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Wildlife Conservation Research Unit, Recanati-Kaplan Centre, Zoology, University of Oxford, Oxford, UKHans Bauer & Claudio Sillero-ZubiriEvolutionary Ecology Group, Biology, University of Antwerp, Antwerp, BelgiumHans BauerLaboratory for Applied Ecology, Natural Resource Conservation, University of Abomey-Calavi, Cotonou, BeninAristide Comlan TehouDepartment of HydroSciences and Environment, University Iba Der Thiam, Thiès, SénégalMallé GueyeDirection de la Faune, de la Chasse et des Parcs et Réserves, Ministère de l’Environnement de la Salubrité Urbaine et du Développement Durable, Niamey, NigerHamissou GarbaDirection de la Faune et des Chasses, Ministère de l’Environnement et du Développement Durable, Ouagadougou, Burkina FasoBenoit DoambaNational Parks Directorate, Ministry of Environment and Sustainable Development, Dakar, SenegalDjibril DiouckThe Born Free Foundation, Horsham, UKClaudio Sillero-Zubiri More

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A review of existing despiking proceduresAmong despiking algorithms for raw, high-frequency, EC data, a popular approach was developed by Vickers and Mahrt6 (hereinafter VM97). The method consists in estimating the sample mean and standard deviation in overlapping temporal windows whose width in time is 5 min. The temporal window slides point by point, and any data point whose value exceeds (pm 3.5 sigma) (sample standard deviation) is flagged as a spike. The method is highly sensitive to the masking effect (where less extreme spikes go undetected because of the existence of the most extreme spikes), a reason for which the procedure is iterated increasing by 0.1 the threshold value at each pass, until no more spikes are detected.A revised version of the VM97 procedure was proposed by Metzger et al.14 (hereinafter M12), who suggested replacing the mean and standard deviation by more robust estimates, such as the median and the median absolute deviation (MAD), respectively. The authors found that this method reliably removed spikes that were not detected by VM97, showing a superior performance.To reduce the high-computational burden attributable to the windowed computations prescribed by the VM97 algorithm, Mauder et al.7 (hereinafter M13) proposed to estimate median and MAD over the whole flux averaging period (usually 30 or 60 min). M13 suggested to consider as spike those observations exceeding (pm 7cdot)MAD. Such an approach was selected as candidate method in the data processing scheme at the ICOS ecosystem stations15.Starkenburg et al9 recommended the approach developed by Brock16 (hereinafter BR86) as the best method for despiking EC data. This algorithm is currently implemented in the processing pipeline adopted by the National Ecological Observatory Network (NEON, https://www.neonscience.org). It is based on a two-stage procedure, where the first step consists in extracting the signal by means of a rolling third-order median filter which replaces the center value in the window with the median value of all the points within the window; the second step aims at identifying spikes by analyzing the histogram of the differences between the raw signal and the median filtered signal. Specifically, the differences are initially binned into 25 classes. Then, the first bins with zero counts on either side of the histogram are identified and points in the original signal that exceed the empty bins are flagged as spikes. If no bin with zero counts is found, then the number of bins is doubled (for example from 25 to 51, with one bin added ensuring to retain an odd number because the mean of differences, which is expected to be close to zero, should fall into the central bin of the histogram). The procedure is iterated by increasing the number of bins until the bin width is not less than the acquiring instrument resolution.The proposed despiking algorithmFigure 1Flowchart of the proposed despiking algorithm.Full size image
In order to define a modeling framework suitable for the representation of a sequence ((x_t)_{t in Z}) of observed raw EC data indexed by time t and contaminated by spikes, we assume a component model as follows:$$begin{aligned} left. begin{aligned} x_t&= mu _t + v_t + s_t,\ end{aligned}right. end{aligned}$$
(1)
