Philippa Kaur

Etymology‘Candidatus Celerinatantimonas neptuna’ (nep.tu’na L. fem. n.), pertaining to Neptunus (L. masc. n. Neptune), the Roman god of the seas and the Neptune grass, Posidonia oceanica.SamplingA P. oceanica meadow at 8 m water depth and nearby sandy sediments in Fetovaia Bay, Elba, Italy13 were sampled between June 2014 and September 2019; individual sampling months and years are indicated in the sections below and/or in the figures and tables. In May 2017, a P. oceanica meadow at the island of Pianosa, Italy was also sampled. All of the samples were obtained via SCUBA diving.Complete plants of P. oceanica were carefully separated from the meadow by hand and stored in seawater-filled containers until arrival at the shore-based laboratory. Sediment for use in the laboratory-based aquaria was scooped into containers from nearby sandy patches. Seawater was pumped through a hose (placed at about 0.5 m above the P. oceanica meadow) into several 50 l barrels onboard the boat and was later used in the laboratory for the aquarium and the incubation experiments.The sediment within the seagrass meadow was sampled with stainless steel core tubes (length, 50 cm), which were drilled into the sediment by divers, and the cores were briefly stored at 22 °C (ambient temperature, September 2019) in a seawater-filled barrel until further processing at the shore-based laboratory.Porewater nutrient samples were obtained using stainless steel lances41 at intervals of around 10 cm. Water column nutrient samples were obtained from above the seagrass meadow at the start or end of sampling. Nutrient samples were collected in 15 ml or 50 ml centrifuge tubes and were stored in a cooler box until further processing.Nutrient measurementsWater column nutrients were measured during several sampling campaigns as indicated in Extended Data Table 1a. Ammonium (NH4+) concentrations were measured fluorometrically42 in the nearby shore-based laboratory, and the remaining water was frozen (−20 °C) for later analyses of nitrate (NO3−), nitrite (NO2−), phosphate (PO43−) and silicate (SiO44−) using an autoanalyser (QuAAtro, Seal Analytical). Porewater samples were obtained in June 2019 and were processed the same as the water column nutrient samples with the exception that ammonium was not measured on site but at the home laboratory at the same time as the other nutrients. Dissolved inorganic nitrogen (ammonium plus NOx−) concentrations in the porewater were averaged for the upper 20 cm (Extended Data Table 1b).Net primary production measurements using the EC methodNet carbon dioxide (CO2) fluxes were calculated on the basis of oxygen (O2) fluxes determined using the aquatic eddy covariance (EC) method. In this non-invasive approach, turbulence-induced transport is resolved using high-frequency current meters combined with fast O2 microsensors. Under the assumption of stationarity, the instantaneous turbulent flux contributions are calculated by correlating vertical current fluctuations to oxygen fluctuations. Our EC system was equipped with an acoustic Doppler velocimeter (ADV, Nortek) and ultra-fast responding optode microsensors with a tip diameter of 430 µm (t90  More

