An individual tree height increment model for mixed white spruce–aspen stands in Alberta, Canada

Forest Ecology and Management 123 (1999) 41±53

An individual tree height increment model for mixed white spruce±aspen stands in Alberta, Canada Shongming Huanga,*, Stephen J. Titusb a

Forest Management Division, Land and Forest Service, Alberta Environmental Protection, 8th ¯oor, Great West Life Building, 9920-108 Street, Edmonton, Alberta, Canada, T5K 2M4 b Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canada, T6G 2H1 Received 7 July 1998; accepted 6 January 1999

Abstract Individual tree-height increment models were developed for white spruce (Picea glauca (Moench) Voss) and aspen (Populus tremuloides Michx.) growing in the boreal mixed-species in Alberta. The models were formulated based on a selected base function (the Box±Lucas function), and the method of parameter prediction. Height increment was modeled as a nonlinear function of tree height, tree diameter, diameter increment, stand density, relative competitiveness of the tree in the stand, site productivity, and species composition. Since the data from permanent sample plots used in this study were time-dependent and cross-sectional, diagnostic techniques were applied to identify the models' error structure. Appropriate ®ts based on the identi®ed error structure were accomplished using the nonlinear least squares procedures with a ®rst-order autoregressive process. The models were also validated on independent testing data sets representing the population on which the models are to be used. Results showed that the average prediction biases were not signi®cantly different from zero at  ˆ 0.05, suggesting that the ®tted models appropriately described the data and performed well when predictions were made. Biological implications of the variables that affect height increment in mixed-species stands were discussed. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Individual tree model; Height growth; Mixed-species stand; Picea glauca; Populus tremuloides

1. Introduction Individual tree-height increment models predict periodic height growth of each individual tree in the stand. Such models have been used as the `driving force' or main variable for individual tree-based growth and yield models such as PTAEDA (Daniels

*Corresponding author. Tel.: +1-780-422-5281; fax: +1-780427-0084; e-mail: [email protected]

and Burkhart, 1975), TASS (Mitchell, 1975), and SPS (Arney, 1985). Many other individual tree-based models such as FOREST (Ek and Monserud, 1974), ORGANON (Hester et al., 1989), and PROGNOSIS (Stage, 1973; Wykoff et al., 1982; Wykoff, 1986) also treated the height increment model as one of the main components. Two different modelling approaches for height increment have been commonly used. One is the growth-potential independent approach in which height increment is directly expressed as a function of tree and stand characteristics, including the competi-

0378-1127/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 9 9 ) 0 0 0 1 5 - 8

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S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

tiveness of the tree in the stand (Lemmon and Schumacher, 1962; Beck, 1974; Wykoff et al., 1982); The other is the growth-potential dependent approach in which a function that de®nes the potential height growth of competition-free trees is ®rst selected, then a competitive adjustment factor (the modi®er) is used to reduce this potential (Ek and Monserud, 1974; Daniels and Burkhart, 1975). This second approach often involves selecting a `biologically meaningful' base potential height increment function and emphasizes how the modi®er is modelled and how it affects potential height increment. It is common that the modi®er is expressed as a function of the individual tree's attributes, such as crown ratio, crown length, tree diameter, tree height, diameter increment, and the competition with other trees as re¯ected by a competition index, basal area per hectare (ha), or number of trees per ha. Mitchell (1975), Krumland (1982), Arney (1985), and Ritchie and Hann (1986) all used this approach. However, Wykoff (1990) regarded the differences in the approaches are mostly semantic because either approach, if appropriately used, can produce acceptable predictions. Choice between the approaches may simply be a matter of preference or convenience. While most researchers choose to use height increment as the dependent variable and relate it directly to other tree and stand characteristics, height-increment models can also be derived by taking the ®rst derivative of the cumulative height function (Hegyi, 1974; Daniels and Burkhart, 1975). The procedure demonstrated by Daniels and Burkhart (1975) ®rst expressed the average dominant height of the stand (HD) as a function of site index (SI) and age (A): HD ˆ SI  10ÿ5.8650  (1/A - 1/25), then the maximum annual height increment was obtained by taking the ®rst derivative of this equation with respect to age, and a modi®er expressed as a function of crown ratio and competition index was used to adjust this maximum height increment. This indirect derivative method has some advantages if the cumulative height equation includes age as an independent variable or if the form of the equation is not overly complicated. It is also possible that height increment may be obtained from a cumulative height prediction model such as H2 ˆ f(H1, other variables), where H1 and H2 are tree heights at times one and two, respectively, and `other variables' may include tree age, diameter,

