Imputing Mean Annual Change toEstimate Current Forest Attributes

Bianca N. I. Eskelson, Tara M. Barrett and Hoilemariam Temesgen

When a temporal treud in forest conditions is present. standard estimates from paneled forestinventories can be biased. Thus methods that use more recent remote sensing data to improveestimates are desired. Paneled inventory data from national forests in Oregon and Washington,U.S.A., were used to explore three nearest neighbor imputation methods to estimate meanannual change of four forest attributes (basal area/ha, stems/he, voleme/ha, biomass/ha). Therandom Forest imputation method outperformed the other imputation approaches in terms ofroet mean square error, TI1C imputed mean annual change was used to project all panels to acommon point in rime by multiplying the mean annual change with the length of the growthperiod between measurements and adding the change estimate to the previously observedmeasurements of the four forest attributes. The resulting estimates of the mean of the forestattributes at the current point in time outperformed the estimates obtained from the nationalstandard estimator,

Keywords forest inventory and analysis, forest monitoring. national forest inventories, nearestneighbor imputarion, Pacific Northwest, paneled inventory data

1 IntroductionKeeping national inventories of forests updatedto reflect current conditions poses substantiallogistical and accuracy issues (Gillis and Leckie1996). In paneled inventory systems the plot net-work is systematically divided into panels withrotating panels being measured on an annual basis(Reams et al 2005); in a 10 year panel inventory,for example, one-tenth of all plots are measuredeach year and plots of each panel are remeasuredevery 10 years. In theory, a paneled inventorysystem can provide current information whenonly the most recent panel is used (Reams andVan Deusen 1999), but in practice, most forestryapplications are at a spatial scale that require com-bining field plots from multiple years to achievesufficient information (McRoberts 2001, Tomppoet al. 2008). However, combining field plots frommultiple years to estimate current conditions cancause a lag bias when forest conditions are chang-ing over time.

In the United States (US), the national inventoryof forests is collected with a panel system by theForest Inventory and Analysis (FIA) program ofthe US Forest Service. The FIA default estimatoris a moving average (MA) approach (Bechtoldand Patterson 2005) which is known to resultin biased estimates when trend is present (VanDeusen 2002). In the western US, the problemof lag bias is exacerbated by a relatively longremeasurement interval (10 years). shifts in forestmanagement in response to altered economic andsocial conditions, changing climate, and a highhut variable disturbance rate from wildfire. dis-ease, and insects. Thus there is interest in usingremote sensing information to reduce lag bias inestimates of current forest conditions. Combiningremote sensing and other ancillary data with fieldplots has become common for improving forestinventory information (Tomppo et al. 2008).

Combining field and remote sensing data, near-est neighbor (NN) imputation is often used formapping (Finley and McRoberts 2008, Katilaand Tomppo 2002, Ohmann and Gregory 2002),and these methods typically impute point-in-timeplot attributes. In contrast, when using imputationto update paneled inventory data, it is possible toimpute mean annual change (MAC) for a plotFor example, Arner et al (2004) estimated mean

annual net volume change using MA approachesand sampling with partial replacement approachesand McRoberts (2001) imputed the difference inbasal area between two measurements to plotswith missing measurements to update basal areafor plots measured in previous years to the cur-rent point in time. There are few studies in thewestern US that examine alternatives to the MA(e.g., Eskelson et al 2009) and none that we areaware of that impute MAC rather than point-in-time measurements.

The objectives of this study are to: 1) use pan-eled data from the Pacific Northwest (PNW) toestimate mean annual change (MAC) of forestattributes using three nearest neighbor imputa-tion methods; and 2) to estimate current forestattributes from paneled inventory data by updat-ing the most recent measurement with imputedMAC. The results are compared with the esti-mates obtained from the MA estimator and thedata from the current panel.

2 Methods

2.1 Data

In this study, 618 primary sampling units (PSU)from six national forests that were collected aspart of the Pacific Northwest Region's CurrentVegetation Survey (CVS) of the US Forest Serv-ice were used. The particular national forests,sampled between 1993 and 1997 (occasion 1measurement) and remeasured in 2000, werethe Colville (28), Mt. Hood (111), Ochoco (82),Rogue River (70), Wallowa-Whitman (199), andWinema (128) (Fig. 1).

