# Research in the time of corona

What a time? Who would have assumed such an unprecedented global disaster? I just hope that as many readers of this blog as possible have remained unaffected by the virus and can continue now in good health.

I was caught by the German lockdown in mid March when re-uniting with my family near Göttingen. Luckily I had just finished all my teaching a month earlier. Of course, all my conferences and scientific visits got cancelled as they were for most of us. On my way between Umeå and Frankfurt I travelled through ghost airports that were only marginally functioning… Glad that I made it in the end.

What was anticipated to be a month in Germany has become three months now, more or less in total isolation from people other than family. It has been a great time for the family to live through this crisis together, of course, and we have taken much comfort from being together.

Surely everybody felt the strange feeling of working from home without much external contact and the confinement of limited living and working space. It was nice to see colleagues from work in Zoom from time to time and to catch up, but otherwise I felt a bit like a monk working quietly in his cell and not to go much beyond the boundaries of the property. When things relaxed a bit, wearing masks in public has become a common habit now and we almost don’t even notice them any more.

Despite many lives lost everywhere, it has certainly been a convenient time to sit down and think about research and to finish some long-standing projects. Luckily research in forest biometrics and quantitative ecology is quite crisis proof: If you have the data, all you need is one or two computers. So I upgraded my home office a bit and simply cracked on.

I came to Germany with an almost finished research project on the spatial correlation between tree species and tree size diversity in highly diverse Chinese temperate woodlands. This topic has intrigued me for quite a while, as I believe the spatial correlation between tree species and tree size diversity is part of nature’s mechanism to maintain high levels of biodiversity. Understanding more about this relationship will allow us to mimic the natural maintenance of biodiversity and this is crucial to our efforts to stem the tide of worldwide biodiversity loss. I collaborated here with Gongqiao Zhang and Xiaohong Zhang from the Chinese Academy of Forestry.

Next I worked on a new principle of quantifying nearest-neighbour size inequality. The idea for this work spontaneously came to me when reviewing methods of modelling asymmetric growth in individual-based models. One of these methods used trigonometric functions and related to another method that was suggested by Oscar García in 2014. When I remembered this, I reached out to Oscar in Chile and we had a good email discussion on this new index, which I much enjoyed. Together with Janusz Szmyt from Poznan University of Life Sciences and Gongqiao Zhang from the Chinese Academy of Forestry we found that the new nearest-neighbour characteristic is a good indicator of spatial size inequality but is also highly correlated with the growth of the subject tree.

Then I considered the intriguing problem of applying modified approval voting to situations where a number of test persons are asked to mark trees for eventual eviction on a sheet of paper or on a tablet computer. This is a situation common in marteloscope research, where all trees of a forest stand are numbered. I reached out to colleagues at Technische Universität Berlin, Markus Brill and his team, and we figured that part of their voting research can be used to work out how a representative list of trees can be calculated as a synthesis of individual marks of a number of test persons. This is a new approach to crowdsourcing with a view to invite experts in a certain environmental field and synthesising their expert knowledge in a representative way through modified approval voting. Such quantitative croudsourcing is very useful whenever new ways in forest management need to be followed and no best-practice guidelines are available. We applied the novel method to 50 marteloscope data sets kindly provided by Jens Haufe from the Technical Development Department of Forest Research at Ae (Scotland, UK).

I much enjoy this interdisciplinary and international work and I am glad that through digitalisation it is still possible to reach out and collaborate despite Corona. There is so much to gain from this kind of cooperation. Somehow research has carried me through this time of change. Let’s just hope that things will have improved over the summer so that we can meet on campus again.

# Weibull distribution for characterising stem-diameter structure

The Weibull density distribution is known as

$Latex formula$,

where $Latex formula$ is the location, $Latex formula$ the scale and $Latex formula$ is the shape parameter, i.e. the parameters of the Weibull distribution are interpretable, which is always a good property of models.
The cumulative distribution function, i.e. the integral of density function, is much simpler:

$Latex formula$

What does it mean?

The Weibull density distribution allows characterising tree stem-diameter distributions by providing trend curves but more importantly by summarising stem diameters by means of three parameters that can be interpreted.

The shape parameter of the Weibull distribution, $Latex formula$ is of particular interest in this context and the following interpretation aid can be used (Burkhart and Tomé, 2012, p. 198):

$Latex formula$ Skewed to the right
$Latex formula$ Skewed to the left
$Latex formula$ Negative exponential, reverse j-shaped

When $Latex formula$ is less than 1, the distribution is reverse j-shaped found in uneven-aged forest stands and when $Latex formula$ equals 1, a negative exponential distribution results. If $Latex formula$, the Weibull distribution approximates a normal distribution and this value divides left- and right-skewed curves. In general, $Latex formula$ gives bell-shapes typical of even-aged forest stands. The location parameter is directly related to the minimum diameter in a stand (Burkhart and Tomé, 2012, p. 265 ). In the context of diameter distributions, all model parameters must be positive.

Where does it come from?

The Weibull distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin and Rammler (1933) to describe a particle size distribution.

Why is it important?

The Weibull distribution is one of the most flexible and most commonly applied model for tree stem-diameters. The model parameters are comparatively easy to estimate and the distribution is easy to apply.

How can it be estimated?