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Data collection and preparation—Dutch watersStudy specimensBetween 2006 and 2019, 1457 deceased harbour porpoises in The Netherlands were collected for post-mortem investigations and diet analysis, with the necropsies conducted following internationally standardized guidelines69. For this study focussing on female life history, we selected all females  > 115 cm, as smaller animals may be maternally dependent and can be considered young of the year23. Cases in DCC5, which represent carcass remains, and of which reproductive organs were not assessable or present due to scavenging or incompleteness of the carcasses were excluded. The reproductive organs of the female porpoises  > 115 cm (n = 328/1457) were macroscopically inspected to differentiate between immature and mature animals, with the presence of ovarian corporal scars used as indication of maturity70. Exact age was determined by assessing tooth growth layer groups (GLG) for a subsample of cases (n = 154), according to previously described methods71. Data can be found in STab. 13.Health and nutritional statusPorpoises collected for post-mortem investigation were necropsied with the primary aim to determine the animals’ causes of death and their health status, with the quantity and quality of data and results strongly depending on carcass freshness and completeness as well as other logistical and financial factors69. For this study, we established three proxies based on the findings and metrics taken and assessed at necropsy: a proxy for health status and two proxies for nutritional status. For the proxy for health status, we assessed the cause of death among the mature females (n = 199/328) and divided all cases in two categories. In the first category, we placed all mature females which most likely died as a direct result of incidental bycatch (diagnosed based on the presence of encircling imprints or external incisions, the recent ingestion of prey, and the exclusion of other causes of death, for more details see IJsseldijk et al.63), as a direct result of a predatory attack (diagnosed based on the presence of large, sharp-edged mutilations with associated ante-mortem bite lesions bilaterally on the tailstock, extremities or on the head, for more details see Leopold et al.58) or as a direct result of another acute cause, such as sharp forced trauma or dystocia (obstructed labour, full-term foetus) which did not present signs of significant disease or debilitation. All other animals were placed in the second category, with these mature females displaying evidence of general and significant debilitation, including infectious disease (such as significant parasitism, bacterial, viral or mycotic infections) and/or emaciation. Cases that could not be grouped, mostly as a result of decomposition, were excluded from analyses that included this as a parameter.The first proxy of nutritional status was based on the mean blubber thickness, measured during necropsies in a dorsoventral line on the left body flank just cranial to the dorsal fin, at three locations: dorsal, lateral, and ventral. Blubber thickness in small cetaceans has previously been shown to decrease during periods of fasting72,73 and this metric has been used as proxy of nutrition by others45,74,75,76. However, it should be noted that blubber thickness is not always a good reflection of individual health nor cause of death (e.g., animals dying of acute causes could also be debilitated62,63). There is uncertainty to what extent factors such as age, sex and season naturally influence blubber thickness, and this should be accounted for. Since we focus our analyses on mature females, no further correction for age and sex was done. However, to correct for season, we modelled the mean blubber thickness as a function of Julian date using a generalized additive model (GAM) to allow a smooth effect of the predictor variable (Julian date). This captures the sinus-shaped seasonal variation in blubber thickness which naturally occurs as a result of changing water- and air temperature25,72 (SFig. 5). The residuals of that model were thereby indicative of an adult females’ nutritional status independent of season, and hence they were used as the proxy for nutrition (referred to in the main text as: nutritional status using corBT, Model 1 in STab. 1).The second proxy of nutritional status used the categorical variable “nutritional condition” (NCC), which is assigned during necropsies as good, moderate, or poor. Animals in good NCC generally presented a convex outline on a cranial perspective, no signs of muscle atrophy (abundant skeletal musculature) and presented signs of visceral fat. Animals in moderate NCC generally did not have a fully round outline on a cranial perspective, showed possible signs of muscle atrophy and did not present visceral fat. Animals in poor NCC generally had a concave outline on cranial perspective, with visible aspects of vertebrae and/or scapula externally, an hollow appearance caudal to the skull and signs of muscle atrophy (based on IJsseldijk et al.69). Since this categorial differentiation is collinear with the first established proxy of nutritional status (SFig. 5), it was not used in the same modelling procedures. Therefore, models were run twice, first with corBT and secondly with NCC (for an overview see STab. 1).Pregnancy rate and foetus sizeThe pregnancy rate (PR) was calculated as the proportion of pregnant females in the total sample of mature females (following e.g.,70,77). Pregnancy rates were also calculated separately for the animals in the two different health status categories (see above). To avoid missing the presence of very small, early embryos, samples from the period of conception (June–August23) as well as samples from the period of calving (May–June23) were excluded in the PR calculations. All foetuses were measured during necropsy (of the dam) and a proportion of these were also weighed.Mean energetic density of dietsAs a measure of the quality of prey species constituting the diet of harbour porpoises necropsied in The Netherlands, we calculated the mean energy density of their diet (MEDD). Prey were identified from stomach contents, mostly from otoliths; for each individual prey that could be identified, the fresh mass was estimated (using78 and following29) The energy density (ED) is defined as the energy per kilogram of wet weight of prey8,79. ED values for all prey species encountered were taken from the literature (STab. 7). If for a given prey species no value for ED could be found, the ED of a comparable species (mostly same genus), or the mean value of its family, was used. For species for which multiple ED values were available, values were averaged. ED values reported in kcal were multiplied by 4.184 to convert to kJ (following e.g.80). To calculate the mean ED of the diet for a group of porpoises (MEDD, kJ·g−1, see Table 1) we used:$$MEDD=frac{1}{sum_{i=1}^{n}{M}_{i}}sum_{i=1}^{n}({M}_{i}*{ED}_{i})$$
(1)
where i is the prey species and M the reconstructed prey mass in grams (following8). The reconstructed prey mass per species is multiplied by the species-specific ED and the energy sum is divided by the total mass of all prey, resulting in the MEDD.Data analyses and statistical models—Dutch watersData were explored prior to analyses following Zuur et al.81,82. Data exploration and analyses were performed using R version 3.6.383, with packages ggplot2, grid, gridExtra, rsq, glmTMB, mgcv and ggpubr. Several statistical models were developed (for referencing in the text see overview in: STab. 1).Influences on foetus sizeTo identify which variables influence foetus size, we firstly identified the best measure for foetus size. A Generalized Linear Model (GLM) for foetus length and weight was fitted (Model 2), with weight only available for a subset of all foetuses (n = 34). This model indicated a close relationship between length and weight (R2 of 0.8 for foetus length as a function of mass, SFig. 5), and foetus length was therefore used as representative for foetus size in the subsequent analysis, to increase sample size. GLMs with a Gaussian distribution were used (Model 3). The model selection tested for covariates and their influence on foetus length, with the predictor variables: Julian date to account for foetus length which increases throughout gestation, total length of the mother, health status of the mother, nutritional status of the mother. Interactions between length of the mother and her nutritional status were included following data exploration. Only cases with complete observation of all parameters were included (n = 43). A backwards model selection approach was applied with the drop1 function from the R language used to assess which model terms could be excluded83. The best fitting model was selected using Akaike’s Information Criterion (AIC), which provides a relative measure of the goodness of fit of statistical models. Model validation was done to identify potential violations of model assumptions by inspection of normalized residuals and assessment of residual probability plots. Likelihood profile confidence intervals (95%) and odds ratios of the most optimal model were calculated. Models were run twice, first using the first proxy of nutritional status based on blubber thickness corrected for season (corBT) and secondly using the nutritional condition category (NCC), taken into account blubber thickness, visceral fat and muscle mass (for full descriptions, see above).Influences on pregnancyTo identify which variables influence pregnancy, we firstly