competition, site productivity, and species composition, etc. This type of model may appeal to some because it seems to ®t the data well with a high coef®cient of determination (R2) value, usually in the >0.90 range. However, a frequently ignored fact about this type of model is that the high R2-value is almost solely dependent on the high correlation between H2 and H1, which occurs because H1 and H2 are obtained from repeated measurements of the same tree over a relatively short time span (usually 5 to 20 years for permanent sample plots). If the error structure of the model is not accounted for, this high correlation can cause inef®cient parameter estimates and produce some ®t statistics that may be appealing but are not as relevant. These statistics generally in¯ate the `true' power of prediction and marginalize the impact of other variables on height increment. This study developed individual tree height increment models for white spruce (Picea glauca (Moench) Voss) and trembling aspen (Populus tremuloides Michx.) growing in boreal mixed-species stands in Alberta. Annual height increment was directly modelled as a function of tree diameter, tree height, diameter increment, stand density and competition, and site productivity. A species composition measure was also incorporated into the models to re¯ect the effect of species proportion on height increment. While the descriptions of the tree and stand variables that could affect height increment in boreal mixedspecies stands were stressed, the biological implications of the effects of these variables on height increment were also emphasized. Methods for model diagnostics and testing were also applied to ensure that the models were ®tted appropriately and were applicable for making predictions. 2. Model development The height-increment models were developed by selecting a base height-increment function and incorporating tree and stand level variables using the parameter prediction approach (Clutter et al., 1983). The base function was used to describe typical heightincrement curves that are both representative of some common height-growth patterns and compatible with the sigmoidal S-shaped yield curves.

S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

2.1. The base function The two-parameter Box±Lucas function (Box and Lucas, 1959; Seber and Wild, 1989) was selected as the base height increment function: HI ˆ

1 …eÿ2 H ÿ eÿ1 H † 1 ÿ 2

(1)

where HI is the annual height increment (m/year), H the total tree height (m), 1 and 2 the parameters to be estimated, and `e' is equal to 2.7182818. Initially, the tree diameter (D) at breast height was used in place of H, but preliminary analyses indicated that Eq. (1) was more appropriate. This base function of height increment and height is similar in concept to the diameter increment±diameter type function used in modelling individual tree diameter growth (Wykoff et al., 1982; Vanclay, 1994; Huang and Titus, 1995). The base function Eq. (1) was originally used to describe an irreversible chemical reaction in which one substance changes into another, and to develop

43

optimal design methodologies in nonlinear situations (Box and Lucas, 1959; Hamilton and Watts, 1985). It was called the two-term exponential function by Rawlings (1988) and classi®ed as an intrinsically nonlinear compartmental model by Seber and Wild (1989). Several typical graphs of Eq. (1) produced by varying 1 and 2 are shown in Fig. 1. The curves are typical of height increment, which begins at a value of zero, increases steadily to reach a maximum, and then decreases smoothly and asymptotically towards zero. The curves are also directly compatible with the commonly observed sigmoidal S-shaped yield curves in biology, where the yield starts at the origin, reaches a maximum growth at an in¯ection point, and then approaches an asymptote as determined by the genetic nature of the living organism and the carrying capacity of the environment. The maximum annual height increment rate of the tree occurs at: Hˆ

2 ln …1 =2 † 1 ÿ 2

(2)

Fig. 1. Typical graphs of Eq. (1) produced by varying parameters 1 and 2. (1) 1 ˆ 0.05, 2 ˆ 0.20; (2) 1 ˆ 0.20, 2 ˆ 0.15; (3) 1 ˆ 0.10, 2 ˆ 0.12; and (4) 1 ˆ 0.30, 2 ˆ 0.07.

44

S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

Fig. 2. Typical graphs of equation HI ˆ aHb exp(ÿcH) (i.e. y ˆ axb e ÿcx, where y ˆ HI and x ˆ H) produced by varying parameters a, b, and c. (1) a ˆ 0.20, b ˆ 0.80, c ˆ 0.10; (2) a ˆ 0.04, b ˆ 2.10, c ˆ 0.21; and (3) a ˆ 0.02, b ˆ 2.10, c ˆ 0.15.

which is obtained by taking the second derivatives of Eq. (1) and setting the resultant equation equal to zero, and solving for H. A different base function in the form of HI ˆ aHb exp (ÿcH) (i.e. y ˆ axb e ÿcx, and y ˆ HI and x ˆ H) was also examined, where a, b and c are parameters to be estimated (Ratkowsky, 1990). Several typical graphs of this function generated by varying a, b and c are displayed in Fig. 2. These curves are also typical of the height increment, and are compatible with the sigmoidal S-shaped yield curves. However, preliminary analyses indicated that this function, along with several other functions such as y ˆ axb exp exp (ÿbx), and (ÿax), y ˆ axb exp (ÿcx2), y ˆ ax y ˆ exp (a ‡ bx ‡ cx2) frequently seen in modelling diameter increment and silvicultural effect (Wykoff, 1990; Vanclay, 1991; Pienaar and Rheney, 1995), underestimated the height increment for young trees. Therefore, Eq. (1) was used as the base function in this study. If other factors that affect height increment are ignored, yearly height increment can be predicted from Eq. (1) given the height of the tree. If the

variations in site productivity and competition are considered to have signi®cant effects on height increment, they may be incorporated into the equation to provide better height-increment predictions. It is fairly well established that, given all other factors held approximately the same, a tree at a given height is expected to have larger height increment on better sites. The presence of competition on height growth is also evident and measures that re¯ect competition have been commonly incorporated into height increment models (Daniels and Burkhart, 1975; Wykoff et al., 1982; Arney, 1985; Ritchie and Hann, 1986). A tree at a given height is expected to have larger height increment if the tree is more competitive as re¯ected by its crown dimensions and competition indices. The diameter or basal area of a tree can affect height increment as well, because larger diameter trees should be in a more competitive position which should lead to greater height increment ± a phenomenon often referred to as `wood grows wood'. Height growth then becomes asymptotic. The height increment of a tree may also be affected by the species compatibility and