Robinson and Hamann (2010, p. 164ff.) described in detail how the three parameters of the Weibull distribution can be estimated using R and the maximum-likelihood method. The grey box below gives an adaptation of that method:

```dweibull3 <- function(x, gamma, beta, alpha) {
(gamma/beta) * ((x -alpha)/beta)^(gamma - 1) * (exp(-((x - alpha)/beta)^gamma))
}

loss.w3 <- function(p, data)
sum(log(dweibull3(data, p[1], p[2], p[3])))

mle.w3.nm <- optim(c(gamma = 1, beta = 5, alpha = 10),
loss.w3, data = myData\$dbh, hessian = TRUE,
control = list(fnscale = -1))

mle.w3.nm\$par # model parameters

# Check whether the curve looks OK
xx <- seq(10, 60, 1)
hist(myData\$dbh, freq = F, breaks = 50, xlim = c(10, 60))
lines(xx, dweibull3(xx, mle.w3.nm\$par[1], mle.w3.nm\$par[2],
mle.w3.nm\$par[3]), lty = 1, col = "red")```

Since R only provides implementations for the two-parameter version, the code starts with a new function dweibull3() implementing the three-parameter version. This is followed by a maximum-likelihood loss function using the previously defined function dweibull3(). This loss function in turn now forms one of the arguments of the optim() function used for carrying out the regression. myData is a data frame that includes a vector of stem diameters that can be addressed by myData\$dbh.

An alternative to nonlinear maximum-likelihood based regression are percentile estimations. The idea of this approach is to estimate the three parameters of the Weibull distribution from selected points of the distribution, e.g. the 63rd or 95th percentile. The theory of percentile estimators is well explained in Clutter et al. (1983, p. 127ff.). However, percentile estimations can be very valuable where maximum-likelihood regression does not produce any or unsuitable results. Some methods of percentile estimation have also been linked to straightforward sampling methods, so that the parameters of the Weibull function can almost be directly sampled in the field without much effort. Also, percentile methods can be used to identify starting values for nonlinear regression. Below one example method of percentile estimations is provided.

According to Wenk et al. (1990, p. 198f.) and Burkhart and Tomé (2012, p. 265)

$Latex formula$.

$Latex formula$ and can be interpreted as the diameter where approximately 63% of all trees are smaller in diameter. $Latex formula$ , i.e. the minimum diameter in a tree population or forest stand.

Finally, Gerold(1988) suggested that $Latex formula$ can be estimated from $Latex formula$ and the diameter where approximately 95\% of all trees are smaller in diameter.

$Latex formula$

$Latex formula$, $Latex formula$ and $Latex formula$ can be estimated from any empirical diameter distribution but also by employing a simple sampling procedure. Based on a systematic sampling grid approximately ten sample points need to be identified in every forest stand along with the first twelve tree neighbours nearest to each sample point. Römisch (1983) found that $Latex formula$ can be estimated from the diameter of the fifth largest tree out of twelve sample trees and that $Latex formula$ can be estimated from the largest diameter tree out of ten sample trees nearest to the sample point. $Latex formula$ and $Latex formula$ are then calculated as the arithmetic means of all ten samples. $Latex formula$ is the smallest diameter of all sample trees (Gerold, 1988).

Literature

Burkhart, H. and Tomé, M., 2012. Modeling forest trees and stands. Springer, Dordrecht.

Clutter, J. L, Fortson, J. C., Pienaar, L. V., Brister, G. H. and Bailey, R. L., 1983. Timber management. A quantitative approach. John Wiley & Sons, New York.

Fréchet, M., 1927. Sur la loi de probabilité de l’écart maximum. Annales de la Société Polonaise de Mathematique 6: 93-116.

Gerold, D., 1988. Describing stem diameter structure and its development by using the Weibull distribution. Wissenschaftliche Zeitschrift der Technischen Universität Dresden 37: 221-224.

Rosin, P. and Rammler, E., 1933. The laws governing the fineness of powdered Coal. Journal of the Institute of Fuel 7: 29–36.

Robinson , A. P. and Hamann , J. D., 2010. Forest analytics with R. An introduction. Use R! Springer. New York, 339p.

Römisch, K., 1983. A mathematical model for simulating growth and thinnings of even-aged pure stands. PhD thesis Technical University Dresden. Dresden, 197p.

Wenk, G., Antanaitis, V. and Šmelko, Š ., 1990. Forest growth and yield science. Deutscher Landwirtschaftsverlag. Berlin, 448p.

Weibull, W., 1951. A statistical distribution function of wide application. Journal of Applied Mechanics 18: 293–297.

# Visiting BOKU University in Vienna

Long had I intended to pay BOKU University in Vienna another visit and much prevented me from carrying out this plan, too many excuses that couldn’t be ignored or so at least we often think by ourselves. To me BOKU University Vienna – officially the University of Natural Resources and Life Sciences, Vienna, is a place of many friends: Here I found many people I collaborated with over the years, I found mentors and students. Many of them visited me at the various places I have worked in Europe and finally I decided to apply for my habilitation at BOKU, which was awarded to me in 2009. The fact that one of my friends here, Hubert Hasenauer, recently has become the vice-chancellor of BOKU, was another incentive to return. At the same time it is is nice to contribute to increasing the cooperation between BOKU and SLU.

BOKU is a truly lovely place, not too far from Vienna city centre but placed in a slightly quieter part of the city next to a large and very attractive park which seems like a natural extension of the campus. The representative facade of the Wilhelm-Exner building at 82 Peter-Jordan Street offers a special welcome in style. The onset of lovely autumn colours in adjacent vineyards is a particular treat and consuming moderate amounts of Heurigen vine clearly fuels scientific inspiration. It was wonderful that my BOKU mentor and friend Hubert Sterba took me to one of Vienna’s Heurigen restaurants the other day.

Manfred Lexer, acting head of the Institute of Silviculture, and I submitted a guest-professorship proposal to BOKU’s Senate last year and after its approval in December 2018 planned this visit, which I unfortunately had to postpone by one term, as I needed the time in the first half of the year to finish my textbook. At the same time Hubert Hasenauer kindly supported my visit, as we had agreed to meet up again a long time ago.In the end things worked out very well, since I could prepare my course in “Individual-based forest ecology and management” in good time and the accompanying textbook had just appeared in print on time.

