OCCASIONAL PAPER SERIES
NO 6 – July 2010

Models and Limits to Predictability1

By
Oscar Garcia

1

Based on “Dimensionalidad en los modelos de crecimiento”, Cuadernos de la Sociedad Española de Ciencias Forestales
23, 19-25, 2007. Materials reproduced with permission

Dr. García is a Professor and FRBC / West Fraser Endowed Chair in Forest Growth
and Yield within the Ecosystem Science and Management Program, University of
Northern British Columbia, 3333 University Way, Prince George, B.C., V2N 4Z9
Canada.

The correct citation for this paper is:

García, O. 2010. Models and limits to predictability. Natural Resources and Environmental Studies
Institute Occasional Paper No. 6, University of Northern British Columbia, Prince George, B.C., Canada.

This paper can be downloaded without charge from http://www.unbc.ca/nres/occasional.html

García  Models and Limits to Predictability

ii

The Natural Resources and Environmental Studies Institute (NRES Institute) is a
formal association of UNBC faculty and affiliates that promotes integrative
research to address natural resource systems and human uses of the environment,
including issues pertinent to northern regions.
Founded on and governed by the strengths of its members, the NRES Institute
creates collaborative opportunities for researchers to work on complex problems
and disseminate results. The NRES Institute serves to extend associations among
researchers, resource managers, representatives of governments and industry,
communities, and First Nations. These alliances are necessary to integrate
research into management, and to keep research relevant and applicable to
problems that require innovative solutions.

For more information about NRESI contact:
Natural Resources and Environmental Studies Institute
University of Northern British Columbia
3333 University Way
Prince George, BC Canada
V2N 4Z9
Phone: 250-960-5288
Email: nresi@unbc.ca
URL: www.unbc.ca/nres

iii

Occasional Paper No. 6
July 2010

CONTENTS

Abstract ......................................................................................................................................................... 2
Introduction................................................................................................................................................... 3
Models ........................................................................................................................................................... 3
Dynamic Forest Growth Models ................................................................................................................... 4
Aggregation ................................................................................................................................................... 5
Limits to predictability................................................................................................................................... 7
Conclusions..................................................................................................................................................10
References ...................................................................................................................................................11

1

Occasional Paper No. 6
July 2010

Abstract
Models of a same system may differ greatly in
scale and level of detail. The implications of
this are examined, in general and more
specifically in relation to forest growth
models. The nature of modelling is discussed,
distinguishing descriptive and predictive
models, and briefly describing the concepts of
dynamical system and state space. Through
examples,
I
demonstrate
limits
to

García  Models and Limits to Predictability

predictability that can make reliable
predictions impossible at the individual level.
A complete understanding of the functioning
of a system, or its computer simulation, do
not imply being able to predict its behaviour.
Although detailed models are useful for
research
purposes,
low-dimensional
aggregated models are generally more
appropriate for decision-making.

2

Introduction
Growth models are useful as research tools, and
for predicting outcomes in forest management.
Their level of detail, scale, or state space
dimensionality, varies within wide limits. We
shall examine some characteristics and
consequences of the scale used. Much of the
discussion is applicable to models in general, not
just to growth models.

Models
A model (or theory) is a partial representation of
some aspect of “reality”. For instance, a scale
model of a building or of an aircraft (material
models), the manual for a DVD player (verbal
model), or our subjective idea of the results of
acting in a certain way in a given situation
(mental model). Mathematical models are like
verbal ones, but using mathematical language.
They generally have the advantage of being less
ambiguous.
Perhaps
more
importantly,
mathematical models allow the reuse of known
recipes, rules, or previously established
theorems, instead of having to reason starting
from scratch each time. For making any decision
it is necessary to employ some kind of model; it
is the link that connects actions with
consequences.
“An engineer thinks that his equations are an
approximation to reality. A physicist thinks that
reality is an approximation to his equations. A
mathematician doesn't care.”
Anonymous
Although obviously a caricature, this quote
reflects certain attitudes regarding the
perception of models and theories. The last part
is not particularly relevant here, it relates to the
role of the mathematician, as such, in the study
of formal relationships regardless of what they
might represent (i.e. the development of the
’recipes’ previously mentioned). There appear to
be, however, differences in the way of thinking
about the nature of models that may be
associated with different experiences and
traditions. One often talks of scientists

