length.Model evaluationTo evaluate our model, we used two metrics, EOS accuracy and Event-based accuracy. EOS is the accuracy of detecting a mowing event at least once during the season. If the probability of detecting a mowing event at least once during the season is more than 50%, then the field is considered mown, otherwise not mown. Event-based accuracy is used to evaluate how well our model correctly predicts mowing events. The formula for quantifying the binary accuracy is defined as follows:$$begin{aligned} acc = frac{TP + TN}{TP + TN + FP + FN} end{aligned}$$
(9)
where TP is the number of times that the model correctly predicted mowing events, given that the start day of the predicted mowing event is not more than 3 days earlier and not more than 6 days later than the actual start day of the mowing event. Within these 9 days, several mowing events may be predicted. To handle this case, only the first predicted mowing event is considered TP, and every next one is considered an FP. TN is the number of times that the model correctly predicted the absence of mowing events. FP is the number of times that the model incorrectly predicted mowing events. It also includes the number of times that the model correctly predicted mowing events, but the start of the event does not fit into a 9-days time frame with the actual start of some mowing event. FN is the number of times where the model missed actual mowing events.Reject region
Figure 12Calibration plot for proposed model.Full size image
Sometimes the model is not confident enough to give a reliable decision about the state of the field. We cannot expect reliable and confident predictions from inaccurate, incomplete or uncertain data. So, it is better in the cases of uncertainty about the prediction to allow the model to abstain from prediction. In this way, the obtained predictions are more accurate, while human experts could check rejected fields. Given the true positive rate and the true negative rate on the validation set, the reject region technique outputs a probability interval ((t_{low}), (t_{upper})) in which the model abstain prediction, where (t_{low}) and (t_{upper}) are the minimum and maximum probabilities that the model is uncertain about its prediction. Out of this interval, the model is confident about its prediction and predicts afield as mowed if the probability is higher than (t_{upper}) and not mown if the probability is less than (t_{low}). We select (t_{upper}), such that the desired true positive rate is reached. To find (t_{upper}), we sort all positives descending by their predicted probabilities and select the top percentage equal to the true positive rate. We choose (t_{low}) such that the desired true negative rate on validation data is reached. To find (t_{low}), we sort all negatives ascending by their predicted probabilities and select the top percentage equal to the true negative rate.Figure 12 shows the calibration plot for our proposed model. Notably, the predicted probabilities are close to the diagonal, which implies that the model is well-calibrated. More

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Nature Video
19 January 2022

Tiny feather wing beetle reveals new way to fly

How a novel flying technique copes with the weird physics of miniature flight

Adam Levy

Adam Levy

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Flying is a tricky business, but when you are less than a millimetre long, things get even tougher. At such tiny scales the physics of flight changes, and yet insects have evolved strategies to fly in miniature. In this Nature video, we zoom in on new research which is revealing the never-before-seen flight tactics of a species of feather wing beetle.Research Article: Novel flight style and light wings boost flight performance of tiny beetles

doi: https://doi.org/10.1038/d41586-022-00144-0

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