S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

their respective proportions in mixed-species stands. The height increment models developed in this analysis attempt to incorporate other tree and stand variables that have signi®cant effects on height increment. These variables are further described in details. 2.2. Stand density and competition Total basal area per hectare for all species combined in the stand is used as the overall mixed-species stand density measure. The use of this stand-level attribute as a simple and objective measure of stand density has been widespread (Clutter et al., 1983), and for reasons described by Spurr (1952), it is particularly suitable for mixed-species stands with irregular age structures. Other stand density measures, such as spacing index (the average distance between trees divided by the average height of dominant canopy), tree±area ratio (the total area occupied by the trees), and Reineke's stand density index, have mainly been used for evenaged pure species stands (Clutter et al., 1983). The crown-based measure of stand density, crown competition factor (CCF), although commonly used as a measure of overall stand density in growth and yield estimation (Stage, 1973; Arney, 1985; Wykoff, 1990), was not used in this study because measurements of crown dimensions from open-grown trees were not available. In addition to the use of total basal area per hectare to re¯ect the degree of overall crowdedness of trees within mixed-species stand, a diameter-based, distance-independent individual tree competition index (CI) expressed as the ratio of the target tree diameter (D) to the average diameter (AVED) of all trees in the stand (CI ˆ D/AVED), was also used to re¯ect the competitiveness of an individual tree relative to neighbouring trees. Like the use of crown-related variables, such as crown length, crown width, and live crown ratio, competition measures based on relative diameter or basal area are also considered to be directly related to tree vigour (Lorimer, 1983; Martin and Ek, 1984). As the CI gets numerically larger, the tree is in a better competitive position which should result in larger height increment. 2.3. Species composition and site productivity It is typical that in Alberta, the boreal mixed-species stands primarily consist of white spruce (Picea glauca

45

(Moench) Voss) and aspen (Populus tremuloides Michx.). In the early development stage of such a mixed white spruce±aspen stand, the shade intolerant aspen has the competitive advantages over white spruce and exhibits relatively more rapid early height growth. As a result, aspen tends to rapidly establish dominance on the site by occupying the upper layer of the canopy. Shade tolerant white spruce is usually established a short, but distinct, time after the overstorey aspen, and exhibits slow juvenile height growth. As time progresses, the competitiveness of aspen is reduced relative to that of white spruce as individual aspen trees start to die at approximately 60 to 80 years of total age, giving dominance gradually to the more shade tolerant and longer lived white spruce. This dynamic process results in changes, both in wholestand density and the proportions of the species in the stand. A species composition measure is used to re¯ect these changes: SC ˆ

BASUMsp BASUM

(3)

where SC is the species composition of the target species, BASUMsp the total basal area per ha (m2/ha) for the target species, and BASUM is the total basal area per ha for all species combined in the stand. The species composition as de®ned in Eq. (3) has a range of zero to one, with zero indicating there is no target species in the stand and one indicating a pure species stand of the target species. A species composition measure de®ned as SC ˆ Nsp/ N, where Nsp is the number of trees per ha for the target species and N the total number of trees per ha for all species combined, was also tried but was subsequently discarded in this study. A basal area-based species composition measure is usually considered better because it combines the numbers with their respective sizes. Since the most commonly used site productivity measure, site index, applies only to even-aged pure species stands, site index is not used in this study. Instead, the site productivity index (SPI) as determined by the dominant and co-dominant trees' height±diameter relationship, was used as the site productivity measure for boreal mixed-species stands (Huang and Titus, 1993). The use of the height± diameter relationship to characterize the site productivity in uneven-aged and mixed-species stands is common (Stout and Shumway, 1982; Vanclay and

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S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

Henry, 1988; Nicholas and Zedaker, 1992; Vanclay, 1994). Site index and age are not used because the boreal forests in Alberta typically have a mixed species composition with an irregular age structure. Both, the site index and age are often less meaningful in this situation (Wykoff, 1990; Huang and Titus, 1993). In addition to the variables representing stand density and competition, species composition and site productivity, tree diameter and diameter increment, which were known to be closely related to height increment in some other studies (Stage, 1975; Daniels and Burkhart, 1975; Wykoff et al., 1982), were also found to signi®cantly affect the height increment of trees in this study. These variables were incorporated into equations to predict the parameters of the base function, Eq. (1). The procedure is similar to the method of parameter prediction commonly used for a Chapman±Richards or Weibull-type function in which the parameters of the function are related to other tree and stand characteristics, but the form of the original function remains unchanged (Clutter et al., 1983). The appropriate ®nal height-increment model for white spruce was found to be: 1 …eÿ2 HS ÿ eÿ1 HS † (4) HIS ˆ 1 ÿ 2 with

1 ˆ a1 ‡ a2 BASUM ‡ a3 CI

(5)

2 ˆ a4 ‡ a5 DI ‡ a6 SC ‡ a7 D ‡ a8 =SPI

(6)

and for aspen, the appropriate height-increment model was found to be: 1 …eÿ 2 HA ÿ eÿ 1 HA † 1 ÿ 2

