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Combining Messy Phenological Time Series

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Abstract

We describe a method for combining phenological time series and outlier detection based on linear models as presented in Schaber and Badeck (Tree Physiol, 22, 973–982, 2002). We extend the outlier detection method based on Gaussian Mixture Models as proposed by Doktor et al. (Geostatistics for environmental applications, Springer, Berlin, 2005) in order to take into account year-location interactions. We quantify the effect of the extension of the outlier detection algorithm using Gaussian Mixture Models. The proposed methods are adequate for the analysis of messy time series with heterogeneous distribution in time and space as well as frequent gaps in the time series. We illustrate the use of combined time series for the generation of geographical maps of phenological phases using station effects. The algorithms discussed in the current paper are publicly available in the updated R – package “pheno”.

Keywords

Linear models Gaussian mixtures Outliers Robust estimation Station effects 
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7.1 Introduction

Phenology, the science of “the timing of recurrent biological events, the causes of their timing with regard to biotic and abiotic forces, and the interrelation among phases of the same or different species” (Lieth 1974) has a long tradition embedded in biological sciences. Réaumur (1735) already proposed a temperature sum model as explanation for the variation in the onset of phenological phases, such as leaf bud break or initiation of flowering in the spring in temperate ecosystems. Linné described the purpose and methods of phenological observations as early as 1751. Phenological studies played a prominent role in the discovery of mechanisms with which organisms synchronise their development and behaviour with the environmental conditions. Spectacular changes in nature that are associated with the advancement of the seasons (greening of the vegetation, colourful flowering, Indian summer or seasonal migration of animals) as well as their usefulness for the timing of human activities are at the origin of observational time series that date back as far as several centuries (see several chapters in Schwartz 2003 on the history of phenology in different countries). In recent years these data have been discovered and explored for studies in the context of climate change research. Since 1991, publications on phenology as one of the easily detectable biotic responses to climate change have experienced a rapid growth (for review see Parmesan 2006, Rosenzweig et al. 2007 and papers cited therein). The growth rate of papers was higher than in other rapidly growing research domains against a background of a slowly growing number of publications on phenology in general.

Phenological data have specific limitations that have to be considered, when inferences are to be made from their analysis. It must be realized that phenological data origin from observations rather than from exact measurements. To obtain the data phenological observers use instructions that leave room for interpretation. Additionally, the exact location of the observation and therefore the environmental conditions as well as the genotype of plant individuals are usually unknown. These various sources of uncertainty introduce an intrinsic variability to phenological observations that is difficult to quantify (Schaber 2002). Moreover, phenological time series are often incomplete and reveal large data gaps, further complicating their analysis. The problem of the uncertainty of individual time series and gaps is often reduced by averaging a set of phenological time series over a geographical area of interest or a time period of interest (e.g. Estrella and Menzel 2006, Menzel et al. 2006, 2008). This way the resulting time series has less gaps and noise of individual time series is reduced at the cost of local information.

The principal problem associated with the use of average time series is often neglected, but can be demonstrated by a very simple consideration (Figure 7.1): assume we have two phenological stations s 1 and s 2, and s 1 has observations in years y 1 and y 2, whereas s 2 has observations in years y 2 and y 3. Further assume that observations at s 1 are equal, say, c 1 and at s 2 we observe c 2 in both years and c 1 > c 2. Obviously, by averaging we obtain a monotonically decreasing time series {c 1, (c 1+c 2)/2, c 2}. Subsequent trend analysis, which is especially popular for phenological time series (see Schaber 2002 and references therein), would show a negative trend. However, neither station actually shows a trend and thus, the resulting combined time series should also not exhibit a trend. In this simple example the resulting trend is clearly due to the fact that the stations have different observation years and that phenology at one station happens to be earlier than the other.
Open image in new windowFig. 7.1
Fig. 7.1

Illustration how averaging time series for station s1 and s2 can lead to undesired results because of their unequal distribution of observations in time (arbitrary units)

