## Exploring the Clumpiness Factor in FPS: A Dialog

### Below is a question and answer session between Dave Walters and Jim Arney.  Dave is the questioner and Jim provided the responses.  Thanks to Dave and Jim for making this dialog available.

1. How do you calculate clumpiness...the formula

The computation of the Clumpiness Index Method is described, using temporary sample plots systematically distributed across a stand.  Variance is computed using Crown Competition Factor (Krajicek etal, 1961) because it includes all trees.  Stand Density Index (Reineke, 1933) and Relative Density (Curtis, 1982) ignore zero-dbh class trees.

!----- Must be at least two plots for variance estimation

SSX = 0.0

SQX = 0.0

PS = 0.0

Do I = 1, MIN0(500,NP)

PS  = PS  + 1.0

SSX = SSX + SCCF(I)

SQX = SQX + SCCF(I)**2 ! Crown Competition Factor per plot

End Do

!----- Must be at least two plots for variance estimation

SEX = 0.0

If(PS.ge.2.0) Then

STX = SQX - SSX*SSX/PS

SEX = 0.0

If(STX.gt.0.0) SEX = SQRT(STX / (PS *(PS-1.)) )

EndIf

!-----

CVX = 0.0

CCF = SSX / PS

PINDX = 0.0

CLUMP = 0.9

If(CCF.gt.0.0 .and. SEX.gt.0.0) Then

CVX = 100. * SEX / CCF

PINDX = 4. * ATAN(CVX**2 / 100.) / 3.1415927 - 1.0

!               Based on spatial pattern index by Lefkovitch (1966)

If(PINDX.lt.0.0) Then

Clump = 0.90 + 0.10 * ABS(PINDX) ! Degree of uniformity

Else

Clump = 0.90 - 0.60 * PINDX**2   ! Degree of clumpiness

EndIf

Xclp = CCF/420.

If(Clump.lt.Xclp) Clump = Xclp ! Full Stocking constraint

If(Clump.lt.0.27) Clump = 0.27 ! No Stocking constraint

If(Clump.gt.0.99) Clump = 0.99

EndIf

!----------------------------------------------------------

2. Assuming clumpiness is related to (calculated in some fashion from) plot to plot variance, is it not affected by the size and intensity of the plots?
Yes, the plot size does have an impact.  Obviously, the larger the plot, then the more variance in stocking that is included in one plot causing a reduction in between plot variance and thus a clumpiness index indicating a more uniform spatial pattern.  We are after a reliable estimate of stocking at the plot center of each plot.  Therefore, the rule-of-thumb for prism sweeps that obtain an average of 5-9 sample trees is appropriate because it is achieving a reasonable plot size for the local condition of size and frequency of trees.  It is important to pick a plot size (or BAF) and stay with that size throughout the sampling of that stand (polygon).  We are after point-density estimates on a systematic grid across each stand.  The observed density at each point provides the information necessary to compute clumpiness accurately.  For this reason we use Crown Competition Factor estimates of density because all trees (not just greater than zero Dbh) contribute to competition density as compared to Stand Density Index or Curtis' Relative Density.  We developed the specific formulation by simulating sample grids over stem-mapped stands in all ages of natural and planted stands.  This would be a good topic for a Master of Science thesis in Biometrics at some University.

****It is reasonably common to combine stands into other spatial units for a variety of purposes.  I usually comine the processed results based on a weighted (acres) average approach. The other approach is to re-compile plots from the different stands.  Given the importance of the clumpiness variable and the error introduced when combining plots calculated from different plot sizes, you would probably agree that the weighted average is the right approach?

Yes, I recommend that you use a weighted by acres approach when combining stands.  This maintains the integrity of the individual cruises from each original stand.

****Also, there is at least 1 inventory processing program in common use which allows for different BAF's by plot. That would be poor data to try to import to FPS.

Yes.  Using different BAF's within the same stand defeats all of the opportunity to provide statistics on clumpiness, realiability, precison and within-stand variability.  No one should be using that type of cruise compiler.

3. How is clumpiness used in the growth model. Is it calculated once and then simply used to affect starting conditions or is it a multiplier that is used at each period...or is it recalculated each period?
The clumpiness index is used to generate a spatial pattern of trees.  This is useful only to a distance-dependent treelist growth model.  Examples are FPS and TASS (BCFS Research Branch).  It cannot be used by distance-independent treelist growth models.  Examples are CACTOS, CRYPTOS, Prognosis (FVS), SPS and ORGANON.  DFSIM is a stand-average model that cannot assign variance to any parameter.
The actual procedure is to generate a random spatial pattern of tree positions, overlay a grid of sample points and then iteratively pull the tree positions toward random clumping centers (5-12 per acre from stem-mapping experience) until the clumpiness index value is achieved.  An average tree size is assigned to all tree positions to compute individual growing space which is then sorted from small to large openings.  The treelist is then assigned to a tree position sorted on Dbh and Species tolerance.  The exact same seed is provided to the map generator each run of the growth model to provide consistency when comparing alternative silvicultural effects on growth.  Once the stem map is generated, the clumpiness index is never used again in the growth projection.  Differences in yield are directly effected by the resultant growing space of each tree.  In future, we will recompute the clumpiness index for output to the database at each future year (age).  The seed for the random start is available as a optional input parameter that is undocumented to the user.  It again is an opportunity for a Ph.D. dissertation topic in Biometrics at some University.