(7)

1 ˆ b1 D ‡ b2 BASUM ‡ b3 SPI

(8)

2 ˆ b4 ‡ b5 DI ‡ b6 SC

(9)

HIA ˆ with

In Eqs. (4)±(9), the subscripts `S' and `A' indicate white spruce and aspen, respectively, HI the annual height increment (m), H the total tree height (m), CI the tree competition index, D the tree diameter (cm) at breast height, DI the annual diameter increment (cm), BASUM the total basal area per ha for all species in the stand (m2), SPI the site productivity index (m), SC the species composition as defined in Eq. (3), and a1 ÿ a8 and b1 ÿ b6 parameters to be estimated.

The parameter prediction equations (Eqs. (5),(6) and (9)), were identi®ed by ®rst plotting the height increment against each independent variable and examining the possible linear and nonlinear relationships between them, and then arranging combinations of variables in linear and nonlinear equations to predict 1, 2, 1, and 2 in the base functions Eqs. (4) and (7) until the most reasonable residual plot from preliminary nonlinear regression ®ts was obtained and the biological interpretations of the estimated coef®cients were consistent with the general understanding of the height-increment process. A positive coef®cient in Eq. (5) or Eq. (8) indicates a positive effect on height increment, and a positive coef®cient in Eq. (6) or Eq. (9) indicates a negative effect on height increment. 3. Description of the data Data from repeatedly measured permanent sample plots (PSP) used in this study were provided by the Alberta Forest Service. The data were collected over the last three decades and the PSPs were located randomly throughout the inventory areas of the province providing representative information for a variety of density, height, species composition, stand structure, age, and site conditions. Within each 0.1ha PSP, the diameter of every tree, and the height of some selected trees representative of the diameter classes present, were measured. A detailed description of how the data were collected and recorded can be found in the Permanent Sample Plots Field Procedures Manual (Alberta Forest Service, 1990). The original PSP data were summarized to provide additional variables, such as the number of trees per ha, basal area per ha, average height, and average diameter, both for all species combined and by each species in the stand. Summarized data from a total number of 164 PSPs were selected to be used in this analysis. These selected plots have up to ®ve remeasurements. Each nonoverlapping growth period from the remeasurements de®nes a growth interval, that is, the growth intervals are obtained from measurements between the ®rst and second, the second and third, but not the ®rst and third. Because both, height and diameter increments are required for the estimation of Eqs. (4) and (7), only trees with both height and

S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

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Table 1 Summary statistics for white spruce and aspen tree and stand characteristics Variable

Tree DBH (cm) Tree height (m) Annual diameter increment (cm) Annual height increment (m) Number of trees/ha ± all species Average DBH (cm) ± all species Basal area (m2/ha) ± all species Average height (m)-all species Species number of trees/ha Species average diameter (cm) Species basal area (m2/ha) Species average height (m) Site productivity index (m) Species composition a

White spruce

Aspen a

mean

minimum

maximum

SD

27.75 22.38 0.21 0.16 1251.49 19.78 39.81 21.57 753.87 20.80 25.10 21.97 16.17 0.62

2.90 3.20 0.00 0.00 148 5.30 11.55 7.90 5.00 5.70 0.20 3.60 8.22 0.0062

63.30 37.90 0.99 1.15 5580 39.708 83.87 31.70 4914.00 54.90 56.65 31.70 21.28 1.00

10.52 6.11 0.13 0.20 744.58 5.60 8.92 3.90 670.86 6.60 12.11 4.26 2.43 0.25

mean

minimum

maximum

SDa

23.67 19.78 0.22 0.17 1461.95 17.32 32.18 18.46 919.60 20.38 19.89 19.37 18.51 0.65

5.30 5.80 0.00 0.00 222 4.60 11.97 7.3 10 2.8 0.47 6.30 8.46 0.016

64.50 32.90 0.72 1.18 6864 39.70 87.9 28.80 3515 46.10 53.33 30.60 24.50 0.99

11.21 5.39 0.12 0.26 791.88 6.55 11.01 4.60 785.87 9.02 9.26 5.08 3.35 0.26

Standard deviation.

diameter repeatedly measured at least once were used in obtaining the growth intervals. A total number of 1725 growth intervals was obtained from 164 PSPs for white spruce, and a total number of 1383 was obtained for aspen. Annual height and diameter increments were calculated as the differences between the values at the end and beginning of the growth intervals divided by the interval length. Descriptive statistics, including the mean, minimum, maximum, and standard deviation of the tree and stand characteristics at the beginning of the growth period are provided in Table 1. Summary statistics for annual height and diameter increments are also shown in Table 1. For each species, 80% of the observations (1368 for white spruce and 1122 for aspen) were randomly selected, using the SAS random number function RANUNI (SAS Institute Inc., 1987), for the estimation of model coef®cients. The remaining 20% of the observations (357 for white spruce and 261 for aspen) were used to evaluate model performance when predictions were made. 4. Analysis 4.1. Model fitting Preliminary nonlinear least-squares (NLS) ®ts of the height increment models were accomplished using