In general terms, phenological time series are unequally distributed in time and space and simple averaging in order to obtain less noisy and longer time series can lead to artifacts as demonstrated in Schaber (2002) for trends of time series of the International Phenological Gardens as published in Chmielewski and Roetzer (2001). In the above example, a solution is simple; first, we take a general mean a, a=(c 1 + c 2)/2 and correct the time series’ observations according to their deviations from the general mean (i.e. c 1–(c 1–a) and c 2–(c 2–a)), and then take the average. Obviously, the resulting time series is now {a, a, a}, which shows no trend, as we expect from inspection of the single time series (Figure 1).

In general, this process is called combination of time series and has been introduced to phenology by Häkkinen et al. (1995) and was put into the general framework of linear models by Schaber and Badeck (2002).

There are several areas of application where methods for combining phenological time series can be useful and where they have already been applied. One application is to obtain a reliable series out of several messy time series. In this application the focus would be on noise reduction (Häkkinen et al. 1995, Linkosalo et al. 1996, 2000, Linkosalo 1999, 2000, Schaber 2002). Another main application is to construct a long time series for trend analysis. In this application data gap filling is of primary interest (Schaber and Badeck 2005). Additionally, combined time series can also be used to find outliers in individual time series (Linkosalo et al. 2000, Schaber 2002, Schaber and Badeck 2002, Doktor et al. 2005). However, applying combined time series for outlier detection might lead to removal of correct observations, if the between station differences vary strongly at inter-annual time scales due to differences in the temperature trajectories, as already hypothesized by Schaber and Badeck (2002). Doktor et al. (2005) discussed some empirical evidence of cold spells that delay the transition to subsequent phenophases cause systematic deviations of the frequency distribution of dates of phase onset. They also introduced Gaussian mixtures as a tool for the quantification of the inter-annual variation in between station differences. This approach can potentially be integrated into the use of combined time series for outlier detection in order to avoid assignment of false outliers.

In the following, we will shortly introduce the method of combination of phenological times series by different types of linear models and discuss some practical issues. Moreover, we will discuss outlier analyses and show applications. We present the algorithms for integration of Gaussian normals (Figure 7.2) into the outlier detection with combined time series and illustrate the effect of this model improvement (Figure 7.3).
Open image in new windowFig. 7.2
Fig. 7.2

The density function of observed budburst dates of Beech (grey bars) modelled for 1993 and for 1981 each with 3 components (curves). A large scale and consistent warming up in spring time usually produces unimodal distributions, as in 1993. In contrast, strong changes in temperature regimes as experienced in 1981 result in multimodal distributions. Still, even unimodal distributions might not be normally distributed but can be more accurately be described by a Gaussian mixture

Open image in new windowFig. 7.3
Fig. 7.3

Number of detected outliers per year for Beech using the outlier detection algorithm of Schaber and Badeck (2002) (LAD) and using Gaussian Mixture Models (GMM). The mixture components are determined and parameterised by an optimisation algorithm

One useful result of the construction of combined time series is the extraction of station effects, (i.e. the characteristic deviation of the date of phase onset at a given observational station relative to the population of all stations). This result is less sensitive to gaps in the data series and different length of observation periods than the deviation from average values. It can be applied to producing maps of average geographical variation in the onset of a phenological phase. We illustrate this application for the bud break of beech in Germany (Plates 1, 2 and 3).

