# Autoregressive Modeling of Forest Dynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Forests and Stock Markets

#### 1.3. Understanding and Modeling of Forest Patch Dynamics

#### 1.4. Our Contributions

## 2. Materials and Methods

#### 2.1. Data Mining of Quebec Provincial Forest Inventories

^{2}[28]. The database consists of 32,552 plot re-measurements at 11,660 different locations. The Quebec forest inventory is designed to comprehensive describe the patch mosaic of Quebec forests and plots cover the Quebec territory practically uniformly [28]. The GIS-based map of forest inventory plots is published as Figure 1 in [10]. Forest plots cover hardwood and mixed forests in the northern temperate zone (9621 and 7663, respectively) and continuous boreal forests in the boreal zone (11,969 measurements). These forest patches (forest inventory plots) are remeasured every few years, often with irregular time intervals between measurements. The inventory plots were affected by natural and anthropogenic disturbances including fire and harvesting. The statistical analysis of the measurement dynamics and re-measurement intervals are published in [11] (see Figure 2 in Appendix to [11]). In this inventory, each patch observation includes the diameter of each tree, its species, soil moisture, and other characteristics. This is the raw data which is then converted to a more tractable data series. In particular, we are interested in biomass and basal area. Calculations of biomass and basal area were previously done according to [29,30]. The computations of biomass and basal area (as well as other characteristics, such as shade tolerance index, and biodiversity measured by Shannon entropy) are done in [11,14,25,27]. The biomass is this article refers to the plot biomass, which is the sum of biomasses of all trees computed using formulas from [29] (see Section 3.1 in [14] for the details). The code used for this article is available on GitHub repository asarantsev/Quebec.

#### 2.2. Autoregressive Model for Individual Forest Patches

^{2}) random variables. If 0 < a < 1, this sequence y(0), y(1),…exhibits mean reversion to its long-term average m = r/(1 − a). That is, if y(t) > m, then y(t + 1) − m is likely to be smaller than y(t) − m, and vice versa. Examples of earlier use of such models for forest modeling include [31,32,33]. They also include spatial models (incorporating distance between patches). We shall not attempt it here, instead treating every patch as effectively isolated. Building a spatial model for Quebec forests is left for future research.

_{1}) from y(t

_{0}) if t

_{0}and t

_{1}are consecutive years for which this patch was observed. Then we try various a and obtain for each a the maximum likelihood estimate via linear regression. We compare these likelihoods and find the best fit for a. It turns out to be a = 1. That is, this sequence does not actually exhibit any mean reversion, but behaves like a random walk, when each next increment is independent from the past:

^{2}) i.i.d.

#### 2.3. Annual Averages

^{2}) i.i.d.

^{2}(which corresponds to the lack of any existing information), and then compute the posterior distribution from the likelihood. Bayesian techniques are increasingly used in ecology [40], as well as in medical statistics [41] and quantitative finance [42].

## 3. Results and Discussion

#### 3.1. Autoregressive Model for Individual Patches

^{2}). As mentioned earlier, our main difficulty is that we do not have observations for all t and p, only for t $\in $ T(p). Assume t, t + u are subsequent time points in T(p). Then

^{u}y(t)) = D(a, u)r + δp (t + u), δp(t + u) ∼ N(0, σ

^{2}) i.i.d.

C(a, u): = [1 + a

^{2}+…+ a

^{2(u−1)}]

^{−1/2}

D(a, u): = C(a, u)[1 + a +…+ a

^{u−1}]

_{p}(t) the logarithm of biomass, and by y’’

_{p}(t) the logarithm of basal area for patch p and year t. If T(p) = {t

_{0}, t

_{1}, t

_{2}, …} has more than one year, order them in increasing order: t

_{0}< t

_{1}< t

_{2}<… Compute correlation coefficient between

_{p}(t

_{k}) − y′

_{p}(t

_{k}

_{−1}) and y″

_{p}(t

_{k}) − y″

_{p}(t

_{k}

_{−1})

#### 3.2. Yearly Averages, Frequentist Analysis

^{2}) i.d.d.

