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A comparison of multiple linear regression and quantile regression for modeling the internal bond of medium density fiberboard.



Abstract

Multiple linear regression Linear regression

A statistical technique for fitting a straight line to a set of data points.
 (MLR MLR

mixed lymphocyte reaction.

MLR Myocardial laser revascularization, see there
) and quantile quantile

division of a total into equal subgroups; includes terciles, quartiles, quintiles, deciles, percentiles.
 regression (QR) models were developed for the intemal bond (IB) of medium density fiberboard (MDF (1) (Main Distribution Frame) A wiring rack that connects outside lines with internal lines. It is used to connect public or private lines coming into the building to internal networks. ). The data set that aligned the 1t3 of MDF came from 184 independent variables that corresponded to online sensors. MLR models were developed for MDF product types that were distinguished by thickness in inches, i.e., 0.750-inch, 0.6875-inch, 0.625-inch, and 0.500-inch. A best model criterion was used with all possible subsets. QR models were developed for each product type for the most common independent variable of the MLR models for comparison. The adjusted coefficient of determination Coefficient of determination

A measure of the goodness of fit of the relationship between the dependent and independent variables in a regression analysis; for instance, the percentage of variation in the return of an asset explained by the market portfolio return. Also known as R-square.
 ([R.sup.2.sub.a]) of the MLR models ranged from 72 percent with 53 degrees of freedom to 81 percent with 42 degrees of freedom. The Root Mean Square Errors (RMSE) ranged from 6.05 pounds per square inch Noun 1. pounds per square inch - a unit of pressure
psi

pressure unit - a unit measuring force per unit area
 (psi) to 6.23 psi; the maximum Variance Inflation Factor The Variance Inflation Factor (VIF) is a method of detecting the severity of Multicollinearity. More precisely, the VIF is an index which measures how much the variance of a coefficient(square of the standard error) is increased because of collinearity.  (VIF VIF - VHDL Interface Format. Intermediate language used by the Vantage VHDL compiler. "A VHDL Compiler Based on Attribute Grammar Methodology", R. Farrow et al, SIGPLAN NOtices 24(7):120-130 (Jul 1989). ) was 5.6, and all residual patterns were homogeneous. A common independent variable for the 0.750-inch and 0.625-inch MLR models was "Refiner Resin Scavenger %." QR models for 0.750 inch and 0.625 inch indicate similar slopes for the median and average, with different slopes at the fifth and 95th percentiles. "Face Humidity" was a common independent variable for the 0.6875-inch and 0.500-inch MLR models. QR models for 0.6875-inch and 0.500-inch indicate different slopes for the median and average, and instability in IB in the outer fifth and 95th percentiles. The use of QR models to investigate the percentiles of the IB of MDF suggests significant opportunities for manufacturers for continuous improvement and cost savings.

The wood composites industry is undergoing unprecedented change in the forms of corporate divestitures and consolidation, real increases in the cost of raw material and energy, and extraordinary international competition. The forest products industry is an important contributor to the U.S. economy. In 2002, this sector contributed more than $240 billion to the economy and employed more than one million Americans in 22,231 primary wood products manufacturing facilities (U.S. Census Bureau Noun 1. Census Bureau - the bureau of the Commerce Department responsible for taking the census; provides demographic information and analyses about the population of the United States
Bureau of the Census
 2004). Sustaining business competitiveness by reducing costs and maintaining product quality will be essential for this industry. One of the challenges facing this industry is to develop a more advanced knowledge of the complex nature of process variables and quantify the causality causality, in philosophy, the relationship between cause and effect. A distinction is often made between a cause that produces something new (e.g., a moth from a caterpillar) and one that produces a change in an existing substance (e.g.  between process variables and final product quality characteristics in the percentiles of the distribution. Information contained in the percentiles is a key measure for quality and safety concerns. This paper provides quantile regression (QR) statistical methods that can improve business competitiveness in the wood composites industry (Young and Guess 1994, 2002).

Some work has been initiated in data mining and predictive modeling of final product quality characteristics of forest products (Young 1997, Bernardy and Scherff 1997, 1998; Greubel 1999; Erilsson et al. 2000, Young and Guess 2002, Young and Huber 2004, Clapp et al. 2007). Much work has been published on simulating process variables and using theoretical models to predict final product quality characteristics (Barnes 2001, Humphrey and Thoemen 2000, Shupe et al. 2001, Wu and Piao 1999, Xu 2000, Zombori et al. 2001). The use of quantile regression to investigate the percentiles of product quality for wood composites has not been documented in the literature.

A data set from a large-capacity North American North American

named after North America.


North American blastomycosis
see North American blastomycosis.

North American cattle tick
see boophilusannulatus.
 manufacturer of medium density fiberboard (MDF) (1) was obtained in 2002. The data set aligned process measurements from online sensors with the Internal Bond (IB) analyzed during periodic destructive testing In destructive testing, tests are carried out to the specimen’s failure. These tests are generally much easier to carry out, yield more information, and are easier to interpret than nondestructive testing. . For example, online sensor measurements were available for measuring press temperature, press closing time, resin content, moisture, weight, etc. The goal of any wood products manufacturer is to efficiently produce a high quality end product. To this end, it is imperative that the manufacturer has an advanced knowledge of the process and causality.

A common goal for statistical research is to investigate and quantify causality between independent variables (X) and response variables (Y) with a high level of scientific inference. In quantifying causality, de Mast and Trip (2007) note the important distinction between exploratory and confirmatory data analysis, which they attribute to Tukey's (1977) work. As Tukey (1977) pointed out, confirmatory data analysis is concemed with testing a prespecified hypothesis. The purpose of exploratory data analysis Exploratory Data Analysis - (EDA)

[J.W.Tukey, "Exploratory Data Analysis", 1977, Addisson Wesley].
 is hypothesis generation (de Mast and Trip 2007). This research was undertaken in the spirit of exploratory data analysis and hypothesis generation. A significant problem for MDF manufacturers is the quantification of known and unknown sources of variation. The objective of this research was to explore causality of sources of MDF process variation beyond the mean of the distribution of internal bond. The paper directly compares the use of Multiple Linear Regression (MLR) and Quantile Regression (QR) for modeling the IB of MDF. MLR and QR were used on the same MDF data set to model process variables and the process variables level of influence on IB. MLR develops models based on the mean of the response variable (e.g., IB), while QR develops models for any percentile of the response variable. Modeling beyond the mean of IB may greatly improve a MDF manufacturers understanding of the process. An improved understanding of process variables and the process variables' level of influence on IB can help MDF manufacturers identify and quantify unknown sources of process variation. Identifying and quantifying process variation can facilitate continuous improvement and improve competitiveness (Deming 1986, 1993).