the PROC MODEL procedure on SAS/ETS software (SAS Institute Inc., 1988). The Gauss±Newton method using the Taylor series expansion as described in Gallant (1987) and Judge et al. (1988) was used in model ®ttings. To ensure the solution is global rather than local least-squares estimates, different initial values of the parameters were provided for the ®ts. Asymptotic ®t statistics, including the NLS estimates of the parameters, the asymptotic t-ratios, standard errors, and p-values of the estimates, the model root-mean squared error (RMSE) and the Durbin± Watson statistic (DW) for testing the ®rst-order autocorrelation of the error terms, are shown in Tables 2 and 3 for white spruce and aspen, respectively. The interpretations of the ®t statistics in Tables 2 and 3 are similar to those in linear cases, because of the relatively large number of observations used in this analysis. The Durbin±Watson statistic is used to evaluate the possible error correlation. Although the Durbin±Watson test procedure is not exactly applicable for nonlinear regression models, the test is approximately valid, especially for large samples (Amemiya, 1983, p. 355; Seber and Wild, 1989, pp. 318±319). Because the sample size used for this analysis is larger than the maximum sample size (200) shown in the extended Durbin±Watson Table (Judge et al., 1988, pp. 991± 994), critical values selected are those recorded for the maximum sample size. For white spruce, the

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S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

Table 2 Fit statistics for white spruce height-increment model Eq. (4) Equation

Eq. (4)

Method

NLS

a

NLS with AR(1) errors

Parameter

Estimate

Asymptotic SE

t-ratio

p-value

RMSE b

DW c

a1 a2 a3 a4 a5 a6 a7 a8

0.089485 ÿ0.00172570 0.048962 0.221253 ÿ0.161879 ÿ0.034426 ÿ0.00196558 0.454981

0.01939 ÿ0.0003895 0.01343 0.01971 0.02178 0.01266 0.0003109 0.24568

4.61 ÿ4.43 3.65 11.23 ÿ7.43 ÿ2.72 ÿ6.32 1.85

0.001 0.0001 0.0003 0.0001 0.0001 0.0066 0.0001 0.0642

0.19219

1.641

a1 a2 a3 a4 a5 a6 a7 a8 d

0.079334 ÿ0.00170770 0.055419 0.219686 ÿ0.152172 ÿ0.035352 ÿ0.00193545 0.400101 0.181556

0.01900 0.0003888 0.01321 0.02167 0.02169 0.01408 0.0003083 0.26573 0.02406

4.18 ÿ4.39 4.20 10.14 ÿ7.01 ÿ2.51 ÿ6.28 1.51 7.55

0.0001 0.0001 0.0001 0.0001 0.0001 0.0121 0.0001 0.1323 0.0001

0.18911

2.018

RMSE b

DW c

a

Nonlinear least squares. Root mean squared error. c Durbin±Watson statistic. d First-order autoregressive AR(1) parameter. b

Table 3 Fit statistics for aspen height increment model Eq. (7) Equation

Eq. (7)

a

Method

a

Parameter

Asymptotic SE

t-ratio

p-value

NLS

b1 b2 b3 b4 b5 b6

0.00292227 ÿ0.00230944 0.00847480 0.206881 ÿ0.189355 ÿ0.027525

0.0012095 0.0005943 0.0017647 0.01625 0.02308 0.01583

2.42 ÿ3.89 4.80 12.73 ÿ8.20 ÿ1.74

0.0158 0.0001 0.0001 0.0001 0.0001 0.0822

0.23549

1.529

NLS with AR(1) errors

b1 b2 b3 b4 b5 b6 d

0.00242251 ÿ0.00248709 0.00987930 0.206976 ÿ0.181396 ÿ0.033944 0.235825

0.0013389 0.0007048 0.0023087 0.01883 0.02348 0.01874 0.02658

1.81 ÿ3.53 4.28 10.99 ÿ7.72 ÿ1.81 8.87

0.0706 0.0004 0.0001 0.0001 0.0001 0.0703 0.0001

0.22906

2.040

Nonlinear least squares. Root mean squared error. c Durbin±Watson statistic. d First-order autoregressive AR(1) parameter. b

Estimate

S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

Durbin±Watson statistic of 1.641 is smaller than the lower bound (1.697) of the critical values for the eight-parameter height-increment model (Eq. (4) and Table 2), indicating a signi®cant ®rst-order autocorrelation for the error terms of the white spruce height-increment model. For aspen, the Durbin± Watson statistic of 1.529 is also smaller than the lower bound (1.718) of the critical values for the six-parameter height-increment model (Eq. (7) and Table 3), indicating a signi®cant ®rst-order autocorrelation for the error terms of the aspen heightincrement model. The autocorrelated errors for both these heightincrement models are typical of the data from PSPs where the plots are repeatedly measured over regular or sometimes irregular time intervals. The ith observation of the height increment model Eq. (4) or Eq. (7) with ®rst-order autoregressive errors AR(1) can be written as: HIi ˆ f …xi ; † ‡ "i ;

where "i ˆ "iÿ1 ‡ i

(10)

where the height increment HIi, i ˆ 1,2,. . .,n, is expressed as a nonlinear function of the independent variable matrix xi, "i are the random errors that follow as AR(1) process,  the autoregressive parameter, and i the independent and identically distributed random errors with mean zero and constant variance 2 . From Eq. (10), the (i ÿ 1)th observation of the height increment model can be written as: HIiÿ1 ˆ f …xiÿ1 ; † ‡ "iÿ1