7.2 Linear Models of Phenological Time Series

7.2.1 Linear Models

It is reasonable to assume that over a climatologically sufficiently homogeneous region, (e.g. middle Europe or Central North America), the phenological development of certain phases is consistent concerning years and stations. This means that a year, which is particularly late, should be late for all stations, and a station that is particularly late, because for example, it is situated on a top of a mountain, should be late in all years. Putting this in mathematical terms, we say that the effect of year y i, i=1,…, n and the effect of station s j, j=1,…, m are independent and additive, such that
$$o_{ij} = a + y_i + s_j + \varepsilon _{ij},\,\, i = 1,...,n\ {\rm{ and }}\ j = 1,...,m.$$
(7.1)
where o ij is the observation in year i at station j and a is the general mean. ɛ ij is an error term that is usually assumed to be homoscedastically normally distributed around zero with some variance σ 2, that is
$$\varepsilon _{ij} \propto N(0,\sigma ^2 )$$
(7.2)
In statistics, (7.1) is called a linear two-way crossed classification model. Usually, additional conditions are imposed that assure that a unique solution exists, such as setting
$$\tilde y_i = a + y_i\ and\ \sum\nolimits_{j = 1}^m {s_j = 0}.$$
(7.3)

We call \(\tilde y_i\), i = 1,…, n the combined time series.

Given observations o ij, that may have missing data, (7.1) and the conditions (7.3), it is essentially straightforward to estimate the \(\tilde y_i\) and s j using least-square optimisation (i.e. minimizing the sum of squared residuals SSR),
$${\rm{SSR}} = \sum\limits_{i,j}^{} {\varepsilon _{ij}^2 }.$$
(7.4)

7.2.2 Fixed and Mixed Effects Models

Depending on the type of analysis we are interested in, we can treat the year and station effects differently, which has consequences for the type of estimation procedure we apply. For instance, when we are mainly interested in the combined time series \(\tilde y_i\), we can refrain from estimating the specific station effects but rather consider the stations to be randomly distributed. Thus, we treat the s j as a random variable
$$s_j \propto N(0,\sigma _s^2 )$$
(7.5)
and estimate the variance component \(\sigma _s^2\), rather than station effects s j. This is called the mixed model, because we have one fixed and one random effect. For this type of analysis special estimation and analysis procedures exist (Searle 1987, Milliken and Johnson 1992, Pinheiro and Bates 2000). Examples of the application of mixed models to obtain reliable phenological time series can be found in Schaber and Badeck (2002, 2005).

On other occasions, we might be interested in the year effects as well as in the specific station effects in order to identify stations that are particularly late, for instance. In this case, we would treat both effects as fixed. In Section 7.3.2 we will give an example. For details on linear models and the large theoretical body that comes with it, please refer to Rencher (2000), Searle (1971, 1987) and Milliken and Johnson (1992) and the literature cited therein.

7.2.3 Practical Issues and the R Pheno-Package

As already indicated, linear models constitute an entire field in statistics and calculations are far from being as easy as just calculating an average. The large theoretical body that comes with the theory of linear models can even be an obstacle rather than being helpful for phenological applications. Therefore, the authors wrote the software package “pheno: auxiliary functions for phenological data analysis” (Schaber 2007) that was designed to make calculations of combined phenological time series and station effects as easy as possible. This software is freely available as a package for the free statistical computing environment R (R Development Core Team 2007). The user has just to provide a table with three columns (observation, year, station) to a function corresponding to the analysis of interest, without having to worry about the calculations. All subsequent examples were calculated using the pheno-package.

One especially useful feature of the pheno-package is that it automatically handles large data sets. To illustrate the problem, we refer to the example in the following Section 7.2.4. In order to calculate the average time series of beech and the station effects for Germany over the years 1951–2004, we considered 74,996 single data points from 2,318 stations. For calculation of the fixed year and station effects, this involves the inversion of a 74,996 × (2,319 + 53 + 1) matrix. With the usual 8-byte number coding the matrix itself occupies around 1.4 GB. With the extra storage needed for matrix inversion, even nowadays most personal computers would exceed their working storage capacity with this operation. Fortunately, the matrices involved mainly consist of zero-entries, such that the application of sparse matrix algorithms saves a great deal of computational and storage resources. Sparse matrix algorithms are provided in other R-packages such as SparseM and quantreg (Koenker and Ng, Koenker 2006) and are already integrated in the R-pheno package. This way, combined time series for whole Germany can be computed on a regular personal computer.