Panel data is a special case of inventory datawith measurements taken at different times. Apanel system was imitated with the available databy randomly assigning 25% of the PSUs to panel4 (P4; 154 PSUs) and the remaining 75% of thePSUs (464 PSUs) to panel 1 (PI), panel 2 (P2),and panel 3 (P3) based on their year of occasion1 measurement (Table 1).

In the CVS inventory, circular PSUs (1 hain size) are established on a regular grid withsquare spacing (5.47 km). Each circular PSU issubsampled by a satellite system of five second-

ary sampling units (SSU). Each SSU consists ofthree fixed-area circular, nested plots with radiiof 3.6 m, 7.3 m, and 15.6 m in which trees withdiameter at breast height (DBH in cm) smaller

than 12.7 cm, with 12.7 cm<DBH>33 cm, andDBR greater or equal to 33 cm are measured.respectively. For a detailed description of theCVS inventory see Max et al, (1996). Live treeswith DBH of 12.7 cm or larger were used in thisstudy and height models developed in Barrett(2006) were employed to fill in missing heights(HT in m). Gross cubic-meter volume and totalgross oven dry weight biomass were calculatedwith volume and biomass equations from theUS Forest Service (USDA 2000). For each PSU,basal area in m2 per ha (SA), stems per ha (SPH),volume in m3 per ha (VOL), and biomass in tonsper ha (BIOT) were calculated and summarized(Table 2). MAC for SA, SPH, VOL, and BIOTwere calculated by dividing the difference of theobserved values in 2000 and the observed value atthe occasion 1 measurement by the growth period

2.2 Nearest Neighbor Imputation

Nearest neighbor (NN) imputation methods aredonor-based methods. Variables of interest arethose forest attributes that are only measured ona subset of PSUs (e .g., MAC of BA, SPH, VOL,and BIOT). Ancillary variables are the attributesthat are measured on all PSUs. In this study, satel-lite, climate. and topography data as well as themost recent measurements of BA., SPH, VOL, andBIOT, which.were taken at measurement occasion1 (between 1993 and 1997), constitute the avail-able ancillary data. Reference data are the PSUsfor which both variables of interest and ancillaryvariables are available (PSUs in P4). Target dataare the PSUs for which only the ancillary vari-ables are available (PSUs in P1, P2, and P3). Thereference PSUs constitute the pool of potentialPSUs which could be selected to impute the MACdata for the target PSUs.

The most similar neighbor (MSN) method(Moeur and Stage 1995), the gradient nearestneighbor (GNN) method (Ohmann and Gre-gory 2002), and the randomforest (RF) method(Crookston and Finley 2008) have been shown toprovide reasonable imputation results for forestattributes (Moeur and Stage 1995, Temesgen etal, 2003, LeMay and Temesgen 2005, Hudaket at. 2008) and for mapping forest composi-tion and structure (Ohmann and Gregory 2002,Ohmann et al 1007). MSN, GNN, and RF wereconducted using the yalmpute R package ver-sion 1.0-9 (Crookston and Finley 2008). The

length (GPL) between the two measurements.Thematic Mapper (TM) images from 2000 from

the national land cover database 2001 (USDI /USDA 2009) were used as ancillary data. Thisdata set includes normalized imagery bands 1to 5 and band 7 (TM1, TM2, TM3, TM4, TM5,TM7), three commonly used band ratios (band 4to band 3, band 5 to band 4, and band 5 to band7), the normalized difference vegetation index(NDVI), the Tasseled Cap (TC) transformationsof the 6 axes (Kauth and Thomas 1976), and amodel of percent canopy cover. Normalizationmethods and development of derived layers aredescribed in Homer et al. (2004). Values foreach plot were calculated by overlaying the TMattributes, which had a 30 m resolution, with thecenter points of the five SSUs and calculating amean of the five values.