where (mu _t) denotes the low frequency component (signal); (v_t) the deviations from the signal level (residuals) whose variability ((sigma _t^2)) is allowed to change slowly over time; and (s_t) the spike generating mechanism which is zero most of time but occasionally generates large absolute values.To achieve unbiased estimates of both the signal and the scale parameter ((sigma _t)) when data are contaminated by errors, the use of robust estimators is required. One of the most popular measures of robustness of a statistical procedure is the breakdown point, which represents the proportion of outlying data points an estimator can resist before giving a biased result. The maximum breakdown point is 50%, since, if more than half of the observations are contaminated, it is not possible to distinguish between the distribution of good data and the distribution of outlying data. Described in these terms, the arithmetic mean has a breakdown point of 0% (i.e. we can make the mean arbitrarily large just by changing any of the data point), whereas the median has a breakdown point of 50% (i.e. it becomes biased only when 50% or more of the data are large outliers).The proposed despiking procedure (hereinafter RobF) makes use of robust functionals whose breakdown point is 50% and consists in three stages (see Fig. 1). In the first step the signal ((mu _t)) extraction is carried out by means of the repeated median (RM) regression technique10,17. The second step involves the estimation of the time-varying scale parameter (sigma _t) by means of the (Q_n) estimator12. A detailed description of the robust functionals will be provided in the following sections. Spikes are detected in the third step, through the examination of outlier scores calculated as:$$begin{aligned} z_t=frac{x_t-mu _t}{sigma _t}. end{aligned}$$
(2)
Any values of (|z_t|) exceeding a pre-fixed threshold value ((z_{th})) is considered as spike. The choice of the threshold value should be based on the outlier scores data distribution which can vary across time. In this work (z_{th}) was set equal to 5 which means that for Normal- and Laplace-distributed data there is a 1 in 3.5 million and 1 in 300 chance, respectively, that an anomalous value is the result of a statistical fluctuation over the spectrum of plausible values. Once detected, spikes are removed and replaced by (mu _t) estimates obtained by the RM filter.Repeated median filterThe idea underlying moving time window based approaches is that of approximating the signal underlying observed data by means of local estimates that approximate the level of data in the center of the window.To this end, we fit a local linear trend11 of the form$$begin{aligned} mu _{t+i}=mu _t+ibeta _t, quad i=-k,ldots ,k, quad mathrm {to} quad {x_{t-k},ldots ,x_{t+k}}, end{aligned}$$
(3)
where k is the parameter defining the time window of length (n=2k+1), whereas (mu _t) and (beta _t) are estimated by means of the RM filter10 as$$begin{aligned} left. begin{aligned} tilde{mu }_t^{RM}&=medbigl (x_{t-k}+ktilde{beta }_t,ldots ,x_{t+k}-ktilde{beta }_tbigr ),\ tilde{beta }_t^{RM}&=med_{i=-k,ldots ,k} Bigl (med_{j=-k,ldots k,j ne i} frac{x_{t+i}-x_{t+j}}{i-j}Bigr ). end{aligned}right. end{aligned}$$
(4)
The only parameter required for the application of the RM filter is k, which controls how many neighbouring points are included in the estimation of (mu _t). Its choice depends not only on the time series characteristics, but also on the situations a procedure needs to handle. For despiking purposes, k has to be chosen as a trade-off problem between the duration of periods in which trends can be assumed to be approximately linear and the maximum number of consecutive outliers the estimator allows to resist before returning biased results.Results of previous studies18 for the evaluation of the RM filter performance in the removal of patches of impulsive noise showed that the RM resists up to 30% subsequent outliers without being substantially affected. Therefore, the minimal window width should be larger than at least three times the maximal length of outlier patches to be removed.To this end, the optimal time window width selection is carried out through a preliminary analysis of the data distribution. Specifically, the time series is subject to a preliminary de-trending procedure, where trend is approximated by a 5-degree polynomial function whose parameters are estimated via iterated re-weighted least squares (IWLS) regression. The optimal window width is then set equal to 4 times the maximum number of values exceeding (pm 3cdot s_g) in 30 s intervals, where (s_g) is the (global) standard deviation estimated by the (Q_n) estimator on de-trended data. To prevent cases where few or no data exceed the threshold values, a minimum window width of 5 s is imposed (i.e. 51 time steps for data sampled at 10 Hz acquisition frequency).