Another prerequisite for the application of linear models is that the time series be connected or overlapping. For many stations this is usually not a problem, but for few data (stations or years) it is recommendable to check (Schaber 2002). There are procedures within the R pheno-package that test for connectivity and automatically extract connected sets of time series.

7.2.4 Outlier Detection

As already mentioned in the introduction, obtaining phenological data is often an error-prone process (Schaber 2002, Schaber and Badeck 2002). Therefore, a proper outlier detection method is indispensable. One of the few types of errors that can be detected is the so-called month mistake. Schaber and Badeck (2002) developed a method to detect month mistakes with combined time series. The usual least-square estimation of combined time series is sensitive to outliers. Therefore, Schaber and Badeck (2002) recommended applying a robust estimation procedure that minimizes least absolute deviations (LAD) (7.6),
$${\rm{LAD}} = \sum\limits_{i,j}^{} {\left| {\varepsilon _{ij} } \right|},$$
(7.6)
before applying the classical least-square estimation. Residuals \(\varepsilon _{ij}\), (i.e. the difference between observed and predicted values), that are estimated to be larger than 30 days are considered as month mistakes and are removed. The details of the procedure are described in Schaber and Badeck (2002) and are also implemented in the R pheno-package.

7.2.5 Gaussian Normals

The assumption that year effects and station effects are independent may not always hold true. This is the case when the inter-station differences vary due to the annual weather trajectories. Years with retarded phase onset in a subset of phenological observation stations due to a cold spell as compared to years without intermittent cold spells are an example of this class of interactions. As an extension to the outlier detection with LAD estimation, frequency distributions of observed budburst dates can be characterised and modelled using Gaussian Mixtures Models (GMM) (Figure 7.2). In many years, observations of a single species can be approximated by a probability density function (pdf), which consists of one, two or more underlying distributions. These can be mainly attributed to changing weather situations within spring, which alter the phenological pace. GMM quantify the number and type of the underlying distributions and thereby allow distinguishing years with different temporal evolution of budburst dates in a quantitative manner. The models can describe distributions with unknown underlying patterns and have the property of being able to represent any distribution of natural observations (Gilardi et al. 2002). A mixture distribution with continuous components has a density of the form (Poland and Shachter 1994):
$$f(x) = p_1 f_1 (x) + ... + p_n f_n (x)$$
(7.7)

Where x is the probability to have an observation at a certain day, \(p_1 ,...,\,p_n\) are positive numbers summing up to one and \(f_1 (x),...,\ f_n (x)\) are the component densities (7.7). To determine potential outliers one has firstly to analyse the uni- or multi-modal frequency distribution to identify the main underlying components (mixtures) and their describing parameters mean, standard deviation and weight (\(\mu _k ,\sigma _k ,p_k\)).

For each year i, all observations o ij are related to a component’s mean \(\mu _k\). The component k a an observation o ij is most related to is determined based on the frequencies f ijk of component k at the observation day o ij :
$$f_{ijk} = \frac{{np_k}}{\sigma _k \sqrt{2\pi}}{\rm{e}}^{-{\frac{{(o_{ij} - \mu _k )^2 }}{{2\sigma _k^2 }}}},$$
(7.8)
where n is the total number of observations of the analysed year i. Then,
$$k_a = \arg \mathop {\max }\limits_k f_{ijk}$$
(7.9)
An observation is declared to be an outlier if
$$\left| {o_{ij} - \mu _{k_a } } \right| \ge 30$$
(7.10)

An optimisation algorithm is applied on the minimisation of several (here maximum four) Gaussian Mixture functions. Due to the authors’ experience from phenological data analysis it is very unlikely that changes in temperature regimes with a sustained impact on the phenological evolution happen more than three times within the period the plant population is experiencing budburst, at least in Central Europe. Akaike’s Information criterion (Akaike 1974) is applied to choose the most appropriate model, balancing between model complexity (number of components) and model fit. The parameterised mixture components are used for outlier detection in order to reduce the number of falsely detected outliers in years showing bi- or multi-modal distributions (i.e. in years with a high variability of observed phenological events).