3

discovering natural laws, with the implication
that such models have an independent
existence. On the other hand, one can argue
that humans can not comprehend all the details
of a reality in which everything is connected to
everything else; they can only reason through
representations based on artificial classifications
and definitions, ignoring the less important
interactions. Models would have more to do
with the structure of the human brain than with
the reality “out there” (assuming that such
reality exists!). At any rate, at least for the type
of models in which we are interested, it seems
more appropriate to think in this way, in which it
makes no sense to speak of a true model.
Quoting G. E. P. Box slightly out of context, “All
models are wrong, but some are useful.”2
Useful for what? There is controversy about the
superiority of various types of models (see for
instance the special issue edited by Mohren and
Burkhart 1994). Part of the problem is that the
criteria for what makes a model superior are not
clearly specified. One of the dichotomies is the
use of models either as research tools, or as
management tools. In research, the use of
models is primarily descriptive; we are
interested in understanding the functioning of a
system, in synthesizing previously isolated facts,
and in generating questions to guide future
studies. The model is a working hypothesis, and
major advances are achieved when the
hypothesis fails. In management, the purpose is
primarily predictive, to predict future behaviour,
possibly in response to alternative treatments or
events. Precision and accuracy in forecasting
takes precedence over qualitative explanation.

2

Economists have yet another view of models: they
speak of “market failures”, it is reality which is
wrong.

Occasional Paper No. 6
July 2010

Figure 1: Terrestrial motion models: (a) Mechanistic, causal. (b) Empirical.

For example, Figure 1(a) is a mechanistic model
that helps to understand the motion of the
planets. It could be used to predict when it will
become dark today. However, the empirical
model of Figure 1(b), which has little or nothing
to do with the functioning of the solar system,
may be more accurate and convenient. Note in
passing that the size and distance proportions in
Figure 1(a) are far from real, and precisely that
makes the model more understandable; realism
in a model is not necessarily a virtue.
As explained later, another aspect related to
model use is the most appropriate level of detail
or complexity, or the dimensionality of the state
space in dynamic models.

Dynamic Forest Growth Models
The most traditional forest stand models are
yield tables. These describe how volumes per
hectare, mean diameter, dominant height,
number of trees, or other variables, change as
functions of time. Sufficient in many situations,
their use is more problematic when there are
silvicultural stand thinnings, or in other cases
where a stand has deviated from the nominal
trajectory provided in the table. Isaac Newton
introduced an approach more flexible than
García  Models and Limits to Predictability

modelling functions of time directly: a dynamic
model describes the rate of change of certain
variables that define the state of the system.
The state trajectories are synthesized by
integration or accumulation of the rates of
change, represented by differential or difference
equations (Figure 2). Other variables of interest
can be estimated from the state at any given
time (Luenberger 1979, Garcia 1994, Garcia
2005b).
A system can be represented by dynamic
models with various levels of detail or
resolution, which differ in the dimension of the
state vector (number of state variables). In
particular, the standard classification of growth
models (Goulding 1972, Munro 1974, Vanclay
1994) reflects differences in dimensionality
(Garcia 1988). Whole-stand models use a few
aggregate variables such as basal area,
dominant height, number of trees. Individualtree models, on the contrary, use tens to
thousands of state variables, including sizes for
each of the trees in a stand or sample plot.
Individual-tree models may be distancedependent, which use tree coordinates to
calculate competition indices (Staebler 1951,
Newnham 1964, Mitchell 1975), or distance-

4

independent, which ignore spatial structure
(Goulding 1972, Stage 1973).

Aggregation
With greater availability of computing power,
there has been an increasing emphasis on the
development of individual-tree models. There is
a tendency to regard aggregation at the stand
level as unnecessary and obsolete. On the other
hand, self-thinning theories, which can be

interpreted as two-dimensional dynamic
models, rather imprecise for managed stands
(Garcia 2005a), remain popular. But models of
intermediate dimensionality are now rare in the
research literature, although still widely used in
forest management practice. Something similar
has happened more recently in population
ecology modelling (e.g. Grimm 1999), and in
other areas such as physics, economics, and
sociology (Garcia 2001).

Figure 2: Example of dynamic growth model with two state variables. The arrows representing the
rates of change for the two variables can be followed to generate a trajectory starting from any
initial state (Garcia 2005a).