({{Q}}_n) scale estimatorBeyond the ability of the filter adopted for signal extraction, the effectiveness of a despiking strategy depends also on the robustness of the scale parameter, (sigma _t), which is of fundamental importance for the outlier scores derivation. Raw EC time series cannot be assumed to be identically distributed as variability may vary over time as the effect of changes in turbulence regimes and heterogeneity of the flux footprint area. In such situations, global estimates of the scale parameter are unrepresentative of the local variability. Consequently, the spike detection procedure becomes ineffective. To cope with this feature, the scale parameter (sigma _t) was estimated in rolling time windows whose width was set equal to those adopted for the signal extraction. As a robust estimates of (sigma _t), we used the (Q_n) estimator12$$begin{aligned} Q_n=2.2219{|x_i-x_j|;i0), then the process (X_t) is said to be integrated of order d, meaning that (X_t) needs to be differenced d times to achieve stationarity. To allow heteroskedasticity, we assume that (varepsilon _t= sigma _t e_t), where (e_t) is a sequence of independently and identically distributed variables with mean 0 and variance 1 and (sigma _t^2) is the conditional variance allowed to vary with time.The latter was simulated by means of a CGARCH process, which can be written as:$$begin{aligned} left. begin{aligned} sigma _t^2&=q_t + sum _{i=1}^r alpha _i (varepsilon _{t-i}^2 – q_{t-i}) + sum _{j=1}^s beta _j (sigma _{t-j}^2 -q_{t-j})\ q_t&=omega + eta _{11} q_{t-1} + eta _{21} (varepsilon _{t-1}^2 – sigma _{t-1}^2), end{aligned}right. end{aligned}$$
(7)
where (omega), (alpha _i), (beta _j), (eta _{11}), (eta _{21}) are strictly positive coefficients; (q_t) is the permanent (long-run) component of the conditional variance allowed to vary with time following first order autoregressive type dynamics. The difference between the conditional variance and its trend, (sigma _{t}^2 – q_{t}), is the transitory (short-run) component of the conditional variance. The conditions for the non-negativity estimation of the conditional variance23 are related to the stationary conditions that (alpha _i + beta _j < 1) and that (eta _{11} < 1) (such quantities provide a measure of the persistence of the transitory and permanent components, respectively).Model order specification and parameter estimation were performed by analyzing real EC data (more detail are provided in the “Results and discussion” section). With this modelling framework, we simulated 18,000 values as in EC raw data sampled at 10 Hz scanning frequency within a 30-min interval. Simulations were executed in the R v.4.0.2 programming environment by using the tools implemented in the rugarch package24.Once simulated, synthetic time series were intentionally corrupted with 180 spiky data points (1% for a sample size of 18000). Two macro-scenarios were considered. In the first scenario (S1), isolated or consecutive spike events of short duration were generated. In particular, 180 spike locations were randomly selected in such a way to obtain 30 single spikes, 30 spikes as double (consecutive) events, and 30 spikes as triple (consecutive) events. In the second scenario (S2), instead, time series were contaminated by impulsive peaks of longer duration. To this end, spike locations were carried out by randomly selecting five blocks of 50 consecutive data points. Once located, spikes were generated by multiplying the corresponding time series values (after mean removal) for a factor 10 in such a way to have magnitude similar to those commonly encountered on real, observed EC data. To simulate consecutive spike events as imposed by S2 scenario, generated spiky data points were taken in absolute term. Each scenario was permuted 99 times.MetricsThe ability of the despiking algorithms was assessed by comparing the number of artificial spikes inserted into the time series with the number of spikes identified by the method. More particularly, by referring to the (2times 2) confusion matrix as reported in Table 1, a valid despiking procedure maximizes decisions of type true positive (TP) while, at the same time, keeping decisions of the types false negative (FN) and false positive (FP) at the lowest levels possible. This trade-off can be measured in terms of Precision and Recall, which are commonly used for measuring the effectiveness