As Mosteller and Tukey (1977) note in their influential text, as recently cited by Koenker (2005): " ... the regression curve Noun 1. regression curve - a smooth curve fitted to the set of paired data in regression analysis; for linear regression the curve is a straight line
regression line
 gives a grand summary for the averages of the distributions corresponding to the set of Xs ... and so regression often gives a rather incomplete picture. Just as the mean gives an incomplete picture of a single distribution, so the regression curve gives a correspondingly incomplete picture for a set of distributions."

Methods

MLR was used to study the relationship between various independent variables and the mean or average of the distribution for a response variable with an important goal of making useful predictions of the response variable. There is a plethora of literature on regression analysis In statistics, a mathematical method of modeling the relationships among three or more variables. It is used to predict the value of one variable given the values of the others. For example, a model might estimate sales based on age and gender.  and MLR, and many tomes are available on the method (Box and Cox 1964, Draper and Smith 1981, Neter et al. 1996, and Myers 1990, Kutner et al. 2004, etc.). For situations where the data are drawn from reasonably homogeneous populations and the response (Y) is normally distributed, traditional methods such as MLR can yield insightful analyses. The usefulness of MLR can breakdown quickly if these stringent assumptions are not met. MLR has three important assumptions: 1) linearity of the coefficients; 2) normal or Gaussian distribution for the response errors (e); and 3) the errors e have a common distribution. In many industrial settings when modeling a quality characteristic such as IB, these assumptions may not be valid.

QR is an approach that allows us to examine the behavior of the response variable (Y) beyond its average of the Gaussian distribution, e.g., median (50th percentile), 10th percentile, 80th percentile, 90th percentile, etc. Examining the behavior of the regression curve for the response variable (Y) for different quantiles with respect to the independent variables (X) may result in very different conclusions relative to examining only the average of Y. In regard to the IB of MDF, examining the lower percentiles using QR may be more important for understanding IB failures (or very strong IBs) and be more beneficial for continuous improvement and cost savings.

Relational database relational database

Database in which all data are represented in tabular form. The description of a particular entity is provided by the set of its attribute values, stored as one row or record of the table, called a tuple.
 

An automated relational database was created by aligning real-time process sensor data with IB readings (Young and Guess 2002). The real-time process data were collected with Wonderware IndustrialSQL[TM] 8.0 (www.wonderware.com). The readings were combined with IB by product type at the instant when a panel was extracted from the production line for testing. The process data were collected using a median value Noun 1. median value - the value below which 50% of the cases fall
median

statistics - a branch of applied mathematics concerned with the collection and interpretation of quantitative data and the use of probability theory to estimate population
 from the last 100 sensor values (e.g., for most of the 184 different sensor variables this represented a 2- to 3-minute time interval). The process data were collected and stored using IndustrialSQL. The lag times corresponding to the time required for the product to travel through the process from the point where a given parameter has an influence to the point where the panel was extracted for IB destructive testing were taken into account. A unique number (idnum) was generated when the panel was extracted from the process, and was later used to match process data with corresponding IB results.

When the IB results were matched with the process data, the combined data were recorded in two tables that appeared in a combined SQL SQL
 in full Structured Query Language.

Computer programming language used for retrieving records or parts of records in databases and performing various calculations before displaying the results.
 database, i.e., a relational database of realtime sensor data and destructive test lab data. The real-time relational database was automatically updated as new lab samples were taken using Microsoft Transact SQL code with Microsoft SQL "Jobs" and "Stored Procedures."

The names used in this manuscript associated with the process variables for the online sensors were nondescriptive at the request of the manufacturer and given the terms of a legal confidentiality agreement. Definitions for the names of the process variables were not allowed under the terms of the legal confidentiality agreement.

Classical linear regression

The classical first-order simple linear regression Simple linear regression

A regression analysis between only two variables, one dependent and the other explanatory.
 model has the form (Draper and Smith 1981),

[Y.sub.i] = [[beta].sub.o] + [[beta].sub.1] [x.sub.1i] + [[epsilon].sub.i], [1]

where [Y.sub.i] is the value of the response variable in the ith observation,

[[beta].sub.o] is the intercept parameter,

[[beta].sub.1] is a slope parameter,

[X.sub.1i] is the value of the independent variable in the ith observation,

[[epsilon].sub.i] is a random error term of the ith observation with mean

E([[epsilon].sub.i]) = 0 and variance [[sigma].sup.2]{[[epsilon].sub.i]} = [[sigma].sup.2] with the error terms being independent and identically distributed, i = 1, ... ,n. Most practitioners use multiple linear regression (MLR) first-order models of the form:

[Y.sub.i] = [[beta].sub.o] + [[beta].sub.1] [x.sub.1i] + [[beta].sub.2] [x.sub.2i], + [[beta].sub.3] [x.sub.3i] + [[beta].sub.k] [x.sub.ki] + [[epsilon].sub.i] [2]

where [Y.sub.i] is the value of the response variable in the [i.sup.th] observation,

[[beta].sub.o] is the intercept parameter,

[[beta].sub.k] is the slope parameter associated with the [k.sup.th] variable,

[X.sub.ki] is the [k.sup.th] independent variable associated with the ith observation,

[[epsilon].sub.i] is a random error term with mean E([[epsilon].sub.i]) = 0 and variance [[sigma].sup.2]{[[epsilon].sub.i]} = [[sigma].sup.2], with the error terms being independent and identically distributed,

i = 1, ... ,n.

The least squares method least squares method

Statistical method for finding a line or curve—the line of best fit—that best represents a correspondence between two measured quantities (e.g., height and weight of a group of college students).
 is a common method in simple regression Noun 1. simple regression - the relation between selected values of x and observed values of y (from which the most probable value of y can be predicted for any value of x)
regression toward the mean, statistical regression, regression
 and MLR and is used to find an affine af·fine  
adj. Mathematics
1. Of or relating to a transformation of coordinates that is equivalent to a linear transformation followed by a translation.