In addition to the satellite data, the follow-ing climate and topography variables were usedas ancillary data (Table 2): Natural logarithmof annual precipitation (mm) and mean annualtemperature (oC) [Data source; DAYMET DailySurface Weather Data and Climatological Sum-maries (Thornton et al 1997, Thornton andRunning 1999)], Elevation (EL in m) and its trans-formations EL2 and In(EL) [Data source: CVSinventory], slope (%), and the transformationscosine (aspect) * slope and sine(aspect) * slope[Data source: 30 m digital elevation model usingArc Workstation GRID surface functions andcommands (Environmental Systems ResearchInstitute 1991)].

similarity between reference and target PSUs isdefined using a weighted Euclidean distance forMSN and GNN:

where W is the weight matrix, Xi is a vector ofstandardized values of the ancillary variables forthe ith target PSU; and Xj is a vector of standard-ized values of ancillary variables for the ith refer-ence PSU. The ancillary variables for both targetand reference PSUs were standardized using themean and variance of the ancillary variables of thereference PSUs resulting in standardized variableswith. zero mean and unit variance, For MSN, theweight used isW=T^2T, where T is the matrixof standardized canonical coefficients for theancillary variables and ^2 is the diagonal matrixof squared canonical correlations between ancil-lary attributes and variables of interest (Moeurand Stage 1995).

For the gradient nearest neighbor method(GNN) the weights are assigned by employing aprojected ordination of the ancillary data basedon canonical correspondence analysis CCCA)(Ohmann and Gregory 2002).

The RF method is an extension of classificationand regression tree (CART) methods (Breiman2001). The data and variables are randomly anditeratively sampled to generate a forest of clas-sification and regression trees. If two PSUs tendto end up in the same terminal nodes in a forestof classification and regression trees, they areconsidered to be similar. The RF distance measureis one minus the proportion of trees where a targetPSU is in the same terminal node as a referencePSU (Crookston and Finley 2008, Hudak: et al.2008).

2.3 Estimation Procedures

BA, SPH, VOL, and BIOT were the variables ofinterest (Y). The observed mean value of Yin 2000(YOBS), based on the observed Y values from all618 PSUs, was used as the best available estimateof the true mean of Y:

where nl is the number of PSUs in PI), with t-3,t-2, and t-1 being the years 1993/1994, 1995, and1996/1997, respectively. In the following, thisMA(4) estimator will be referred to as MA.

Instead of filling in the missing values for P1 ,P2, and P3 with their previous measurements,as was done in the MAcalculation, MSN, GNN,and RF were explored to impute the MAC of thevariables of interest for the PSUs in P1, P2, andP3. The R package yalmpure (Crookston andFinley 2008) was used to impute MAC. The PSUsin P4 were the reference data in the imputationprocess. The imputed MAC was then used toupdate the variables of interest for each PSU inP1, P2, and P3 to the year 2000 by multiplyingit with the GPL and adding it to the occasion 1measurement as follows:

where Yi is the observed Y value of the ith PSU,and n4 is the number of PSUs in P4.

The M .A estimator, the FIA default method,was also used to estimate the mean value of Yfor the year 2000:

where Y is the observed Y value of the ith PSUin 2000 and n=618.

The 618 PSUs were randomly split into P4 (154PSUs) and P1, P2, and P3 (remaining 464 PSUs)500 times. For each of the 500 realizations ofrandomly splitting the data, the following estima-tors were used to estimate the mean value of thevariables of interest for the year 2000.

Using the data from P4 only, the mean valueof Y for the year 2000 (SAMPLE25 estimator)was estimated as:

ith PSU at time t-3 (1993/1994), t-2 (l995), andt-1 (1996/1997) for P1, P2, and P3, respectively.MACimp,i is the imputed MAC for the ith PSUwith imp refeniag to me NN imputation methodused. GPLi is the growth length period between

the occasion 1 measurement and me remeasure-men! in 2000.

The mean value of Y lO! me year 2000 was thenestimated as follows:

where IMP refers to the NN imputation methodused and Yimp.i is the imputed Y value for the ithPSU as described in Eq. 5.

Two sets of ancillary variables were tested forthe NN imputation methods: The first set includedthe available climate, topography, and satellitedata (Data set A) and the second set consistedof the previous measurements of the variablesof Interest that were taken at measurement occa-sion 1 from 1993 to 1997 (Data set B: BAocc1,SPHoccl, VOLoccl, BIOTocc1).

