EasySDE User Guide - University of Northern British Columbia
Contents
1. 1 A a eX Ve 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 In 3 often Ho and to are taken as zero forcing the curve to go through the origin More generally they may represent some initial size at a certain age This is a typical sigmoid curve The parameter a is the asymptote b is a speed factor scaling the time axis and cis 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 management treatments 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 labeled in various ways a common one being by the height reached at some specified reference or base 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 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 parameterization any or2. cal point of view Often although not always the estimate of either or Om turns out to be zero Apparently the data does not provide sufficient information to separate the different variance components Of course the problem would go away if either one of the two sources of variability is ig nored as has been done in all the other methods I know off At any rate this does not seem to have an appreciable impact on the estimates for the parameters of interest a b c to and might be taken as a warning against the temptation of introducing even more realistic error structures Actu ally measurement sampling error in stem analysis is small relative to that in sample plots although the risk of bias is higher and there it might not be a bad idea to fix om 0 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
3. is the most likely top height at a base age among any hypothetical stands of a given species etc that could grow in a site And similarly for other ages At least in this view the site index base age is essentially arbitrary a de vice for assigning a number to each curve in the family Therefore mod elling 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 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 1 I could have said expected instead of most likely Mean and mode converge to the same value as sample size increases so presuming an infinite number of hypothet ical stands it makes no difference in the definition There are differences in estimation situations as discussed later a way that crossings occur n appropriate stratification should be used Alternatively multidimensional indicators might be possible although little or no research in this direction appears to have been done n experimental model variant with two local parameters is included in this package 2 3 Estimation Rational parameter estimation requires some sort of
4. on the graph window saved in various formats etc En ter set terminal and help terminal to see the output formats available 15 SitePlot 0 9 Figure 3 Graphing with SitePlot 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 The gnuplot executable is Wganuplot exe 16 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 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 pdf Garc a O 1983 A stochastic differential equation model for the height growth of forest stands Biometrics 39
5. 1059 1072 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 http web unbc ca garcia unpub dfsite pdf Garc a O 1999 Height growth of Pinus radiata in New Zealand New Zealand Journal of Forestry Science 29 1 131 145 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 Management 71 3 217 225 Richards F J 1959 A flexible growth function for empirical use Journal of Experimental Botany 10 290 300 Seber G A F Wild C J 1989 Nonlinear Regression Wiley 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 17
6. Brownian motion process a b a H V2bow t 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 The factor 2b does not appear in Garc a 1983 If I remember correctly it was introduced here to simplify the programming or maybe for some other obscure reason called a stochastic differential equation SDE To keep things simple we neglect any correlations across plots 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 hy H 0m6i 6 where the e are independent standard normal variables and om is another parameter to be estimated This makes the error slightly 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 og 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 c Cm and oo In a specific model version or variant some of these parameters may be fixed often zero global or local They may also be some function of other new global and local parameters examples later Actually a lo
7. EasySDE User Guide Oscar Garcia University of Northern British Columbia garciaQunbc ca July 2003 Contents 1 What 2 Theory 2 1 Growth height and site A A ee A Ae BE te GeO A 2 3 Estimation ia a ee Pe ee Se ei Ta 2 4 Model selection and hypothesis testing 3 Program usage 3 1 Datatormat pi A hae 3 2 Model specification 2 ee 3 3 Pre proGessing si ep oa a e ee ew Re aa aE S gA R n O Ta ol e RA Ee al 3 09 ny el e as A A E T 4 Graphing 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 remeasurement data The old software was not particularly user friendly which undoubtedly limited its popularity EasySDE greatly simplifies the process Its ease of use should compare favourably to most other site mod elling procedures In short various parameterizations 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 shortly Data can be two or more height age measurements on any number of sample plots Measurement intervals can vary and may be of any length Either permanent sample plot
8. Forests 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 1 appears The items in it are as follows EasySDE 0 9 Model 7 BLOCAL Gl Ll G2 0063 64 0 0 Figure 1 Stage 1 Model specification window Title One line of optional descriptive information to be printed at the top of the program output 10 Binary files If checked the data and local parameter estimates from the last run are used The model version in the new run must use the same local parameter s File Name of the input data file 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 such as 10 or 100 make easy the scaling back Model Click to select a model version 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 09 ALOCBH is for data using breast height age it forces a height of 1 3 at age zer
9. all of a b and c can be fixed 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 ts in 3 S all 1 He acje Ve 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 expected trends as up to here With actual stand data 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 been thinking of site index as the actual height 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
10. cal 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 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 It in based on calculating the probability under the model of obtaining the observed data Considered as a function of the parameters substituting the given data this is the likelihood function The ML estimates are the parameter values that maximize the likelihood function 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 parameterizations are essentially 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
11. 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 Garc a 1999 3 Program usage 3 1 Data format The data must be arranged in a text file with three columns plot number age and top height Plot numbers must be integers with up to 9 digits The other values are free format and columns may be separated by any number of spaces and or tabs 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 test dat for an example 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 A note on stem analysis data Often heights are interpolated for every year using methods such as those described by Dyer and Bailey 1987 Only values for the actual crosscuts should be used here Because in general the tree tip corresponding to the 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 Jim Goudie BC Ministry of