2. Of or relating to the geometry of affine transformations.
 function that best fits a given set of data. (2) Recall that a strength of the least squares method is that it minimizes the sum of the n squared errors (SSE (1) An earlier full-screen editor in OS/2.

(2) (Streaming SIMD Extensions) A series of additional instructions built into Pentium CPU chips for improved multimedia performance by performing mathematical operations on multiple sets of data at the
) of the predicted values on the fitted line ([[??].sub.i]) and the observed value (y). (3)

[n.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  (i=1)] [([y.sub.i] - [[??].sub.i].sup.2] [3]

Model building and best model criteria

Model building.--Model building using MLR is quite popular due to the refinement of user-friendly, inexpensive statistical software and real-time data warehousing See data warehouse.

data warehousing - data warehouse
. Many in the forest products industry use MLR as a basic method for data mining. A popular model building method for MLR is "stepwise regression In statistics, stepwise regression includes regression models in which the choice of predictive variables is carried out by an automatic procedure.[1][2][3] ." In this paper stepwise regression was used to develop first-order linear models of the IB for MDF.

Stepwise regression was introduced by Efroymson (1960). This method is an automated procedure used to select the most statistically significant variables from a large pool of explanatory variables. The method does not take into account industrial knowledge about the process, and therefore other variables of interest may be later added to the model if necessary. Three approaches can be used in stepwise regression: 1) backward elimination; 2) forward selection; and 3) mixed selection. The backward elimination method begins with the largest regression, using all variables, and subsequently reduces the number of variables in the equation until an acceptable model is developed (Draper and Smith 1981). The forward selection procedure attempts to achieve a similar conclusion working from the other direction, i.e., starting with one variable and inserting variables in turn until the regression is satisfactory (Draper and Smith 1981). The order of insertion is determined by using the partial correlation coefficient Correlation Coefficient

A measure that determines the degree to which two variable's movements are associated.

The correlation coefficient is calculated as:
 as a measure of the importance of variables not yet in the equation (Neter et al. 1996). The basic procedure is to select the most correlated independent variable (X) with Y and find the firstorder linear regression equation. This continues by finding the next most correlated independent variable (X) with Y, and so forth. The overall regression is checked for significance; improvements in the [R.sup.2] value and the partial F-values for all independent variables in the model were noted. The F-values are used for the F-test, which is used to calculate the one sided probability of the likelihood that two variances are different. The partial F-values are compared with an appropriate F percentage point, and the corresponding independent variables are retained or rejected from the model according to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 whether the test is significant or not significant. This continues until a suitable first-order linear regression equation is developed; see Kutner et al. (2004), Neter et al. (1996), and Myers (1990).

In stepwise regression it is important to note that the user specifies the probabilities ([alpha]) for an independent variable (X) "to stay" and also the probabilities "to leave" the model. The mixed selection procedure is a combination of the aforementioned procedures. In this paper, the mixed stepwise regression procedure was used with a best model criteria.

Best model criteria.--There is much literature written on "Best Model Criteria" in model building using MLR. We use SAS (1) (SAS Institute Inc., Cary, NC, www.sas.com) A software company that specializes in data warehousing and decision support software based on the SAS System. Founded in 1976, SAS is one of the world's largest privately held software companies. See SAS System. [R] Business Intelligence and Analytics Software (www.sas.com). For our model we used the following seven criteria in selecting the best model of IB. The criteria include: 1) maximum adjusted [R.sup.2.sub.a]; 2) minimum Akaike's Information Criterion There are a number of statistics that can act as an information criterion. They include:
  • Akaike's information criterion
  • the Bayesian information criterion, also known as the Schwarz information criterion
  • Hannan-Quinn information criterion
 (AIC); 3) Variance Inflation Factor (VIF) < 10; 4) significance of p-value < 0.05 for selected independent variables; 5) residual pattern analysis; 6) absence of heteroscedasticity (i.e., equal variance of residuals); and 7) no bias in the residuals, i.e., E([[epsilon].sub.i]) = 0.

Adjusted [R.sup.2], or [R.sup.2], is a better measure for building models with the potential of a large number of independent variables than the Coefficient of Determination ([R.sup.2]). [R.sup.2] will always increase as an additional independent variable is added to the model, where [R.sup.2.sub.a] will only increase if the residual sum of squares In statistics, the residual sum of squares (RSS) is the sum of squares of residuals,



In a standard regression model , where a and b
 decreases. [R.sup.2.sub.a] minimizes the risk of "over-fitting" and penalizes for model saturation, i.e., the model is penalized pe·nal·ize  
tr.v. pe·nal·ized, pe·nal·iz·ing, pe·nal·iz·es
1. To subject to a penalty, especially for infringement of a law or official regulation. See Synonyms at punish.

2.
 if additional independent variables do not reduce the residual sum of squares. The formula for [R.sup.2.sub.a] is:

[R.sup.2.sub.a] = 1 - (1 - [R.sup.2] (n-1/n-p-1), 0 [less than or equal to] [R.sup.2.sub.a] [less than or equal to] 1 [4]

where, [R.sup.2.sub.a] = 1 - [n.summation (i=1)] [([Y.sub.i] - [[??].sub.i]).sup.2]/[n.summation (i=1)] [([Y.sub.i] - [[??].sub.i]).sup.2] = 1 - SSE/SSTO, 0 [less than or equal to][R.sup.2][less than or equal to] 1

[5]

The important AIC statistic is calculated as follows:

AIC = n 1n (SSE/n) = 2p [6]

where n is the number of observations, and p is the number of independent variables.

The goal is to balance model accuracy and complexity. This is achieved by finding the minimum value of AIC (Akaike 1974).

VIF is a diagnostic tool used to check the impact of multicollinearity in the MLR model. The VIF is calculated for each independent variable and is computed as follows:

([VIF.sub.k]) = [(1 - [R.sup.2.sub.k].sup.-1] [7]

where [R.sup.2.sub.k] is the coefficient of multiple determination for [X.sub.k] when regressed on the remaining p - 2 predictors in the model. High levels of multicollinearity (VIF > 10) can falsely inflate inflate - deflate  the least squares estimates; therefore, lower VIF values are desired (Kutner et al. 2004).