of set-based retrieval25. For any given threshold value, the Precision is defined as the fraction of reported spikes that truly turn out to be spikes:$$begin{aligned} text {Precision}=frac{text {TP}}{text {TP}+text {FP}}, end{aligned}$$ (8) while the Recall is correspondingly defined as the fraction of ground-truth spikes that have been reported as spikes:$$begin{aligned} text {Recall}=frac{text {TP}}{text {TP}+text {FN}}. end{aligned}$$ (9) Table 1 Confusion matrix.Full size tableAs a measure that combines Precision and Recall, we consider the balanced F1-Score, which is the harmonic mean of the two indices above-mentioned, and given by:$$begin{aligned} text {F1-Score}=2 cdot frac{text {Precision} cdot text {Recall}}{text {Precision} + text {Recall}}. end{aligned}$$ (10) We have (0le text {F1-Score} le 1) where 0 implies that no spikes are detected and 1 indicates that all, and only, the spikes are detected. The closer to 1 the F1-Score index, the greater the effectiveness of the despiking method.In addition to the previous outlined metrics, a comparison between variances of (simulated) uncorrupted time series and the one estimated after the application of the despiking procedure has been performed.For an overall evaluation of the performance of the despiking algorithms, the Friedman test26 using a significance level (alpha =0.05), followed by a post-hoc test based on the procedure introduced in Nemenyi27 was applied. The Friedman test is a non-parametric statistical test, equivalent to repeated-measures ANOVA, which can be used to compare the performances of several algorithms28. The null hypothesis of the Friedman test is that there are no significant differences between performances of all the considered algorithms. Provided that significant differences were detected by the Friedman test (that is the null hypothesis is rejected) the Nemenyi test can be used for pairwise multiple comparisons of the considered algorithms. Nemenyi test is similar to the post-hoc Tukey test for ANOVA, and its output consists of a critical difference (CD) threshold. In order to do that, ranks are assigned to algorithms. For each data set, the algorithm with the best performance gets the lowest (best) average rank. The mean performance of two despiking algorithms is judged to be signifycantly different if the corresponding average ranks differ by at least the critical difference (the graphical output of Nemenyi test was implemented using tools provided in the R package tsutils (https://CRAN.R-project.org/package=tsutils)). More

Systematic paleontologyOrder Rodentia Bowdich, 182131Family Ischyromyidae Alston, 187632Genus Asiomys Qi, 198733Asiomys dawsoni Qi, 198733Figure 3A–EMaterial. Fragment of right calcaneus (IVPP V24417), early Middle Eocene, Huheboerhe, Irdin Manha Formation, Erlian Basin, China.Description. The bone is damaged and most probably that of a juvenile as it shows loss of the tissue in the extremities of the bone such as the calcaneal tuber and calcaneal eminence, which are usually less calcified in juveniles. The bone is relatively large (Table 1), with an elongated calcaneal tuber and a relatively short body (Fig. 3A–D). The sustentaculum tali is partly damaged; it has a subcircular articulation facet, which was probably more extended craniocaudally than mediolaterally. The caudal margin of the sustentaculum tali is inclined cranially, similar to the condition seen in species A and more than in species B (Fig. 3A). The sustentacular facet overlaps about one-half of the craniocaudal reach of the ectal facet. The groove for the ‘spring ligament’ (sensu Szalay and Decker34), which runs along the medial edge of the sustentaculum tali, is poorly pronounced. Likewise, the calcaneal groove for the tendon of the flexor fibularis muscle is shallow and poorly marked, most probably due to poor preservation. The ectal facet is relatively wide and similarly shaped as in species B (below). The peroneal process is completely damaged.Table 1 Measurements (in mm) of ischyromyid calcanei from the early middle Eocene of the Erlian Basin, Nei Mongol, China.Full size tableFigure 2Linear measurements of the calcaneus. Abbreviations: AEW, ectal facet anterior width; BL, calcaneal body length; BW, calcaneal body width; CCL, calcaneocuboid facet length; CCW, calcaneocuboid facet width; CL, calcaneus length; CMT, calcaneus maximum thickness; CW, calcaneal