Obviously, this method is more conservative as the one based on LAD estimates. LAD estimation assumes that each year the observations are distributed around one general mean (the year effect) whereas applying Gaussian mixtures we assume that there might be several means. We detect only outliers at the margins of the whole Gaussian mixture and consequently less than before (Figure 7.3).

Interestingly, even a unimodal distribution could be more accurately defined by a Gaussian mixture (Figure 7.2). In fact, there was not a single year between 1951 and 2004 where the distribution of observations could be described by a single normal distribution (P < 0.01, Shapiro-Wilk test).

7.3 Applications

7.3.1 Gaussian Normals

The two outlier detection methods are compared with respect to the number of observations declared as outliers in each year, respectively. As expected GMM identifies, in general, fewer outliers (Figure 7.3). This, however, comes at a cost of false negatives (declaring an observation not to be an outlier when it actually is).

7.3.2 Station Effects

We calculated the fixed effect model (1) with constraints (2) for whole Germany for the years 1951–2004 for beech budburst without month-mistakes. We considered only stations that had at least 20 observations. After the removal of 433 outliers according to the robust estimation method we considered 74562 observations from 2318 stations. In Plate 1 we present a map of the calculated station effects plus the general mean m=120 (30th of April in non-leap years) in day of the year (DOY). To our knowledge, this is the first time that a consistent map for the characteristic timing of a specific phenological phase for such a large region is presented. Note that for this application the underlying trends (see Schaber and Badeck 2005) have not been removed.

The underlying assumption that the observations within the relatively large geographic space of Germany (357,092 km2) are elements of a unimodal population is illustrated with Figure 7.2 (curve for year 1993). In many cases a station net well distributed over a geographical space with continuous gradients of environmental conditions will result in such a distribution. However, the distribution may be different from unimodal, if a geographical domain is made up by two sub-domains with very different environmental conditions (Figure 7.2, year 1981).

The maps of the station effects (Plate 1) and the interpolated station effects by external drift krigging (EDK) (Plate 2) illustrate phenological responses to

  1. 1.

    climatological differences between regions at similar elevation (e.g. 50–100 [m] asl): the northern lowlands of Saxony are phenologically later than the Muensteraner Becken and the Northern Upper Rhine valley. The average March and April temperatures (1951–2003) are 3.94 and 8.27 °C, respectively in Saxony at 15 stations at 12.4–13.9 longitude and 51.4–51.9 latitude. They are 5.19 and 8.62 °C, respectively in the vicinity of Muenster at 11 stations at 7.0–7.9 longitude and 51.7–52.2 latitude. They are 6.26 and 9.91 °C, respectively in the Northern Upper Rhine valley at 7 stations at 8.3–8.45 longitude and 49.3–49.9 latitude,

     
  2. 2.

    the lapse rate across elevational gradients (the higher, the later),

     
  3. 3.

    the combined influence of the inverse lapse rate of early spring (see Table 1 and Figure 2 in Doktor et al. (2005)) and general climatological gradients between east and west Germany (northern lowlands: the closer to the sea the later at similar elevation).

     

The difference map (Plate 3) between station effects and station averages shows a slight general bias towards later combined station effects especially in the eastern part of Germany. These differences might be due to gaps in the time series, which are particularly common in this part of Germany. An indication that this is indeed the case is the fact there is a slight negative tendency between difference and number of observations per station (P<0.07).

7.4 Summary

Phenological data are messy data. Their analysis calls for appropriate methods that can deal with their inherent uncertainties as well as correct for effects due to their heterogeneous distribution in time and space. Simple averaging as a method to accommodate noise and gaps is likely to lead to erroneous results especially when the ratio of gaps to total number of observations is high or when a low number of observation series is averaged. The application of linear models to obtain combined time series constitutes an adequate method to handle gaps and noise in individual time series.

