Ecology

AffiliationsCAS Key Laboratory for Plant Diversity and Biogeography of East Asia, Kunming Institute of Botany, Chinese Academy of Sciences, Kunming, ChinaHong QianResearch and Collections Center, Illinois State Museum, Springfield, IL, USAHong QianDepartment of Biology, University of Missouri–St. Louis, St. Louis, MO, USARobert E. RicklefsUniv. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LECA, Laboratoire d’Ecologie Alpine, Grenoble, FranceWilfried ThuillerAuthorsHong QianRobert E. RicklefsWilfried ThuillerCorresponding authorCorrespondence to
Hong Qian. More

The BNNs demonstrated that the average yields of cacao farmer groups, in Sulawesi over distinct time periods, are closely associated with the ENSO OI patterns 9 to 25 months before harvest. The ENSO OI short term pattern explained slightly less (69%) of the variation in the average yield than the long term pattern (77%). We consider both these levels of prediction to be high, however, the short term pattern level was simpler and was used for further analyis. The linear regression between predicted and actual yields indicates that the model will tend to underestimate cacao productivity at high yields (e.g. in excess of 100 kg ha−1 month−1).The predictions made by the BNNs indicated that cacao yields are substantially impacted by ENSO conditions, which accords with prior observations21. The fertilizer response varied according to the ENSO profile: the greatest predicted response was in the Neutral ENSO profile with a smaller response under the MinCent ENSO profile, especially when unfertilized yields were low, and essentially no response under the MaxCent ENSO profile. Hence, the analysis provides insights into the appropriate fertilizer regime for distinct ENSO OI patterns in the period 9 months before harvest. We also note that recent methods to improve prediction of future ENSO OI patterns make it possible to predict them with reasonable accuracy for up to 1 year3. Thus, it is possible to relate average cacao crop performance and management practices directly to ENSO patterns in a given region without the need for weather data when the following conditions are met: (1) data exist on crop performance in any given site over time with distinct management practices; and (2) the weather patterns are driven by ENSO OI. We have used cacao as proof-of-principle, and suggest that this principle can readily be applied to other crops.A great advantage that Bayesian methods have over other machine learning approaches is that they can utilise variance based probability distributions to predict the likelihood of any given outcome. The model was used to predict the most likely monthly yield and expected standard deviation from each farm group under a specific ENSO profile when either fertilized or unfertilized. The standard deviations attained across all predicted responses was remarkably low, typically less than 1 kg ha−1 per month. Both the construction of the model and the subsequent predictions were based upon the mean yield data from 10 farms in each group at each monthly harvest under a single management type. As a result, all variations in yield across those 10 farms would have been excluded from the network constructed. As a consequence, while the predictions returned by the model might precisely reflect the mean response from each group, the limited input data will mean that the range of possible outcomes under any predicted scenario is likely to be underestimated. Up to now we have established proof-of-principle stage, the next stage will be first to improve the assessment of the predicted probability distributions and then to develop channels for communicating the results of the analysis to farmers followed by appraisal of their opinions and use of the information provided. Options for improving estimates of the probability distribution include both incorporating all observations from within each group, to ensure that farm-to-farm variance is adequately captured, and to extend the observations across more seasons to ensure that the variability of response to contrasting ENSO profiles is better represented.The analysis presented here is based on the average yields for each group of farmers. However, previous analysis indicates much variation in yield within the farmers groups20. Furthermore, those farmers with higher average yields tended to maintain their yield advantage relative to those with lower yields, even when conditions were adverse. This supports the view that the differences in yield between the high average yield and the low average yield farmers are due to management skills, rather than more favorable soils and weather conditions20. This suggests that if the average yields of individual farmers relative to the mean of all farmers are known, then the ENSO predictions can be used to predict their yield levels, and also their response to fertilizer applications.The demonstration that on farm yields and response to one management variable, fertilizer, can be linked directly to ENSO OI data supports the view that, in the future, with cacao or other crops, data on farm yields obtained with distinct management practices can be coupled with ENSO OI data to both determine probable crop yields and also to define differential crop response to management at specific sites under distinct ENSO OI patterns without the need for accurate weather data. The ENSO OI data exists, what is often lacking is data on yield with distinct management