Residual pattern analysis is key criteria for assessing model quality. Residuals should not have any pattern when plotted against Y, any X, or as a function of time. Heteroscedasticity is tmequal variance of residuals and indicates that the model is ill-founded. Residual plots should not have any slope or bias with a E([[epsilon].sub.i]) = 0.

SAS code for mixed stepwise regression.--When modeling manufacturing processes, it is important to consider the most recent data first, i.e., this data will be most informative for continuous improvement (Deming 1986, 1993). SAS code was used to develop the mixed stepwise regression MLR models for the four product types using the previously described Best Model Criteria. MLR models for all possible subsets were explored using the most recent data and then moving backward in time. Initial models were developed for the 50 most recent data records and additional models were developed for each additional record moving backward in time through the data. The aforementioned best model criteria were used in selecting the best model from the subsets provided by SAS.

Quantile regression

QR is intended to offer a comprehensive strategy for completing the regression picture (Koenker 2005). It is different from the MLR approach in that it takes into account the differences in behavior a characteristic may have at different levels of the response variable by weighting the central tendency measure. Also, this method uses the median as the measure of central tendency rather than the mean. The nonparametric median statistic may offer additional insight in the analysis of data, especially when compared to the parametric mean or average statistic.

[FIGURE 1 OMITTED]

The QR model does not require the product characteristics or the response variable (IB in this study) to be normally distributed and does not have the other rigid assumptions associated with MLR. The first-order QR model has the form (Koenker 2005),

[Q.sub.yi],<[TAU tau
n.
Symbol The 19th letter of the Greek alphabet.


tau (tou),
n
]|x)= [[beta].sub.o] + [beta].sub.i][x.sub.i] + [F.sup-1.sub.u]([TAU]) [8]

where, [Q.sub.yi], is the conditional value of the response variable given [TAU] in the [i.sup.t]h trial, [[beta].sub.o] is the intercept, [[beta].sub.i] is a parameter, [TAU] denotes the quantile (e.g., [TAU] = 0.5 for the median), [x.sub.i] is the value of the independent variable in the [i.sup.th] trial, [F.sub.u] is the common distribution function (e.g., normal, Weibull, lognormal log·nor·mal  
adj. Mathematics
Of, relating to, or being a logarithmic function with a normal distribution.



log
, other, etc.) of the error given [TAU], E(F.sup.-1.sub.u]([TAU])) = 0, for i = 1, ... ,n, e.g., [F.sup.-1](0.5) is the median or the 0.5 quantile.

Just as we can define the sample mean as the solution to the problem of minimizing a sum of squared residuals, we can define the median as the solution to the problem of minimizing a sum of absolute residuals (Koenker and Hallock 2001). The symmetry of the piecewise linear Piecewise linear may refer to:
  • Piecewise linear function
  • Piecewise linear manifold
 absolute value function implies that the minimization of the sum of absolute residuals must equate the number of positive and negative residuals, thus assuring that there were the same number of observations above and below the median (Koenker and Hallock 2001). Minimizing a sum of asymmetrically weighted absolute residuals yields the quantiles (Koenker and Hallock 2001). Solving

min [n.summation (i=1)] [[rho].sub.[TAU]] (y.sub.i] - [xi] [9]

where the function [[rho].sub.[TAU]], e.g., in Eq. [9], is the tilted absolute value function appearing in Figure 1 that yields the [[TAU].sup.th] sample quantile as its solution (Koenker and Hallock 2001). To obtain an estimate of the conditional median function in quantile regression, we simply replace the scalar scalar, quantity or number possessing only sign and magnitude, e.g., the real numbers (see number), in contrast to vectors and tensors; scalars obey the rules of elementary algebra. Many physical quantities have scalar values, e.g.  [xi] in Eq. [9] by the parametric function [xi] ([x.sub.i], [beta]) and set [TAU] to 1/2. (4) To obtain estimates of the other conditional quantile functions, replace absolute values by [[rho].sub.[TAU]](.), e.g., Eq. [9], and solve

[??]([TAI]) = min [n.summation over (i=1)] [[rho].sub.[TAU] (Y.sub.i]-[xi]([x.sub.i], [beta])). (10)

For any quantile [TAU] [member of] (0,1). The quantity [??]([TAU]) is called the [[TAU].sup.th] regression quantile.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Results and discussion

The internal bonds of four different product types of MDF were analyzed. Each product type represents a different board thickness in inches (i.e., 0.750-inch, 0.625-inch, 0.6875-inch, and 0.500-inch). All possible subset MLR models were explored for the four product types using [R.sup.2.sub.a] as a key indicator for determining the best subset model (Figures 2, 3, 4, and 5). The [R.sup.2.sub.a] for all possible subsets was an indicator of a MDF manufacturer's stability in reproducing product quality from one production run to the next, i.e., product types where the [R.sup.2.sub.a] changes slowly as more records were added moving back in time may indicate less volatility in IB between production runs, and also that changes in processes occur less frequently between production runs of the product type. Once acceptable MLR models were obtained (i.e., using the best model criteria), commonalities in the independent variables were explored among the four product types.

Product types 0.750-inch and 0.625-inch

For the 0.750-inch product type a MLR model was developed with an [R.sup.2.sub.a] of 75 percent, 50 degrees of freedom and 11 parameters. The RMSE of the model was 7.69 psi and the maximum VIF for any independent variable was 5.03. Residual patterns for the MLR model were homogeneous (Table 1). Recall the RMSE estimates the SD of the residual error (Mensuration) See Error, 6 (b).

See also: Residual
, which is the square root of the MSE MSE Mouse (computer)
MSE Materials Science & Engineering
MSE Mean Squared Error
MSE Mean Square Error
MSE Master of Science in Engineering
MSE Manufacturing Systems Engineering
MSE Mechanically Stabilized Earth
 which is the SSE divided by the degrees of freedom.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

For the 0.625-inch product type a MLR model was developed with an [R.sup.2.sub.a] of 72 percent, 53 degrees of freedom and 11 parameters. The RMSE of the model was 6.05 psi and the maximum VIF for any independent variable was 5.60. Residual patterns for the MLR model were homogeneous (Table 1).