The Institute of Silviculture has a long tradition in eco-physiological gap modelling (such as the PICUS model) but also with statistical individual-oriented modelling, such as the MOSES model. During my visit I am attempting to add to this rich modelling expertise my experience in point-process inspired individual-based modelling. Currently I am using some of Hubert Hasenauer’s Norway spruce – Scots pine data that he used for his MOSES model to develop a new, advanced interaction-field-based model for mixed-species forests. The model will be based on relative growth rates and attempts to better describe the simultaneous allocation of biomass to height and stem-diameter growth.

At the same time I am cooperating with Xiaohong Zhang from the Chinese Academy of Forestry who currently spends a year at BOKU’s Institute of Silviculture and is attending my course in “Individual-based forest ecology and management”. Together we are investigating the interaction between Quercus mongolica and Pinus koraiensis in semi-natural woodlands in China’s Jilin province.

On Tuesday, 22 October, I will give a scientific talk on “Understanding forest development through interaction: Individual-based models in forestry” as part of the Institute’s Science Afternoon.

It is good that there is funding for such academic visits. They clearly make a big difference to the life of researchers and students. Everybody involved in them gains experience, motivation and inspiration, things that are so essential to our daily work. I wished more SLU students and researchers would use the opportunity to visit BOKU.

# Individual-based methods in forest ecology and management

How is it to write a textbook? That was the question I was wondering about when approaching my forties. A colleague once mentioned to me, his mentor had told him that forty years of age is the magic threshold when researchers really begin to see the holistic picture of their research as if standing on a mountain top and seeing the horizon at last. Was I at that stage and saw how everything was connected? A good question at the time and difficult to answer. To some degree, yes. I felt ready and got curious and started asking around when and why other academics felt compelled to write a textbook. I personally had felt this strong vocation for a long time without being able to say where it had actually come from. So I was interested to learn why others had decided to write a textbook. The answers varied as was to be expected. Some authors were invited to join a senior colleague, others were working in a field that was not well covered at the time and thought there was an urgent need to fill this gap in the literature. One colleague told me he felt committed to writing a substantial textbook in forest ecology to boost research cooperation and was richly rewarded by the warm reception the book had received. There was also a distinctive (almost geographic) pattern (I am not revealing where…) of academics nearing retirement or writing textbooks in their retirement that often did not make it beyond the first edition. Of course, scientific skills, knowledge and insights keep growing all the time and beyond the age of 40 or even beyond retirement there are still many scientific breakthroughs and insights to expect from most academics. And universities are well advised to make good use of their capital of emeriti professors…

It seemed many textbook authors did not engage in lengthy analyses or even did pros and cons to deliberate whether they should or should not engage in writing their book. Most authors of the better books simply felt the need, the personal urge to embark on this adventure. And an adventure it truly is. For a start you often get discouraged by your own organisations to write a book, since articles are rated more highly. I have seen this at a number of universities. This is a shame, since textbooks are the only way to show the big, holistic picture, i.e. how all the small peer-reviewed research papers relate and together make a big, intriguing story. Also, of course, textbooks are a crucial contribution to academic education and ensure that all our good ideas are passed on to the next generation of researchers and practitioners.

Once I had emancipated myself from these concerns my mind opened up to comments from those colleagues from across the world who had nudged me for years and suggested I should write a textbook that would include my research in spatial forest structure, growth analysis, individual-based modelling and human tree selection behaviour. Most of these well-meaning colleagues advised: “Write the book now, not later.” So I sat down and considered the task. For years I had been teaching courses on these subjects to BSc, MSc and PhD students. While I was still pondering the contents and structure of the book, I was awarded my SLU chaired professorship at Umeå and I sadly had to put the book plans on hold while building up my new professorship and research group, which, of course, was also fun and very exciting. Then after four years the opportunity arose to get relieved of some duties and to free up time. This was the chance to engage in writing the textbook and to finish it. The last four years my group and I managed to publish many good research paper that would lent good authority to the book. So one fine sunny afternoon in September 2018 in my office at Umeå the book “Individual-based methods in forest ecology and management” was finally conceived and nine months later we submitted the manuscript. Pavel Grabarnik from the Russian Academy of Sciences in Pushchino had joined the book project on my invitation. Pavel and I go back a long way and have discussed questions of ecological modelling for more than ten years. It was a particularly nice experience for us to work so closely together for such an intense period and rewarding task. The forword was kindly written by Dan Binkley, a recent Wallenberg professor at SLU. During his visits to Umeå Dan and I shared inspiring discussions and reflected on individual-based forest ecology and management.

Looking back the writing process was really fun. It was an eye-opener also for us authors. Better than ever before we saw and understood how seemingly different concepts and ideas were related and how big and important the field of individual-based forest ecology and management actually is. The writing process was a true academic quest and every hour spent was worth it. The most dreary and nerve-racking time clearly was checking the proofs this summer but that was not our experience alone and is part of the process of writing a book.

Well, now here it is. One of these greenish books with a little bishop on the front cover. The text provides essential information on theories and concepts of individual-based forest ecology and management and introduces point process statistics for analysing plant interactions. This is followed by methods of spatial modelling with a focus on individual-based models. The book is complemented by key concepts of modern plant growth science. Finally new methods of measuring, analysing and modelling human interaction with trees in forest ecosystems are introduced and discussed. For better access and understanding, all methods introduced in this book are accompanied by example code ready to use in the statistical software R and by worked examples.

