Error Propagation in Forest Planning Models

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We evaulate the impact of measurement error on three components of a forest planning model: initial inventory, first period harvest and silvicultural activities chosen. Calculated initial inventory was off by as much as 15%, and significantly different objective function values and activity schedule also resulted.

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  • Error Propagation in Forest Planning Models

    1. 1. Error Propagation in Forest Planning Models Karl R. Walters and Sean J. Canavan
    2. 2. Outline <ul><li>Measurement errors </li></ul><ul><li>Characterization of errors </li></ul><ul><li>Traditional Methods </li></ul><ul><li>Two Stage Error Distribution (TSED) method </li></ul><ul><li>Case Study in error propagation </li></ul><ul><li>Discussion of results </li></ul>
    3. 3. Measurement Error (ME) in Forestry <ul><li>Arises when there is a difference between observed and actual value for an attribute </li></ul><ul><ul><li>Sampling error (only portion of population measured) </li></ul></ul><ul><ul><li>Grouping error (model calibrated to one level of precision and then applied to a different one) </li></ul></ul><ul><ul><li>Mensuration error (a flaw in the measurement process) </li></ul></ul>
    4. 4. Measurement Error (ME) in Forestry <ul><li>Demonstrated Consequences: </li></ul><ul><ul><li>Biased estimates of tree and stand attributes </li></ul></ul><ul><ul><li>Biased model parameters and predictions </li></ul></ul><ul><ul><li>Decreased precision of model predictions </li></ul></ul><ul><ul><li>Heteroskedastic prediction errors </li></ul></ul><ul><ul><li>Skewed distributions </li></ul></ul><ul><ul><li>Biased fit statistics </li></ul></ul>
    5. 5. Measurement Error (ME) in Forestry <ul><li>Distribution of errors provides the means for evaluation and possible correction methods </li></ul><ul><li>Specified by either: </li></ul><ul><ul><li>Probability distribution function (PDF) or, </li></ul></ul><ul><ul><li>Cumulative distribution function (CDF) </li></ul></ul>
    6. 6. Normal (0,1) PDF x 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 -5 -4 -3 -2 -1 0 1 2 3 4 5 f(x)
    7. 7. Normal (O,1) CDF x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 F(x) = P(X < x)
    8. 8. Traditional Methods <ul><li>Errors assumed to be normally distributed </li></ul><ul><ul><li>Assume μ ( δ x ) = 0 </li></ul></ul><ul><ul><li>Assume μ ( δ x ) = constant other than 0 </li></ul></ul><ul><ul><li>Assume μ ( δ x ) = f(x) </li></ul></ul><ul><li>In turn each of these separated by </li></ul><ul><ul><li>Assume σ ( δ x ) = constant </li></ul></ul><ul><ul><li>Assume σ ( δ x ) = variable </li></ul></ul>
    9. 9. Traditional Methods <ul><li>Unbiased error doesn’t necessarily average out </li></ul><ul><ul><li>Basal area example </li></ul></ul><ul><ul><ul><li>20 cm tree, 314.16 cm 2 </li></ul></ul></ul><ul><ul><ul><li>0.5 over, 330.06 cm 2 </li></ul></ul></ul><ul><ul><ul><li>0.5 under, 298.65 cm 2 </li></ul></ul></ul><ul><ul><li>Average = 314.56 cm 2 </li></ul></ul><ul><li>Normal easy to model </li></ul><ul><ul><li>Is it appropriate choice? </li></ul></ul>
    10. 10. Normal (O,1) CDF x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 F(x) = P(X < x)
    11. 11. Fitted CDF Equation 0.00 0.20 0.40 0.60 0.80 1.00 -1.5 -1 -0.5 0 0.5 1 1.5 Error Size Cumulative Probability  Pr(  = 0)  Pr(  < 0)  Pr(  > 0) P(X = x ) = Pr(  < 0)*Negative Error CDF  < 0 Pr(  < 0) + Pr(  = 0)  = 0 Pr(  < 0) + Pr(  = 0) + Pr(  > 0)*Positive Error CDF  > 0
