EasySDE User Guide - Forest Growth and Yield
Contents
1. the dotted blue curve Our approach uses equation 3 with to Ho being the current point and gives the red continuous curve The stand is still growing on site 20 so that predicted growth should be higher than for a stand of the same age and height but on site 17 Similarly projections for stands that overshoot their nominal trajectories should be lower than those for stands on higher site qualities that stay the course Warning As with any single number measure the concept of site index can fail if pushed too far It can be a good indicator of relative productivity within biogeoclimatic regions where growth patterns and limiting factors are similar Otherwise not just the level but also curve shape may vary in such a way that crossings occur An appropriate stratification should be used Alternatively multidimensional indicators might be possible although little or no research in this direction appears to have been done An experimental model variant with two local parameters is included in this package 2 3 Estimation Rational parameter estimation requires some sort of model for the variabil ity of the observations Experience with regression suggests that moderate deviations from the assumptions e g additive errors uniform variance independence normality etc may not be overly critical for the quality of the estimates A rough characterization of the error structure is nevertheless desirable It is clear for instan2. Chapman Richards following Pien aar and Turnbull 1973 It will be convenient to use a power transformation H with c 1 m and write it as dH dt The equivalence can be verified by differentiating on the left hand side and re arranging Integrating and solving for H as explained above gives the height age equation b a H 2 H all 1 H a e to Ve t 3 I have omitted the subscript 1 to simplify We shall also refer to t as age although it only needs to be a relative time or nominal age shifting by a constant does not make any difference For a whole growth curve often Ho and to are taken as zero in 3 forcing the curve to go through the origin When using breast height age t 0 sometimes 0 5 and Hy 1 3 m This is a typical sigmoid curve The parameter a is the asymptote b is a speed factor scaling the time axis and c is a shape parameter determining the relative height of the inflection point 2 2 Site index Top height growth is widely used as an indicator of site productivity The advantage over more direct measures such as volume production is that it is less affected by stand density It is assumed that height age curves e g 3 vary across sites Depending on site quality the curves lie at different levels without intersecting each other The curves may be labelled in various ways a common one being by the height reached at some specified reference or base
3. The Gi stand for global parame ters and the Li for local parameters A number of models are available in EasySDE ALOCAL has a as local parameter and is forced to pass through the origin It produces anamorphic site index curves A TO to A TOSO include non zero origin parameters and or the additional initial variance go BLOCAL to B TOSO are similar but taking b as the local parameter The resulting curves are polymorphic Actually proportional along the t axis LINEAR and POWER are more general models where curve shape varies in a more flexible way with site At the cost of an extra parameter Unfortunately only a numerical solution of equation 3 for the local as a function of site is possible so they are not as convenient for many uses They can be useful for assessing the adequacy of the simpler models and as a last resort if the others are not good enough These models were used in Garc a 1996 ABLOCALS has both a and b as local parameters so it does not produce traditional site index curves It may be useful for hypothesis testing and experimentation Graphing the relationship between the estimated a and b might suggest appropriate re parametrizations ALOCBH to ABLOCBH are similar to the above but using breast height age They force a height of 1 3 m at age 0 5 the standard used in BC If your breast height definitions are different the easiest may be to transform your data temporarily and adjust the results la
4. age the site indez Mathematically this corresponds to a one parameter family of curves That is the curve equations for different sites differ only in the value of one scalar 5 parameter For instance in 2 and 3 one of the parameters a b or c may change with site quality while the others are common to all sites More generally this may happen after a re parametrization any or all of a b and c may be assumed to be functions of some other parameter that varies with site We call a site dependent parameter local being specific to a sample plot and the others global pertaining to the model as a whole The relationship between the local parameter and the conventional site index S is found by substituting the base age t in 3 S all 1 Ho a e toto 4 In simpler cases 4 can be algebraically solved for the local parameter and this substituted in 3 to obtain a height age equation indexed by S A numerical solution may be needed in other instances All this is fairly straightforward in a deterministic world or when dealing with predicted trends as up to here With actual stand observations we need to be more precise Implicit in the traditional concept seems to be the idea that site curves and site index represent some sort of average over hypothetical stands that might grow in the site That is site index is a property of the site More recently some researchers have thought of site index as the actual he