width; EL, ectal facet length; TEW, ectal facet total width; TL, tuber calcanei length; TT, tuber calcanei thickness; TW, tuber calcanei width; TWM, tuber calcanei width in mid-length. (Figure created in Corel Draw X4 (v.14.0.0.567) by Łucja Fostowicz-Frelik).Full size imageFigure 3Ischyromyid calcanei from the early middle Eocene of the Erlian Basin, Nei Mongol, China. (A–E), Asiomys dawsoni (IVPP V24417), right calcaneus, juvenile?; (F–K), species A (IVPP V24416), right calcaneus, adult; (L–Q), species B (IVPP V24418), right calcaneus, adult. In: A, F and L, dorsal; B, G and M, medial; C, H and N, lateral; D, I and O plantar; J and P caudal; E, K and Q, cranial views. Explanatory line drawings (right side) show important morphological features. Note sustentacular facet marked pale yellow. Scale bar equals 10 mm. (Photographs taken by Łucja Fostowicz-Frelik; drawings created in Corel Draw X4 (v.14.0.0.567) by Łucja Fostowicz-Frelik).Full size imageThe calcaneal tuber is strongly compressed, but it resembles in shape those of species A and B. A long groove for the calcaneofibular ligament is impressed on its lateral side.The anterior plantar tubercle is large and swollen, similar to that in species A, and touches the brim of the calcaneocuboid surface. The latter, only slightly damaged laterally, is round in outline, without a distinct pit, and inclined about 20–30°.Systematic remark: The fossil was associated with Asiomys dentition found in the same spot. We attribute specimen IVPP V24417 to Asiomys dawsoni, based on this fact and its distinctive size (Asiomys being the largest rodent in the assemblage). Asiomys is the only ischyromyid rodent known from the basal strata of the Irdin Manha Formation of Huheboerhe.Genus indet.Species AFigure 3F–KMaterial. Right calcaneus (IVPP V24416), early Middle Eocene, Irdin Manha Escarpment, Irdin Manha Formation, Erlian Basin, China.Description. The right almost complete calcaneus of an adult specimen is relatively large (Table 1), comparable in length to the calcaneus of a coypu (Myocastor coypus) or Asiatic brush-tailed porcupine (Atherurus macrourus). The bone has a characteristically elongated calcaneal tuber and rather short body (Fig. 3F–I). The calcaneal tuber is quite slender in comparison with the structure found in the coypu and porcupines. The shape of the bone resembles most closely the calcaneus of Paramys wortmani (see35: Fig. 12B), although in Paramys the calcaneal tuber is more compressed mediolaterally.The sustentaculum tali is large and eminent, reaching far medially and tapering, although its medial end forms a blunt edge parallel to the long axis of the bone. This medial edge also bears a well-marked but not deep groove of the calcaneonavicular (or ‘spring’) ligament (Fig. 3G). The sustentacular facet (facies articularis talaris media in Fostowicz-Frelik36: Fig. 12B2) is round, with only slight anteroposterior compression. It occupies almost the whole dorsal surface of the sustentaculum, encroaching slightly onto the calcaneal body. In that it differs from Notoparamys and Paramys wortmani, which both have a much more medially placed sustentacular facet, which does not encroach on the calcaneal body. The range of the sustentacular facet overlaps less than one-third of the ectal facet (posterior facies articularis talaris in Fostowicz-Frelik36: Fig. 12B2) on its anterior and medial sides. The calcaneal eminence is slightly longer than that in Marmota and Sciurus, in proportions closer to that of porcupines and of similar size as in Paramys wortmani. The ectal facet is wide, long, and has a distinctly helical course, even more strongly marked than in North American ischyromyids (see Rose and Chinnery35: Fig. 12A). It is, however, inclined more strongly mediolaterally than in Notoparamys and Paramys, and faces strongly medially. On the dorsal side of the calcaneal eminence, posterolateral to the ectal facet, there is a flattened rough area (finely pitted), marking the place of attachment of the lateral collateral ligaments binding the distal fibula and the astragalus with the calcaneus and stabilizing the astragalocalcaneal joint.A calcaneal body is short and stocky with poorly marked tendon ridges at the dorsal surface. A large peroneal process is partly damaged at its lateral margin. The process is placed closer to the cuboid surface than the sustentaculum tali. The position of