coded all mature, pregnant females as 1 and all mature, non-pregnant females as 0. Next, GLMs with a binomial error distribution and logit link were used (Model 4) to test the influence of included covariates on the likelihood of pregnancy. Only cases with complete observations of all parameters were included (n = 65). The predictor variables included in the saturated model were age, year to assess temporal variance, month to assess seasonal variance, health status (proxy, categorical), and nutritional status. Interactions were added following data exploration: between health and nutritional status, between the health status and year and health status and month. Model selection, validation and interpretation was conducted following the protocol previously described above. Models were run twice, first using the first proxy of nutritional status based on for season corrected blubber thickness (corBT, numerical) and secondly using the nutritional condition category, taking into account blubber thickness, visceral fat and muscle mass (NCC, categorical) (for full descriptions, see above).Age at sexual maturityThe age at sexual maturity (ASM), or age at 50% maturity, was determined using binomial logistic regression models. Maturity, coded as 1 for mature females and 0 for immature females, was modelled as a function of age (in years) to assess ASM (n = 154, Model 5). The model was fitted using a binomial error distribution and logit link, as is appropriate for binary data and the ASM was estimated by calculating the negative of the slope over the intercept.Assessment of porpoise life history and environmental condition globallyThe life history response variables assessed were PR and ASM, which were obtained from 17 different studies. The earliest study was conducted between 1941 and 1943, but the majority of the studies were performed between 1980 and 2019 (including the present study). The environmental predictor variables used were quality of diet, expressed as mean energy density of diet (MEDD), cumulative human impact (CHI) with data on climate change, fishing, land-based pressures, and other human activities, and lastly chemical pollution expressed as polychlorinated biphenyls (PCBs). Fifteen diet studies were used, ranging from 1985 to 2019 (including this study). One comprehensive study was used to obtain the CHI information for the year 2008. A total of 21 studies reporting PCB levels in harbour porpoises, conducted in the period 1971–2019 (including the present study) were collated. Details below and in STab. 1.Life historyFor PR the following were tabulated: (1) the number of pregnant females out of the total number of mature females in each study, (2) the determined conception period and whether this was accounted for in the calculation of the PR, (3) the method to assess pregnancy, which was either based on the presence of a foetus or presence of a corpora lutea (CL), and (4) the source of the specimens: either directly from fisheries, strandings including trauma cases, or a combination thereof (STab. 3). For ASM we provide: (1) how ASM was assessed in each study, and (2) the standard error (SE) or confidence interval (CI), if reported (STab. 4).Energy density of preyA literature search was performed for diet studies from stomach contents of porpoises from or near the study areas where PR and ASM were determined. When multiple diet studies were available the study was selected that best corresponded to the time frame at which PR and ASM were calculated. For the diet studies which reported the reconstructed prey mass in grams we used formula (1) (STab. 8). When the prey mass was reported as a percentage of relative abundance in terms of estimated biomass of prey (%M), we multiplied %M by the ED of the prey species and divided the total %M (STab. 9), using:$$MEDD=frac{1}{sum_{i=1}^{n}{mathrm{%}M}_{i}}sum_{i=1}^{n}({mathrm{%}M}_{i}*{ED}_{i})$$
(2)
For the studies where the %M was presented in a bar chart, we measured the %M using digital callipers.Cumulative human impactAn ecosystem-specific, multiscale spatial model containing high resolution data on the intensity of human stressors and their impact on marine ecosystems was developed by Halpern et al.12,14 as part of their Ocean Health Index project. CHI values derived by this model are based on fourteen stressors related to human activities from four primary categories: (1) land-based drivers, including nutrient pollution runoff, organic chemical pollution runoff (pesticides), direct impact of humans (density of coastal human populations), and light; (2) five types of (commercial) fishing, including commercial demersal destructive, commercial demersal non-destructive high bycatch, commercial demersal non-destructive low bycatch, pelagic high bycatch, pelagic low bycatch, and artisanal; (3) climate change, including sea surface temperature, ocean acidification and sea level rise; and (4) shipping. Extensive descriptions of these drivers are published in the methods and supplementary material of Halpern et al.12,14, including information on the origin and validation of the data.For this study we used the global CHI dataset that is publicly available via the Knowledge Network for Biocomplexity14. Data on CHI was based on the year 2008. We extracted the CHI scores for each of our study areas at ~ 1 km2 resolution and calculated the min, max, mean and median values. To do so, we defined our study areas using the standard georeferenced marine regions as published under the Flanders Marine Institute84. In most cases we combined two or more regions from the database to get full coverage of the study area, but for the study areas where the marine regions did not provide full coverage, we used a manually created polygon. The list of regions is given in STab. 15. For areas with more than one life history study (Denmark and The Netherlands) we used the newest studies since these provided the better match to the time of the CHI score calculation (STab. 2).Chemical pollutionPolychlorinated biphenyls were not included in the list of organic polluters by Halpern et al.14. However, PCBs have been specifically associated with reproductive impairment in many marine mammal species16,17,18,30,50, therefore the correlation with life history parameters for this industrial organic pollutant was assessed separately. Data was retrieved from the International Whaling Commission’s (IWC) ‘POP Contaminants Trend Explorer’ tool, hosted on the portal of the Sea Mammal Research Unit (SMRU, University of St. Andrews, Scotland). This tool is established under the IWC Scientific Sub-Committee on Environmental Concerns (IWC SC/68A 2019) as part of the IWC Pollution 2020 Initiative and includes data from scientific publications from the 1970s–2000s34,43. The database was provided by the tool manager and included data restricted to adult males, to reduce the bias of biotransfer of chemicals, which occurs during gestation and lactation in females35. The tool reports PCB concentrations in blubber, which is the most commonly assessed tissue in marine mammals for studying the burden of the highly lipophilic and stable PCB compounds47. PCB concentrations that were measured in porpoises in the same areas from which life history parameters were verified with the literature and included. In addition, the literature was searched for PCB analyses of harbour porpoises published in the 2010s, as well as own institutional databases, and data added to align time frame, where possible, with time frame of conducted life history studies (STab. 2).The presentation of concentrations of pollutants was based on either wet weight (ww) or lipid weight (lw). To allow comparison, the datapoints need to be converted to one common unit, with lw most frequently reported. Studies reporting only ww or dry weight were not included. Studies reporting ww and percentage of lipids (%lipids) were converted to lw, using:$$lw=frac{ww}{%lipids}*100$$
(3)
The datapoints were converted to mg/kg lw for all studies and the mean ∑TotalPCB is reported per area.The variance of the sum of congeners reported ranged from ∑6PCBs up to ∑99PCBs, with several older studies reported Aroclor mixtures. Data per congener was however largely not available in literature. We therefore present two mean ∑PCB datapoints: firstly including all studies regardless of the sum of congeners or mixtures (referred to as PCB1), and secondly limited to studies reporting ∑17-99PCBs (referred to as PCB2).Statistical models for global assessmentFor the analyses we restricted to study areas with complete observations of the environmental conditions to compare models. A GLM fitted with a binomial distribution and logit link was used to determine the effect of environmental conditions on pregnancy rates (Model 6). The response variable was the number of pregnant females (Npreg) in the total number of females (Ntotal) (grouped binomial data, STab. 3). Since the differences between study areas can be large because of unknown effects, an individual normal random effect for area was added on the logit scale. Another GLM was conducted to determine the effect of the three environmental conditions on age at sexual maturity (Model 7) fitted with a Gaussian distribution and weighed by sample size (Ntotal) (STab. 4). This model was applied twice using two individual predictor functions: first, with the predictor variables MEDD, CHI and PCB1 and secondly with the predictor variables MEDD, CHI and PCB2. The latter restricted the analyses to a smaller number of study areas due to missing data, but it reduced some of the bias because of very small ( More