    12. 12. Fits of Error Type Probabilities 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0.5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 Dbh (inches) Probability P(e > 0) P(e = 0) P(e < 0)
    13. 13. Empirical Dbh Error CDF Surface Cumulative Probability Dbh (inches) Error (inches) 0.25 0.50 0.75 1.00 0.0 20.0 30.0 40.0 -0.6 0.0 0.6 1.2 1.8 10.0
    14. 14. Fitted Dbh CDF Surface
    15. 15. Forest Modeling Experiment <ul><li>Inventory data (tree lists) from PNW covering a range of forest types </li></ul><ul><li>Apply corrections to measured dbh and ht derived from TSED analysis </li></ul><ul><ul><li>None, dbh only, ht only, dbh & ht </li></ul></ul><ul><li>Project growth through Organon </li></ul><ul><li>Use yield tables in mock forest planning exercise for a TIMO client </li></ul>
    16. 16. Example Forest <ul><li>Southwest Washington </li></ul><ul><li>69,000+ ac </li></ul><ul><li>Species group(8), site (3),BA(4),stocking(3) </li></ul><ul><li>Elevation(3),slope(3), operability(2) </li></ul><ul><li>Regen(2spX2dens), PCT(4), fert, CT(2), prune </li></ul>
    17. 17. Model Parameters <ul><li>NPV maximization (pseudo-delivered price) </li></ul><ul><ul><li>Delivered prices for 10 products </li></ul></ul><ul><ul><li>Average logging costs ($/mbf) by equipment type, average hauling cost ($/mbf) </li></ul></ul><ul><ul><li>Road const & maint ($/mbf), sev. taxes ($/mbf) </li></ul></ul><ul><li>5-yr planning periods </li></ul><ul><li>30 period planning horizon </li></ul>
    18. 18. Unconstrained <ul><li>Any differences between the four runs are due to yield coefficients </li></ul><ul><li>Not confounded by constraints </li></ul>
    19. 19. Unconstrained <ul><li>Initial Inventory </li></ul><ul><ul><li>Dbh_err 14.7% high </li></ul></ul><ul><ul><li>Others < 0.2% high </li></ul></ul><ul><li>All periods </li></ul><ul><ul><li>Dbh_err maintains differential </li></ul></ul>
    20. 20. Unconstrained <ul><li>Period 1 Harvest </li></ul><ul><ul><li>Dbh_err 5.05% high </li></ul></ul><ul><ul><li>Others < 0.76% high </li></ul></ul>
    21. 21. Unconstrained 4,838,760 5,142,948 4,925,347 5,180,696 Total Harvest 33,076 32,206 33,355 28,352 HW saw logs 4,403 4,275 4,428 4,333 RC saw logs 18,073 16,731 47,399 16,736 WW oversize logs 591,469 613,119 615,886 606,089 WW saw logs 224,367 220,707 232,815 221,610 DF oversize logs 3,467,413 3,756,908 3,480,189 3,795,724 DF saw logs 499,959 499,000 511,276 507,851 DF export logs Dbh_Ht Err Ht_Err Dbh_Err Clean 50 Year Summary
    22. 22. Unconstrained <ul><li>Period 1 regeneration </li></ul><ul><ul><li>Clean plants significantly more ac to DF 550 tpa </li></ul></ul><ul><li>Period 1 PCT </li></ul><ul><ul><li>Clean performs significantly more PCT </li></ul></ul><ul><ul><li>Only 450tpa PCT </li></ul></ul><ul><ul><li>Others use 537tpa also </li></ul></ul>
    23. 23. Unconstrained 36,766 36,144 36,423 35,213 commercial thin 49 19 49 - precomm thin 537tpa 36 - 19 266 precomm thin 436tpa - - - - precomm thin 360tpa - - - - precomm thin 300tpa 18,142 18,142 18,142 17,039 slashing - - - - planted NF 680tpa 5,656 5,656 5,656 5,108 planted NF 550tpa 19,381 20,504 19,381 30,342 planted DF 550tpa 30,912 29,820 30,850 19,586 planted DF 450tpa 55,948 55,980 55,886 55,036 clearcut Dbh_Ht Err Ht_Err Dbh_Err Clean 50 Year Summary
    24. 24. Nondeclining yield <ul><li>Nondeclining yield constraint on total sawlog </li></ul><ul><li>Smooths volume and revenue spikes </li></ul>