(11)

Multiplying by  on the sides of Eq. (11), then subtracting the results from Eq. (10) gives a nonlinear equation with uncorrelated errors HIi ˆ HI…iÿ1† ‡ f …xi ; † ÿ f …xiÿ1 ; † ‡ i

(12)

A ®ve-step procedure (Judge et al., 1988) was used to ®nd the NLS estimate of the parameter matrix  in Eq. (12): 1. Fit Eq. (10) by ordinary NLS without considering its error structure, obtain the estimate of the parameter NLS. 2. Calculate the residuals "i ˆ HIi ÿ f(xi,NLS), and estimate  by fitting "i ˆ "iÿ1 ‡ i. 3. Find the estimate of  by applying ordinary NLS to the following model: HIi ÿ HI…iÿ1† ˆ f …xi ; † ÿ f …xiÿ1 ; † ‡ ei

(13)

49

4. Replace NLS in Step (2) by  from Step (3), then re-estimate  by fitting "i ˆ "iÿ1 ‡ i. 5. Replace  in Step (3) by  from Step (4), then reestimate  using ordinary NLS method. This process is iterated until  is converged. The resulting ®t statistics for white spruce and aspen are also shown in Tables 2 and 3. The Durbin±Watson statistics (2.018 and 2.040) for both these models are greater than their respective upper bounds (1.841 and 1.820) of the critical value, indicating that autocorrelation for the adjusted error terms is not signi®cant and the AR(1) speci®cation is appropriate for the error terms of the white spruce and aspen height-increment models. The height-increment models for white spruce and aspen were also diagnosed for possible unequal error variances. Because the ordinary residuals are intrinsically not independent and do not have a common variance, studentized residuals were used to account for these problems (Rawlings, 1988; Neter et al., 1990). The plots of studentized residuals against the predicted values of height increment from the preliminary NLS ®ts (without accounting for the AR(1) error terms) and the NLS ®ts accounting for the AR(1) error terms showed homogeneous bands of data points, with zero studentized residuals across the centre of these points. The error terms of the models were therefore considered to be identically distributed with an equal error variance. If unequal error variances are present, weighted NLS procedures as demonstrated by Huang et al. (1992), and Huang and Titus (1994, 1995) could be used. 4.2. Diagnostics for multicollinearity The tree and stand variables used in the heightincrement models may somehow be inter-correlated. Multicollinearity can exist in such models because of inter-correlated variables. Collinearity statistics were obtained (using the keyword COLLIN with the PROC MODEL procedure) to detect the presence, severity, and form of multicollinearity. For white spruce, the eigenvalues of the correlation matrix for the set of independent variables were arranged from the largest (5.0044) to smallest (0.0354), and the square root of the ratio of the largest-to-smallest eigenvalue, the condition index,

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S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

was calculated to be: (5.0044/0.0354)1/2 ˆ 11.89, which is <30, the proposed critical value for moderate multicollinearity (Belsley et al., 1980). This indicates that multicollinearity is not a serious problem for the white spruce height-increment model (Eq. (4)). For aspen, the eigenvalues of the correlation matrix for the set of independent variables were also obtained and arranged from the largest (3.8543) to smallest (0.0816), and the condition index was found to be: (3.8543/0.0816)1/2 ˆ 6.87, which also indicates that multicollinearity is not a serious problem for the aspen height-increment model (Eq. (7)).

Similarly, using the independent testing data set for aspen and the estimated results from Table 3, the mean and the standard deviation of the aspen prediction bias were found to be 0.004252 and 0.244923, respectively. The standard error of the estimated mean prediction bias was 0.2449232/(261)1/2 ˆ 0.01516. A t-test of the hypothesis that the mean prediction bias was zero gave t ˆ 0.2805 which, with 260 degrees of freedom, was not signi®cant at the  ˆ 0.05 level, indicating that the mean prediction bias for aspen was also not signi®cantly different from zero. The mean squared error of prediction for aspen was calculated to be:

4.3. Model testing

MSEPaspen ˆ …261ÿ1†0:2449232 =261 ‡ 0:0042522

The independent testing data sets were used to evaluate model performance when predictions were made. For white spruce, the actual height-increment values from the testing data set were compared to the predicted height-increment values using Eq. (4) with the estimates of parameters from NLS with AR(1) errors in Table 2. The bias of the prediction was obtained by subtracting the predicted height increment from the actual height increment. The mean () and the standard deviation (s) of the prediction bias were found to be 0.015198 and 0.175266, respectively. The standard error of the estimated mean prediction bias was 0.175266/(357)1/2 ˆ 0.009276. A t-test of the null hypothesis, namely that the mean prediction bias was zero, was conducted according to the method described by Rawlings (1988). The calculated t ˆ 1.6384 which, with 356 degrees of freedom, is not signi®cant at the  ˆ 0.05 level, indicating that the mean prediction bias was not signi®cantly different from zero. The mean-squared error of prediction (MSEP) for white spruce was calculated as (Rawlings, 1988): MSEP ˆ

…n ÿ 1†s2 …357 ÿ 1†0:1752662 ‡ 2  n 357 (14) ‡ 0:0151982 ˆ 0:030863

where the squared prediction bias term 2 contributed 0.75% of the MSEP. The square root of MSEP gave 0.1757, which is fairly close to the RMSE value obtained on the model ®tting data set (Table 2). This is an indication that the model predicted the white spruce data reasonably well (Montgomery and Peck, 1992).