SAMPLE25, MA, and the three imputationmethods were compared based on the estimatedoverall means of the variables of interest in 2000(see Eqs. 3, 4, and 6). The basis of evaluationwas accuracy, as expressed by the root meansquare error (RMSE), and bias calculated as themean difference between the estimated and theobserved (Eq, 2) mean values. RMSE and biaswere approximated using a random sample ofm = 500 realizations of splitting the data. BothRMSE and bias were expressed as percent of theobserved mean for each variable of interest:

mated bias (in original units) of me SAMPLE25,MA, and the NN imputation methods was sig-nificantly different from zero, and the p-valueswere reponed.

The MAC estimates based on MSN, GNN, andRF imputation were compared to the observedMAC also using the approximated RMSE andbias from m=500 iterations of randomly splittingthe data. Two-sided t-tests were performed toexamine whether the estimated bias (in originalunits) of the three M-l imputation methods usingdata sets A and B for imputing MAC was sig-nificantly different from zero, and the p-valueswere reported.

3 Results

MSN imputation provided similar results forMAC estimates for both sets of ancillary vari-ables with the variance contributing most to theRMSE. When BAocc1, SPHoccl, VOLocc1 andBIOTocc1 were used as ancillary variables, thep-values of the two-sided t-test testing the unbi-asedness of the estimator ranged from 0.7041 to0.9453 and the p-values for the estimates basedon climate, topography, and satellite data rangedfrom 0.1556 to 0.4581 indicating that MSN impu-tation provided unbiased MAC estimates withboth ancillary data sets (Table 3). GNN estimatesof MAC were significantly biased (p<0.000l)and the large bias (>66%) contributed most tothe RMSE. Using climate, topogrnphy, and satel-lite data as ancillary variables resulted in smallerbias and hence, smaller RMSE values than usingBAoccl, SPHocc1 VOLoccl, and BIOTocc1 asancillary variables (fable 3). In (terms of RMSE,RF using climate, topography, and satellite data asancillary variables provided tile best estimates of

where Yj is the estimate of Y for the jth iterationof data splitting using either SAMPLE25, MA,or the NN imputation methods. Two-sided t-testswere performed to examine whether the esti-

MAC. However, these were biased for BA, VOL,BIOT (p<0.0007) and suggestive, but inconclu-sively unbiased for SPH (p=0.0770). RF usingBAoccl, SPHocc1, VOLocc1, and BIOToccl asancillary data provided unbiased MAC estimates(p>0.l) for all four variables of interest andsmaller RMSE values than those achieved byeither MSN or GNN imputation (Table 3).

For estimating current BA and SPH the RMSEvalues of MA were about half the size of thoseobserved for SAMPLE25. For estimating currentVOL and BIOT the RMSE values for MA wereabout a third of those observed for SAMPLE25.SAMPLE25 results were unbiased (p>0.5). MAresults were precise but biased (p<0.000l) andthe bias contributed most to the RMSE. Theopposite was true for the unbiased SAMPLE25estimates, where the variance contributed most tothe RMSE (Table 4).

MSN resulted in biased estimates of currentBA, SPH, VOL, and BIOT (p<0.0047) whenBAoccl, SPHoccl, VOLoccL and BIOTocclwere used as ancillary variables. When climate,topography, and satellite data were used as ancil-lary variables, estimates were biased for BA(p=0.0021), unbiased fer SPH and VOL (p>0.1),and suggestive, hut inconclusively unbiased forBIOT (p=0.460). For both sets of ancillary van-ables, the MSN estimates were similar in terms

of RMSE and outperformed the MA estimates interms of both bias and RMSE (Table 4).

For the GNN estimates of current forestattributes, bias contributed most to RMSE. WhenBAocc1, SPRoccl, VOLocc1, and BIOTocclwere used as ancillary variables, the bias andhence the RMSE was larger than for the ancillaryvariable set including climate, topography, andsatellite data. For both sets of ancillary variables,bias and RMSE were larger than for any of theother estimators (Thhle 4).

RF estimates of the current variables of interestwere biased (p<0.0016) when BAoccl, SPHocc1,VOLocc1, and BIOToocl were used as ancillaryvariables. When climate, topography, and satel-lite data were used as ancillary variables, RFprovided suggestive, but inconclusively unbiased(p=0.0421) and unbiased (p =0.(975) estimatesof current BA and SPH, respectively. and biasedestimates of VOL and BIOT (p <0.00(1). RMSEvalues were smallest when climate, topography,and. satemte data '.'Wereused as ancillary variables.For both sets of ancillary variables, RF imputationoutperformed the MA estimates both in terms ofbias and RMSE (Table 4).