CORRESPONDENCE
02 November 2021

For NGOs, article-processing charges sap conservation funds

Kevin A. Wood

 ORCID: http://orcid.org/0000-0001-9170-6129

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Julia L. Newth

 ORCID: http://orcid.org/0000-0003-3744-1443

1
&

Geoff M. Hilton

 ORCID: http://orcid.org/0000-0001-9062-3030

2

Kevin A. Wood

Wildfowl & Wetlands Trust, Slimbridge, UK.

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Wildfowl & Wetlands Trust, Slimbridge, UK.

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Geoff M. Hilton

Wildfowl & Wetlands Trust, Slimbridge, UK.

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The shift from a ‘reader pays’ to an ‘author pays’ model of scientific publishing presents a financial threat to environmental non-governmental organizations (eNGOs). Many of these support, conduct and publish applied research on real-world solutions to the planet’s most pressing challenges. Funded mainly by donations, eNGOs must now choose between taking conservation action and publishing more research papers.

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Nature 599, 32 (2021)
doi: https://doi.org/10.1038/d41586-021-02979-5

Competing Interests
All three authors are current employees of the Wildfowl & Wetlands Trust, an environmental non-governmental organization that is actively involved in undertaking and publishing research.

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Data setsMonthly climate data of maximum and minimum temperature, dewpoint temperature, and precipitation at a 1/24th degree horizontal resolution from 1950 to 2020 was acquired from the Parameterized Regression on Independent Slopes Model (PRISM)44. Monthly surface downward shortwave radiation and 10-m wind speeds at a 0.25-degree horizontal resolution were acquired from ERA-545 for the same period and bilinearly interpolated to the PRISM grid. Monthly data for the same variables from a single ensemble member from each of 30 climate models participating in the Sixth Coupled Model Intercomparison Project (CMIP6) were acquired from the historical climate experiment for 1950–2014 and from the SSP2-45 experiment for 2015–2050 and interpolated to a common 1.0-degree horizontal resolution grid (Supplementary Table 4).Following Abatzoglou and Williams, we calculated three proxies of aridity using monthly climate data: mean vapor pressure deficit (VPD), Penman-Monteith reference evapotranspiration (ETo), and climatic water deficit (CWD46, defined as ETo minus actual evapotranspiration3). We modified ETo to account for potential reduced stomatal conductance due to increasing atmospheric carbon dioxide, which reduces surface resistance to evapotranspiration. We made this modification following the method of Yang et al.47. Importantly, the effect of CO2 on surface resistance at the scale of the western US is highly uncertain and this method derives the strength of this effect from earth system models. Each index was calculated as follows. At each grid cell, we calculated mean Mar–Sep VPD, the sum of Mar–Sep ETo, and Jan-Dec CWD; each of these time series was standardized to the 1991–2020 baseline using z-score transformations to create a fuel aridity index f for each grid cell. The regionally averaged fuel aridity index F was calculated by first taking the average of f over grid cells that have a majority of land classified as forest or woodland in the LANDFIRE environmental site potential product48. We then re-standardized F relative to the 1991–2020 reference period and applied equidistant quantile mapping49 to each model. The latter ensures that the distributions of modeled Z match those of observed Z for the 1991–2020 period while preserving changes in Z from this reference period. Herein we used CWD for F because it presents a more balanced view of precipitation and atmospheric demand than VPD or ETo alone, exhibits strong links to the forest-fire area over the observational record, and has more conservative increases in fire under future climate (Supplementary Fig. 2). The variance explained in forest-fire area when defining F as VPD, ETo, and detrended CWD is presented in Supplementary Table 1. We note that our approach does not explicitly incorporate daily meteorology such as the number of dry days or critical fire-weather patterns10 beyond that already included in F.Burned area data from wildland fires were acquired from Monitoring Trends in Burn Severity (MTBS) during 1984–201850 and from the version 6 MODIS burned area dataset during 2001–202051. The forested burned area was aggregated by lands classified as forest or woodland48. MTBS includes primarily fires ≥404 ha that comprises >95% of burned area in the region52. We further excluded areas in the unburned-to-low burn severity class53 as well as fires classified as prescribed burns in MTBS. Further, we did not include forested area treated by prescribed fire as a contemporary area for prescribed fire is more than an order of magnitude less than that of forest-fire area41. Forest-fire area estimates for 2019–2020 were obtained using adjusted burned areas from MODIS based on a linear model that relates MODIS and to the MTBS forest-fire area time series during the overlapping 2001–2018 period26.Experimental designWe focus on macroscale climate–fire models operating at the scale of the entire western US forested area. While there is value in spatially refined models, efforts to parameterize empirical relationships at localized scales can be limited by the stochastic nature of ignitions and fire weather—particularly in locations with long fire return intervals with zero-inflated distributions of annual burned area. Strong interannual relationships between fuel aridity and strain on national fire suppression resources shared across the region highlight the implicit value in considering larger spatial scales54. The macroscale approach is further justified because the leading mode of variability in fuel aridity across forested land is a commonly signed regionwide pattern that is strongly correlated (r2 = 0.79) to the logarithm of forest-fire area (Supplementary Fig. 3).Static modelFollowing previous empirical models of annual forest-fire area3,25, we first consider a static model of western US annual forest-fire area (FFA) based on F (fuel aridity) of the form:$${{{{{rm{log }}}}}}left({{{{{{mathrm{FFA}}}}}}}(t)right)={alpha }_{{{{{{mathrm{s}}}}}}}+{beta }_{{{{{{mathrm{s}}}}}}}Fleft(tright)+{{{{{rm{varepsilon }}}}}},$$
(1)
where t is the year, αs and βs, are regression coefficients, and ε represents an error term. We use annual CWD for F as it accounts for precipitation and atmospheric demand, exhibits strong interannual relationships with FFA, and provide more conservative estimates of projected changes in aridity and thus area burned than other aridity metrics such as VPD3,7,12. The error term ε is drawn from the population of the log-residual of observed minus modeled FFA. This error term represents variability not captured in the FFA–F relationship (e.g., extreme fire-weather conditions, human ignitions) that is important for the full distribution of FFA.Dynamic modelsThe contemporary climate–fire relationship in Eq. 1 should persist with increased F until increased burned area and severity cause fuel limitations15. Fire-fuel feedbacks that alter the climate–fire relationship primarily occur through temporary reduction of fine fuels; such feedbacks can reduce the burning potential for approximately three decades post-fire38,55. Further, longer-lived reductions in the forest-fire area can occur when forests do not recover from fire and instead transition to non-forest vegetation that can still carry fire. However, constraints on the area burned imposed by fire-fuel feedbacks are weakened by concurrent drought, which allows the fire to propagate across sparser fuels, and can markedly shorten the window of reduced burning18.We incorporate these effects through a term L, which represents the fraction of contemporary forested land that is incapable of carrying fire in a predominately forested environment in a given year, in a dynamic model of the form:$${log }left(frac{{{{{{{mathrm{FFA}}}}}}}}{1-Lleft(tright)}right)={alpha }_{{{{{{mathrm{d}}}}}}}+{beta }_{{{{{{mathrm{d}}}}}}}Fleft(tright)+{{{{{rm{varepsilon }}}}}},$$