_{1}+…+ ξ

_{t}, ξ

_{i}∼ N (µ, ρ

^{2}) i.i.d.

_{1}+…+ ξ

_{t}), t = 0, 1, 2,…

^{2}/2)); Var[m(t)] = m

^{2}(0) exp(2µt + σ

^{2}t) (exp(σ

^{2}t) − 1)

#### 3.3. Yearly Averages, Bayesian Analysis

_{1}(t), …, x

_{n}(t). For these values, we performed the analysis above: the posterior mean µ(t) and the posterior variance v(t) were generated. The simulation of µ(t) and v(t) was performed N = 1000 times. In Figure 6, we have histograms of 1000 simulations for µ(t) and σ(t) for t = 0 (year 1970), for both biomass and basal area. Hence we obtained 38 sequences of N numbers: µ

_{1}(t), …, µ

_{N}(t), t = 1970, …, 2007. The average growth rate based on simulated results is

_{i}(t) = µ

_{i}(t) −µ

_{i}(t − 1), t = 1, …, T, i = 1, …, N. Assuming these are N (g, σ

^{2}) i.i.d. we estimate g and σ

^{2}as $\widehat{g}$ in Equation (8) and ${\widehat{\sigma}}^{2}$:

^{2}for biomass and basal area:

## 4. General Discussion

#### 4.1. Towards Autoregressive Theory of Forest Dynamics

#### 4.2. Future Research

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

quantile-quantile (for a plot) | |

AR | autoregression |

FIA | USDA Forest Service Forest Inventory and Analysis Program |

VARIMA | vector autoregressive integrated moving average model |

USDA | the United States Department of Agriculture |

GIS | Geographic Information Systems |

ARIMA | autoregressive integrated moving average model |

i.i.d. | independent and identically distributed (random variables) |

## Appendix A. Maximal Likelihood and Minimal Standard Error

_{i}= c

_{0}(a) + c

_{1}(a)x

_{i1}+ c

_{2}(a)x

_{i2}+…+ c

_{d}(a)x

_{id}+ ε

_{i}(a), ε

_{i}(a) ∼ N (0, σ

^{2}) i.i.d.

^{2}) is given by

^{2}. To maximize f from (A3), we need to first minimize the standard error s

^{2}(a) by computing it for each a and then choosing an appropriate a; then to choose the σ which maximizes (A3) for given s

^{2}(a), which turns out to be (after standard calculus exercise): σ

^{2}= s

^{2}(a).

## Appendix B. Background on Autoregressive Models and Random Walk

_{0}, x

_{1}, x

_{2}, …):

_{n}= r + ax

_{n−1}+ ε

_{n}, ε

_{n}∼ N (0, σ

^{2}) i.i.d.

^{2}), with

_{n}converge to that of the limiting distribution: E (x

_{n}) → m, Var (x

_{n}) → ρ

^{2}. Thus this time series exhibits mean reversion: If x

_{n}> m then x

_{n+1}is more likely to decrease compared to x

_{n}than to increase.

_{n}

_{+1}− x

_{n}are independent for different n. This sequence does not have a limit as n → ∞. The expectation E (x

_{n}) = E (x

_{0}) is constant. But the variance Var (x

_{n}) = Var (x

_{0}) + nσ

^{2}.

## Appendix C. Background on Bayesian Inference

_{1}, …, x

_{N}∼ N (m, σ

^{2}). Denote the variance by σ

^{2}= v. Random variables x

_{1}, …, x

_{n}are independent, and N (m, v) has density

^{−1}. This is an infinite measure:

_{1}, …, x

_{n}given m, v is

^{−1}has marginal Gamma distribution with shape n/2 and expectation 1/S; and v has inverse Gamma distribution with shape n/2. The conditional distribution of m given v is normal. The unconditional (marginal) distribution of m is Student (t-distribution). A Student distribution has heavier tails than a normal distribution, which implies more uncertainty, resulting from our Bayesian estimation framework.