Common independent variables for the 0.750-inch and 0.625-inch MLR models were highlighted as bold in Table 1. "Refiner Resin Scavenger %" and "Core Water to Wood" were common for both 0.750-inch and 0.625-inch product types. It was surprising to see the scaled estimates for "Refiner Resin Scavenger %" differ in sign for each product type. (5) The "Refiner Resin Scavenger %" has a negative scaled estimate of approximately -9.12 psi on IB for 0.750-inch while the "Refiner Resin Scavenger %" has a positive scaled estimate of approximately 8.40 psi on IB for 0.625-inch. This may indicate that "Refiner Resin Scavenger %" was an important source of variability between the two product types that the manufacturer needs to further investigate.

[FIGURE 6 OMITTED]

"Core Water to Wood" has a large scaled estimate for both product types and has a negative influence on IB. The influence of "Core Water to Wood" as measured by the scaled estimate was -21.04 psi for 0.750-inch and -10.87 psi for 0.625-inch. This may reflect a difference in scale for this process variable as related to the refining process for different product types that have varying throughput levels at the refiner.

To examine the influence of "Refiner Resin Scavenger %" beyond the mean effect on IB, QR was explored for this common parameter for both 0.750-inch and 0.625-inch. (6) We find that the influence of "Refiner Resin Scavenger %" on the lower percentiles of IB was quite different than the midrange midrange Epidemiology The halfway point or midpoint in a set of observations; for most data, MR is calculated as the sum of the smallest observation and the largest observation, divided by 2; for age data, one is added to the numerator; a midrange is usually  and higher percentiles (Figs. 6 and 7). The dashed line represents the MLR fit, the solid dark line represents the median fit, and the gray lines correspond to the 5th, 10th, 25th, 75th, 90th, and 95th percentiles, respectively. For the 0.750-inch product type (Fig. 6), the slopes of the percentiles were quite different depending on percentile. The median and average have similar slopes. The 5th percentile (possible IB failures) and 95th percentile (extreme IB strength) behave quite differently than the inner percentiles. This may be helpful to a MDF producer in analyzing occurrences of IB failures, i.e., Why does IB decrease at a faster rate for the lower percentiles? What were the other operational settings and factors occurring during these events?

For the 0.625-inch product type (Fig. 7), the slopes of the percentiles were extremely different depending on percentile and on scale of the level of"Refiner Resin Scavenger %." The median and average have similar slopes. However, for percentiles above the 50th percentile (median) the effect of "Refiner Resin Scavenger %" has a stronger positive influence on IB the higher the percentile. For percentiles below the 50th percentile (median) the effect of "Refiner Resin Scavenger %" has a stronger negative influence on IB the higher the percentile. This may indicate that other factors were influencing IB in concert with "Refiner Resin Scavenger %" or that the quality of the "Refiner Resin Scavenger %" itself was varying. The QR analysis for the common parameter "Refiner Resin Scavenger %" indicates opportunities for additional root cause investigation by the manufacturer in sources of variability in "Refiner Resin Scavenger %" that influence IB.

Although only one independent variable was used for illustration purposes, the quantile regression algorithm in R can also be applied to multiple independent variable models. Further analysis was conducted to examine the differences between the MLR and QR median fits. For the 0.750-inch product type (Table 2), the largest discrepancies between coefficients occur in "Dryer 1 Fan Current," "Dryer 2 Fan Current," and "Core Water to Wood." The percent differences were 39.84 percent, 34.37 percent, and 28.2 percent, respectively.

[FIGURE 7 OMITTED]

For the 0.625-inch product type (Table 3), the largest discrepancies between coefficients occur in "Shavings shavings

curly wafers of wood produced when trimming wood with a plane; used as bedding for horses. See also sawdust.
 Raw Weight," "Relative Ambient Humidity," and "Weight Actual." The percent discrepancies were 42.96 percent, 16.16 percent, and 12.78 percent, respectively. These discrepancies reflect significant differences between modeling the mean and the median (50th percentile) of IB. These differences may illustrate the risk of incorrect decision-making about process variables that influence the mean of IB when the distribution is not Gaussian. Incorrect decisions lead to higher operating targets, unexpected IB failures and ultimately higher overall productions costs. Further analysis could be conducted for other IB quantiles that may be invaluable to the producer for understanding low or failing IBs. A comparison of the 10th and 90th percentiles of the coefficients (Tables 2 and 3) may also give a good method for the practitioner on the relative comparisons of the influence of a process variable on lB. The discrepancies in coefficients highlight the importance of examining the percentiles of a distribution.

Product types 0.6875-inch and 0.500-inch

For 0.6875-inch a MLR model was developed with an [R.sup.2.sub.a] of 81 percent, 42 degrees of freedom and 13 parameters. The RMSE of the model was 6.23 psi, and the maximum VIF for any independent variable was 4.54. Residual patterns for the MLR model were homogeneous (Table 4).

For 0.500-inch a MLR model was developed with an [R.sup.2.sub.a]of 75 percent, 43 degrees of freedom, and 10 parameters. The RMSE of the model was 6.57 psi, and the maximum VIF for any independent variable was 5.55. Residual patterns for the MLR model were homogeneous (Table 4).

"Face Humidity" was the common independent variable for both 0.6875-inch and 0.500-inch MLR models (Table 4). It was surprising to see the scaled estimates for "Face Humidity" differed in sign for each product type. The "Face Humidity" has a negative scaled estimate of -10.02 psi on IB for 0.6875-inch while the "Face Humidity" has a positive scaled estimate of 4.81 psi on IB for 0.500-inch. This may signify that "Face Humidity" was an important source of variability acting on IB that the manufacturer needs to investigate.

To examine the influence of "Face Humidity" beyond the mean effect on IB, QR was explored for this common parameter for both 0.6875-inch and 0.500-inch. The average and median fits for 0.6875-inch for "Face Humidity" have different slopes, which may indicate lack of normality normality, in chemistry: see concentration.  in the response variable IB. We found that influence of "Face Humidity" on the outer 5th and 95th percentiles of IB was quite different than the inner percentiles (Fig. 8). This may indicate more volatility in IB for the MDF producer for this product type in the presence of changes in "Face Humidity." For the 0.500-inch product type (Fig. 9), the slopes of the IB percentiles were very similar for all of the percentiles for "Face Humidity." The median and average have different scales, which may also indicate non-normality in the response variable IB. The QR analysis for 0.500-inch may indicate that this product type has less volatility in IB in the presence of changes in "Face Humidity" when compared to the 0.6875-inch product type. It may also indicate that the product was easier to make between production runs in the presence of changes in "Face Humidity." The QR models for "Face Humidity" may reveal an opportunity for further root cause analysis by the manufacturer.