practices. To obtain this type of information in heterogeneous growing environments using traditional Randomized Control Trials is simply not possible. However, we suggest that schemes, such as those to collect the cacao data we have here with distinct management treatments superimposed on farmers fields20, can be used. Furthermore, even without superimposing management practices, simply monitoring crop performance, weather and the variation in management practices of farmers can be used to relate yield to variation in weather patterns and management28,29,30. However, this is only effective if the data of a large number of cropping events is brought together for analysis, which requires social organization and the willingness to share data28. Our experience with cacao indicates that small farmers are willing to share data, but an external agency is required to manage the overall process of data collection and compilation20. Similar experiences with CropCheck and in Australia and Chile support this point of view31,32. The value of shared information through formation of farmer groups is well established33,34 and we suggest that the methodology described here could be implemented through farmer groups. Hence, through monitoring of crop performance and management coupled with Bayesian based machine learning tools and currently available ENSO OI information and predictions, farmers and agronomists can adjust management practices, in this case fertilizer applications, according to ENSO profiles. This will require social organization and support for the collection, compilation and analysis of the data; however, we believe it offers a route to provide farmers with an improved and cost effective knowledge base, derived from sparse data resources, to better manage their crops.Social organization is not only required for the collection of data to be analysed, but also for the disemination to farmers of the knowledge generated though its interpretation. Current tendencies of providing farmers with the basis to make better decisions recognise the restrictions of the linear model for extension and tend towards active farmer participation in the interpretation of data through such mechanisms as farmers field schools35, formation of farmers groups (see for example Montaner 200434) and innnovation networks (see for example Klerkx et al. 201036, Wood et al. 201437, World Bank, 200838). Further development of farmers´organizations and innovation networks will be required to effectively deploy the concepts presented in this paper.The principles developed here could be applied to other crops, such as coffee, olive and oil palm, and this type of analysis could be extended to other regions, such as Africa where data on crop response to management and weather variation is sparse. At the same time, we note that additional information on, inter alia, crop management, topography and soil types could substantially improve the predictive power of the networks. Furthermore, these machine learning techniques can be used to mine existing big data sets collected by large commercial interests, to discover relationships between environment, management and crop production, and thereby supplement, at low cost, the findings generated by formal controlled scientific experiments. In the case of small farmers, social organization and external support will be required.There are several caveats on the use of this proposed methodology. First, the relationship between the ENSO phenomenon and the weather patterns will be specific to each location or recommendation domain. Hence, models and inferences for management cannot be readily transferred from one recommendation domain to another. Furthermore, the definition of the area that comprises a recommendation domain is not simple. Thus, whilst we consider the principles developed here to be universal, the models themselves will be specific to each recommendation domain, which are currently still difficult to define but new approaches are becoming increasingly available to do so (e.g. Rubiano et. al. 201618; Rattalino Edreira et al. 201817).A further complication of the suggested approach is the lack of understanding of the underlying mechanisms that establish the associations. This deficiency limits the ability to identify the specific causes of different crop productivities, and thus limits our ability to resolve these unidentified problems.Growers decisions on how much to invest in their crop production practices depends on the expected prices of the commodities they produce: when prices are expected to be high, they will invest more, and when prices are low they may even abandon their crops. It has not escaped our notice that the predictive power of the machine learning resources would also provide the cacao industry as a whole with insights into the fluctuations in future cacao supply and hence prices. This would allow farmers and others in the cacao supply chain to minimize uncertainty and better manage the overall industry. The experiences strongly support the idea that machine learning is a useful tool in our armoury opening the opportunity to utilize information from on farm performance coupled with publicly available data to improve agricultural management. More