## Appendix D. Empirical Data

Year Gap | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Quantity | 1 | 2 | 65 | 915 | 1381 | 3334 | 1923 | 2214 | 2543 | 2677 | 1972 | 694 |

Year Gap | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |

Quantity | 922 | 533 | 391 | 569 | 390 | 123 | 8 | 66 | 8 | 17 | 22 | 22 |

Year Gap | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 |

Quantity | 4 | 17 | 14 | 9 | 6 | 5 | 12 | 1 | 0 | 5 | 2 | 0 |

**Table A2.**Empirical means x(t) and variances S(t) of biomass and basal area logarithms, for each year.

Year | Number of Observations | Yearly Mean of Biomass Logarithm | Yearly Variance of Biomass Logarithm | Yearly Mean of Basal Area Logarithm | Yearly Variance of Basal Area Logarithm |
---|---|---|---|---|---|

1970 | 522 | 1.81 | 0.77 | 1.67 | 0.48 |

1971 | 1216 | 2.05 | 0.8 | 1.9 | 0.5 |

1972 | 1286 | 1.97 | 0.67 | 1.87 | 0.43 |

1973 | 335 | 1.66 | 0.59 | 1.64 | 0.39 |

1974 | 304 | 1.68 | 0.55 | 1.66 | 0.36 |

1975 | 902 | 1.97 | 0.66 | 1.96 | 0.54 |

1976 | 1883 | 2.17 | 0.61 | 2.02 | 0.4 |

1977 | 422 | 1.82 | 0.37 | 1.81 | 0.25 |

1978 | 1319 | 2.77 | 0.7 | 2.53 | 0.48 |

1979 | 1339 | 2.68 | 0.77 | 2.52 | 0.54 |

1980 | 1047 | 2.2 | 0.7 | 2.13 | 0.52 |

1981 | 396 | 1.86 | 0.7 | 1.8 | 0.51 |

1982 | 8 | 1.98 | 0.12 | 1.97 | 0.12 |

1983 | 98 | 2.23 | 0.66 | 2.06 | 0.43 |

1984 | 358 | 2.65 | 0.57 | 2.32 | 0.36 |

1985 | 629 | 2.38 | 0.75 | 2.15 | 0.47 |

1986 | 665 | 2.24 | 0.62 | 2.07 | 0.4 |

1987 | 732 | 2.22 | 0.62 | 2.05 | 0.42 |

1988 | 604 | 1.98 | 0.56 | 1.92 | 0.34 |

1989 | 1597 | 2.49 | 0.95 | 2.38 | 0.58 |

1990 | 723 | 1.46 | 0.47 | 1.54 | 0.34 |

1991 | 581 | 2.02 | 0.61 | 1.92 | 0.38 |

1992 | 1782 | 2.67 | 0.68 | 2.54 | 0.46 |

1993 | 865 | 2.88 | 0.77 | 2.65 | 0.55 |

1994 | 647 | 3.21 | 0.67 | 2.93 | 0.48 |

1995 | 625 | 2.85 | 0.77 | 2.66 | 0.52 |

1996 | 858 | 2.57 | 0.86 | 2.43 | 0.68 |

1997 | 2247 | 3.08 | 0.81 | 2.82 | 0.57 |

1998 | 977 | 2.6 | 0.68 | 2.49 | 0.49 |

1999 | 905 | 2.11 | 0.67 | 2.13 | 0.54 |

2000 | 101 | 2.62 | 1.0 | 2.46 | 0.74 |

2001 | 756 | 2.47 | 0.38 | 2.45 | 0.3 |

2002 | 309 | 2.32 | 0.47 | 2.32 | 0.39 |

2003 | 3414 | 3.08 | 0.85 | 2.83 | 0.61 |

2004 | 19 | 2.55 | 0.29 | 2.51 | 0.23 |

2005 | 641 | 2.71 | 0.73 | 2.57 | 0.57 |

2006 | 599 | 3.42 | 0.81 | 3.08 | 0.53 |

2007 | 841 | 2.67 | 0.81 | 2.55 | 0.62 |

## References

- Levin, S.A. Fragile Dominion: Complexity and the Commons; Perseus Publishing: Cambridge, MA, USA, 1999. [Google Scholar]
- Levin, S.A. Complex adaptive systems: Exploring the known, the unknown and the unknowable. Am. Math. Soc.
**2003**, 40, 3–19. [Google Scholar] [CrossRef] - Strigul, N. Individual-based models and scaling methods for ecological forestry: Implications of tree phenotypic plasticity. In Sustainable Forest Management; Garcia, J., Casero, J., Eds.; InTech: Rijeka, Croatia, 2012; pp. 359–384. [Google Scholar] [CrossRef]
- Botkin, D.B. Forest Dynamics: An Ecological Model; Oxford University Press: Oxford, UK, 1993. [Google Scholar]