5

Occasional Paper No. 6
July 2010

Because of their flexibility, conceptual simplicity,
and the representation of interaction
hypotheses in the most natural and intuitive
way, individual-tree models, individual-based
models, or multi-agent models, are particularly
attractive as research tools. Even for these
purposes, however, high dimensionality has
drawbacks that may make it advisable to
complement the individual-based models with
aggregate models. One of the disadvantages
might be called the “so what?” effect: observing
the results of complex simulations is not always
very illuminating. A reductionist approach may
also lose sight of so-called emergent properties.
There is a growing and confusing literature on

these properties, that tend to be described as
“the whole being greater than the sum of the
parts”; in reality, they generally relate to the fact
that correctly modelling individuals is usually
easier than modelling the interactions between
them. As stated by Levin and Pacala (1997):
“individual-based models have the advantage
that they are closer in detail to real systems;
that advantage is also a disadvantage in that
they retain all the details that may hide what is
really important at broader scales”. For these
and other reasons, there is growing interest on
deriving aggregate models from individualbased models (Garcia 2001).

Figure 3: Histograms of 20 random samples of size 50 obtained from the distribution shown.

García  Models and Limits to Predictability

6

In the case of prediction for decision-making,
the limitations of individual-based models are
more serious. One relates to spatial correlations
in the size and growth of neighbouring trees
caused by competition, by similarities of microsite, or by other factors. Tree sizes not being
independently distributed on the ground, the
concept of size distribution used by distanceindependent models presents problems, and
parameter estimation can have significant
biases. It has also been found that the effects of
micro-site often produce positive spatial
correlations, masking any negative correlations
due to competition, contrary to the assumptions
in current distance-dependent models (Garcia
2006). A second problem with the application of
highly detailed models is that often the initial
state is not known with sufficient precision,
which obviously does not allow a reliable
prediction of future states. For instance, even
assuming independence, it is known that
obtaining reasonable estimates of higher
moments, or of the shape of a probability
distribution, requires very large samples
(Kendall and Stuart 1976). Often this high
variability (Figure 3) is not appreciated in the
applications, and one can be mislead as to the
credibility of the projections. A third problem

with the use of complex models for making
predictions is discussed in more detail next.

Limits to predictability
This limitation of individual-based models goes
beyond growth modelling, and has to do with
what can and can not be predicted. To illustrate,
think of the circles of Figure 4 as particles, balls,
or pucks moving in the plane. An individual is
thrust in a certain direction. It is easy to
calculate the trajectory (Figure 4a). Now, what
happens if we change slightly the launching
angle? (Figure 4b). Even with an uncertainty of
one millionth of a degree in the initial angle, the
result becomes completely unpredictable after a
few bounces: the “butterfly effect”, Chaos
Theory, sensitive dependence on initial
conditions.
The situation in forest growth modelling might
not be as bad as this; or perhaps it might.
Experiments with some models suggest that
when altering the starting diameter of one of
the trees by a few millimetres, the difference
increases and spreads quickly to the rest. It may
not be possible to project individual diameters
as well as is generally believed.

Figure 4: Trajectories in the plane. Effect of uncertainty in the initial angle.

7

Occasional Paper No. 6
July 2010

What can be done? If Figure 4 represented a
gas, the behaviour of the whole could be
approximated by the equation PV = kT: pressure
times volume is proportional to temperature.
Note that these variables are properties of the
aggregate, they do not exist at the molecular
level. In fact, pressure and temperature are
related to the mean and variance of the
velocities. For an ideal gas, Statistical Mechanics
is able to derive the aggregate equation from
the dynamics of the individual molecules. In
solids, the relationships between properties of
the ensemble and molecular properties are still
topics of research.
When designing a bridge or car component, in

principle one could model the trajectory of all
the individual molecules. In practice, one would
probably use an average position (centre of
gravity), and apply an aggregate model
proposed by Isaac Newton in the seventeenth
century: d2 x / d t2 = F / m. This is an empirical
model, based on observations, without any
theoretical basis. And it is an approximation, it
fails when going too fast or too small, although
within a certain range it is pretty good.
Figure 5 depicts a pinball machine. Explanation
for the younger audience: a ball is shot through
the channel on the right-hand side, and drops
down the slope colliding with various objects.

Figure 5: Pinball machine. Knowing its functioning does not guarantee predictability.

García  Models and Limits to Predictability

8

Figure 6: Microsoft Pinball, a pinball computer simulation. Simulating is not predicting.