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Experimental data of Russian wheat aphid (RWA) development and survivalThe RWA development and survival data are from our laboratory experiments which involve the observations of 1800 RWA individuals in a factorial arrangement of five temperature and five barley plant-growth stage regimes with a total of 25 treatments. Each treatment has 72 RWA individuals as replicates. The experiment was designed to investigate the influence of temperature and barley plant-growth stage treatments on RWA development, survival and reproduction in controlled environment growth chambers. Temperature treatments were 8–1 °C, 17–10 °C, 23–16 °C, 28–21 °C, and 33–26 °C, fluctuating on a 14:10-h rectangular-wave cycle. The photoperiod was 14 vs. 10 (light vs. dark) for all treatments, with the higher constant temperature during the light phase and the lower temperature during the dark phase. Mean temperatures weighted by photoperiod were 5.1 °C, 14.1 °C, 20.1 °C, 25.1 °C and 30.1 °C, respectively. Barley plant-growth stages were two-leaf, tillering, flag leaf, inflorescence and soft dough, respectively corresponding to 12, 23, 39, 59, and 85 on the Zadoks (1974) scale38. More detailed information on the experiment design can also be found in Ma (1997) and Ma & Bechinski (2008a, 2008b)1,2,39.For analysis, we divided the life cycle of the RWA into 9 stages: first to fifth instar (abbreviated as 1st-5th), pre-reproductive period (from the time of last molting until the first nymphal production, designated Pre_R), immature period (1st + 2nd + 3rd + 4th + 5th, designated immature), mature (immature + R_age, designated mature), adult (from the time of last molting until death, designated adult). We also treat lifespan as a special variable, i.e., time from birth to death, designated lifespan. There is a response time T associated with each RWA stage and the lifespan; T is either development time (for individuals that successfully developed from one stage to the next), or death time (for individuals that died within the stage), depending on the state indication variable (short as ‘state variable’ or ‘state’). The unit for time (T) is calendar day (24 h). For stages other than adult and lifespan, if the state indication variable takes a value 1, then T is developmental time; if state is 0, then T is the death time or other censored time (e.g. lost accidentally in observation). In contrast, for the adult and lifespan stages, if state is 1, then T is death time of an individual; if state is 0, then T records the time when observation stopped due to some laboratory handling accident before the individual naturally died. Further information on the laboratory experiments is also described in Ma (1997), and Ma & Bechinski (2008a, 2008b)1,2,39.Unified survival analysis approach to insect development and survivalIssues of censoring in entomological researchCensoring occurs when the failure times of some individuals within the observation sample cannot be observed. Censoring is often unavoidable in time-to-event studies. A patient in a clinical trial may be withdrawn from the study after a period of participation; similarly, insects under observation may be lost tracks due to accidental events such as operational faults. Such kind of censoring belongs to the so-termed random censoring. In other cases, observing all individuals for the full time course to failure (such as the occurrence of death) is too costly or unacceptable, leading to the so-terms right censoring. In other situations, the process may have been going on but unnoticed prior to formal study, and consequently a starting point has to be selected, such as the exposure to some newly discovered risks, or the occurrence of a new infectious disease. This last category of censoring is known as left censoring.All three censorings exemplified above may occur in entomological experiments. Whereas the censorings discussed so far might be avoided or minimized, we realize that, in the study of insect populations, even with a perfect experiment design being perfectly executed, at least two types of natural censoring mechanisms seem uncontrollable thanks to the very nature of insect development and instarship. Two examples are presented here: (i) In a life table study, when a cohort of insects is observed, the insect development (molting, emergence, etc.) or survival (death) are typical examples of time-to-event or failure time random variable. This is not the focal point of our arguments. The point is that some insect individuals may die and never emerge from the observed instar or stage. From the perspective of observing insect development, the data may be censored due to the “premature” death events. How long it would have taken for those prematurely dead individuals to complete their developments is hardly knowable. (ii) It is well known that the number of instars in an insect species may be different among individuals of the same population; one may never know the exact number of instars an individual can potentially experience if it dies prior to reaching adult stage. For example, in the case of RWA, the majority of individuals has 4 instars, but 2, 3, 5 are also possible. If a RWA nymph died before reaching the adult stage, we may never know how many instars this prematurely dead individual would pass through. Hence, unless zero mortality in immature stages is possible, censoring in studies of insect developments occurs naturally and is incontrollable. Therefore, insect development and survival are dependent in the sense that in order to develop to the next stage, an insect individual must survive through the current stage. Such kind of dependence can