the sustentaculum tali and the proportions of the calcaneal body of specimen IVPP V24416 resemble rather closely the calcaneus of Paramys wortmani (see35).The calcaneal tuber is not ‘pinched’ at its dorsal side but moderately compressed, thus there is no coracoid ridge posterior to the ectal facet. At the lateral side of the tuber, there is a long groove for the calcaneofibular ligament running askew, towards the dorsal surface of the calcaneal tuber. The groove for the calcaneofibular ligament is more weakly expressed than in the North American paramyines and arboreal sciurids, but similar to that of Marmota.The caudal surface of the calcaneal tuber is subcircular (only slightly more extended dorsoplantarly than mediolaterally, see Fig. 3 and Table 1). The groove for the calcaneal tendon (= Achilles tendon) is deep and placed asymmetrically at the surface (Fig. 3J). Also, the medial process of the calcaneal tuber is much better developed and extending medially.The plantar surface of the bone is almost straight with a delicate flexure cranially to a well-developed plantar heel process (Fig. 3G). The anterior plantar tubercle is relatively large, swollen, but shifted medially, towards the sustentaculum tali. It is placed very close to the cuboid surface, almost touching its margin; such location and the medial shifting resembles the condition in some ground squirrels, e.g., Cynomys (see Fostowicz-Frelik et al.8: Fig. 3D–F). The anterior plantar tubercle is also somewhat flattened and inclined medially and forms a well-marked calcaneal groove for the tendon of the flexor fibularis muscle.The calcaneocuboid articular surface is semicircular, slightly wider mediolaterally than long dorsoplantarly, which distinguishes species A from Marmota and paramyines (see35). It is almost transversally positioned, not inclined, as in most of the rodent taxa (coypu and porcupines included), and gently concave; it is also confluent and level with the cuboid pit, forming one round surface at the cranial end of the bone.Genus indet.Species BFigure 3L–QMaterial Right calcaneus (IVPP V24418), early Middle Eocene, Daoteyin Obo, Irdin Manha Formation, Erlian Basin, China.Description The bone is complete, slightly larger than in species A (Table 1), matching in length the calcaneus of the coypu. Its overall structure is very similar to the calcaneus of Paramys (either P. wortmani or P. taurus, see Rose and Chinnery35: Fig. 12B, C). It has a long and strong calcaneal tuber and a relatively strong but short calcaneal body (Fig. 3L). The tuber is more compressed mediolaterally than in species A; thus, the caudal surface of the tuber is extended more dorsoplantarly than mediolaterally (Fig. 3P). The attachment for the calcaneal tendon forms a rounded concavity at the caudal side of the tuber, and is more horizontally and symmetrically located at the surface than in species A. The lateral surface of the calcaneal tuber bears a marked scar from the calcaneofibular ligament, although the scar is convex, not concave as in species A and in other compared taxa (e.g., Cynomys).The sustentaculum tali is large and round; it is located relatively close to the calcaneal body, not extending as far medially as in the North American paramyines (see35). It is slightly longer anteroposteriorly and located more caudally (closer to the ectal facet) than in species A. Thus, the sustentacular surface overlaps ca. one-half of the cranial part of the ectal facet. The medial edge of the sustentacular shelf bears a deep groove for the ‘spring ligament’.The ectal facet is large, equally wide throughout its length, long and helical, although its course is straighter along the proximodistal direction than in species A. The ectal surface faces mediodorsally, with a slightly weaker medial component than in species A. The dorsal surface of the tuber, just caudal to the ectal facet, is not typically ‘pinched’ into a sagittally oriented crest, but it is, nevertheless, more mediolaterally compressed than in the species A, similar to Marmota.The calcaneal body forms about one-third of the bone length. Its dorsal surface is carved by deep longitudinal marks indicating the position of the extensor digitorum brevis muscle (Fig. 3). A middle-size peroneal process is located cranially at the calcaneal body. It is strong and long anteroposteriorly, reaching almost the edge of the calcaneocuboid surface. Its