ˆ 0:059776 The square root of MSEPaspen gave 0.2445, which is again close to the RMSE value obtained on the aspen model ®tting data set (Table 3). 5. Discussion and conclusion The height-increment models, as expressed in Eqs. (4) and (7), provide individual tree height-increment predictions for white spruce and aspen grown in the boreal mixed-species stands in Alberta. Most of the asymptotic t-ratios for the estimated coef®cients of the ®tted models are signi®cant at the  ˆ 0.05 level (Tables 2 and 3). Because of the mathematical properties of the chosen base function and the use of the parameter prediction method, both models have the ability to assume various shapes with different parameter values and produce satisfactory curves under most circumstances. All curves produced by the models assume biologically reasonable shapes that closely mimic the biological process of height growth, and provide realistic height-increment predictions in the cases where the models are extrapolated beyond the range of the original data. The ®tted height-increment models re¯ect some commonly held beliefs and interesting facts about the height increment of white spruce and aspen in the boreal mixed-species stands in Alberta. While the following discussion attempts to address the effect of some individual variables, it is important to recognize that height increment is jointly determined by the combination of all the variables.

S. Huang, S.J. Titus / Forest Ecology and Management 123 (1999) 41±53

The signi®cant negative coef®cients a2 and b2 for total basal area per hectare in Eqs. (5) and (8) (Tables 2 and 3) suggest that stand density has a signi®cant negative effect on white spruce and aspen height increment. If other conditions remain approximately the same, height increment is reduced as stand density increases, and dense stands result in smaller height increment. This agrees with the results observed by Wykoff et al. (1982) for mixed conifers of the northern Rocky Mountains and by Arney (1985) for coastal Douglas-®r, although a different stand density measure (i.e. crown competition factor) was used in their studies. The reason for a reduced height increment with increased stand density may relate to the fact that trees in dense stands have reduced crown development because the available growing space is limited. This, in turn, restricts the trees' ability to utilize the photosynthetic potential for height growth and reduces the trees' growth ef®ciency (Gilmore and Seymour, 1996). The coef®cients that represent the effects of species composition, a6 and b6 in Eqs. (6) and (9), are all negative (Tables 2 and 3). Because a negative coef®cient in Eq. (6) or Eq. (9) indicates a positive effect on the dependent variable, negative a6 and b6 values suggest that the effects of species composition are positive on both, white spruce and aspen height increment. This means that for two mixed-species stands growing in similar conditions, the one with more white spruce will have larger white spruce height increment, and the one with more aspen will have larger aspen height increment. The reason for this is not readily apparent. A possible explanation is that, under most circumstances, the height increment of both species is directly related to the relative abundance of the species in the stands. The ability of a particular species to compete with other species in the stand is stronger if the relative proportion of this particular species is larger. The increasing presence of white spruce reduces the relative competitiveness of aspen in the stands and the increasing presence of aspen reduces the relative competitiveness of white spruce. The signi®cant negative coef®cients, a5 and b5 in Eqs. (6) and (9), indicate that a large diameter increment leads to a large height increment. This is rather straightforward as increased diameter growth is commonly expected with increased height growth. Absolute height-increment rate was found positively related

51

to tree size (Larocque and Marshall, 1993). Because the height-increment models are expressed as a function of diameter increment, many variables that directly affect diameter increment can also indirectly affect height increment. The estimated coef®cients a3 and a7 in Eqs. (5) and (6), and b1 in Eq. (8), re¯ect a positive competition index and diameter effect on height increment, which means that for two trees growing under similar conditions, the one with a larger diameter or a larger CI value is in a better competitive position, and will have a larger height increment. The result of increased height increment with increased tree diameter is usually thought to be directly related to the increased competitiveness for larger diameter trees (Lorimer, 1983; Martin and Ek, 1984). If the other factors remain the same, stronger competitors, as measured by their relatively larger diameters or diameter ratios (i.e. CI), will have larger height increments. The ®tted models re¯ected this phenomenon. They may also af®rm that the asymmetric or one-sided competition commonly observed in even-aged plant monocultures (Harper, 1977; Weiner and Thomas, 1986), is equally applicable for white spruce and aspen growing in unevenaged and mixed-species stands. Mixed-species multicohort stands can be rather complex in their age structures, crown structures, and development patterns (Fajvan and Seymour, 1993). De®ning a measure of site for uneven-aged and mixed-species stands, and incorporating it into a model is a rather intricate problem. This problem is precisely the one that led researchers like McLintock and Bickford (1957), Stout and Shumway (1982), Wykoff et al. (1982), Vanclay and Henry (1988), and Nicholas and Zedaker (1992) to develop heightdiameter and other ecologically based relationships as alternatives to the traditional site index in attempting to provide accurate and realistic site productivity measures for these stands. The use of site productivity index (SPI) determined by the dominant-codominant height±diameter relationship is by no means an ultimate solution. However, the positive coef®cient values for a8 and b3 re¯ect the effect of site productivity on height increment and reveal that better sites support faster height increment. This is true for both, white spruce and aspen. While the height-increment models developed in this study show the effects of some tree and stand