    25. 25. Nondeclining Yield <ul><li>Inventory </li></ul><ul><ul><li>Dbh Err usually has highest inventory </li></ul></ul><ul><ul><li>Dbh_Ht Err usually lowest </li></ul></ul>
    26. 26. Nondeclining Yield <ul><li>Period 1 Harvest </li></ul><ul><ul><li>Dbh Err 0.5% lower </li></ul></ul><ul><ul><li>Ht Err 1.8% higher </li></ul></ul><ul><ul><li>Dbh_Ht Err 2.7% lower </li></ul></ul><ul><li>Period 9 Harvest </li></ul><ul><ul><li>Dbh Err 7.6% lower </li></ul></ul><ul><ul><li>Ht Err 8.1% higher </li></ul></ul><ul><ul><li>Dbh_Ht Err 0.7% higher </li></ul></ul>
    27. 27. Nondeclining Yield 3,498,616 3,624,749 3,561,097 3,654,132 Total Harvest 35,169 33,695 34,992 33,844 HW saw logs 5,815 5,767 6,181 6,099 RC saw logs 27,815 27,137 57,946 27,880 WW oversize logs 798,669 803,591 819,808 824,113 WW saw logs 351,540 316,824 352,489 322,816 DF oversize logs 1,636,676 1,764,109 1,631,819 1,761,552 DF saw logs 642,932 673,628 657,862 677,828 DF export logs Dbh_Ht Err Ht_Err Dbh_Err Clean 50 Year Summary
    28. 28. Nondeclining Yield <ul><li>Period 1 regeneration </li></ul><ul><ul><li>Only plants DF </li></ul></ul><ul><ul><li>Densities chosen relatively consistent </li></ul></ul><ul><li>Period 1 PCT </li></ul><ul><ul><li>Dbh_Ht_err significantly more acres, all densities </li></ul></ul><ul><ul><li>Clean is about half acres </li></ul></ul><ul><ul><li>Ht_err is mostly 450tpa </li></ul></ul>
    29. 29. Nondeclining Yield 22,493 21,884 21,600 21,368 commercial thin 228 19 1941 - precomm thin 537tpa 537 - 367 266 precomm thin 436tpa 161 256 - 716 precomm thin 360tpa 74 56 105 56 precomm thin 300tpa 18,142 17,094 17,145 17,094 slashing - - - - planted NF 680tpa 5,019 5,019 5,019 5,019 planted NF 550tpa 18,704 21,878 16,636 20,809 planted DF 550tpa 21,085 17,931 23,117 18,978 planted DF 450tpa 44,808 44,828 44,772 44,806 clearcut Dbh_Ht Err Ht_Err Dbh_Err Clean 50 Year Summary
    30. 30. Discussion <ul><li>Initial inventory off by as much as 14.7% </li></ul><ul><ul><li>Nightmare scenario in acquisition due diligence </li></ul></ul><ul><li>Unconstrained runs show similar results but </li></ul><ul><ul><li>Clean run did more PCT than others </li></ul></ul><ul><ul><li>Clean run used equal densities of DF </li></ul></ul><ul><ul><li>Clean produced more sawlog over 50 yrs </li></ul></ul>
    31. 31. Discussion <ul><li>Flow constraint created bigger differences </li></ul><ul><ul><li>Significant differences in total volume and in product composition </li></ul></ul><ul><ul><li>Significant differences in silvicultural regimes chosen </li></ul></ul>
    32. 32. Discussion <ul><li>Measurement errors </li></ul><ul><ul><li>Conventional wisdom assumes MEs cancel out over the long run… NOT TRUE!!! </li></ul></ul><ul><ul><li>Effects are apparent immediately </li></ul></ul><ul><ul><ul><li>Not limited to small, consistent variation </li></ul></ul></ul><ul><ul><li>Can pronounce differences in merchandising </li></ul></ul><ul><ul><ul><li>Log length is becoming the dominant parameter in price determination </li></ul></ul></ul>
    33. 33. Discussion <ul><li>Optimization models can amplify ME effects </li></ul><ul><ul><li>Sensitive to prices tied to yields </li></ul></ul><ul><ul><li>Tries to capitalize on erroneous differences in yield to maximize revenue </li></ul></ul><ul><ul><li>Inappropriate silvicultural regimes chosen </li></ul></ul><ul><li>Not only objective function is changed </li></ul><ul><ul><li>Timing and activities also changed </li></ul></ul><ul><ul><li>Plan is off-track and analysis becomes suspect </li></ul></ul>
    34. 34. Thank you. Any questions?

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