lateral edge shows a deep groove for the tendon of the peroneus longus muscle, while its dorsal surface forms a groove for the peroneus brevis muscle tendon (Fig. 3). Species B differs from the ground squirrels in the shape and location of the peroneal process, which is less extended laterally in species B than e.g., in marmots, although it is relatively much larger than in the coypu and porcupines.The anterior plantar tubercle looks less swollen than in species A; it is located at the very margin of the calcaneocuboid surface and as in species A is shifted medially (Fig. 3O, Q). The calcaneocuboid surface is slightly inclined (ca. 25°) anteromedially, which distinguishes the bone from species A, Marmota, and Notoparamys, which all have the calcaneocuboid facet almost transversal and perpendicular to the long axis of the calcaneus. In this respect, the calcaneocuboid surface resembles more closely the calcaneus of Paramys taurus (Rose and Chinnery35: Fig. 12C). The calcaneocuboid surface is almost round, slightly wider mediolaterally, resembling that of species A. A relatively small calcaneal pit (extending only to a half of the anterior plantar tubercle base, see Fig. 3Q), smaller but deeper than in species A, forms a shallow sink at the medial side of the surface, cranially to the sustentaculum tali.PCA analysisA Principal Component Analysis (PCA) was performed based on 14 measurements of the calcaneus. The analysis included the calcaneal measurements of five ischyromyid species (two described here as species A and B, and three comparative species from North America) and 16 extant large rodent species (Supplementary Table S1). The extant taxa represent six basic types of locomotor adaptations found in rodents: ambulatorial (terrestrial generalists), amphibious (swimming), arboreal (tree climbing), cursorial (four-pedal runners), ricochetal (bipedal jumpers), and semi-fossorial (burrowing).Principal Components 1 and 2 (PC1 and PC2) represent 87.48% and 5.75% of the variance, respectively, whereas Principal Components 3–4 represent further 4% of the variance (Supplementary Table S2). All the variables are positively correlated with PC1 and their loadings are very balanced (Fig. 4). Thus, it implies that the PC1 represents a proxy for the size of the bone. PC2 is most strongly correlated with the length of the calcaneal body, BL (-0.86) and more weakly correlated with the width of the cuboid facet (CCW) and anterior width of the ectal facet (AEW), 0.31 and 0.21, respectively (Fig. 4). The correlation with the length of the calcaneal body is an especially important factor for estimating an animal’s vertical jumping ability; the species with elongated calcaneal bodies are generally better jumpers (see8,36). The strong negative correlation of the length of the calcaneal body in the second component is illustrated by grouping the species with a strong jumping locomotor repertoire (e.g., squirrels and chinchillas) towards the left side of the plot (Fig. 4). Incidentally, this phenomenon does not concern the calcanei of ricochetal species (see the position of Pedetes versus that of Sciurus and Chinchilla: Fig. 4), where the mechanics of a jump are differently realized, and the stabilisation and relative stiffness of the ankle joint plays the most important role (thus, the calcaneal body and calcaneal tuber are more similar in size).Figure 4Principal component analysis of 14 metric parameters of rodent calcanei. The morphospace including paramyid calcanei from Nei Mongol in yellow circle. Lines connecting all data points represent a minimum spanning tree (MST) based on a Euclidean distance matrix. The loadings of the Components 1 and 2 shown at the corresponding axes. Strictly fossil taxa marked in red and pink, extant in black. (Figure created in Corel Draw X4 (v.14.0.0.567) by Łucja Fostowicz-Frelik).Full size imageIn the plot of PC1 against PC2, ischyromyids do not cluster together. Instead, the PCA morphospace is divided into two (or even three) broad groups of ischyromyid locomotor adaptations: the ambulatorial species and those with more pronounced jumping or cursorial ability. Chinese taxa fall among typically large ambulatorial rodents, such as the coypu (Myocastor) and porcupines (Atherurus and Hystrix). Closest to them there is the North American ischyromyid Quadratomus, which is somewhat shifted towards the cursorial species and