- Shugart, H.H. A Theory of Forest Dynamics: The Ecological Implications of Forest Succession Models; Springer: Berlin, Germany, 1984. [Google Scholar]
- Pacala, S.W.; Canham, C.D.; Silander, J.A., Jr. Forest models defined by field measurements: I. The design of a northeastern forest simulator. Can. J. For. Res.
**1993**, 23, 1980–1988. [Google Scholar] [CrossRef] - Pastor, J.; Sharp, A.; Wolter, P. An application of Markov models to the dynamics of Minnesota’s forests. Can. J. For. Res.
**2005**, 35, 3011–3019. [Google Scholar] [CrossRef] - Moorcroft, P.; Hurtt, G.; Pacala, S.W. A method for scaling vegetation dynamics: The ecosystem demography model (ED). Ecol. Monogr.
**2001**, 71, 557–586. [Google Scholar] [CrossRef] - Strigul, N.; Pristinski, D.; Purves, D.; Dushoff, J.; Pacala, S. Scaling from trees to forests: Tractable macroscopic equations for forest dynamics. Ecol. Monogr.
**2008**, 78, 523–545. [Google Scholar] [CrossRef] - Liénard, J.F.; Strigul, N.S. Modelling of hardwood forest in Quebec under dynamic disturbance regimes: A time-inhomogeneous Markov chain approach. J. Ecol.
**2016**, 104, 806–816. [Google Scholar] [CrossRef] - Strigul, N.; Florescu, I.; Welden, A.R.; Michalczewski, F. Modelling of forest stand dynamics using Markov chains. Environ. Model. Softw.
**2012**, 31, 64–75. [Google Scholar] [CrossRef] - Pacala, S.W.; Canham, C.D.; Saponara, J.; Silander, J.A., Jr.; Kobe, R.K.; Ribbens, E. Forest models defined by field measurements: Estimation, error analysis and dynamics. Ecol. Monogr.
**1996**, 66, 1–43. [Google Scholar] [CrossRef] - Liénard, J.; Strigul, N. An individual-based forest model links canopy dynamics and shade tolerances along a soil moisture gradient. R. Soc. Open Sci.
**2016**, 3, 150589. [Google Scholar] [CrossRef] [PubMed] - Liénard, J.F.; Gravel, D.; Strigul, N.S. Data-intensive modeling of forest dynamics. Environ. Model. Softw.
**2015**, 67, 138–148. [Google Scholar] [CrossRef] - Caswell, H. Matrix Population Models: Construction, Analysis, and Interpretation; Sinauer Associates: Sunderland, MA, USA, 2001. [Google Scholar]
- Wu, J.; Loucks, O.L. From balance of nature to hierarchical patch dynamics: A paradigm shift in ecology. Q. Rev. Biol.
**1996**, 70, 439–466. [Google Scholar] [CrossRef] - Watt, A.S. Pattern and process in the plant community. J. Ecol.
**1947**, 35, 1–22. [Google Scholar] [CrossRef] - Levin, S.A.; Paine, R.T. Disturbance, patch formation, and community structure. Proc. Natl. Acad. Sci. USA
**1974**, 71, 2744–2747. [Google Scholar] [CrossRef] [PubMed] - Scholl, A.E.; Taylor, A.H. Fire regimes, forest change, and self-organization in an old-growth mixed-conifer forest, Yosemite National Park, USA. Ecol. Appl.
**2010**, 20, 362–380. [Google Scholar] [CrossRef] [PubMed] - McCarthy, J. Gap dynamics of forest trees: A review with particular attention to boreal forests. Environ. Rev.