(2)
where the response of log(FFA) to fuel aridity reduces as a function of L. We present various potential forms and strengths of fire-fuel feedbacks in L that are guided by the ecological literature and account for post-fire tree regeneration failure, fuel limitations imposed by recent fire history, and waning of fuel limitations during drought18,22,23,24. L is influenced by semi-permanent limitations due to failure of post-fire forest regeneration (Lrf), and temporary limitations due to recent fire history (Lf):$$Lleft(tright)={L}_{{{{{{{mathrm{rf}}}}}}}}left(tright)+{L}_{{{{{{mathrm{f}}}}}}}(t).$$
(3)
Importantly, L is poorly constrained and likely varies in geographically and temporally complex ways18,34. For example, L can differ for a fixed fraction of recently burned forest. A relatively small L implies weak feedbacks allowing forests to more easily reburn. A relatively large L implies strong feedbacks, for example, where heterogeneous fire effects create patch mosaics that constrain fire spread even though there is ample fuel. Finally, the age threshold for L may decrease with continued climate change, with some indications that recent fires burned through forests , 2end{array}right.,$$
(4)
where μ is set at 0.1 (Eq. 4 is plotted in Supplementary Fig. 4a). Hence, the fraction of forested land that is semi-permanently ineligible to carry forest fire because previously burned forest did not regenerate as forest (Lrf) is the cumulative sum of the product of annual FFA and ρ since 1984:$${L}_{{{{{{{mathrm{rf}}}}}}}}left(tright)=mathop{sum }limits_{i=1984}^{t}frac{rho left(tright){{{{{{mathrm{FFA}}}}}}}(t)}{T},$$
(5)
where T refers to the contemporary area of forested land48. Note that Eq. 4 and μ can be modified to account for the diversity of species-specific responses at local-to-regional scales given the acknowledgement that some species are more resilient than others and local plant water stress alters regeneration probabilities58,59. Overall, Lrf as parameterized here resulted in values approaching Lrf ~0.01 by 2050, suggesting that the inability of trees to regenerate post-fire is a minor contributor to fire-fuel feedbacks through mid-century. Modifications to the parameters in Eq. 4 resulted in only minor differences in projected FFA (Supplementary Table 3).Temporary fire-fuel feedbacks L
f
Most studies in forested environments show strong fire-fuel feedbacks in the first 5–10 years post-fire55,60. This temporary fire-fuel feedback, which we refer to here as Lf, tends to wane after 10 years60, with the longevity τ of the fire-fuel feedbacks varying geographically, from as short as ~15 years in warmer sites in the southwest to over ~30 years in cold mesic systems in the northern Rockies18. Herein, we use a baseline τ = 30 years, which results in a conservative estimate of future area burned.We consider two forms for how Lf incorporates information on annual fire histories over the previous τ years: a constant feedback and a fading feedback. These forms of Lf are defined below in Eqs. 6 and 7 and plotted in Supplementary Fig. 4c.In the case of the constant feedback, the effect of burned area on Lf remains constant over the τ years following fire. At the scale of the whole western US forested area, the constant form, therefore, assumes that the transient limitation is simply proportional to the total FFA over the preceding τ years:$${L}_{{{{{{mathrm{f}}}}}}}left(tright)=gamma mathop{sum }limits_{i=-tau }^{-1}frac{{{{{{{mathrm{FFA}}}}}}}(i)}{T}.$$
(6)
In Eq. 6, parameter γ represents the strength of the feedback, described in more depth below.The fading feedback form of Lf more heavily weights the contribution from recent FFA compared to older FFA. At the scale of the whole western US forested area, this form applies constant weight to FFA in the five most recent years given strong fire-fuel feedbacks of recent fires, and increasingly reduces the contributions from prior years based on a sinusoid function:$${L}_{{{{{{mathrm{f}}}}}}}left(tright)=gamma frac{mathop{sum }nolimits_{i=-5}^{-1}{{{{{{mathrm{FFA}}}}}}}left(iright)+mathop{sum }nolimits_{i=-tau }^{-6}{{{{{{mathrm{FFA}}}}}}}left(iright)ast left[1-{cos }frac{pi left(-i-5right)}{tau -5}right]/2}{T}.$$
(7)
Given the uncertainty in the efficacy of the fire-fuel feedback, we present results using both the constant and fading formulations for the temporary fire-fuel feedbacks.We additionally considered three different fuel-limitation strengths γ in Eqs. 6 and 7 to account for direct and indirect potential effects of past fires: γ = 0.5, referred to as weak; γ = 1, referred to as moderate; and γ = 1.5, referred to as strong. For the weak (γ = 0.5) fuel-limitation case using the constant feedback model, the fractional forested area ineligible to burn is only half of the total area burned in the past 30 years, indicating that half of recent burned areas can reburn. For the strong-constant fuel-limitation case, the forested area ineligible to burn post-fire exceeds the total recent burned area by 50%. An example of a strong fuel limitation is a burn mosaic with reduced connectivity that constrains the ability of subsequent fire spread into the adjacent forest that did not burn in the previous τ years. We considered higher values of γ, but these yielded degraded cross-validation skills when modeling the historical period (Supplementary Table 2).Longevity of fire-fuel feedbacks during droughtFinally, some temporary fuel limitations can be overcome during extreme fire-weather conditions and during periods of drought. For example, while reduced fuel loads in a post-fire landscape serve as an effective barrier for fire propagation under moderate fuel aridity, the fire spread probability increases with increasing F34. Studies have found that the longevity of fire-fuel feedbacks was a third shorter during periods of extreme drought than in periods without drought stress18,34. For example, there is evidence of short-interval (95% of the iterations had bias CE  > 0, >95% of the iterations had r  > 0, and the inner 95% of the simulations included a bias of 0.Supplementary Table 2 shows that the static model and many of the dynamic models have significant cross-validated skills. However, skill decreased in the dynamic models as the feedback strength increases. While the weak dynamic feedback models had similar cross-validation skill as the static model, dynamic models with very strong feedbacks (γ ≥ 2) had sizeable underpredictions in FFA by up to 46% for the validation period. Hence, we excluded such parameters from the further analysis given that such results were incongruent with the observational record.Three statistical metrics of annual variability of FFA were calculated for both static and dynamic models. First, we used generalized extreme value theory to estimate recurrence intervals for FFA greater than equal to that of the 2020 fire season. Second, we calculated the interquartile range (IQR) in modeled FFA to examine changing interannual variability. Lastly, we examined the percent of years with modeled FFA below the 1991–2020 observed median as a measure of quiescent fire years. Calculations were performed separately for each climate model for 1991–2020 and 2021–2050. More