The application of Bayes statistics is an alternative way of analysing messy phenological datasets (see e.g. Dose and Menzel 2004). Future work should compare Bayes statistics to the methods discussed in the current paper and address the respective sensitivity to assumptions about priors and underlying distributions as well as to the types of errors and data gaps.

The approach of Gaussian mixtures to consider station x year effects can be further developed by assigning stations to tentative mixture components before checking for outliers or including mixed terms in the linear model (1).

With ongoing efforts to expand the databases of phenological observations by data mining it is very likely that more data sets with sparse data and data gaps will become available in the near future. For example, see the instructive account of the spatial and temporal coverage of the Japanese cherry flowering time series and the step-wise expansion of the data base (Aono and Kazui 2007). The methods described with the current paper are available as an R-package. The routines within this R pheno-package allow for the construction of combined time series that can serve for time series analyses. They can be applied for outlier detection. The calculation of station and year effects facilitates geo-statistical analyses of geographic patterns in the onset of phenological phases as well as their relation to weather pattern in specific years.

Notes

Acknowledgments

We want to express our thank to Roger Koenker, one of the authors of the R packages quantreg and SparseM that implement procedures to calculate quantile regression for robust estimates and sparse matrix algorithms, who was a tremendous help in incorporating these packages into the pheno-package. We also thank Achim Glauer for his continuous support in maintenance of the phenological database at the Potsdam Institute of Climate Change Research, and the German Weather Service DWD for making the data available to us.