**2001**, 9, 1–59. [Google Scholar] [CrossRef] - Bugmann, H. A review of forest gap models. Clim. Chang.
**2001**, 51, 259–305. [Google Scholar] [CrossRef] - Dubé, P.; Fortin, M.; Canham, C.; Marceau, D. Quantifying gap dynamics at the patch mosaic level using a spatially-explicit model of a northern hardwood forest ecosystem. Ecol. Model.
**2001**, 142, 39–60. [Google Scholar] [CrossRef] - Kohyama, T.; Suzuki, E.; Partomihardjo, T.; Yamada, T. Dynamic steady state of patch-mosaic tree size structure of a mixed dipterocarp forest regulated by local crowding. Ecol. Res.
**2001**, 16, 85–98. [Google Scholar] [CrossRef] - Hanson, P.J.; Weltzin, J.F. Drought disturbance from climate change: Response of United States forests. Sci. Total Environ.
**2000**, 262, 205–220. [Google Scholar] [CrossRef] - Liénard, J.; Florescu, I.; Strigul, N. An Appraisal of the Classic Forest Succession Paradigm with the Shade Tolerance Index. PLoS ONE
**2015**, 10, e0117138. [Google Scholar] [CrossRef] - Van Wagner, C.E. Age-class distribution and the forest fire cycle. Can. J. For. Res.
**1978**, 8, 220–227. [Google Scholar] [CrossRef] - Liénard, J.; Strigul, N. Linking forest shade tolerance and soil moisture in North America. Ecol. Indic.
**2015**, 58, 332–334. [Google Scholar] [CrossRef] - Perron, J.; Morin, P. Normes d’inventaire Forestier: Placettes-échantillons Permanents; Ministry of Forests: Quebec, QC, Canada, 2011. [Google Scholar]
- Jenkins, J.; Chojnacky, D.; Heath, L.; Birdsey, R. National-Scale Biomass Estimators for United States Tree Species. For. Sci.
**2003**, 49, 12–35. [Google Scholar] - Woodall, C.W.; Heath, L.S.; Domke, G.M.; Nichols, M.C. Methods and Equations for Estimating Aboveground Volume, Biomass, and Carbon for Trees in the US Forest Inventory; USDA Forest Service: United States Department of Agriculture: Madison, WI, USA, 2010. [Google Scholar]
- Fleming, R.A.; Barclay, H.J.; Candau, J.N. Scaling-up an autoregressive time-series model (of spruce budworm population dynamics) changes its qualitative behavior. Ecol. Model.
**2002**, 149, 127–142. [Google Scholar] [CrossRef] - Lichstein, J.W.; Simons, T.R.; Shriner, S.A.; Franzreb, K.E. Spatial autocorrelation and autoregressive models in ecology. Ecol. Monogr.
**2002**, 72, 445–463. [Google Scholar] [CrossRef] - Fox, J.C.; Bi, H.; Ades, P.K. Modelling spatial dependence in an irregular natural forest. Silva Fenn.
**2008**, 42, 35. [Google Scholar] [CrossRef] - Fama, E.F. Random walks in stock market prices. Financ. Anal. J.
**1995**, 51, 75–80. [Google Scholar] [CrossRef] - Barrett, A.; Rappoport, P. Price-Earnings Investing. JP Morgan Asset Manag. Real. Returns
**2011**, 1, 1–12. [Google Scholar] - Ioannidis, C.; Peel, D.A.; Peel, M.J. The time series properties of financial ratios: Lev revisited. J. Bus. Financ. Account.
**2003**, 30, 699–714. [Google Scholar] [CrossRef] - Campbell, J.Y.; Shiller, R.J. Valuation ratios and the long-run stock market outlook: An update. Tech. Rep. Natl. Bur. Econ. Res.
**2001**. [Google Scholar] [CrossRef] - Davis, J.; Aliaga-Díaz, R.; Thomas, C.J. Forecasting Stock Returns: What Signals Matter, and What do They Say Now; The Vanguard Group: Valley Forge, PA, USA, 2012. [Google Scholar]
- Goyal, A.; Welch, I. Predicting the equity premium with dividend ratios. Manag. Sci.