I hope you will enjoy it. Check it out! My first course based on the new book will be given at BOKU University, Vienna in October this year. We thank the team at Springer for the excellent cooperation and for their patience.

And please academics, don’t give up despite the odds! Live your dreams and write your textbooks. Academia needs every single one of them.

# Potential growth and quantile regression

What does it mean?

AGR = potential AGR $Latex formula$ modifier(s)

In many plant modelling applications growth processes are modelled in such a way that maximum growth or the growth of dominant plants is reduced by mitigating agents such as the interaction with other plants and factors of the physical environment such as light, water, temperature and nutrients. Dominant plants can for example be thought of as open-grown trees growing on their own in the open landscape without interacting with any other trees so that at least stem diameter growth can be considered as maximum. This strategy is referred to as the potential-modifier approach. (AGR is absolute growth rate, see Pommerening and Muszta, 2016.)

Where does it come from?

The potential-modifier approach was first published by Newnham (1964) and Botkin et al. (1972) as part of empirical and so-called gap models.

Why is it important?

Potential growth defines an upper limit of growth for a given species on a given site. This avoids unrealistic model estimations, as can be the case when estimating AGR directly from plant size, interaction and factors of the abiotic environment. Usually only the model parameters of the function defining potential growth change when adapting the model to new species and sites whilst the parameters relating to the modifiers stay the same. This increases the robustness of the overall growth model and allows easier adaptations.

How can it be used?

Measurements from the above-mentioned open-grown trees can be used for defining potential AGR. However, since this involves substantial additional sampling efforts, another strategy is to apply quantile regression (Koenker and Park, 1994; Cade and Noon, 2003) to the data collected for parametrisation. Quantile regression is not so different from conventional regression. Instead of fitting functions of the usual 0.5-quantile of the observed data (mean regression), larger or smaller quantiles are selected. For the application to the potential-modifier approach usually upper quantiles of 0.95 or 0.975 are used.

For example, assuming a dependency on size tree stem-diameter, AGR can be described using the first derivative of the Chapman-Richards growth function (Pienaar and Turnbull, 1973; see also my earlier blog on this function):

$Latex formula$,

where y in this case is tree stem diameter but otherwise could be any plant size characteristic. A, k and p are model parameters.

R code

In R, it is quite straightforward to apply quantile regression thanks to the quantreg package. In preparation AGR and size data should be compiled in a common data frame. Then we install and load the quantreg package:

`install.packages("quantreg", dep = T)library(quantreg)`

Next we define the AGR function to be used for describing potential growth and this we again use the aforementioned first derivative of the Chapman-Richards growth function, where time is exchanged for size (stem diameter dbh in this case):

`dpot <- function(dbh, xA, xk, xp) {                   return(xA *  xk * xp * exp(-xk * dbh) * (1 - exp(-xk * dbh)) ^      (xp - 1)) }`

Finally we can use the quantile regression routine, which in turn uses function dpot:

`nlsout <- nlrq(AGR ~ dpot(dbh, A, k, p), data = TreeList, start = list(A = 54.1, k = 0.01, p = 1.19), tau = 0.975, trace = TRUE)`

From the syntax we can see that the difference to common regression procedures such as nls is minimal. The main difference is parameter $Latex formula$ which defines the quantile. A model summary can be obtained in the same way as from nls which also allows retrieving the model parameters:

`summary(nlsout)A <- summary(nlsout)\$coefficients[1]                         k <- summary(nlsout)\$coefficients[2] p <- summary(nlsout)\$coefficients[3] `

And here is an example result:

Cade and Noon (2003) noted that quantile regression also has a general place in data analysis providing a more complete view of possible causal relationships between variables in ecological processes, where mean regression techniques would fail to identify relationships between explanatory and response variables. The additional advantage is that one can directly estimate rate parameters for changes in the quantiles of the distribution responses conditional on the predictor variables.

Literature

Botkin, D. B., Janak, J. F., Wallis, J. R., 1972. Some ecological consequences of a computer model of forest growth. The Journal of Ecology 60: 849.

Cade, B. S., Noon, B. R., 2003. A gentle introduction to quantile regression for ecologists. Frontiers in Ecology and the Environment 1: 412-420.

Koenker, R., Park, B. J., 1994. An interior point algorithm for nonlinear quantile regression. Journal of Econometrics 71: 265-283.

Pienaar, L. V. and Turnbull, K. J., 1973. The Chapman-Richards generalization of von Bertalanffy’s growth model for basal area growth and yield in even-aged stands. Forest Science 19: 2-22.

Pommerening, A. and Muszta, A., 2016. Relative plant growth revisited: Towards a mathematical standardisation of separate approaches. Ecological Modelling 320: 383-392.

Newnham, R. M., 1964. The development of a stand model for Douglas fir. Ph.D thesis, University of British Columbia, Vancouver, 201p.

# Forest biometrics – what lies ahead?

The emancipation of forest science indeed took a long time. Forest academies starting off as teaching institutions for forestry staff more than 200 years ago carried out limited research to support state and private forest management. The spirit of this set up continued until recently with teaching that was more practical than in other fields of natural sciences and research that was predominantly industry oriented and dominated by forest management and planning. This fundamentally changed in most western countries towards the 1990s when forest science started to be considered and reviewed like any other subject area in natural and social sciences. This development is still ongoing in Europe and throughout the world. It has led to a fundamental change in subject areas within forest science: Academic fields such as soil science, genetics and plant physiology that until then played a modest role went right to the top whilst formerly dominating fields such as forest management lost much of their importance. Some academic fields were abandoned. At other places forest science was shut down at respective universities or merged with other fields beyond recognition. Whether all of these changes were to the better, is another matter …