12. corporating the distributions implied by 6 Tf you must see Garc a 1983 or Seber and Wild 1989 for the gory details A modified full Newton method using first and second derivatives is used to improve efficiency and reliability in minimizing the negative of the log likelihood over hundreds of variables parameters The implementation makes use of the special structure of the problem with partitioning strategies to take advantage of sparsity in the matrix of second derivatives For details see Garc a 1980 and 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 paper in theoretically interesting but practically less important limitations of the model Some have misunderstood this and dismissed the whole ap proach as useless so let us try to keep things in perspective No model is perfect nor it should be 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 second issue may be a little more troubling at least from a methodologi
13. d increasing ages If there are no error messages the number of plots is displayed and we can go to the next stage by clicking Next The contents of the message window can be saved to a file through the Save button If you must know the front end runs DataPrep exe which reads the input data file and generates intermediate files with the data DATA binary the locals PARS binary and other info CTRL 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 and results in REPORT a text file There is a little more flexibility when running these programs manually specifically a list of plot specific local initial estimates can be used see the advanced notes 13 34 Run The same window is then used to display the iteration log as the optimiza tion 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 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 The run should be possible to be aborted by clicking the Quit button but it isn t Otherwise the program runs for up to a maximum of 200 iterations If this happ
14. ens 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 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 stan dard error Mean correlations among local parameters Mean correlations between local and global parameters t As mentioned at the end of Section 2 3 often one of the variance estimates turns out to be zero Take with a pinch of salt 14 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 respective factors Can also be obtained from the SitePlot program see below Before exiting the user is prompted to optionally save the report if not already saved 4 Graphing SitePl
15. er of growth rings Therefore it can hardly have a direct causal effect Only accumulated changes reflected in the current state of the stand can be causes In principle physiological tissue aging might have some influence on growth rates For the most part however tree size must be the dominant variable Starting from some initial height Ho at time to accumulation i e inte gration of the growth rate in 1 produces the height H at some other time t This can be calculated taking advantage of 1 being a separable differential equation dH dt f E m dH Ea mE t t to Ho f pa e If we can analytically integrate the left hand side and then solve for H1 we obtain a formula for H as a function of t to and Ho A flexible and commonly used model has f H nH kH It was pro posed in this form by von Bertalanffy in the 1930 s not just with m 2 3 as is often thought von Bertalanffy 1938 1949 and was popularized for plant growth by Richards 1959 Introduced in forestry by Pienaar and Turnbull 1973 for some reason it is sometimes called the Chapman Richards model It will be convenient to use a power transformation A with c 1 m and write it as dH Cc C ap dla HO 2 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 over age equation H all
16. model for the variability of the observations Experience suggests that moderate deviations from the assumptions e g additive errors uniform variance independence normality etc in regression 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 instance that the deviations of a stand from the model of equation 3 would tend to increase with time and that they are not sta tistically independent This because the error due among other things to weather fluctuations accumulates over time Ordinary least squares is therefore relatively inefficient Seber and Wild 1989 Ch 7 review ap proaches 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 us ing standard statistical packages We try 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 for non overlapping intervals Technically what is called a Wiener white noise or
17. o before scaling BLOCAL to BLOCBH are similar but taking b as the local parameter The resulting curves are polymorphic Actually proportional along the t axis LINEAR and COMP are more general models where curve shape varies in a more flexible way with site At the cost of an extra parameter Unfortu nately only a numerical solution for the local as a function of site is possible so they are not as convenient for many uses They can be useful for assess ing the adequacy of the simpler models and as a last resort if the others do not work COMP was used in Garc a 1996 Finally 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 parameterizations 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 11 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 Advanced and Programmer Notes It is recommended to start with the simplest model in each class ALOCAL or BLOCAL and possibly use the estimated parameters as starting point for the more general ones The Binary files option is useful there Initial estimates These are
18. or stem analysis data can be used Some methodological background is given in the next section EasySDE user 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 adventurous for added flexibility you might want to run directly the underlying programs Or even hack into the code or use some of its components for other purposes Details are contained in the Advanced and Programmer Notes included 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 1989 pp 354 356 and Appendix C but what follows should be enough to get you started Extensive experimental results are reported in Garcia 1999 Similar methods were used by Rennolls 1995 2 Theory 2 1 Growth height and site On a given site and under average weather conditions the expected height growth depends on the current height dH T f H a where H 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 numb
19. ot aids in plotting the data and site index curves Running SitePlot brings up the window in Figure 3 It generates commands for gnuplot a free graphing program included The first part is similar to EasySDE The Title appears at the top of the graph Only models 1 to 12 are supported Scaling factors and scaled parameter values can be used It is possibly to plot just data just 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 Section 2 2 It 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 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 the 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 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 right click
20. the starting points for the likelihood opti mization iterations They must correspond to the specified scaling if any For the local s a same value is initially used for all plots 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 If possibly try stepping through models of increasing complexity using previ ous 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 lies usually between 0 5 and 1 try 0 7 Try the reciprocal of the base age for b The 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 Model window the window in Figure 2 appears 12 f EasySDE 0 9 Data Preparation of x Using 53 plots P Save _Save Help Quit Next Figure 2 Stage 2 Data pre processing window No user input is required The input data file is checked for consistent plot numbers an
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