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2D mandible outlinesIn17, elliptical Fourier analysis (EFA) is employed to investigate the lateral shape difference between 106 fossil mandibles of 5 groups: A. robustus ((n=7)), H. erectus ((n=12)), H. heidelbergensis ((n=4)), H. neanderthalensis ((n=22)), and H. sapiens ((n=61)). In the presented work, the authors apply the shape-changing chain method to the same dataset. Twelve samples were suppressed, including all 7 A. robustus samples, 4 H. neanderthalensis samples, and one H. sapiens sample. Therefore, 94 mandible profiles of 4 groups of the ancient human are analyzed: H. erectus ((n=12)), H. heidelbergensis ((n=4)), H. neanderthalensis ((n=18)), and H. sapiens ((n=60)). The dataset is in the form of Cartesian coordinates of points along the mandible boundary. Note that the shape-changing chain method does not require pre-alignment of curves or removal of the size factor. However, the mandible profile dataset the authors obtained had already lost the information of the original sample sizes. Therefore, only normalized mandibular shapes are compared herein. Figure 5 illustrates the mean shape of each group of profiles by aligning all profiles in the set using a standard Procrustes superimposition (PS) which includes translation, scaling, and rotation of the profiles27.Figure 5Mean mandibular shapes of samples from H. erectus (red circles), H. heidelbergensis (blue triangles), H. neanderthalensis (green squares), and H. sapiens (black diamonds).Full size imageUsing the relative angle method ((k=150), (T=20^circ)), five apices are reserved, constituting six sub-profiles for each profile. An illustration of the location of the apices on a mandible profile is shown in Fig. 6. Each sub-profile is then matched with a shape-changing chain individually. The determination of segment type vector of each sub-profile refers to the growth mechanism of mandible proposed by Enlow et al.28. As shown in Fig. 6, the mechanism of mandible growth involves bone resorption (indicated by the arrows pointing towards the mandible contour) and bone deposition (indicated by the arrows pointing out of the mandible contour). Although the whole mandible’s displacement direction is forwards and downwards, the reconstruction of the ascending limb is generally backwards and upwards. G-segments and C-segments are employed to approximate the growing portions in target profiles and characterize the difference in profile lengths.Figure 6A profile ((j=1)) from the H. erectus group: Five apices (red circle) are located using the relative angle method ((k=150), (T=20^circ)) and divide the profile into six sub-profiles. The arrows represent the growth pattern of the mandible28.Full size imageNote that the growths of the inferior edge of the mandibular body and the posterior edge of the mandibular ramus are more significant than the rest parts of the mandible profile. The 94 mandible profiles are then matched with a shape-changing chains using the following scheme. The segment vectors for the first, second, third, and sixth sub-profiles are defined as (left[{text{MGM}}right]) alike, and the segment vectors for the fourth and fifth sub-profiles are both defined as (left[{text{MCGM}}right]), where the C-segments and G-segments are used to capture the difference in arc lengths. Therefore, the overall segment vector is$$mathbf{V}=left[text{M G M M G M M G M M C G M M C G M M G M} , right]text{,}$$where there are a total of 20 segments—12 M-segments, 2 C-segments, and 6 G-segments. After the segment type vector is defined, the shape-changing chain is generated to match the target mandible profiles and then is optimized for each sub-profile. The maximum and mean error of all profiles of the final matching result are ({E}_{text{max}}=8.0863) and (overline{E }=0.6009) units, respectively. Figure 7 shows the best (a), the average (b), and the worst match (c) according to ({tilde{E }}_{j}). Note that in the worst match, the G-segment at the condyle (head) of the mandible causes the largest matching error. This is because the third primary segmentation point (between the two M-segments that follow) identified using the relative angle method for this specific profile is not at the tip of the condyle as the majority of the profiles.Figure 7The fitting result of 94 human mandibles. (a) The best match (the 4th profile—H. erectus, ({tilde{E }}_{4}=0.3924)); (b) The match with error closest to (overline{E }) (the 6th profile—H. erectus, ({tilde{E }}_{6}=0.6006)); (c) The worst match (the 13th profile—H. heidelbergenis, ({tilde{E }}_{13}=1.0771)).Full size imageThe orientation difference between two neighboring segments reflects the rotational angle between them, and thus are employed in the statistical analysis in the next step. Denote the direction of a vector (mathbf{u}={left{{u}_{x},{u}_{y}right}}^{T}) as (angle left(mathbf{u}right)), then the orientation change between the ({e}{text{th}}) and the ({(e+1)}{text{th}}) segments on the ({j}{text{th}}) profile is calculated as the difference between the direction of the last piece on the ({e}{text{th}}) segment and the direction of the first piece on the ({(e+1)}{text{th}}) segment$${sigma }_{j}^{e}=angle left({overline{mathbf{z}} }_{{j}_{2}}^{e+1}-{overline{mathbf{z}} }_{{j}_{1}}^{e+1}right)-angle left({overline{mathbf{z}} }_{{j}_{{m}_{j}^{e}+1}}^{e}-{overline{mathbf{z}} }_{{j}_{{m}_{j}^{e}}}^{e}right), forall e=1,dots ,q-1 j=1,dots ,p.$$
(10)
In the mandible example, 19 angular variables are generated from 20 segments. As in17, a stepwise discrimination analysis (DA) is conducted (in IBM SPSS 22) to figure out the relationship among the four homo groups. DA is a supervised classification method and returns (g-1) canonical components among (g) groups of samples29. Figure 8 shows the convex hull of four homo genus plotted with the first and the second canonical components. The three main groups: H. erectus, H. neanderthalensis, and H. sapiens, are separated from each other in the direction of the first canonical component. H. heidelbergensis and H. neanderthalensis have an overlap in the direction of the second canonical component. In stepwise DA, leave-one-out cross-validation (LOOCV) is applied to verify the stability of the linear model. As a result, the prediction accuracy is 91.5% and the cross-validation accuracy is 80.9%. This DA result suggests that the shape-changing chain method is useful in analyzing 2D shapes. The classification matrices of original prediction and LOOCV are presented in Table 2, showing the details of discrimination of the four mandibular shape groups.Figure 8Canonical plot of the 94 human mandibles from four groups (H. erectus, H. heidelbergensis, H. neanderthalensis, and H. sapiens) based on the orientation changes between segments (19 variables).Full size imageTable 2 Classification matrices of the original DA and cross-validated prediction of 94 human mandibles.Full size tableNote that the classification results as shown in Fig. 8 and Table 2 are in accordance with the results obtained with EFA in17. The high misclassification rate of H. heidelbergensis and its distribution on the canonical plot are also in keep with the mainstream opinion that H. heidelbergensis is a chronospecies evolving from H. erectus and is considered as the most recent common ancestor (MRCA) between H. sapiens and H. neanderthalensis. In the work of Lestrel et al. based on EFA, 20 harmonics are employed to match 106 mandibular shapes, producing 82 Fourier descriptors17. Then, 12 distances from the centroid to specified points on each mandible’s contour are used in statistical analysis. Compared to their study, the shape-changing chain method generates only a total of 28 variables (20 orientations of all segments and 8 arc lengths of C-segments and G-segments). The differences of orientations between neighboring segments is then calculated and generates 19 variables to be analyzed in stepwise DA. Table 3 shows a comparison of the variables generated in the approximation of curves and used for statistical analysis with the shape-changing chain method and EFA. The shape-changing chain method performs a satisfying approximation result of the mandibular shapes with much fewer variables compared with EFA.Table 3 Numbers of variables used in the shape-changing chain method and in EFA17 for fitting and analyzing human mandible profiles.Full size table2D leaf outlinesLeaf classification is a typical problem that has been studied with various methods, such as artificial neural networks (ANN)30, image moments31, and EFA9. In addition, many leaves have a symmetrical shape creating issues for effective EFA12. Using the shape-changing chain method, the fitting result reveals the growth of portions on the contour and the rotation between them. This kind of information can be used in statistical analysis. Although other methods which also make use of non-shape information (size, color, etc.) have been very convenient and efficient in recognizing leaf genera, leaf matching and classification remains a problem to test the ability of the shape-changing chain method to fit and compare profiles with complicated and largely varying shapes. In this example, nine groups of 145 leaves are studied (see the groups and the number of samples in each group in Table 3). The original scanned and binarized images of the nine genera of leaves are shown in Fig. s1. The contours are traced using the Moore-Neighbor method32 and then smoothed with the MATLAB cubic spline interpolation (see Fig. s2). All leaf profiles of their original sizes are analyzed. The arc lengths of the profiles range from 1141.5 units to 8433.1 