5. in all the other methods I know of At any rate this does not seem to have any appreciable impact on the estimates for the parameters of interest a b c tp and might be taken as a warning against the temptation of introducing even more realistic error structures 2 4 Model selection and hypothesis testing Models with the same number of parameters can be compared and chosen according to the highest value of the computed likelihood or of its logarithm the log likelihood In comparing models with different number of parame ters extra flexibility needs to be penalized Various theories suggest adding penalties of one half to three log likelihood units per parameter Akaike s information criterion AIC perhaps the most popular prescribes one unit Note that the likelihood value by itself is essentially meaningless it is the relative values or differences in the logarithms that are important For hypothesis testing one approach considers a log likelihood difference of about two units as significant More orthodox is a likelihood ratio test It is based on the fact that twice the log likelihood difference is asymptotically distributed as a x with degrees of freedom equal to the difference in number of parameters For examples see Garcia 1999 2005 3 Program usage 3 1 Data format The data must be arranged in a text file with three columns plot number age and height Plot numbers must be integers with up to 9 digit
6. predicted height growth depends on the current height dH A 0 where A is top height and t is time or age It is usually assumed that within limits other stand variables such as density do not significantly affect top height It might be argued that growth also depends on age But in fact age as elapsed time since birth does not have a physical presence at any given time other than perhaps in the number of growth rings Therefore it can hardly have a direct causal effect Only accumulated changes reflected in the current condition state of the stand can be causes In principle ageing of meristems might have some influence on growth rates For the most part however tree size should be the dominant variable Starting from some initial height Ho at time to accumulation i e integra tion of the growth rate in 1 produces the height HA at some other time fi This can be calculated taking advantage of 1 being a separable differential equation dH dt f E Mm qH ti a dt t to m FE a If we can analytically integrate the left hand side and then solve for H we obtain a formula for H as a function of t to and Ho A flexible and commonly used model has f H 7H KH It was pro posed in this form by von Bertalanffy 1938 1949 not just with m 2 3 as it is often said and was popularized for plant growth by Richards 1959 In forestry the model is sometimes called
7. 0 i l Site curves ALOCAL a 45 Predicted for S I 20 A Predicted by ADA GADA ne 40 we 35 F i e mais gs o es I 20 4 15 F 10 10 5 L 0 7 Age years Figure 1 Dashed green and dotted blue site index curves Continuous red projec tions for off curve site index 20 stands To illustrate some of the conceptual subtleties assume a model with a as the local parameter and height zero at age zero ALOCAL in Section 2 4 The base age is 25 years b 0 04 and c 0 7 In Figure 1 the dashed green curves are site index curves That is heights predicted at birth by equation 3 with ty Hy 0 and various values of the site parameter a Now suppose that for whatever reason e g unfavourable weather or heavy weed competition a 10 year old stand growing on site 20 happens to be 15 short of the normal height for that age and site See the point at age 10 just below the site index 20 curve What should be the future predicted height The popular ADA GADA methods Bailey and Clutter 1974 Cieszewski and Bailey 2000 use a transition equation similar to 3 but obtained by eliminating the local parameter in the site index function 7 That greatly simplifies estimation no local parameters to deal with but it assumes that given the current stand condition growth is independent of site quality The ADA GADA projection is the site index curve that passes through the current point
8. 90 300 Seber G A F Wild C J 2003 Nonlinear Regression Wiley Interscience also Wiley 1989 von Bertalanffy L 1938 A quantitative theory of organic growth Inquiries on growth laws II Human Biology 10 181 213 von Bertalanffy L 1949 Problems of organic growth Nature 163 156 158 21