Forest biometrics is a newcomer in forest science. It modestly started off as forest mathematics at different places in Europe and North America some 100-150 years ago for teaching a minimum of essential quantitative skills. As many academic fields initially it was supposed to fill a support role for the engineering parts of forest science teaching and research. However, since the 1960s/70s this view started to become increasingly outdated and the subject area successfully established itself at most universities as a research field in its own right in the same way as other disciplines.

Some chairs in forest biometrics have since then specialised on forest growth and yield modelling or general statistics. With some notable exceptions this is typically the case in North America. In Europe, however, there is an increasing trend for forest biometrics to play an important role in quantitative ecology and ecosystem modelling. This niche is also partly occupied by other research institutions outside forest science and the collaboration with these has been very fruitful and inspiring.

The Forest Faculty at SLU was recently ranked as a leading international forest science institution. As a professor representing SLU this was really nice to read. The Faculty has surely deserved this for all their hard work and at the same time ours is one of the last few forest science only faculties in Europe. In 2024 SLU will host the IUFRO world congress at Stockholm, the birthplace of the SLU Faculty of Forest Sciences and next year we will launch a new MSc degree in Forest Ecology and Sustainable Forest Management.

Along with other academic fields forest biometrics has an important role to play to help the Faculty to maintain this position in the world. To strengthen forest biometrics and in recognition of the achievements of the current chair, the professorship in forest biometrics at SLU recently moved from the Department of Forest Resource Management to the Department of Forest Ecology and Management.  The chair has received a warm welcome at the new department and I am most grateful to my new colleagues. I will continue my research in individual-based methods of forest ecology involving point process statistics, individual-based modelling, tree growth analysis and the analysis of interaction between humans and trees. The organisational change reflects current trends elsewhere in Europe and allows a better integration of quantitative ecology in the wider academic field of forest ecology. SLU therefore have made an important decision towards shaping the future of forest science and maintaining its mission in Sweden, Europe and the world.

# Modelling size distributions

Analysing and studying size distributions has been important to population ecology and beyond for a long time. Empirical size distributions give important clues about current size structure of a given population and often even allow conclusions about the prevailing ecological processes. Negative exponential stem-diameter distributions are, for example, often associated with forests that are exposed to some level of disturbance and bell-shaped stem-diameter distributions can often be found in forest plantations. Therefore computing size distributions is often one of the first data-exploration tasks in ecological studies as well as in studies related to forest management. Other important contexts include:

• Sustainability of timber resources in small-scaled forest ownership,
• Silvicultural controlling,
• Decision support for forest management,
• Training and education,
• Maintaining a certain forest structure for recreational forests,
• Important starting point for analyses in point process statistics.

Empirical size distributions can be produced using histograms and bar plots. These approximate the probability density function, which can be modelled using for example the Weibull distribution model but also the beta, gamma or normal distributions. These are parametric models and there is also the possibility to produce non-parametric trend curves based on kernels. In that case no particular model assumptions are made and applying non-parametric trend curves is a good strategy for initial data exploration to prepare the selection of suitable parametric models.

All of these methods and models have in common that they provide static information about size distributions. However, size distributions are dynamic and we are often interested in how these size distributions change with time. Of course, it is possible to model populations bottom-up at the level of individuals and then to summarise each simulation step by size distributions. Often the dynamics of the size distribution itself is of interest, researchers and stakeholders are, for example, interested in the demographic processes of certain size variables. This research approach is then closely related to other demographic studies, e.g. in animal or human populations. Typical processes involved in a dynamic size distribution are shown in the schematic graph below.

Back in the early 2000s silviculturists in the UK were very interested in these dynamic size distributions for deriving thinning guides for Continuous Cover woodlands. I remember well that I first started to discuss them in greater detail with Hubert Sterba (BOKU University, Vienna), when he first visitied me at Bangor University in 2002. In 2012 I had the opportunity to exchange views with Jean-Philippe Schütz (ETH, Zürich) whilst working in Switzerland.

In Schütz and Pommerening (2013) we used such a demographic approach to model the equilibrium conditions for a Douglas fir single-tree selection system in North Wales. The equilibrium conditions were identified in this work to inform and guide forest management. The Douglas-fir plots in Artist’s Wood (Gwydyr Forest) in North Wales have always been my favourites during my time at Bangor University and Jean-Philippe and I intended to provide evidence that single tree selection systems are possible with Douglas fir. The modelling approach was based on Schütz (2006). Brzeziecki et al. (2016) refined this approach and applied it to natural stands in Bialowieza Forest, where no forest management takes place. Here our research objective was to model equilibrium size distributions for the main tree species, in order to find out about future species composition trends as a contribution to current discussions about this important virgin forest.