***OK. So, FPS is truly a stochastic model being used deterministically?  Have you evaluated the differences observed if you change the seed?  I agree that this would be an interesting student project.  With all of the assumptions made here (choosing 1 of an infinite number of starting spatial arrangements, average tree size, assignment of trees to positions....is it really worth it for purposes of accurate predictions? Or, is inclusion of this variable more useful from an educational perspective as a vehicle to explore forest dynamics?

Yes, I have evaluated differences with alternative values of the random seed.  The default value is mid-range among the resulting distributions.  The random start (seed) is similar to using a random start in laying out a systematic grid of plots.  The location of subsequent plots is deterministic once the first is located.  In this case, a random pattern is generated and subsequently spatially aggregated to achieve the degree of clumpiness desired across the stand.  The particular pattern is not important, but the ability to produce a representative range of spatial openings is important.  The range is highly dependent on the degree of clumpiness found in the stand.  Resulting growth is highly dependent on both this range of spatial openings and the range of tree sizes found in the stand.

Distance-Independent growth models assume all trees have exactly the same amount of growing space and that if one tree dies (or is cut) then all trees get more growing space (Trees on wheels).  This assumption is much more difficult to swallow, especially in a mixed-species, mixed-size class stand.

4. Related to 3, I have observed that there is a trend in clumpiness in my inventory data with generally smaller, younger stands having LOWER clumpiness values and larger, older stands having higher clumpiness values....somewhat of a trend towards uniformity.  Those results appeal to my sense of what "should" happen in the forest.  Does this happen in the model somehow?
Yes, young stands can have highly variable starting values of clumpiness depending on method of stand establishment, site preparation, brush, etc.  In the growth model there tends to be more mortality in the clumps as the stand develops.  Also the natural regeneration model of FPS fills in the openings with new trees over time depending on the size of opening, tolerance of the species and degree of brush and animal control in effect at the time of regeneration and initial growth.  These are all topics of discussion and local validation in our upcoming calibration workshops sponsored by the Forest Biometrics Research Institute at the University of Montana.  These trends to uniformity, their rate of change, correlation to species tolerance are all highly localized within even one species.  Regional growth models averaged over large datasets will never achieve a precise estimate of growth on a given tree farm or forest, but a locally calibrated Species Library from direct observations on one tree farm does have that capacity.  The calibration workshops will move us analytically in that direction.

***I believe you are agreeing that as a stand matures in the real world, clumpiness will decline (the number will approach 1.0). Would it not be more appropriate to recalculate clumpiness in the model to reflect the same changes over time? In fact, if you don't do this, is there not the potential for creating a bias in some fashion in model predictions?

Once the spatial pattern is initially generated, the clumpiness index is no longer used.  We could recalculate it at any point in time to observe the changes in degree of clumpiness due to mortality, thinning and ingrowth.  What is used is the actual locations and sizes of the list of trees in the stand.  Model predictions are the summation of the result of all tree growth in relation to size and growing space of each tree in the list.

5. SPS/FPS - is clumpiness the same or different?
As discussed in Question 3 previously, SPS (distance-independent, treelist model) is different than FPS (distance-dependent, treelist model) in its native internal model structure.  SPS only tallies the number of non-stocked plots in a stand to compute a percent of openings.  That is used as a clumpiness parameter to compact the average stand density.  For example, 6 of 10 plots are stocked resulting in a clumpiness value of 0.60 and all trees pushed into 0.60-acres for computation of stand density for growth.
I originally designed the FPS structure in 1970-72 and used a variation of it until about 1980.  At that time everyone was into intensive plantation silviculture, there would be full-stocking, uniform spacing and size with short rotations.  Therefore, I put the distance-dependent structure on the shelf and developed a distance-independent structure (SPS) since natural spatial patterns and all-age structures were history.  By 1995 the various State regulations for green-tree retention, mixed-species, mixed-age and natural regeneration silviculture changed the demands on the type of growth model that could be used with confidence for long-range planning.  I dusted off the original distance-dependent structure and build the FPS growth model.  During 1995-2002 we have successfully conducted a large number of regional cooperative analyses to calibrate this model to most species and regions of the West.  There are in excess of 20,000 permanent research plots with up to 50-years of remeasurement in our database files from which the Regional Species Libraries were developed.  The most robust permanent plots for model calibration are those with stem maps and all trees measured for both Dbh and Height.