The theory is well known, there is no mystery
about its operation, but can we predict the
trajectory of the ball? Understanding or
explaining does not imply being able to predict.
Figure 6 is an even better example: Microsoft
Pinball. It is a fairly realistic computer
simulation; apparently it contains no stochastic
elements, only calculations based on physical
laws. Given the time that the keyboard space
bar is held down, the movement of the ball is
perfectly predetermined. Can we predict it? It is
sometimes argued that some complex process
model is not yet very precise because of lack of
knowledge about the functioning of some
components, or of the values of certain

9

parameters; with further research it would
become useful for management. In reality, a
model can be very useful for understanding
things better, but there are inherent limitations
to predictability beyond a certain level.
A final example is shown in Figure 7. It is a
device sometimes used in teaching probability.
Steel bearings fall through a grid of pegs into the
bottom compartments. It is hopeless trying to
predict the fate of any of the balls individually. It
is possible, however, to predict reasonably well
the average final position, and to some extent,
its variance. With an adequate sample it may be
possible to get some idea of the distribution.

Occasional Paper No. 6
July 2010

Figure 7: Device for demonstrating the binomial or normal distribution. Individual-level predictions are virtually
impossible, but some statistical summaries can be predicted.

Descriptive models, used primarily for research,
should generally be mechanistic and detailed,
with high dimensionality in their state space,

although some aggregated models can also be
useful. Detailed process models contribute
indirectly to the improvement of managementoriented models, but in these the priorities are
different. For decision-making it is preferable to
link decisions to consequences as directly as
possible.
Contrary to what is sometimes
thought, the management of complex systems
requires simple models.

García  Models and Limits to Predictability

10

Conclusions

References
García, O., 1988. Growth modelling - A (re)view. New Zealand Forestry 33 (3), 14-17.
García, O., 1994. The state-space approach in growth modelling. Canadian Journal of Forest Research
24, 1894-1903.
García, O., 2001. On bridging the gap between tree-level and stand-level models. In: Rennolls, K. (Ed.),
Proceedings of IUFRO 4.11 Conference “Forest Biometry, Modelling and Information
Science”,University of Greenwich, June 25-29,
2001.(http://cms1.gre.ac.uk/conferences/iufro/proceedings).
García, O., 2005a. TADAM: A dynamic whole-stand approximation for the TASS growth model. The
Forestry Chronicle 81 (4), 575-581, (Errata: 81(6), 815, 2005).
García, O., 2005b. Thinking about time. In: Naito, K. (Ed.), The Role of Forests for Coming Generations Philosophy and Technology for Forest Resource Management. Japan Society of Forest Planning
Press, Utsunomiya, Japan, pp. 47-54.
García, O., 2006. Scale and spatial structure effects on tree size distributions: Implications for growth
and yield modelling. Canadian Journal of Forest Research 36 (11), 2983-2993.
Goulding, C. J., 1972. Simulation techniques for a stochastic model of the growth of Douglas-fir. Ph.D.
thesis, University of British Columbia.
Grimm, V., 1999. Ten years of individual-based modelling in Ecology: What have we learned and what
could we learn in the future? Ecological Modelling 115, 129-148.
Kendall, M., Stuart, A., 1976. The Advanced Theory of Statistics, 4th Edition. Vol. 1: Distribution Theory.
Griffin.
Levin, S. A., Pacala, S. W., 1997. Theories of simplification and scaling of spatially distributed processes.
In: Tilman, D., Kareiva, P. (Eds.), Spatial Ecology. The Role of Space in Population Dynamics and
Interspecific Interactions. Princeton University Press, Princeton, New Jersey, Ch. 12, pp. 271-295.
Luenberger, D., 1979. Introduction to Dynamic Systems; Theory, Models and Applications. Wiley, New
York.
Mitchell, K. J., 1975. Dynamics and simulated yield of Douglas-fir. Forest Science Monograph 17, Society
of American Foresters.
Mohren, G., Burkhart, H., 1994. Contrasts between biologically-based process models and
management-oriented growth and yield models. Forest Ecology and Management 69, 1-5.
Munro, D. D., 1974. Forest growth models: A prognosis. In: Fries, J. (Ed.), Growth Models for Tree and
Stand Simulation. Royal College of Forestry, Research Note 30, Stockholm, Sweden, pp. 7-21.
Newnham, R. M., 1964. The development of a stand model for Douglas fir. Ph.D. thesis, The University
of British Columbia.
Staebler, G. R., May 1951. Growth and spacing in an even-aged stand of Douglas-fir. Master's thesis,
School of Natural Resources, University of Michigan.

11

Occasional Paper No. 6
July 2010

Stage, A. R., 1973. Prognosis model for stand development. Research Paper INT-137, USDA Forest
Service, Int. Northwest For. and Range Exp. Sta., Ogden, Utah.
Vanclay, J. K., 1994. Modelling Forest Growth and Yield: Applications to Mixed Tropical Forests. CABI
International, Wallingford, UK, 312 p.

García  Models and Limits to Predictability

12