The modelling approach is simple and effective. As an important component it is (1) necessary to model the outgrowth rate $Latex formula$ for each diameter class $Latex formula$. This is a function of (2) size-dependent absolute growth rate $Latex formula$, which needs to be modelled in the next step. Finally (3) size-dependent mortality rate $Latex formula$ has to be determined.

Using these “ingredients” the number of trees expected at steady state can be calculated for every size class starting from a known value of $Latex formula$:

$Latex formula$

Here $Latex formula$  and $Latex formula$. Using such a dynamic size-class model many questions in population ecology and forest management can be answered.

There have been many alternative models such as the $Latex formula$-factor model and its derivatives (see for example Cancino and Gadow, 2002), but they have been found to be too inflexible and not based on real, observed growth and mortality processes.

Literature

Brzeziecki, B., Pommerening, A., Miścicki, S., Drozdowski, S. and Żybura, H., 2016. A common lack of demographic equilibrium among tree species in Białowieża National Park (NE Poland): evidence from long-term plots. Journal of Vegetation Science 27, 460-469.

Cancino, J. and Gadow, K. v., 2002. Stem number guide curves for uneven-aged forests development and limitations. In: Gadow, K. v., Nagel, J. and Saborowski, J. (eds.), 2002. Continuous cover forestry. Assessment, analysis scenario. Kluwer Academic Publishers. Dordrecht, pp. 163-174.

Schütz, J.P., 2006. Modelling the demographic sustainability of pure beech plenter forests in eastern Germany. Annals of Forest Science 63, 93-100.

Schütz, J.P. and Pommerening, A., 2013. Can Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) sustainably grow in complex forest structures? Forest Ecology and Management303, 175-183.

# Reconstruction of spatial woodland structure

Reconstruction is commonly understood as the process and the result of re-establishing something that (at least partially) does not exist any longer or of re-establishing the unknown. There are a range of established reconstruction techniques for example in archaeology, in forensics (e.g. face recognition), in medicine (e.g. implants) and computing (data reconstruction). Also statistical imputation, i.e. the process of estimating missing observations and spatial interpolation in geostatistics are related to reconstruction. Without doubt reconstruction is a fascinating research field with many different applications.

Another important, more ecological application is the reconstruction of spatial forest structure. All data, that are collected in forest ecosystems, have a temporal as well as a spatial dimensions. The properties of the whole forest ecosystem, e.g. wood production, habitat and recreational values, to a large degree depend on the underlying ecosystem structure, particularly on its microstructure. This microstructure is typically shaped by physiological and ecological interactions, but the microstructure also influences interactions. These structure-property relationships play a crucial role in providing ecosystem goods and services and in the maintenance of biodiversity. Usually data related to spatial forest structure are available only on a sample basis but subsequent research requires full information which then needs to be reconstructed.

Reconstruction can even be employed for habitat modelling: Imagine a set of summary statistics that well describe requirements of a certain, endangered animal species. These can then be used to modify an existing landscape at the computer to meet these requirements. In considering this option, we have actually moved on from reconstruction to construction, because an active change of landscape structure is modelled. Bäuerle and Nothdurft (2011) used spatial reconstruction to model habitat trees.

Another important purpose of reconstruction is the testing of competing summary characteristics. There are usually a number of statistics describing the same aspect of spatial forest structure or of a certain material. Which one is better, which one should be used in a given analysis? One way of shedding light into this is by reversing the analysis. The competing summary characteristics  are used in separate simulations for reconstructing the original data from the results provided by these summary characteristics. This synthesis of a given analysis is nothing else than reconstruction and is applied a lot in materials science.

So how does it work?

Well, it turns out (re)construction is not difficult at all. First you need one or more summary characteristics describing the structure you intend to reconstruct. This can be a simple index based on a mean, it can be a histogram or a function. You can also choose a number of different summary characteristics.

Then you select a stochastic optimisation method. For this other researchers and I have successfully used the simulated-annealing method. This method was first developed in physics, it relates to thermodynamics, particularly to how the energy state of for example metals change when they anneal. The reconstruction algorithm is based on simulated annealing. One way to start is by randomly dispersing tree locations in a given observation window. Then iteratively  one of these points is randomly selected and shifted to a new random location within the observation window. For this change the summary characteristic(s) are re-calculated and compared with a target, e.g. the same summary characteristic for some ideal of a reference. If the change leads to a better approximation of the target, it is made permanent otherwise it is rejected and the old state is restored (improvements-only algorithm). In either case, this is followed by the random selection and shift of another point. This process stops when the difference between target and observed characteristic(s) becomes very small or after a certain number of iterations (Torquato, 2002).

In previous publications I employed this method to reconstruct the structure of a whole forest stand from sample data by interpolating the unmeasured structure between the sample plots (Pommerening and Stoyan (2008). This even let to the astonishing result that in some cases certain summary characteristics were better estimated from the reconstruction than from the sample although no additional information was added. Nothdurft et al. (2010) in fact corrected a biased sampling design through spatial reconstruction. Together with Estonian colleagues we also used reconstruction for simulating off-plot edge correction buffers (Lilleleht et al., 2014). It is also possible to include existing, measured objects in the reconstruction through conditional simulation (Pommerening and Stoyan, 2008).