Although only one independent variable was used for illustration purposes, the quantile regression algorithm in R can also be applied to multiple independent variable models. Further analysis was conducted to examine the differences between the MLR and the QR median, 10th and 90th percentile fits. For the 0.6875-inch product type (Table 5), the largest discrepancies between the coefficients of median and average fits occur in "Face Humidifier humidifier,
n a device for adding moisture to dry air inside the home to help counteract the reduction in saliva that often occurs as a result of hyposalivation, radiation therapy, or other treatments that cause xerostomia.
 Temperature," "Core Scavenger Resin Flow," and "Dryer Mass Flow." The percent discrepancies were 66.53 percent, 23.16 percent, and 16.14 percent, respectively. For the 0.500-inch product type (Table 6), the largest discrepancies between the coefficients of median and average fits occur in "Face Humidity," "Mat Shave Off Target," and "Refiner S Steam Flow." The percent discrepancies were 36.58 percent, 22.39 percent, and 15.60 percent, respectively. A comparison of the 10th and 90th percentiles (Tables 5 and 6) of the coefficients gives a good method for relative comparisons of the influence of a process variable on IB. The discrepancies in coefficients highlight the importance of examining the percentiles of a distribution.

A significant finding of the research was the identification of similar regressors across all product types for both the MLR and QR models of the median. Results suggest that the regressors "Core Scavenger Resin %," "Face Humidity," and "Core Water to Wood" require further root cause investigation. The variable "Core Scavenger Resin %" was related to the application rate of scavenger resin and its influence on IB which was not contrary to the literature (Maloney 1977 and Suchsland and Woodson 1986). However, "Face Humidity" was related to the amount of humidity in the face fiber layer of MDF during mat formation, and "Core Water to Wood" was an indicator of the amount of [H.sub.2]O added during the defribration process of wood to fiber. The significant influence of both of these process variables in the MLR and QR models has not been previously documented in the literature. This finding further strengthens the justification for conducting more exploratory research Exploratory research is a type of research conducted because a problem has not been clearly defined. Exploratory research helps determine the best research design, data collection method and selection of subjects.  when investigating sources of variation in the MDF manufacturing process.

Further hypothesis-based research using a designed experiment would provide additional quantification of the causality of these sources of variation relative to IB. Even though industrial experimentation is difficult given the many possible nuisance sources of variation and potential measurement error, the inductive inductive

1. eliciting a reaction within an organism.

2.


inductive heating
a form of radiofrequency hyperthermia that selectively heats muscle, blood and proteinaceous tissue, sparing fat and air-containing tissues.
 research described in this paper provides insight for hypothesis generation, i.e., hypothesis tests and deductive de·duc·tive  
adj.
1. Of or based on deduction.

2. Involving or using deduction in reasoning.



de·duc
 scientific reasoning may provide more detailed scientific inference of causality.

The exploratory analysis of this research further highlighted that a narrow-view or focus only on the mean of the distribution may lead to incorrect conclusions, operational inefficiency and ultimately higher cost of manufactured product. Further analysis could also be conducted to examine each quantile (e.g., 1st, 5th, 50th, 99th, etc.) with respect to similar variables, e.g., "Core Scavenger Resin %," "Face Humidity," and "Core Water to Wood." A more detailed examination of each quantile using a designed experiment may also provide additional validation of important sources of variation for the manufacturer.

Conclusions

Multiple Linear Regression models (MLR) and Quantile Regression (QR) models were developed and compared for the Internal Bond (IB) of Medium Density Fiberboard (MDF). The models were developed from a manufacturing data set for a North American MDF producer. Models were developed for MDF product types that were distinguished by thickness in inches, i.e., 0.750 inch, 0.6875 inch, 0.625 inch, and 0.500 inch. MLR models have stringent assumptions and investigate causality of the mean of the response. QR models do not have the stringent assumptions and limitations of MLR and allow for the investigation of causality beyond the mean for any percentile of the response (e.g., 1st, 25th, 50th, 80th, 99th, etc.)

Common independent variables or regressors for the 0.750-inch and 0.625-inch MLR models were "Refiner Resin Scavenger %" and "Core Water to Wood." The scaled estimates for "Refiner Resin Scavenger %" differed in sign for each product type. The influence of "Refiner Resin Scavenger %" on the lower percentiles of IB was quite different than the midrange and higher percentiles. The research identified opportunities for additional investigation using designed experimentation on the "Refiner Resin Scavenger %" source of variation.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

"Face Humidity" was common for both 0.6875-inch and 0.500-inch product types. The scaled estimates for "Face Humidity" differ in sign for each product type and the average and median fits for 0.6875-inch for "Face Humidity" have different slopes, suggesting a lack of normality in the response variable IB. The QR analysis for 0.500-inch may indicate that this product type has less volatility in IB in the presence of changes in "Face Humidity" when compared to the 0.6875-inch product type. This may suggest the product 0.500-inch was easier to manufacture across production runs in the presence of variability in "Face Humidity" The discrepancies in regressors and the magnitude of scaled estimates further highlighted limitations of investigating only the mean of the distribution.

The research questions generated from the study relate to identifying significant and common sources of variation in MDF manufacture beyond the mean of the distribution of the response variable IB. Important research questions relate to the relative influence of "Core Scavenger Resin %," "Face Humidity," and "Core Water to Wood" in the manufacture of the IB of MDF. Additional designed experimentation in an industrial setting may lead to additional scientific inference and quantification of causality for these process variables.

Quantile regression methods can improve forest products manufacturers' knowledge of process variation. Improved knowledge of process variation facilitates variation reduction and costs savings, both vital for the long-term sustained business competitiveness of this important forest products industry.

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  • John C. Wiley, American ambassador
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(1) "Large-scale production of MDF began in the 1980s. MDF is an engineered wood product formed by breaking down softwood softwood

Timber obtained from coniferous trees (mainly of the pine and fir families). With the exception of bald cypress, tamarack, and larch, softwood trees are evergreens.
 into wood fibers, often in a defibrator (i.e. "refiner"), combining it with wax and resin, and forming panels by applying high temperature and pressure (http://en.wikipedia.org/wiki/Medium-density_fibreboard). MDF has become one of the most popular composite materials in recent years. MDF is uniform, dense, smooth, and free of knots and grain patterns, and is an excellent substitute for solid wood in many applications. Its smooth surfaces also make MDF an excellent base for veneers and laminates. Builders use MDF in many capacities, such as in furniture, shelving shelv·ing  
n.
1. Shelves considered as a group.