can be thus distinguished as differently specialized (more cursorial). Two other North American ischyromyids, Ischyromys and Reithroparamys, are grouped with Chinchilla and Ondatra, respectively, which may imply some jumping and slightly scansorial locomotor adaptations for Ischyromys and those of typical agile generalist species for Reithroparamys.Although the sample is limited, the results of the PCA analysis point to general differences in the structure of the calcaneus, and thus, locomotor specialisation, between Asian and North American ischyromyid species. Moreover, Asian species seem to differ less from each other than the North American ones do, reflecting the overall greater species diversity and coverage of a wider niche spectrum of the North American ischyromyids.Functional and paleoecological implicationsThe studied calcanei add to our knowledge on the functional aspects of locomotion of ischyromyid rodents. Proximal tarsal morphology has been recently used to interpret the locomotor behavior of some extinct rodents (see e.g.,8,37,38,39). In the scheme of locomotor categories of Samuels and Van Valkenburgh40, attributions proposed for early ischyromyids fit into generally terrestrial41, arboreal42 or a mixture of those two35.A relatively short calcaneal body, widely spread sustentaculum tali, and a large peroneal process observed in most ischyromyid species (including these studied herein) indicate rather poor cursoriality. Instead, their ankle joint structure allows for a large freedom of foot movements in different planes. A medially extended sustentaculum tali together with a long and helically twisted ectal facet indicate a large degree of sliding between the calcaneus and astragalus along their articular facets, which makes possible a great degree of foot torsion resulting in foot eversion and inversion. This effect is further enhanced by an extended calcaneocuboid facet that is gently concave and oriented perpendicularly to the long axis of the calcaneus in species A.Such adaptations are helpful for both clinging to branches and adjusting to uneven or inclined substrate during climbing. A great degree of freedom of movement may be helpful also during burrowing, when the hind legs are used to push forward loose soil out of a burrow or an animal is forced to maintain a crouched posture, when it digs with its forelegs and head. Nevertheless, as much as the calcaneal structure may suggest some burrowing ability in ischyromyids (see Rose and Chinnery35), the rest of the postcranial skeleton known from the more complete specimens of North American representatives41 does not support fossorial adaptations. In particular, a long tail in the pre-Oligocene North American (see e.g., Paramys or Reithroparamys in Wood41: figs. 8 and 44, respectively) suggests some arboreal adaptations or at least occasional climbing, as such a tail greatly enhances balancing on uneven terrain. In contrast, typically fossorial mammals have reduced tails43.The overall morphology of dental and mandibular remains16,18 of Asian ischyromyids is similar to that of their North American counterparts16,19. As complete or even partial postcranial skeletons are unknown for the Asian ischyromyids, we can surmise their general locomotor adaptations based on calcaneal morphology which, although not in striking contrast with their North American counterparts, shows some differences.Overall, the calcaneal morphology of Chinese ischyromyids is closest to that of ground squirrels and especially porcupines (both Atherurus and Hystrix) and the coypu; the similarity to the last one is supported also by the PCA analysis. The calcaneal morphology and proportions may therefore reflect their locomotion behavior as generalized terrestrials, with a somewhat limited ability to climb (a rare but observed behavior in Hystrix) and to dig burrows (as does Atherurus43). A transverse and gently concave calcaneocuboid facet of species A facilitates foot rotation along the long axis, useful on an uneven, rocky terrain or while traversing branches, when an animal needs a flexible foot for a better grip (see Chester et al.44). On the other hand, the lack of both a characteristically bent calcaneal tuber and posteriorly located peroneal process in all ischyromyids (except for Notoparamys, see Rose and Chinnery35) argues against the arboreal adaptations characteristic of tree squirrels. More