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system an attempt to say something about (or attain power over) what is incompletely understood (or under-controlled; Jones 2009; Szary 2015). For the purpose
of this chapter, model boundaries illustrate the situations under which a model is
likely valid to some degree; ideally, likelihood and degree of validity are mathematically established. Model boundaries can be biological (e.g., specific to ecosystems,
species, life stages), physical (e.g., specific to latitudinal, geological, hydrological,
and topographical extents, or specific in time and place), or political (e.g., specific
to regions or countries). Further, models can be classified as reductionist or systembased, quantitative or conceptual, correlative or mechanistic, static or dynamic, and
hybrids thereof (Horning et al. 2010). Model boundaries and modeling approaches
should be determined early in the planning phase and reported. Some model limitations are likely beyond the researcher’s control, such as instances in which data can
only be acquired within certain political units. However, the choice of model to
describe a particular system should be made deliberately; and the modeling approach
should determine data collection, and not vice versa.
The data needed to investigate a phenomenon of interest with spectroscopy
depend not only on the research question, the modeling approach, and model boundaries but also on the aim of the analysis (e.g., model calibration, validation, interpretation) and the level of spectral data acquisition (leaf-level spectroscopy, proximal,
airborne, satellite RS; see Sect. 15.3., Gamon et al. Chap. 16). Data for model calibration and validation should match the conceptual model’s boundaries (e.g., the
model’s temporal and spatial scale) and the modeling approach. For example, while
quantitative and correlative models are ideally based on relatively uncorrelated or
orthogonal variables, conceptual and mechanistic models ideally include all variables relevant for a particular study system. Drawing inferences from models and
applying them to make predictions are only justifiable when model accuracy has
been assessed (Horning et al. 2010); a model can give a very accurate description of
a particular system, but one would not know until its accuracy is assessed.
Model calibration describes the process of determining the values of parameters
so that model outputs fit the observed data. Internal validation refers to testing a
model’s ability to explain the data used to populate the model. One common method
for this is cross-validation. During cross-validation, the data set is split into calibration and (internal) validation data, the calibration data are used to fit the model, the
model coefficients are applied to the validation data, and predicted and measured
values from the validation data are statistically compared to assess model fit. Usually
the data are split repeatedly, and model statistics (and often model parameters) are
averaged across the number of splits. In k-fold cross-validation, k indicates the
number of random data subsets or splits. One subset is omitted from model calibration and used for validation, and the process is repeated until all subsets have been
left out once. For small data sets, leave-one-out cross-validation is particularly useful; here, only one sample is omitted from model calibration and used for validation,
and the process is repeated until all samples have been left out once. In contrast to
internal validation, external validation refers to a model’s ability to predict observations not used for model development (Dahabreh et al. 2017), which is critical for
A. K. Schweiger
system an attempt to say something about (or attain power over) what is incompletely understood (or under-controlled; Jones 2009; Szary 2015). For the purpose
of this chapter, model boundaries illustrate the situations under which a model is
likely valid to some degree; ideally, likelihood and degree of validity are mathematically established. Model boundaries can be biological (e.g., specific to ecosystems,
species, life stages), physical (e.g., specific to latitudinal, geological, hydrological,
and topographical extents, or specific in time and place), or political (e.g., specific
to regions or countries). Further, models can be classified as reductionist or systembased, quantitative or conceptual, correlative or mechanistic, static or dynamic, and
hybrids thereof (Horning et al. 2010). Model boundaries and modeling approaches
should be determined early in the planning phase and reported. Some model limitations are likely beyond the researcher’s control, such as instances in which data can
only be acquired within certain political units. However, the choice of model to
describe a particular system should be made deliberately; and the modeling approach
should determine data collection, and not vice versa.
The data needed to investigate a phenomenon of interest with spectroscopy
depend not only on the research question, the modeling approach, and model boundaries but also on the aim of the analysis (e.g., model calibration, validation, interpretation) and the level of spectral data acquisition (leaf-level spectroscopy, proximal,
airborne, satellite RS; see Sect. 15.3., Gamon et al. Chap. 16). Data for model calibration and validation should match the conceptual model’s boundaries (e.g., the
model’s temporal and spatial scale) and the modeling approach. For example, while
quantitative and correlative models are ideally based on relatively uncorrelated or
orthogonal variables, conceptual and mechanistic models ideally include all variables relevant for a particular study system. Drawing inferences from models and
applying them to make predictions are only justifiable when model accuracy has
been assessed (Horning et al. 2010); a model can give a very accurate description of
a particular system, but one would not know until its accuracy is assessed.
Model calibration describes the process of determining the values of parameters
so that model outputs fit the observed data. Internal validation refers to testing a
model’s ability to explain the data used to populate the model. One common method
for this is cross-validation. During cross-validation, the data set is split into calibration and (internal) validation data, the calibration data are used to fit the model, the
model coefficients are applied to the validation data, and predicted and measured
values from the validation data are statistically compared to assess model fit. Usually
the data are split repeatedly, and model statistics (and often model parameters) are
averaged across the number of splits. In k-fold cross-validation, k indicates the
number of random data subsets or splits. One subset is omitted from model calibration and used for validation, and the process is repeated until all subsets have been
left out once. For small data sets, leave-one-out cross-validation is particularly useful; here, only one sample is omitted from model calibration and used for validation,
and the process is repeated until all samples have been left out once. In contrast to
internal validation, external validation refers to a model’s ability to predict observations not used for model development (Dahabreh et al. 2017), which is critical for
A. K. Schweiger