On my website pommerening.org you can find example code in R and C++ for construction, where you can essentially model a spatial point pattern that leads to a certain pre-set value of the aggregation index by Clark and Evans. Along similar lines it is also possible to set a certain species mingling index (see one of my previous blogs) for an existing point pattern and then to swap pairs of trees of different species. Here the point locations remain the same, only the species marks are re-allocated (see below for some results). Applications to remote-sensing and other image data are also known. There is no limit really to anybody’s resourcefulness to invent new ways of spatial reconstruction.Example of species mingling construction: A random point pattern was simulated using a Poisson process in a window of 100 x 100 m. The two species (red and yellow were randomly assigned to the points with a probability of 0.5, left). Through construction the species dispersal was optimised to achieve a mingling index J of -0.20, where among the four nearest neighbours heterospecific points attract each other (right). Note that the point locations have remained the same.

Interested? Any questions? Don’t hesitate to get in touch.

Literature

Bäuerle, H. and Nothdurft, A., 2011. Spatial modeling of habitat trees based on line transect sampling and point pattern reconstruction. Canadian Journal of Forest Research 41, 715-727.

Lilleleht, A., Sims, A. and Pommerening, A., 2014. Spatial forest structure reconstruction as a strategy for mitigating edge-bias in circular monitoring plots. Forest Ecology and Management 316, 47-53.

Nothdurft, A., Saborowski, J, Nuske R. S. and Stoyan, D., 2010. Density estimation on k-tree sampling and point pattern reconstruction. Canadian Journal of Forest Research 40, 953-967.

Pommerening, A. and Stoyan, D., 2008. Reconstructing spatial tree point patterns from nearest neighbour summary statistics measured in small subwindows. Canadian Journal of Forest Research 38, 1110–1122.

Torquato, S., 2002. Random heterogeneous materials. Interdisciplinary applied mathematics 16, Springer, New York.

Tscheschel, A. and Stoyan, D., 2006. Statistical reconstruction of random point patterns. Computational & Data Analysis 51, 859-871.

# Tree interactions and individual-based modelling

Many scientific papers deal with aspects of plant interactions including competition and facilitation. It is very likely that plant interactions play an important role in structuring our ecosystems and this is indeed a prolific research field that has kept many researchers busy for the last hundred years. In the absence of a deeper understanding of the physiological mechanisms involved in interactions, plant performance (in terms of growth and yield) was often used as an indicator of plant interaction patterns. Given the same environmental conditions, a lack of performance may be an indicator of competition and surplus performance may indicate dominance or facilitation, but this does not always need to be the case.

Stem growth of trees can, for example, be considered a performance variable. A common assumption based on ecological theories is that trees with few competitors, i.e. growing in quite open conditions, respond more in terms of growth than those that grow in tight neighbourhoods. Research in tree mechanics, however, suggests that what is interpreted as competition effects, has more to do with a tree’s need to adjust to environmental challenges, particularly to withstand the forces of wind. A tree protected by neighbouring trees in a tight neighbourhood does not need to invest much in stem growth because it is sheltered from the wind, whilst another tree growing in more open conditions is strongly affected by wind. Considering this the growth response merits a very different interpretation.

Promising research is currently underway attempting to uncover the physiological receptors and processes related to interaction. This will undoubtedly put us into a better situation to review and select appropriate criteria for identifying the mode and intensity of interactions.

In this context, individual-based models assume an important role in exploring plant interactions. This model type is in fact specialised on handling interactions between individuals or agents.  The behaviour of a system, for example a forest ecosystem, is the result of individuals acting and interacting in a given environment. The same applies to the projection of growth or mortality, which no longer is the result of statistical estimations but a consequence of individual “decisions” and interactions. Thus following this modelling philosophy interaction is handled in a very mechanistic way and can also be linked with physiological receptors and processes. Individual-based modelling is also very intuitive and creative. Many software packages such as NetLogo and the Java Agent-Based Modelling Toolkit have been developed to support this modelling approach, however, R and Python are also good choices to begin with. Individual-based modelling is not limited to any particular field of science and shares interesting links with physics and point process statistics.

An excellent opportunity for exploring the world of A/IBMs is a workshop that my group is offering at Umeå (Sweden) on 21/22 August 2018, see our workshop website. In this workshop we will have a few keynote talks followed by talks of the workshop participants. Every participant can highlight anything they consider important, e.g. why they are interested in IB modelling, what they perceive as challenge or programming problems they ran into and want to share. The workshop is not limited to any particular ecosystem or agent and I believe the exchange across (eco)systems and specific applications can be very inspiring and eye-opening. If you haven’t signed up yet please consider the opportunity and get in touch!

# Behind the paper “Do large trees tend towards high species mingling?”

It often occurred to me that it is a pity to be able to read finished research papers only. As we all know, they have to be very concise and can only report a selection of the most positive outcomes of everybody’s research. However, this is only the tip of the iceberg, isn’t it? There usually is much more to it. Much delight, of course, but also struggle, much frustration when the results turn out a different way than expected or when the reviewers fail to see the point of the paper. In fact in summer this year I have received an interesting email where an eminent researcher stated: “It seems crazy, but my experience has been the more novel my ideas, the more likely they are to get rejected from publication.” Many of his colleagues responded to this by saying: “Me too”. I think this is encouraging and disappointing at the same time. It highlights one of the failures of the current peer-review system but also tells us that we have to try even harder to tell the fascinating stories of our research.