**2003**, 49, 639–654. [Google Scholar] [CrossRef] - Ellison, A.M. Bayesian inference in ecology. Ecol. Lett.
**2004**, 7, 509–520. [Google Scholar] [CrossRef] - Gurrin, L.C.; Kurinczuk, J.J.; Burton, P.R. Bayesian statistics in medical research: An intuitive alternative to conventional data analysis. J. Eval. Clin. Pract.
**2000**, 6, 193–204. [Google Scholar] [CrossRef] [PubMed] - Rachev, S.T.; Hsu, J.S.; Bagasheva, B.S.; Fabozzi, F.J. Bayesian Methods in Finance; John Wiley & Sons: Hoboken, NJ, USA, 2008; Volume 153. [Google Scholar]
- Craven, B.D.; Islam, S.M. A model for stock market returns: Non-Gaussian fluctuations and financial factors. Rev. Quant. Financ. Account.
**2008**, 30, 355–370. [Google Scholar] [CrossRef] - Waggoner, P.E.; Stephens, G.R. Transition probabilities for a forest. Nature
**1970**, 225, 1160–1161. [Google Scholar] [CrossRef] [PubMed] - Stephens, G.R.; Waggoner, P.E. A half century of natural transitions in mixed hardwood forests. Bull. Conn. Agric. Exp. Stn.
**1980**, 783, 44. [Google Scholar] - Usher, M.B. Markovian approaches to ecological succession. J. Anim. Ecol.
**1979**, 48, 413–426. [Google Scholar] [CrossRef] - Logofet, D.; Lesnaya, E. The mathematics of Markov models: What Markov chains can really predict in forest successions. Ecol. Model.
**2000**, 126, 285–298. [Google Scholar] [CrossRef] - Liénard, J.; Harrison, J.; Strigul, N. US forest response to projected climate-related stress: A tolerance perspective. Glob. Chang. Biol.
**2016**, 22, 2875–2886. [Google Scholar] [CrossRef]

**Figure 4.**Histograms for biomass (measured in 10

^{3}kg/ha) and basal area (measured in m

^{2}/ha) in 2007.

**Figure 5.**Histograms for biomass (measured in 10

^{3}kg/ha) and basal area (measured in m

^{2}/ha) in 2019 superimposed upon 2007.

**Figure 6.**Histograms of N = 1000 simulations for means and variances of the logarithm for biomass and basal area, 1970. The biomass is measured in 10

^{3}kg/ha, and the basal area is measured in m

^{2}/ha.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rumyantseva, O.; Sarantsev, A.; Strigul, N. Autoregressive Modeling of Forest Dynamics. *Forests* **2019**, *10*, 1074.
https://doi.org/10.3390/f10121074

**AMA Style**

Rumyantseva O, Sarantsev A, Strigul N. Autoregressive Modeling of Forest Dynamics. *Forests*. 2019; 10(12):1074.
https://doi.org/10.3390/f10121074

**Chicago/Turabian Style**

Rumyantseva, Olga, Andrey Sarantsev, and Nikolay Strigul. 2019. "Autoregressive Modeling of Forest Dynamics" *Forests* 10, no. 12: 1074.
https://doi.org/10.3390/f10121074