2. Material for shelves.

3. An incline; a slope.


shelving
Noun

1. material for shelves

2.
, laminate flooring This article or section needs sources or references that appear in reliable, third-party publications. Alone, primary sources and sources affiliated with the subject of this article are not sufficient for an accurate encyclopedia article. , decorative molding, and doors. MDF can be nailed, glued, screwed, stapled, or attached with dowels, making it a versatile product" (www.wisegeek.com/what-is-mdf.htm).

(2) An affine (from the Latin, affinis, "connected with") subspace Noun 1. subspace - a space that is contained within another space
mathematical space, topological space - (mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional"
 of a vector space vector space

In mathematics, a collection of objects called vectors, together with a field of objects (see field theory), known as scalars, that satisfy certain properties.
 (sometimes called a linear manifold manifold

In mathematics, a topological space (see topology) with a family of local coordinate systems related to each other by certain classes of coordinate transformations. Manifolds occur in algebraic geometry, differential equations, and classical dynamics.
) is a coset In mathematics, if G is a group, H a subgroup of G, and g an element of G, then
gH = is a left coset of H in G, and
Hg = is a right coset of H in G.
 of a linear subspace This article is about linear subspaces of an abstract vector space. For subspaces of Rn, see Euclidean subspace.

The concept of a linear subspace (or vector subspace
. A linear subspace of a vector space is a subset that is closed under linear combinations, e.g., linear regression equation of a linear subspace (http://mathworld.wolfram wolfram: see tungsten. .com/AffineFunction.html. 2006).

(3) There has been a dispute about who first discovered the method of least squares Noun 1. method of least squares - a method of fitting a curve to data points so as to minimize the sum of the squares of the distances of the points from the curve
least squares
. It appears that it was discovered independently by Carl Friedrich Gauss (person) Carl Friedrich Gauss - A German mathematician (1777 - 1855), one of all time greatest. Gauss discovered the method of least squares and Gaussian elimination.  ( 1777 to 1855) and Adrien Marie Legendre (1752-1833), that Gauss started using it before 1803 (he claimed in about 1795, but there is no corroboration of this earlier date), and that the first account was published by Legendre in 1805, see Draper and Smith (1981).

(4) Variants of this idea were proposed in the mid-eighteenth century by Boscovich and subsequently investigated by Laplace and Edgeworth (Koenker and Hallock 2001).

(5) Scaled estimate is a helpful statistic in MLR models in that it illustrates the relative influence of independent variables on the response variable. The scaled estimate is the influence that an independent variable has on the response variable when the independent variable moves one-half its range used in the model.

(6) It is important to note that multiple parameter models can be built using quantile regression, but for the purposes of illustration we chose to look only at the single parameter ease.

The authors are, respectively, Research Associate Professor, Forest Products Center; former Graduate Research Assistant, Forest Products Center and Dept. of Statistics, Operations, and Management Sci. (currently Statistician with Eastman Chemical Company Eastman Chemical Company is a United States based chemical company, engaged in the manufacture and sale of chemicals, plastics and fibers. Eastman has 16 manufacturing sites in 10 countries, supplying its products throughout the world. ); and Professor, Associate Professor, and Associate Professor, Dept. of Statistics, Operations, and Management Sci.; all at the Univ. of Tennessee, Knoxville, Tennessee “Knoxville” redirects here. For other uses, see Knoxville (disambiguation).
Founded in 1786, Knoxville is the third-largest city in the state of Tennessee, behind Memphis and Nashville, and is the county seat of Knox CountyGR6.
 (tmyoungl@utk.edu, fguess@utk.edu, bensmail@utk.edu, rleon@utk.edu). This research was partially supported by The Univ. of Tennessee Agri. Expt. Sta. McIntire Stennis E112215 (MS-75); USDA Special Wood Utilization Grants R112219-150 and Rl12219-184. Funding was also provided by Univ. of Tennessee, Dept. of Statistics, Operations, and Management Sci. This paper was received for publication in May 2007. Article No. 10352.
Table 1.--MLR models for product types 0.750-inch and 0.625-inch.

               0.750-inch
                                 Scaled
Parameters                      estimate   p-value

Face MDF temperature             -12.565   < 0.0001
Dryer S fiber moisture           -10.906   < 0.0001
Refiner resin scavenger %         -9.118   < 0.0001
Core dryer outlet temperature     18.498     0.0092
Press position time               19.926   < 0.0001
Dryer 1 fan current               23.662   < 0.0001
Dryer 2 fan current              -25.384   < 0.0001
Refiner S chip level              10.666   < 0.0001
Refiner S feeder screw speed       9.294     0.0017
Core water to wood               -21.043   < 0.0001
ESP milliamps                    -11.714   < 0.0001

       Important regression statistics

[R.sub.a] (2,3)                    0.751646
d.f. (4)                          50
P (5)                             11
[VIF.sub.max] (6)                  5.0315819
RMSE (7)                           7.697272
Residual pattern                  Homogeneous

               0.625-inch
                                 Scaled
Parameters                      estimate   p-value

Shavings raw weight              -15.872   < 0.0001
Refiner resin scavenger %          8.396    0.00080
Core grinding steam flow          12.720   < 0.0001
Core resin to wood %              22.473   < 0.0001
Dryer mass flow                   10.642   < 0.0001
Resin water tank temperature     -21.556   < 0.0001
Core refiner feeder screw          4.868     0.0110
 speed
Core water to wood               -10.872     0.0077
Face humidifier temperature       13.583   < 0.0001
Relative ambient humidity          5.858     0.0100
Weight actual                     12.205   < 0.0001

      Important regression statistics

[R.sub.a] (2,3)                    0.723694
d.f. (4)                          53
P (5)                             11
[VIF.sub.max] (6)                  5.603058
RMSE (7)                           6.051464
Residual pattern                  Homogeneous

(3) Adjusted coefficient of determination.

(4) Degrees of freedom.

(5) Number of explanatory variables.