be addressed ideally with conditional probability models in survival analysis including Cox proportional hazards models, which is applied to modeling the development and survival of RWA in this paper.With most statistical methods other than survival analysis, censoring presents a dilemma. If a researcher chooses to exclude the censored observations, the resulting sample may be too small to conduct proper statistical analysis. Even if the resulting sample is large enough, the parameters estimated are biased, and there is no well-established statistical procedure to quantify the bias, because there is no guarantee that the censored individuals can be represented by the remaining sample. On the other hand, if the censored individuals are included, although compartmental modeling (using a probabilistic approach, or invoking differential equations) might be useful so that the numbers of individuals in the different instars (compartments) are modeled over time with estimation of the rates of transition, there are no objective procedures to process their “partial” lifetime information, which again may introduce bias without even knowing the degree of bias introduced.How significant can the difference be between the two schemes—one with censored individuals excluded before applying survival analysis, and the other processed with survival analysis directly (i.e., the partial information of the censored individual is preserved)? Ma (1997, 2010)1,15 and Ma & Bechinski (2008b, 2009b)2,14 treated the prematurely dead RWA individuals as censored with survival analysis approach, and found that the difference between two treatments: survival analysis vs. excluding the premature dead individuals range from 4%–25% in the estimate of median development, depending on the severity of censoring (death rates)15. The treatment resolves the previously identified dilemma because survival analysis has developed effective procedures and methods (based on the asymptotical theory or more recently on the counting stochastic process) that can properly extract the partial information in those censored data.While the previous type of censoring due to premature death or variable instarships seems more likely to occur in insect demography and phenology studies under laboratory conditions, there is another subtle but hardly avoidable censoring mechanism, which is termed as interval censoring in survival analysis and it may be more likely to be encountered in field insect research such as life table study by sampling insect population periodically. This type of censoring is required because sampling is discrete and linear, whereas the process under investigation is continuous and possibly nonlinear with respect to time, which makes the precise recording of the survival times for all individuals in an experiment impossible. With interval censored data, each event time is then only known to lie in some interval, and the precise time to an event is often not known due to the limited sampling points.Ma & Bechinski (2008b, 2009b, Ma 1997, 2010) argued that1,2,14,15, whenever time-to-event data is in concern, survival analysis can be harnessed to perform the two fundamental statistical analyses in lieu of traditional statistical methods, that is: (i) hypothesis testing—replacing procedures such as significance testing, ANOVA, life tables analysis etc.; (2) model-parameter estimation—replacing conventional regression modeling. The prevalence of time-to-event data as one of the two major categories of time-dependent process data (the other is time-series data as noted previously), as well as the fact that survival analysis is developed to study time-to-event variables with observation censoring, make a very strong case for entomologists to adopt survival analysis as an appropriate statistical tool. In a previous study, we demonstrated the use of survival analysis for hypothesis testing and life table analysis (Ma 2010)15. In the present paper, we demonstrate the second area—model-parameter estimation. Specifically, we try to show that survival analysis offers a unified approach to model both insect development and survival.Survival Analysis and Proportional Hazards Model (PHM)Survivor, hazards and probability density functionsGiven response time (survival or failure time) T of a subject, three functions are usually used to describe the random variable (T): the survivor function, the probability density function, and the hazard function.The survivor function S(t) is defined as the probability that T is at least as great as a value t; that is,$$S(t) = P(T ge t)quad t > 0.$$
(1)
The survivor function is actually 1′s complement of the distribution function of random variable (T), that is, S(t) = 1–F(t), where F(t) is the distribution function of T.The probability density function (p.d.f) of T is$$f(t) = mathop {lim }limits_{{Delta t to 0^{ + } }} frac{{P(t le T le t + Delta t)}}{{Delta t}} = – frac{{dS(t)}}{{dt}}.$$
(2)