***Aren't most of your permanent research plots with stem maps progeny tests?

No,  most are stands of natural origin with intensive silvicultural treatment regimes.  Examples are the Levels-of-Growing-Stock (LOGS) studies (includes plantations), USFS Wind River plantation spacing trials, BLM Oregon thinning trials, Crown Zellerbach thinning/fertilization trials, Weyerhaeuser thinning/fertilization trials, MacMillan-Bleodel thinning/fertilization trials, Simpson thinning trials, Roseburg thinning/fertilization trials, etc.  In other words, approximately 800 of the 2,500 permanent plots used to build DFSIM and SPS.

6. The affect of clumpiness appears to be "slight" between about .85 and 1.0. Lower than .85 it increases exponentially (well, not quite). Is this by design.
Long-term yields essentially decrease proportionally to the value of the clumpiness index due to the ever increasing size of openings in the stand.  Actually within the growth model the border trees are growing out into the openings and increase in size relatively faster than similar size trees within the closed canopy.  This is all by design based on observations from natural stand permanent plots.  However, there are no multipliers involved, it is the result of the initial spatial arrangement of the trees based on the clumpiness index and range of tree sizes in the initial stand.

*** OK.  Perhaps what I am seeing (an increase in effect over time...i.e., a 20 year projection may show X difference between 2 clumpiness values but a 40 year projection may show X*X difference....not quite that extreme.) is due to the natural feedback loops in the model...you start low, you simply spiral lower.   It is hard for me to get a feel for whether this is 'right' or not!

It is not a single, fixed regression model with a deterministic prediction of clumpiness.  There is no multiplier effect.  The important distinction is to remember that this treelist distant-dependent growth model is iterative.  After each growth step the resulting tree sizes and growing spaces are re-evaluated.  If one tree dies (or is cut) then more growing space is provided to the trees in the near proximity resulting in higher growth and crown ratios for those specific trees.  If a clump (or whole stand) becomes too dense, then mortality increases.  If an opening increases then nearby trees gain growing space and retain long crowns.  It is actually self-correcting, which is not possible with a single regression model.

****NEW Question - I believe it is a fairly standard situation where an industrial owner inherits or acquires summarized data...stand tables with species by dbh detail or less.  If the area is small, it may be reasonable to simply put that area on the block for re-inventory soon. However, that may not always be feasible and one might have to use the inherited data for some time. Perhaps even use it for some long-term forecasting.  In those situations, clumpiness is not likely to be a variable that they have and the data to calculate it is clearly not available (no plot data).  What should they do?  I have been using some experience tables based on "complete" data in other areas. I have looked at predicting clumpiness from more readily available variables with poor success.

It is very difficult to predict clumpiness as a function of other stand parameters such as age, site, density or species.  This is because clumpiness is a function of past history of silvicultural treatment (or lack thereof).  As such 100 Douglas-fir trees that are 10 years old on a site index of 90-feet have a wide range of potential yields and sizes at 80 years.  A major reason is the degree of clumpiness of these 100 trees across the acre.  From my experience when you only have stand averages from Western WA and OR then default to 0.85 for clumpiness.  For interior BC, WA, OR, ID, MT default to 0.75 for clumpiness.  This is based on photo classification of polygons of similar species, size and density resulting in stands approximately 25-35 acres in size on average with a minimum of five acres in size.  These polygons will contain rock outcroppings, brush and/or wetlands to a minor degree.  Therefore, the stand is not uniformly stocked from border to border.  A systematic grid of cruise plots across a stand should pick up this variability.  The FPS cruise compiler determines the clumpiness index for this stand from the variance of density among these plots.

****NEW QUESTION - Have you considered and/or evaluated the model which results from removal of clumpiness from FPS?   (I guess that is just SPS, perhaps?). Does it really help that much in terms of prediction accuracy? Do you have any results showing the improvements?

Essentially, removing clumpiness from FPS would be defaulting to every tree in the stand having the same growing space.  You would then probably want to adjust the growing space every time a tree died.  This results in the trees moving after each growth interval to equalize their spatial openings.  Yes, this is essentially SPS (and ORGANON, Prognosis, FVS, Cactos, Cryptos, and all treelist distance-independent models).  I have results of comparisons, and they are significant.  However, the improvements are smallest in single-species, even-aged plantations with short rotations and no thinnings.  The reason should be obvious by now.

Now think about the species, size, structure of current forest inventories that we are attempting to manage and the type of silviculture (green-tree retention, shelterwood, seed-tree, selection) that is being imposed on these stands.  We must use a model with the flexibility to handle these structures and regimes if we hope to assess the sustained yield and value flow derived from managing these kinds of forests.

*****

Hopefully this review is useful.  I have analyzed many datasets over many species and regions, the trends are very similar in general.  I believe this to be a robust model structure, but encourage further research into various components as identified earlier.  I am fortunate to have worked intensively on growth models in seven western States and three Canadian Provinces.  This has given me insight into regional trends not provided to researchers assigned to one USFS Experiment Station or University Research Cooperative for most of their career.