9. EasySDE User Guide Oscar Garc a University of Northern British Columbia garciaQunbc ca September 2009 Contents 1 What 2 Theory 21 Growth height and site 2424402 22404 464 445 Ded BUE WO 10 As ds DE E eo has G 235 Estimations ca EE A AA E AR 2 4 Model selection and hypothesis testing 3 Program usage Onl Dalai m alee eos ica taa ha ya Lone ha 3 2 Model specification E EA ad Be 3 3 BPre procesing ak ne Sintec ee ee ee ee ee ee ee eis 11 11 3 5 Results 3 6 Behind the scenes 4 Graphing The Guide is definitive Reality is frequently inaccurate Douglas Adams The Restaurant at the End of the Universe 1 What EasySDE is essentially a GUI front end to some old software for developing forest height growth or site index models Fairly sophisticated and statis tically efficient methods are used making good use of almost any kind of repeated measurements data The old software was not particularly user friendly which undoubtedly limited its appeal EasySDE greatly simplifies the process Its ease of use should compare favourably to most other site modelling procedures In short various parametrizations of the Richards equation can be used For estimation environmental and measurement error sources are modelled through a stochastic differential equation SDE All parameters are esti mated simultaneously by maximum likelihood don t panic all this will be explained shor
10. In addition we recognize that an observed stand top height h may differ from the real H due to measurement and sampling error For mathematical convenience we assume h Hi gai 6 where the e are independent standard normal variables and om is another parameter to be estimated This makes the error somewhat higher for larger heights standard error approximately proportional to H which seems reasonable Finally it might happen that growth perturbations are higher at the seedling stage than later on An optional additional parameter co may be used to represent extra variability over some time previous to the first measurement The general model has then 8 basic parameters a b c to Ho 0 om and go In a specific model version or variant any of these parameters may be either fixed e g zero global or local They may also be some function of other new global and local parameters examples later Actually a local parameter represents many parameters one for each sample plot Typically one needs to estimate hundreds of parameters Although for most applications only values for the globals among the first five above are used 2 The factor v 2b implies larger effects for higher growth rates when b is local and is constant The data consists of a number of sample plots each with a sequence of age height observation pairs t hi We estimate the parameters by the method of maximum likelihood ML I
11. Section 2 2 The original units are used no scaling The equation is in gnuplot format which is fairly standard except for using for exponentiation On the lower part of the window labels and ranges for the Age and Height axes can be given or changed Any strings can be used for the labels If a 18 SitePlot 2 0 Title imeasc il Data file Fo ese Scaling Age ho Height ho ID a DOE Gp a Gn lo Eo Model None data only Parameters aim am Age axis Height axis Site index Label Age years Label Top Height m From ho To 30 By 5 From lo To From lo To E Index age 50 Grid Curves line type Aa Figure 4 Graphing with SitePlot range limit is left blank it is calculated by gnuplot automatically from the data and or function values The age limits must be given when not plotting data For plotting site index curves one needs the index age the range from lowest to highest site index and the site index interval between curves Finally we may choose to draw a grid or not and different line types for the curves can be selected enter test in gnuplot to see the types available 19 If everything goes well on clicking OK gnuplot is executed and the graph is displayed The user is left in gnuplot where the graph can be embellished printed on Windows right click on the graph saved in various formats etc Enter set terminal and help termin
12. al to see the output formats available For instance entering set output filename eps followed by set term post eps 20 and replot produces a good quality Encapsulated PostScript file Gnuplot is an excellent general purpose graphing program well worth learning See www gnuplot info There is a good brief tutorial at www duke edu hpgavin gnuplot html The gnuplot commands are left in the file SitePlot plt This can be loaded from gnuplot possibly after manual editing On Windows the gnuplot exe cutable is wgnuplot exe References Bailey R L Cieszewski C J 2000 Development of a well behaved site index equation jack pine in north central Ontario Comment Canadian Journal of Forest Research 30 1667 1668 Bailey R L Clutter J L 1974 Base age invariant polymorphic site curves Forest Science 20 155 159 Cieszewski C J Bailey R L 2000 Generalized algebraic difference ap proach Theory based derivation of dynamic site equations with polymor phism and variable asymptotes Forest Science 46 1 116 126 Dyer M E Bailey R L 1987 A test of six methods for estimating true heights from stem analysis data Forest Science 33 3 13 Garc a O 1980 A stochastic differential equation model for the height growth of forest stands Presented at the 5th Australian Statistical Con ference Sydney http web unbc ca garcia unpub oz80 paf Garc a O 1983 A stochastic differential equation