(6) Maximum variance inflation factor.

(7) Root mean square error.

Table 2.--MLR and OR models for product type 0.750-inch.

                                        Coefficients

0.750-inch                  MLR        QR        QR 10th      QR 90th
Variables                 average    median     percentile   percentile

Intercept                 40264.84   34655.89    40679.39     44452.38
Face mdf temperature         -0.27      -0.27       -0.33        -0.09
Dryer S fiber moisture       -5.10      -4.87       -5.76        -2.33
Refiner resin scavenger
 percent                  -1314.71   -1535.49    -1488.00     -1373.25
Core dryer outlet
temperature                   1.91       1.63        1.97         1.46
Press position time           1.96       2.21        2.02         1.93
Dryer 1 fan current          75.06      53.67       78.09        95.56
Dryer 2 fan current         -65.80     -48.93      -67.03       -77.23
Refiner S chip level          4.00       3.16        3.37         6.14
Refiner S feeder screw
 speed                        0.31       0.32        0.31         0.33
Core water to wood         -835.05    -651.32     -825.34      -957.74
ESP milliamps                -0.16      -0.17       -0.15        -0.19

Table 3.--MLR and QR models for product type 0.625-inch.

                                       Coefficients

0.625-inch                 MLR        QR        QR 10th      QR 90th
Variables                average    median     percentile   percentile

Intercept               -1029.56   -2063.15      1896.63     -1745.51
Shavings raw weight        -1.55      -1.09        -1.38        -1.64
Refiner resin             949.74    1084.13       588.90       889.98
 scavenger %
Core grinding steam         0.34       0.37         0.38         0.28
 flow
Core resin to wood %       12.03      10.73        13.25        10.17
Dryer mass flow             0.68       0.76         0.65         0.67
Resin water tank           -1.69      -1.81        -1.49        -2.01
 temperature
Core refiner screw          0.14       0.14         0.26         0.04
 speed
Core water to wood       -133.48    -127.01      -150.40      -105.60
Face humidifier             1.22       1.31         0.97         1.80
 temperature
Relative ambient            1.22       1.42         0.56         2.39
 humidity
Weight actual             157.15     139.34       130.54       130.00

Table 4.--MLR models for product types 0.6875-inch and 0.500-inch.

              0.6875-inch

                                 Scaled
Parameters                      estimate    p-value

Face scavenger resin %            25.479   < 0.0001
Dryer mass flow                   -8.192     0.0005
Core humidifier temperature      -10.683     0.0037
Face fiber mat moisture           26.949   < 0.0001
Mat shave off level              -15.408   < 0.0001
Refiner S chip level              14.655   < 0.0001
Refiner S grinding steam flow     21.066   < 0.0001
Refiner S screw speed             -5.873     0.0030
Core scavenger resin flow         -6.914     0.0225
Dryer ESP outlet temperature     -13.138   < 0.0001
Face humidity                    -10.016     0.0031
Press open time                    5.471     0.0056
Face humidifier temperature       19.560   < 0.0001

             Important Regression Statistics

[R.sup.2.sub.a]                    0.808614
d.f.                              42
P                                 13
[VIF.sub.max]                      4.5371586
RMSE                               6.233895
Residual pattern                  Homogeneous

              0.500-inch

                                 Scaled
Parameters                      estimate    p-value

Core total weight                 -5.191     0.0112
Mat shave off target               6.823     0.0020
Press preposition time            10.060     0.0015
Weight target                      7.938     0.0194
Core blow line pressure           19.091   < 0.0001
Face digester pressure            -9.494     0.0004
Core resin pressure              -11.273     0.0013
Refiners steam flow               -7.452     0.0078
Core refiner screw speed         -21.777   < 0.0001
Face humidity                      4.811     0.046

             Important Regression Statistics

[R.sup.2.sub.a]                    0.747666
d.f.                              43
P                                 10
[VIF.sub.max]                      5.5493187
RMSE                               6.573086
Residual pattern                  Homogeneous

Table 5. - MLR and OR models for product type 0.6875-inch.

                                        Coefficients

0.6875-inch                 MLR        QR        QR 10th      QR 90th
Variables                 average    median     percentile   percentile

Intercept                 -1231.75   -1556.92     -692.31     -1693.05
Face scavenger resin %      280.75     314.06      227.50       345.93
Dryer mass flow              -0.61      -0.53       -0.69        -0.85
Core humidifier
 temperature                 -1.54      -1.57       -1.86        -2.16
Face fiber mat moisture      24.50      22.78       27.19        17.18
Mat shave off level         -16.18     -15.63      -17.63       -16.17
Refiner S chip level          1.91       1.84        1.68         2.24
Refiner S grinding steam
 flow                         0.04       0.04        0.04         0.03
Refiner S screw speed        -0.18      -0.19       -0.21        -0.09
Core scavenger resin
 flow                        -3.76      -3.05       -5.72        -0.29
Dryer ESP outlet
 temperature                 -0.68      -0.63       -0.74        -0.73
Face humidity                -4.81      -4.84       -6.10        -2.38
Press open time               0.34       0.30        0.45         0.26
Face humidifier
 temperature                  2.38       3.96        2.93         4.29

Table 6.--MLR and OR models for product type 0.500-inch.

                                       Coefficients

0.500-inch                  MLR        QR      QR 10th      QR 90th
Variables                 average    median   percentile   percentile

Intercept                  -225.75   -173.05   -305.28       -86.06
Core total weight            -0.07     -0.08     -0.03        -0.09
Mat shave off target          9.52      7.78      9.45         9.95
Press preposition time        0.93      0.90      1.36         0.60
Weight target               158.76    153.68    219.71        56.18
Core blow line pressure       1.65      1.71      1.69         0.82
Face digester pressure       -2.06     -2.21     -2.18        -1.72
Core resin pressure          -0.12     -0.13     -0.13        -0.07
Refiner S steam flow         -0.01     -0.01     -0.01        -0.002
Core refiner screw speed     -0.55     -0.55     -0.41        -0.36
Face humidity                 2.37      1.74      0.31         4.58
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Author:Young, Timothy M.; Shaffer, Leslie B.; Guess, Frank M.; Bensmail, Halima; Leon, Ramon V.
Publication:Forest Products Journal
Article Type:Report
Geographic Code:1USA
Date:Apr 1, 2008
Words:8284
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