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variables on white spruce and aspen height increment ± and this may provide some information about the ecology and silviculture of white spruce and aspen growing in boreal mixed-species stands in Alberta ± the models were primarily developed as main components in an individual tree-based, distance-independent mixed-wood growth simulator. They provide individual tree height-increment predictions in natural stands, and are not appropriate in regenerated juvenile stands. Applications and extensions of these models within the context of natural disturbance models should also be studied further (Gilmore, 1997). As the models developed in this study are based on functions that relate height increment to tree and stand variables, but do not involve individual tree or stand age explicitly, they can be applied to even- and uneven-aged stands. Input variables for the modes are simple tree and stand variables that are readily obtainable from ordinary inventories, and are compatible with the data-collection process in Alberta. Potential users of the models, with limited information or alternative measures for some of the variables appearing in the models, may consider re-®tting other forms of the models by replacing the variables that are not available, or by dropping the variables if the cost to obtain the variables is too high relative to the amount of additional variation explained by adding such variables into the models. In either case, a ¯exible base height±increment height function, such as the Box± Lucas function that closely mimics the biological process of height growth, should be retained, and the method of parameter prediction used in this study is recommended for incorporating the effects of other tree and stand variables. Acknowledgements This research was based on work conducted while Shongming Huang was a Ph.D. candidate and receiving ®nancial support from Alberta Forest Development Research Trust Fund and the Department of Forest Science, University of Alberta. Data were collected by the Forest Management Division of the Alberta Land and Forest Service. Special thanks are due to Mr. Dave Morgan and Mr. Tom Lakusta for providing valuable help and cooperation. The reviews and comments by the members of the thesis super-

visory and examining committees, Drs. Wiktor L. Adamowicz, Douglas P. Wiens, S. Ellen Macdonald, Jerome N. Sheahan, Jack D. Heidt, and Harold E. Burkhart, are gratefully acknowledged. References Alberta Forest Service, 1990. Permanent sample plots: field procedures manual. Timber Management Branch, FMOPC 83-03, Alberta Forest Service, Edmonton, Alberta, pp. 102. Amemiya, T., 1983. Non-linear regression models. In: Griliches, Z., Intriligator, M.D. (Eds.), Handbook of Econometrics, vol. I. North-Holland, Amsterdam. pp. 333±389. Arney, J.D., 1985. A modelling strategy for the growth projection of managed stands. Can. J. For. Res. 15, 511±518. Beck, D.E., 1974. Predicting growth of individual trees in thinned stands of Yellow-poplar. In: Fries, J. (Ed.), Growth Models for Tree and Stand Simulation. Royal Coll. of For., Stockholm. pp. 47±55. Belsley, D.A., Kuh, E., Welsch, R.E., 1980. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. John Wiley & Sons, New York. pp. 292. Box, G.E.P., Lucas, H.L., 1959. Design of experiments in nonlinear situations. Biometrika 47, 77±99. Clutter, J.L., Fortson, J.C., Pienaar, L.V., Brister, G.H., Bailey, R.L., 1983. Timber Management ± A Quantitative Approach. John Wiley & Sons, New York. pp. 333. Daniels, R.F., Burkhart, H.E., 1975. Simulation of individual tree growth and stand development in managed loblolly pine plantations. Div. For. Wild. Res., VPI and State Univ., Blacksburg, VA, FWS-5-75. pp. 69. Ek, A.R., Monserud, R.A., 1974. FOREST: A computer model for simulating the growth and reproduction of mixed species forest stands. Sch. of Nat. Res., Univ. of Wisc.-Madison, R2635. pp. 85. Fajvan, M.A., Seymour, R.S., 1993. Canopy stratification, age structure, and development of multicohort stands of eastern white pine, eastern hemlock, and red spruce. Can. J. For. Res. 23, 1799±1809. Gallant, A.R., 1987. Nonlinear Statistical Models. John Wiley & Sons, New York. pp. 610. Gilmore, D.W., 1997. Ecosystem management ± a needs driven, resource-use philosophy. For. Chron. 73(5), 560±564. Gilmore, D.W., Seymour, R.S., 1996. Alternative measures of stem growth efficiency applied to Abies balsamea from four canopy positions in central Maine, USA. For. Ecol. Manage. 84, 209± 218. Hamilton, D.C., Watts, D.G., 1985. A quadratic design criterion for precise estimation in nonlinear regression models. Technometrics 27, 241±250. Harper, J.L., 1977. Population Biology of Plants. Academic Press, London. pp. 892. Hester, A.S., Hann, D.W., Larsen, D.R., 1989. ORGANON: Southwest Oregon growth and yield model user manual. For. Res. Lab., Coll. of For., Oregon State Univ. pp. 59.

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