units, the areas of the leaves range from (6.1671times {10}^{4}) units2 to (1.4615times {10}^{6}) units2.Applying the relative angle method, a number of apices are recognized on each leaf contour. These apices are the primary segmentation points that determine the boundaries of sub-profiles on leaf contours. Note that the shapes of leaves from different groups vary significantly, therefore the point interval and angle threshold used for locating apices varies from group to group. For some groups, the numbers of apices identified on different samples may be different too. Table 4 shows the parameters used for identifying apices as well as the minimum and maximum numbers of apices identified on samples for each group.Table 4 Parameters used for identifying apices on leaf contours and the number of apices identified for each group.Full size tableIn order to maintain homology, supplementary segmentation points are added to divide all sample profiles into the same number of portions. There is no need to add more segmentation points on the profile that contains the most number of apices (red oak, (j=101)), therefore the total number of segmentation points on each profile is determined to be 34, dividing each profile into 35 portions. In order to reduce the matching error, supplementary segmentation points are distributed as evenly as possible in sub-profiles formed by the primary segmentation points (original apices) using a method developed based on a genetic algorithm (GA). In this problem, the locations of the supplemented segmentation points on the ({j}{text{th}}) profile are determined through the fitness function determined as follows$${F}_{j}=sum_{e=1}^{q}{left({k}_{j}^{e+1}-{k}_{j}^{e}-frac{{N}_{j}-1}{q}right)}^{2}.$$
(11)
In Eq. (11), the number of pieces contained in the ({j}{text{th}}) profile (({N}_{j}-1)) divided by the number of portions (q) yields the average number of pieces in each portion. (({k}_{j}^{e+1}-{k}_{j}^{e})) is the number of pieces contained in the ({e}{text{th}}) portion confined by the ({e}{text{th}}) and the ({(e+1)}{text{th}}) segmentation points on the ({j}{text{th}}) profile. After encoding the locations of all segmentation points in the GA and several rounds of optimization based on a certain scale of crossover and mutation, the set of supplementary segmentation points that minimizes the fitness function, Eq. (11), is determined. The original apices (red circles) and supplementary segmentation points (green circles) distributed on samples from different groups are shown in Fig. 9. In this example, each profile is finally divided into 35 portions.Figure 9The original apices (red circles) and supplementary segmentation points (green circles) on leaf contours. (a) Cherry, (b) Dogwood, (c) Gum, (d) Hickory, (e) Mulberry, (f) Red maple, (g) Red oak, (h) Sugar maple, (i) White oak. For each group, the sample that contains the most original apices is presented.Full size imageThe length of each portion varies among profiles, thus M-segments are not applicable. In addition, some portions still contain local burrs and sharp corners, which would not be matched well by C-segments. Therefore, each portion is matched by a G-segment, and the segment vector contains 35 G-segments. The maximum and mean error of 145 leaf profiles are ({E}_{text{max}}=60.6063) and (overline{E }=8.7062) units, respectively. Figure 10 shows the best, the average, and the worst matching results of the leaves according to ({tilde{E }}_{j}). More matches of nine genera of leaves are illustrated in Fig. s3. The result show that given the distribution of apices (primary segmentation points that determine sub-profiles), the GA strategy can automatically determine the distribution of supplementary segmentation points along a profile. With the segmentation points generated from this process, the shape-changing chain matches the leaf contours with small error compared to the random segmentation in the previous study.Figure 10The fitting results of 145 leaves. (a) The best match (the 62nd profile—hickory, ({tilde{E }}_{62}=0.8748)); (b) The average match (the 88th profile—red maple, ({tilde{E }}_{88}=4.0866)); c The worst match (the 110th profile—red oak, ({tilde{E }}_{107}=9.7866)).Full size imageFor classification analysis, 34 orientation differences between neighboring segments are calculated using Eq. (10). Three more variables are employed: The number of primary segmentation points, the number of burrs (detected using the relative angle method with (k=50) and (T=30^circ)), and the arc length of each profile. This sums up to a total of 37 variables. A stepwise DA is performed to classify the 145 leaf samples, and 22 out of the 37 variables are selected for analysis. The variances of the first three canonical functions are 73.5%, 13.3%, and 7.5%, which add up to 94.3% in total. Figures 11 and 12 illustrate the 2D and 3D canonical plots of the nine genus of leaves based on the first three canonical components. The plots show that gum, red maple, and white oak are distinctively separated from other groups. Cherry and mulberry are partially overlapped in the directions of canonical Roots 1 and 2 for their similar overall shapes and serrated edges. There is also an overlap between dogwood and hickory in the directions of canonical Roots 1 and 3 for their similar shapes and smooth edges. The prediction accuracy is 98.6%, and the leave-one-out cross-validation is 97.9%. Only two samples of cherry are misidentified as mulberry, and one sample of hickory is discriminated as dogwood. The DA results reveal that the shape-changing method is capable of fitting a large number of profiles that have complicated shapes and different sizes, as well as generating useful variables for statistical analysis. The leave-one-out cross-validation accuracy suggests that this method is also effective with fewer variables. In addition, the shape-changing chain method enables direct observation and comparison of variables that have physical meanings, such as the relative angles between segments.Figure 11The 2D Canonical plots of nine genus of leaves based on 22 variables.Full size imageFigure 12The 3D Canonical plot of nine genus of leaves based on 22 variables.Full size image3D cranial suture curvesThe shape-changing chain method is now applied to 3D suture curves on human infants’ skulls from a study of coronal synostosis18,19. The dataset contains 63 samples categorized into 4 groups, including left unicoronal synostosis (LUCS, (n=8)), right unicoronal synostosis (RUCS, (n=19)), bicoronal synostosis (BCS, (n=16)), and unaffected cases ((n=20)). The original data of each sample consist of 209 anatomical landmarks and curve semilandmarks located on the skull surface, especially along some anatomical lines as sutures. In this work, three curves that characterize the skull deformation are selected for analysis: the coronal suture curve, the lambdoid suture curve, and the sagittal curve which is comprised of anatomical landmarks and curve semilandmarks located on the metopic suture, the sagittal suture, and the mid-line on the occipital bone. Figure 13 shows the three suture curves on a skull surface.Figure 13The location of the coronal suture (magenta), sagittal curve (blue) and the lambdoid suture (red) on a human infant skull. The intersection points between sutures, P1 and P2, divide the sagittal curve into three sub-profiles and the lambdoid suture into two sub-profiles.Full size imageBCS occurs when the coronal sutures on both sides of the skull fuse prematurely, causing the overall head shape to become broad and short. In this case, the relative location of the lambdoid suture on the skull will move forward compared to the unaffected cases, but its shape is not affected as obviously as the coronal suture or the sagittal curve which are directly affected by coronal synostosis. In order to investigate the relative location and orientation in addition to the shape of the suture curves, a standard Procrustes superimposition is performed on the original data so that all skulls represented by the 209 landmarks and semilandmarks are scaled to the same size and aligned. Figure 14 illustrates the mean shapes of the sagittal curves and the lambdoid sutures of each group. It can be observed that for BCS cases, the sagittal curve is shorter in the anterior–posterior direction, the coronal suture becomes wider in the left–right direction, and the lambdoid suture is longer and positioned relatively forward. These differences are in accordance with the overall wider and shorter BCS skull shape. As for LUCS and RUCS cases, all three curves display a symmetrical shape deformation or orientation change about the skull symmetry plane (X = 0).Figure 14Mean shapes of (a) the sagittal curves, (b) the coronal suture, and (c) the lambdoid sutures of four groups: LUCS (red dotted line), BCS (blue solid line), RUCS (green dotted dashed line), and unaffected cases (black dashed line). Notice the symmetry of the suture curves about the skull symmetry plane (X = 0).Full size imageThe anatomical landmarks P1 and P2 (Fig. 13) which are the intersection points between the sutures, are selected as the primary segmentation points. Thus, the sagittal curve, the coronal suture, and the lambdoid suture are divided into three, two, and two sub-profiles, respectively. Since the coronal suture and the lambdoid suture grow symmetrically about the skull symmetry plane, their segment type vectors for two sub-profiles should be symmetric, respectively. In addition, each segment type vector should contain a G- or H-segment to characterize the growth. The segment type vectors of the three curves are designated as:

Sagittal curve: (left[begin{array}{cc}{text{M}}& {text{G}}end{array}right]), (left[begin{array}{cc}{text{M}}& {text{H}}end{array}right]), (left[begin{array}{cc}{text{M}}& {text{G}}end{array}right]);

Coronal suture:(left[begin{array}{ccc}{text{M}}& {text{G}}& {text{H}}end{array}right]), (left[begin{array}{ccc}{text{H}}& {text{G}}& {text{M}}end{array}right]);

Lambdoid suture: (left[begin{array}{cc}{text{G}}& {text{H}}end{array}right]), (left[begin{array}{cc}{text{H}}& {text{G}}end{array}right]).

In spatial cases, the orientation of each segment is given by 3 parameters, and each G- or H-segment is characterized by an additional length parameter. Therefore, this matching scheme generates 21, 22, and 16 parameters to describe the shape variances for the sagittal curves, the coronal suture curves, and the lambdoid suture curves, respectively. Note that the suture curves are relatively smooth, thus the average value of the maximum error on all segments ({overline{E} }_{j}) is very significant of the matching error of the chain at the ({j}{text{th}}) profile. Therefore, this parameter is chosen to assess the error in this application. Figure 15 shows the best, the average, and the worst matches of the sagittal curves, the coronal sutures, and lambdoid sutures. The overall mean error ((overline{E })) of the sagittal curves, the coronal sutures, and the lambdoid sutures are 0.8728, 0.5060, and 0.3666 units, respectively.Figure 15The fitting results of (a–c) sagittal curves, (d–f) coronal sutures, and (g–i) lambdoid sutures from 63 samples. The left column (a, d, g) is the best match of each group, the middle column (b, e, h) is the average match of each group, and the right column (c, f, i) is the worst match of each group. (a) ({overline{E} }_{s-52}=0.5122) (unaffected); (b) ({overline{E} }_{s-19}=0.8681) (BCS); (c) ({overline{E} }_{s-46}=1.6055) (unaffected); (d) ({overline{E} }_{c-12}=0.2507) (BCS); (e) ({overline{E} }_{c-54}=0.5082) (unaffected); (f) ({overline{E} }_{c-42}=0.9636) (RUCS); g ({overline{E} }_{l-29}=0.1225) (BCS); (h) ({overline{E} }_{l-37}=0.3687) (BCS); (i) ({overline{E} }_{l-28}=1.0997) (RUCS).Full size imageIn order to represent the orientation of the spatial chain, the ({e}{text{th}}) segment at the ({j}{text{th}}) profile is characterized by a unit vector that points from the starting point to the endpoint of the segment as$${mathbf{u}}_{j}^{e}=frac{{overline{mathbf{z}} }_{{j}_{{m}_{j}^{e}+1}}^{e}-{overline{mathbf{z}} }_{{j}_{1}}^{e}}{Vert {overline{mathbf{z}} }_{{j}_{{m}_{j}^{e}+1}}^{e}-{overline{mathbf{z}} }_{{j}_{1}}^{e}Vert }.$$
(12)
Since each vector ({mathbf{u}}_{j}^{e}) contains three Cartesian coordinates, each sagittal curve, coronal suture, and lambdoid suture is thus characterized by 18, 18, and 12 variables, respectively. For lambdoid sutures, its relative location which is characterized by the coordinates of point P2 is also analyzed. Therefore, the total number of variables analyzed for a lambdoid suture is 15. This is much fewer than the 209 landmarks and semilandmarks analyzed in the work of Heuzé et al.19. Stepwise DA is conducted with the variables above, and LOOCV is performed to verify the stability of the linear model. In stepwise DA, 6, 7, and 8 variables are selected to be analyzed for sagittal curves, coronal sutures, and lambdoid sutures, respectively. Figure 16 illustrates the canonical plots of the three set of curves. As shown in Fig. 16, all three set of suture curves display strong separation among four classes on the 2D canonical plots. Besides, the LUCS and RUCS curves are distributed in the opposite directions from the BCS and unaffected ones along the first canonical component, while the BCS and unaffected curves differ in the direction of the second canonical component. These plots confirm the symmetrical shape deformation of the suture curves of LUCS and RUCS cases, and the changes in the lengths and relative locations of the suture curves of BCS as observed in Fig. 14.Figure 16The 2D canonical plots of the suture curves selected from 63 skull samples. (a) The sagittal curves based on 6 variables. (b) The coronal sutures based on 7 variables. (c) The lambdoid sutures based on 8 variables.Full size imageThe original DA prediction accuracy and cross-validated accuracy are both 100% for the sagittal curves, which indicates that the shape difference of the sagittal curves can efficiently distinguish specific diagnosis of coronal synostosis. The original DA prediction accuracy and cross-validated accuracy for the coronal sutures are 98.4% and 96.8%, respectively. There are two cases (BCS and RUCS) misclassified as unaffected case. As for the lambdoid sutures, the original DA prediction accuracy and cross-validated accuracy are both 98.4%. The only misclassified case in both predictions is that one BCS lambdoid suture is categorized as unaffected. This suggests that the coronal suture and the lambdoid suture is subjected to both shape deformation and location transformation due to coronal synostosis.These matching and classification results show that the shape-changing chain is efficient in fitting and analyzing 3D curves with a very moderate number of variables compared to other parametric methods. For example, Zhou et al. employed discrete cosine transform (DCT) to analyze the same three sets of suture curves33. In their work, 12, 6, and 6 harmonics are employed to fit the sagittal curves, the coronal suture curves, and the lambdoid suture curves, resulting in 36, 18, and 18 coefficients to be analyzed. Table 5 shows a comparison of the variables used to match the curves and perform statistical analysis with the two methods. A comparison of the classification accuracies of the two methods is not provided because Zhou et al. employed between-group principal component analysis (bgPCA) while the presented work uses stepwise DA. Note that the variables obtained in DCT are mathematical coefficients which are hard to interpret, while the variables in the shape-changing chain method represent the orientations, lengths, or locations of segments, providing direct information of the variance of the curve shapes.Table 5 Numbers of variables used in the shape-changing chain method and in DCT33 for fitting and analyzing suture curves.Full size table More