13. ble to abort the run by clicking the Quit button but it isn t Otherwise the program runs for up to a maximum of 200 iterations If this happens try a better starting point and or better scaling Clicking Next takes us to 3 5 Results A similar window displays the final results report The report starts with the run identification and the termination condition CONVERGED for a successful run For each plot the following is shown Plot number Number of measure ments Contribution to 2 times the log likelihood large values flag outliers Local parameter s estimate s Their approximate standard error An es timated site index for each plot can be obtained from the local parameter using equation 4 General info follows Number of plots Total number of measurements Log likelihood Then info on the global parameters Estimates Approximate standard errors Matrix of approximate correlations between them Finally for the local parameter s Average of the estimates Mean standard error Mean correlations among local parameters Mean correlations between local and global parameters Note that all results involve the specified scaling factors if any Back scaling can be figured out by substituting in 5 the age and height divided by the 4 As mentioned at the end of Section 2 3 sometimes one of the variance estimates turns out to be zero Take with a pinch of salt 17 respective factors It may also be o
14. btained from the SitePlot program see below It can be a good idea to verify through a run without scaling starting from the final parameter values Before exiting the user is prompted to optionally save the report if not already saved 3 6 Behind the scenes If you must know the front end runs DataPrep exe which reads the input data file and generates intermediate files with the data DATA tmp binary the locals PARS tmp binary and other info CTRL tmp text If everything goes well these are then passed on to SDEfit exe which does the hard work The final local estimates and other information are left in PARS tmp and results in REPORT tmp a text file There is a little more flexibility when running these programs manually a list of plot specific local initial estimates can then be used see the advanced notes 4 Graphing SitePlot aids in plotting the data and site index curves Running SitePlot brings up the window in Figure 4 It generates commands for gnuplot a free graphing program included in the Windows installation The first part is similar to EasySDE The Title appears at the top of the graph Not all models are supported Scaling factors and scaled parameter values can be used It is possibly to plot only data only curves or both Once the parameters are given the site index equation can be displayed with the Show equation button This gives top height as a function of age site index and index age
15. ce that the deviations of a stand from the model of equa tion 3 would tend to increase with time and that they are not statistically independent This because the error due among other things to weather fluctuations accumulates over time Ordinary least squares is therefore rela tively inefficient Seber and Wild 2003 Ch 7 review several approaches to modelling this kind of data and others are found in the forestry literature Many such as autoregressive and mixed effects models trade a somewhat crude representation of error structure for the convenience of using standard statistical packages We use a more realistic stochastic model that requires special purpose software Hopefully EasySDE now makes its application not more difficult than other methods For each sample plot we represent the effects of environmental noise mostly weather as a perturbation added to the right hand side of 2 The perturbation is assumed to be a continuous random process with values that are independent on non overlapping intervals Technically what is called white noise the formal derivative of a Wiener or Brownian motion process w b a H Dove 5 Here w t is the standard mean 0 variance 1 Wiener process and is a new parameter determining the perturbation intensity or variance This is called a stochastic differential equation SDE To keep things simple we neglect any correlations across plots
16. ight reached by a specific stand at the base age a property of the stand Failing to distinguish between these two different definitions which we might dub site site index and stand site index has caused a great deal of confusion and unnecessary controversy see for instance Bailey and Cieszewski 2000 and references therein We take the site site index view Specifically Site index is the top height at a base age predicted at birth for any hypothetical stands of a given species etc that might grow on a sitet At least in this view the site index base age is essentially arbitrary a device for assigning a number to each curve in the family Therefore modelling 1 Predicted refers to some kind of point estimate such as an expected most likely or median value Mean mode and median converge to the same value as sample size increases so presuming an infinite number of hypothetical stands it makes no difference in the definition There are differences for estimation as discussed later Note that heights predicted by 3 for an existing stand and in particular the predicted height at base age differ depending on its current tp Ho For the same reasons that your life expectancy at birth differs from your life expectancy now approaches where results change with the chosen reference age are at best ugly Our methods are strictly base age invariant in the sense of Bailey and Clutter 1974 5