Every research usually has quite a bit of background history that doesn’t get published. Of course, not every single bit of our research is a hit and may be as fascinating as the discovery of the DNA double helix by Watson and Crick. But still, I am sure that there are a great deal of fascinating background stories that would make our lives richer, if they were actually told. Imagine what encouragement and boost in inspiration this could potentially give to young researchers and particularly to PhD students!

In this spirit, feeling committed to my own appeal, I would like to say a few words on our recent paper “Do large forest trees tend towards high species mingling?” published in Ecological Informatics.

The story of this paper probably started off during my field trips to Bialowieza Forest in 2011 and 2016, when Bogdan Brzeziecki and his colleagues from Warsaw University of Life Sciences kindly showed me a number of forest sites where a species change has occurred and large remnant trees of the previous woodland community were dominating clusters of different species (see the photo below showing a remnant Norway spruce in a cluster of young hornbeam trees). The image of these remnant trees somehow stuck in my head, thank you Bogdan, and kept me thinking about the possible underlying processes that let to this situation.

Then in October 2016 I was reviewing somebody else’s manuscript that was submitted for publication. In this work, the author hinted that there may perhaps be a tendency of larger forest trees to be surrounded by heterospecific neighbours in mixed species forests. This really was the trigger and the hint somehow struck me like a lightning: I thought by myself that this would actually be a fascinating research question. I shoved the review papers onto another pile of papers on my desk and dug a bit deeper into ecological theories. From earlier I recalled the Janzen-Connell hypothesis, the herd immunity hypothesis and size-hierarchy processes that describe natural processes of maintaining species coexistence and biodiversity.

One observation in many natural plant populations is that self-thinning leads to local size hierarchies (also referred to as size inequality), where heterospecific stands include dominant trees emerging from a first colonisation cohort, which are often surrounded by patches of smaller sized trees of the same cohort. Small trees of these early colonisers are initially often of the same species as the dominant trees and according to the mechanisms of both the Janzen-Connell and herd immunity hypotheses later decrease in numbers due to self-thinning processes. Eventually the small early colonisers are partially or completely replaced by even smaller individuals of other species from subsequent colonisation cohorts. This combined effect of species and size replacement enforces both local size hierarchies and the mingling of different tree species in a given area or patch and prevents the development of monocultures (see B-D in the figure below).With every new paper that I discovered on this subject the story and research question got even better. It was like tasting new wine all the time. Finally it became clear to me that it would be interesting to study this question for a wide range of different forest types in the temperate and boreal climate zones across the world. In terms of the aforementioned ecological theories, forest ecosystems in these climate zones have to date not been as intensively studied as tropical forest.

It took me a while then to devise a good method for proving the point that there is a trend for larger forest trees towards high species mingling. At first I was thinking about using so-called second-order characteristics from point process statistics. I tried a few things there, but none of the outcomes really was to the point. Then I remembered the species mingling index that was originally devised by my mentor Prof. Klaus von Gadow back in 1993. Of course, this index was about spatial species mingling, so it should actually suit the research question quite nicely.  Still, I needed to establish a relationship between species mingling and size. That line of thinking eventually let me to logistic regression, since I recalled that we often model tree selection probabilities (for thinnings) using logistic regression, where size is an important explanatory variable. Tree mortality modelling is a related topic, where logistic regression is often applied. However, the discrete mingling index can take k + 1 different values depending on the number of nearest neighbours, k. For using logistic regression I thus had to convert the mingling index to a binary variable. Instead of using only k = 1 neighbour I stuck to the idea of including a slightly larger neighbourhood for a better description of species mingling. Finally – after much thinking and trial calculations – I settled on the idea of using k = 3 neighbours leading to four discrete mingling values, i.e. 0, 0.33, 0.67 and 1. I argued that a situation where no or only one neighbour is of a different species can be summarised as “low mingling”. On the other hand, values of 0.67 and 1 indicating two and three heterospecific neighbours describes a situation of “high mingling”. A few validation checks have later shown that this was a good choice. My definition of low and high mingling also allows a straightforward mapping and detection of clusters of high and low mingling in in a forest, see Fig. 2 in the publication. I still remember that this basic idea came to me out of the blue while actually travelling between Umeå and Göttingen.

In January 2017, I gave a first talk on this research to a larger audience at Dresden Technical University (Germany). The discussion following my presentation was interesting and varied. There was some scepticism in the audience, as you often get with new ideas, particularly with regard to the finding that the investigated trend should really hold for many different forest types. This kind of scepticism also came up in the review process. It took much discussion and re-wording to bring the point of the paper across in an appropriate way. Overall we found significant support for the expectation that large trees and trees growing at low local densities often have a tendency towards high species mingling. An explanation for this can be the aforementioned theories, but also disturbances and even forest management. We also found that size differentiation and species mingling are strongly related in local neighbourhoods. This is a very interesting and useful finding for conservation.

Now it would be great to follow these relationships up in detailed studies of specific forest ecosystems using spatio-temporal data. And while working on this topic two new research ideas came to me out of nowhere. An MSc student at Umeå is currently pursuing one of them and the other one is still sitting in my folder of new ideas. Perhaps I will post their background stories here one day, too, who knows.