RF using climate, topography, and satellite dataas ancillary variables provided the best resultsoverall in terms of bias and RMSE for estimatingcurrent BA, SPH, VOL,. and BIOT, followed by

4 DiscussionAfter the start of the second inventory cycle,MAC can be estimated using remeasured plots)(Arner et al. 2004). Considering the year 2000 asthe start of the second inventory cycle, MAC wasestimated in this study. The results of the three NNimputation methods that were explored to imputeMAC of forest attributes showed the same patternthat was observed in all earlier study where thesame three NN imputation methods were used toimpute BA, SPH, VOL, and BIOT (see Eskelsonet al, 2009). RF imputation provided the bestresults in terms of RMSE followed by MSN.The results of this study suggest that GNN PSU-level imputation is not adequate to impute MACof forest attributes. This might be due to the factthat the CCA in the GNN procedure requires theuse of environmental factors for the ordination(Ohmann and Gregory 2002) which might not bepicked up in the ancillary data that was used forthe imputation in this study.

MSN and RF using BAoccl, SPHocc1,VOLocc1,and BIOTocc1 as ancillary variables in terms ofRMSE (Table 4).

In a previous study, RF imputation usingBAocc1, SPHocc1, VOLocc1, and BIOToccl asancillary data for directly imputing current EA,SPH, VOL, and BIOT provided the most accu-rate estimates compared to the SA MPLE25 andMA estimators and MSN and GNN imputation.The results of this PSU-level RF imputation areprovided in Table 4. See Eskelson et al, (2009)for details. The RF imputation in this study out-performs the PSU-level RF imputation from theprevious study in terms of RMSE for EA, SPH,VOL, and BIOT and in terms of bias for BA andSPH fer both sets of ancillary data. For both setsof ancillary data MSN imputation in this studyoutperforms the PSU-level RF imputation fromthe previous study in terms of both RMSE andbias for SPH (Table 4.).

data. We appreciate the help that Jim Alegria,Carol Apple, Bob Brown, and Melinda Moeurprovided in obtaining national forest data, andthank Kurt Campbell for assistance with volumeand biomass equations, We also thank Greg Lattafor creating the map of the national forests andtwo anonymous reviewers for their helpful reviewcomments,

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The performaace of using MAC of forestattributes of one inaventory cycle to predict MACfor the next inventory cycle could not be testedIn this study since only one remeasurement of thePSUs was available for the CVS data. This is anarea of research that should be pursued as soonas multiple remeasurements of the FA annualinveatory are available in the western US.

When BA, SPH, VOL and BIOT were updated tothe year 2000 using MAC estimates obtained fromMSN and RF imputation, the estimates of the meanBA, SPH, VOL, and BIOT in 2000 outperformedthose of the SAMPLE25 and MA estimators in termsof accuracy. These results indicate that updatingthe variables of interest for unmeasured PSUs tothe current point in time using estimated MACfrom MSN or RF imputation should be preferredover using the SAMPLE25 or MA estimators forestimating the current forest a.ttributes.

The estimates of mean BA, SPH, VOL, andBIOT in 2000 using MAC estimates obtainedfrom RF imputaticn also outperformed the esti-mates from PSU-level RF imputation that wasused to directly impute BA, SPH, VOL, andBIOT in 2000 (see Eskelson et al 2009). This isdue to the fact that the approach employing theimputed MACestimates makes use of the previ-ously observed measurements directly, whereasthe PSU-level RF imputation only makes use ofthe previous measurements indirectly by usingthem as ancillary data. Adding a multiple ofestimated MAC to the previously observed meas-urement will result in current estimates that willbe dose to the actual values even if the estimatedMAC values were not perfect, If current BA, SPH,VOL, and BIOT are imputed directly as was donein Eskelson et at (2009), the imputed values callbe either close to the actual values or they canbe completely different, The results of this studysuggest that the approach using imputed MACvalues should be preferred over directly imputingcurreat BA, SPH, VOL, and BIOT

Acknowledgments

We thank Janet Ohmann and Matt Gregory forsharing their ancillary data from the NLCD andDAYMET with us and for their insights on the

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