References

  1. Akaike H (1974) A new look at statistical model identification. IEEE T Automat Contr AC 19:716–723CrossRefGoogle Scholar
  2. Aono Y, Kazui K (2007) Phenological data series of cherry tree flowering in Kyoto, Japan, and its application to reconstruction of springtime temperatures since the 9th century. Int J Climatol 28:905–914, DOI: 10.1002/joc.1594CrossRefGoogle Scholar
  3. Chmielewski F-M, Rötzer T (2001) Response of tree phenology to climate change across Europe. Agricult Forest Meterol 108:101–112CrossRefGoogle Scholar
  4. Doktor D, Badeck F-W, Hattermann F et al. (2005) Analysis and modelling of spatially and temporally varying phenological phases. In: Renard P, Demougeot-Renard H, Froidevaux R (eds) Geostatistics for environmental applications. Proceedings of the fifth European conference on geostatistics for environmental applications, Springer, Berlin, pp 137–148Google Scholar
  5. Dose V, Menzel A (2004) Bayesian analysis of climate change impacts in phenology. Global Change Biol 10:259–272CrossRefGoogle Scholar
  6. Estrella N, Menzel A (2006) Responses of leaf colouring in four deciduous tree species to climate and weather in Germany. Climate Res 32:253–267CrossRefGoogle Scholar
  7. Gilardi N, Bengio S, Kanevski M (2002) Conditional gaussian mixture models for environmental risk mapping. IEEE International Workshop on Neural Networks for Signal Processing (NNSP), pp 777–786Google Scholar
  8. Häkkinen R, Linkosalo T, Hari P (1995) Methods for combining phenological time series: application to budburst in birch (B. pendula) in Central Finland for the period 1896–1955. Tree Physiol 15:721–726PubMedGoogle Scholar
  9. Koenker R (2006) quantreg: Quantile Regression. R package version 4.01, http://www.r-project.org
  10. Koenker R, Ng P, SparseM: Sparse Linear Algebra. R package version 0.71. http://www.econ.uiuc.edu/~roger/research/sparse/sparse.html.
  11. Lieth H (ed) (1974) Phenology and seasonality modelling. Springer, BerlinGoogle Scholar
  12. Linkosalo T (1999) Regularities and patterns in the spring phenology of some boreal trees. Silva Fenn 33:237–245Google Scholar
  13. Linkosalo T (2000) Analyses of the spring phenology of boreal trees and its response to climate change. Dissertation, University of HelsinkiGoogle Scholar
  14. Linkosalo T, Häkkinen R, Hari P (1996) Improving the reliability of a combined phenological time series by analyzing observation quality. Tree Physiol 16:661–664PubMedGoogle Scholar
  15. Linkosalo T, Carter TR, Häkkinen R, Hari P (2000) Predicting spring phenology and frost damage risk of Betula spp. under climatic warming: a comparison of two models. Tree Physiol 20:1175–1182PubMedGoogle Scholar
  16. Menzel A, Estrella N, Heitland W et al. (2008) Bayesian analysis of the species-specific lengthening of the growing season in two European countries and the influence of an insect pest. Int J Biometeorol 52:209–218, DOI 10.1007/s00484-007-0113–8CrossRefPubMedGoogle Scholar
  17. Menzel A, Sparks TH, Estrella N et al. (2006) Altered geographic and temporal variability in phenology in response to climate change. Global Ecol Biogeogr 15:498–504Google Scholar
  18. Milliken GA, Johnson DE (1992) Analysis of messy data. Volume I: Designed experiments. Chapman and Hall, New YorkGoogle Scholar
  19. Parmesan C (2006) Ecological and evolutionary responses to recent climate change. Annu Rev Ecol Evol S 37:637–669CrossRefGoogle Scholar
  20. Pinheiro JC, Bates DM (2000) Mixed-effects models in S and S-Plus. Statistics and computing. Springer, New YorkCrossRefGoogle Scholar
  21. Poland WB, Shachter RD (1994) Three approaches to probability model selection. In: Lopez de Mantaras R, Poole D (eds) Uncertainty in artificial intelligence: proceedings of the tenth conference, Morgan Kaufmann, San Francisco, pp 478–483Google Scholar
  22. R Development Core Team (2007) R: A language and environment for statistical computing. R Foundation for statistical computing, Vienna, Austria. ISBN 3-900051-07-0, http://www.R-project.org
  23. Reaumur RA (1735) Observations du thermomètre faites à Paris pendant l‘année 5 comparées avec celles qui ont été faites sous la ligne, à l‘Isle de France, à Alger et en quelques-unes de nos îles de l‘Amérique. Mémoires de l‘academie royale des sciences Paris, 737–754Google Scholar
  24. Rencher AC (2000) Linear models in statistics. John Wiley, New YorkGoogle Scholar
  25. Rosenzweig C, Casassa G, Imeson A et al. (2007) Assessment of observed changes and re-sponses in natural and managed systems. In: Parry ML, Canziani OF, Palutikof JP et al. (eds) Climate change 2007. Impacts, adaptation and vulnerability. contribution of working group II to the fourth assessment report of the Intergovernmental panel on climate change, Cambridge University Press, Cambridge, UK, pp 79–131Google Scholar
  26. Schaber J (2002) Phenology in Germany in the 20th century: methods, analyses and models (PIK-Report No. 78). PIK, Potsdam, downloadable at: http://www.pik-potsdam.de/research/publications/pikreports
  27. Schaber J (2007) Pheno: Auxiliary functions for phenological data analysis. R package version 1.3. http://www.r-project.org
  28. Schaber J, Badeck FW (2002) Evaluation of methods for the combination of phenological time series and outlier detection. Tree Physiol 22:973–982PubMedGoogle Scholar
  29. Schaber J, Badeck FW (2005) Plant phenology in Germany over the 20th century. Reg Environ Change 5:37–46CrossRefGoogle Scholar
  30. Schwartz MD (ed) (2003) Phenology: an integrative environmental science. Tasks for vegetation science, vol 39. Kluwer Academic Publishers Dordrecht, The NetherlandsGoogle Scholar
  31. Searle SR (1971) Linear models. John Wiley, New YorkGoogle Scholar
  32. Searle SR (1987) Linear models for unbalanced data. John Wiley, New YorkGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Max Planck Institute for Molecular GeneticsComputational Systems BiologyBerlinGermany
  2. 2.Potsdam Institute of Climate Impact ResearchPotsdamGermany
  3. 3.Department of BiologyImperial CollegeLondonUK

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