17. model for the height growth of forest stands Biometrics 39 1059 1072 20 Garc a O 1996 Toward new site index curves for Douglas fir in the Nether lands Working paper Royal Veterinary and Agricultural University Unit of Forestry Fredericksberg Denmark http web unbc ca garcia unbub dfsite pdf Garcia O 1999 Height growth of Pinus radiata in New Zealand New Zealand Journal of Forestry Science 29 1 131 145 Garcia O 2005 Comparing and combining stem analysis and permanent sample plot data in site index models Forest Science 51 4 277 283 Milner K S 1992 Site index and height growth curves for ponderosa pine Western larch lodgepole pine and Douglas fir in Western Montana West ern Journal of Applied Forestry 7 1 9 14 Nigh G D 1995 Compatibility improvements and bias reduction in height age models Research Report 03 British Columbia Ministry of Forests Research Branch Pienaar L V 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 Rennolls K 1995 Forest height growth modelling Forest Ecology and Man agement 71 3 217 225 Reventlow C D F 1879 A Treatise of Forestry Society of Forest History Horshglm Denmark English translation 1960 Richards F J 1959 A flexible growth function for empirical use Journal of Experimental Botany 10 2
18. o s are a little trickier with decent scaling something around 0 05 should be OK Help Displays an abbreviated version of this section Quit As it says Next Advances to the next stage 3 3 Pre processing On clicking Next the information is checked and prepared for use by the main estimation program If there are no missing or incorrect entries in the 15 Model dialog the window in Figure 3 appears EasySDE 2 0 Data Preparation Using 18 plots Figure 3 Stage 2 Data pre processing window No user input is required The input data file is checked for inconsistent plot numbers and decreasing ages If there are no error messages the number of plots is displayed and we can proceed to the next stage by clicking Next The contents of the message window can be saved to a file through the Save button 34 Run The same window is then used to display the iteration log as the optimization proceeds To monitor progress values for the parameters and for the function being minimized which is 2 times the log likelihood are shown for each 16 iteration together with some other more esoteric information Most relevant the contribution to the objective function and the local parameter estimates for a few plots are shown followed by iteration number function and global parameter values This was really more useful in the good old days of over night runs to see if the program had not crashed It should be possi
19. ring count at a crosscut is located some distance above the crosscut crosscut heights may be adjusted as shown by Dyer and Bailey 1987 However as suggested by Milner 1992 and by Jim Goudie quoted by Nigh 1995 a simpler and probably better method is to use the crosscut height reducing the ring count by 0 5 years 3 2 Model specification On running EasySDE the window in Figure 2 appears The items in it are as follows LA EasySDE 2 0 Model Tie I Use previous binary files Daafle Bose Scaling Age ho Height fio ID a b e 69 0 Om to Ho Model 1 ALOCAL Ll Gl G2 0 Ga Ga 0 0 Initial estimates Figure 2 Stage 1 Model specification window Title One line of optional descriptive information to be printed at the top of the program output Binary files If checked the data and local parameter estimates from the previous run are used The model version in the new run must use the same local parameter s File Name of the input data file 13 Scaling The ages and heights in the data file are divided by these values Convergence is usually faster and more reliable if the scaled ages and heights are not too far from unity If the scale factors are different from one the initial estimates and the output correspond to the scaled model Values like 10 or 100 make things easier simplyfying the calculations when converting back Model Click to select a model version
20. s The other values are free format and columns may be separated by any number of spaces and or tabs 11 Each row corresponds to one measurement date with the sequence of plot measurements ordered by increasing age Different plots must be separated by a blank line See the examples in the Work folder Before going any further it is a good idea to plot the data to check for funnies SitePlot may be used for this A detailed description is in Section 4 but for plotting just data it should be self explanatory Example data Two data sets are included in the Work folder bcpine dat Breast height age years and top height m from permanent sample plots of lodgepole pine in the Interior of British Columbia Taken from a file provided with the documentation of VDYP 6 http www for gov bc ca hts vri ip software vdyp vdyp manual vdyp99 html overlay reventlow dat Total age and heights from stem analysis of 18 beech trees in the First Copenhagen District done by Count Reventlow around the year 1800 From Table 11 of Reventlow 1879 Converted to metric and stump age calculated by linear extrapolation a likely underestimate Notes on stem analysis Often heights are interpolated for every year using methods such as those described by Dyer and Bailey 1987 That is a Bad Idea the artificial data can distort the real trends use only values for the actual crosscuts Because in general the tree tip corresponding to the