Conversely,(S(t) = int_{t}^{infty } {f(u)du}) and f(t) ≥ 0 with (int_{0}^{infty } {f(t)dt = 1.})The hazard function specifies the instantaneous rate of failure at T = t, conditional upon survival to time t. It is defined as$$lambda (t) = mathop {lim }limits_{{Delta t to 0^{ + } }} frac{{P(t le T < t + Delta t|T ge t)}}{{Delta t}} = frac{{f(t)}}{{S(t)}}.$$ (3) The relationships among S(t), f(t), and λ(t) are expressed as follows:$$lambda (t) = frac{ - dlog S(t)}{{dt}}$$ (4) $$S(t) = exp left( { - int_{0}^{t} {lambda (u)du} } right)$$ (5) $$f(t) = lambda (t)exp left( { - int_{0}^{t} {lambda (u)du} } right).$$ (6) Proportional hazards model (PHM)Cox (1972, 1975) proposed the proportional hazards model (PHM)16,40$$lambda (t,,z) = lambda_{0} (t)exp (zbeta ) = lambda_{0} (t)exp (beta_{1} z_{1} + beta_{2} z_{2} + ...beta_{n} z_{n} ),$$ (7) where λ(t, z) denotes the hazard function at time t for an individual with the characteristic represented by the covariate vector z of n elements. In entomological research, examples of z may include environmental factors (temperature and plant growth stage in this paper) that influence the development and survival of insects. Here λ0(t) is an arbitrary unspecified baseline hazard function for continuous time t. The hazards function λ(t, z) is a product of an underlying age-dependent risk, λ0(t) (baseline hazard function) and another factor, exp(zβ), which depends on covariates z and the vector β of parameters. Baseline hazard function λ0(t) is the hazard function for individuals on which covariates have “neutral effect”—the values of covariates are equal to either zero or to their averages (an example is shown later) depending on the model form adopted. The PHM estimates the risks of other groups in relation to this baseline. Other specifications of the hazard relationship are possible (e.g., λ(t, z) = λ0(t) + zβ), but the problem with these alternatives is the mathematical possibility of predicting negative hazard rates, which then requires extra constraints on estimation procedures to ensure positive values.The PHM invokes two assumptions. The first is the proportionality assumption, that there is a multiplicative relationship between the underlying hazard function and the log-linear function of the covariates such that the ratio of hazard functions for two individuals with different sets of covariates is constant in time (from which the PHM derived its name). The second assumption is that effects of covariates on the hazard function are log-linear.The conditional (with respect to the covariate vector z) probability density function of T given z for the PHM is$$f(t;,z) = lambda _{0} (t)exp (zbeta )exp left[ { - exp (zbeta )int_{0}^{t} {mathop lambda nolimits_{0} (u)du} } right].$$ (8) where λ0(t) is the baseline hazard function as explained previously, z is the vector of covariates (e.g.. air temperature and crop growth state in this study), and β is the vector of Cox’s PHM regression coefficients (parameters).The conditional survivor function (or simply called the survivor function) of T given z for the PHM is$$S(t;,z) = [S_{0} (t)]^{exp (zbeta )} ,$$ (9) where$$S_{0} (t) = exp left[ { - int_{0}^{t} {lambda _{0} (u)du} } right].$$ (10) S0(t) is called the baseline survivor function; it is computed for the default categories of the covariates (e.g., average temperature and plant growth stage in the case of this study). Therefore, the survivor function of t for a covariate vector z is obtained by raising the baseline survivor function S0(t) to a power. The usefulness of Eq. (9) is that one can predict survivor probabilities under different covariate values.If λ0(t) is arbitrary, this model is sufficiently flexible for many applications. There are two important generalizations that do not substantially complicate the estimation of β, but broadly expanding their applications: the stratified proportional hazards model and the proportional hazards model with time-dependent covariates.In the stratified version, the function λ0(t) is allowed to vary in specific subsets of the data. In particular, the population is divided into r strata wherein the hazard λj(t; z) in the j-th stratum depends on an arbitrary shape function λ0j(t). The model can be written as$$lambda_{j} (t,,z) = lambda_{0j} (t)exp (zbeta )quad j = {1},{ 2}, ldots ,r.$$ (11) This generalization is useful when the covariates do not seem to have a multiplicative effect on the hazard function. Here the range of those variables can be divided into strata where only the remaining regression variables contribute to the exponential factor in Eq. (11).The second generalization to the PHM is to allow the variable z to depend on time itself, without (Eq. 12) or with (Eq. 13) stratification:$$lambda [t,,z(t)] = lambda_{0} (t)exp [z(t)beta ],$$ (12) $$lambda_{j} [t,,z(t)] = lambda_{0j} (t)exp [z(t)beta ]quad j = 1,2, ldots ,r.$$ (13) The estimation of β depends only on the rank ordering of the variable vector z and is invariant with respect to the monotonic transformation on the dependent variable, i.e., survival time. The procedure used to estimate β is to maximize the so-called partial likelihood functions as described by Cox (1975), Kalbfleisch & Prentice (1980, 2002) and Kleinbaum and Klein (2012)19,20,30,40. BMDP™ 2L program, Survival Analysis with Covariates (BMDP 1993)41, was used to construct the proportional hazards models. The input data set was as described in Ma (1997)1. Finally, as an example, we use 1989 air temperature and barley plant-growth stage data from Moscow, ID., reported by Elberson (1992)42, as inputs to run the proportional hazards model for survival of RWA during the entire life cycle (i.e., model for LifeSpan stage). We further used coxphf function in the Survival package of open source R-Project to cross-verify the results from BMDP software. The information of Survival package, which is the cornerstone of R implementation of survival analysis, can be accessed at: (https://cran.r-project.org/web/views/Survival.html). More