21. special structure of the problem with partitioning strategies to take advantage of sparsity in the matrix of second derivatives Details in Garc a 1980 and in the Advanced and Programmer Notes in this package It might be worthwhile to clarify a couple of things that have caused some confusion I probably shot myself in the foot by delving in the Biometrics pa per into theoretically interesting but practically less important limitations of the model Some have misunderstood this and dismissed the whole approach as useless so let us try to keep things in perspective No model is perfect nor should it be striving for perfection has an unacceptable cost in terms of parsimony and estimation efficiency The first quibble was mentioning that 5 cannot be strictly correct the additive random perturbation could con ceivably make the transformed height to go negative True but in reality unlikely and inconsequential The same thing is routinely ignored without a second thought when using regressions with non negative variables The 10 second issue may be a little more troubling at least from a methodological point of view Often although not always the estimate of either o or on turns out to be zero Apparently sometimes the data does not provide suffi cient information to reliably separate the different variance components Of course the problem would go away if either one of the two sources of vari ability is ignored as has been done
22. t is based on calculating the probability under the assumed model of obtaining the observed data Considered as a function of the parameters substituting the given data this is called the likelihood function The ML estimates are the parameter values that maximize the like lihood function Apart from an intuitive appeal ML estimation has a number of nice statistical properties Two characteristics are particularly attractive for this kind of application First no matter how complicated a model the procedure is well defined one simply obtains the likelihood function and finds a maximum Second it is invariant under transformations that is the ML estimate for any function of the parameters equals that function of the parameter ML estimates This is convenient where parametrizations are es sentially arbitrary should we get good estimates for the site index for its logarithm for a local parameter for the height at a certain age Unlike with other methods all these are simultaneously possible and compatible The likelihood function is obtained through integrating the SDE 5 and in corporating the distributions implied by 6 If you must see Garc a 1983 or Seber and Wild 2003 for the gory details A modified full Newton method using first and second derivatives is used for efficiency and reliability in minimizing the negative of the log likelihood over hundreds of variables pa rameters The implementation makes use of the
23. ter These are the models that generally have been found most useful in practice Other variants can be implemented with just a little more effort see the 3 Conjecture there are no truly polymorphic models with a base age invariant ex plicit expression for the indexing parameter in terms of site index 14 Advanced and Programmer Notes Section 4 It is recommended to start with the simplest model in each class e g ALOCAL or BLOCAL and use the estimated parameters as starting point for more general variants The Binary files option can be useful there Initial estimates These are the starting points for the likelihood opti mization iterations They must correspond to the specified scaling if any For the local s a common value is initially used for all plots unless the Binary files option is specified Tab and Shift Tab may be used to move among the entry fields Convergence can be slow or fail if initial estimates are not good enough When possible try stepping through models of increasing complexity using previous estimates as starting points Using Binary files takes advantage of the previous individual plot locals It is also a good idea to try several starting points to partially guard against local optima Some suggestions when starting from scratch For a guess a reasonable height upper bound Parameter c usually lies between 0 5 and 1 try 0 7 Try the reciprocal of the scaled base age for b The
24. tly Data can consist of two or more height age pairs on any number of sample plots The number of measurements per plot and the mea surement interval lengths can vary Either permanent sample plot or stem analysis data may be used Some methodological background is given in the next section EasySDE usage instructions follow This should be more than sufficient for model development using the variants that past experience has shown to be the most useful If you really know what you are doing and feel adventur ous for added flexibility you might want to run the underlying programs directly Or even hack into the code or use some of its components for other purposes Details are contained in the Advanced and Programmer Notes in cluded in this distribution All sources are freely available under the MIT license www opensource org licenses The models and methods are described formally in Garcia 1983 and Seber and Wild 2003 pages 354 356 and Appendix C but what follows should be enough to get you started Extensive experimental results are reported fo instance in Garcia 1999 2005 Related methods were used by Rennolls 1995 The software runs on Microsoft Windows and on Linux Everything here applies equally to both unless stated otherwise The screen pictures were generated on Windows Linux will look slightly different 2 Theory 2 1 Growth height and site On a given site and under average weather conditions the
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