Optimizing Forest Sampling by Using Lagrange Multipliers
- 1 Department of Forestry and Management of the Environment and Natural Resources, Democritus University of Thrace, Orestiada, Greece
Abstract
In two-phase sampling, or double sampling, from a population with size N we take one, relatively large, sample size n. From this relatively large sample we take a small sub-sample size m, which usually costs more per sample unit than the first one. In double sampling with regression estimators, the sample of the first phase n is used for the estimation of the average of an auxiliary variable X, which should be strongly related to the main variable Y (which is estimated from the sub-sample m). Sampling optimization can be achieved by minimizing cost C with fixed var Y , or by finding a minimum var Y for fixed C. In this paper we optimize sampling with use of Lagrange multipliers, either by minimizing variance of Y and having predetermined cost, or by minimizing cost and having predetermined variance of Y.
- J. Penman, M. Gytarsky, T. Hiraishi, T. Krug, D. Kruger, R. Pipatti, L. Buendia, K. Miwa, T. Ngara, K. Tanabe and F. Wagner, “Good Practice Guidance for Land Use, Land- Use Change and Forestry,” Intergovernmental Panel on Climate Change: IPCC National Greenhouse Gas Inventories Program, Institute for Global Environmental Strategies (IGES) for the IPCC, Kanagawa Japan, 2003.
- F. Loetsch, F. Z?hrer and K. Haller, “Forest Inventory,” BLV, Verlagsgesellschaft, 1973.
- I. Doig, “When the Douglas-Firs Were Counted: The Beginning of the Forest Survey,” Journal of Forest History, Vol. 20, 1976, pp. 20-27.
- W. Frayer and G. Furnival, “Forest Survey Sampling Designs—A History,” Journal of Forestry, Vol. 97, No. 12, 1999, pp. 4-10.
- T. Gregoire, “Roots of Forest inventory in North America,” A Paper Presented at the A1 Inventory Working Group Session at the SAF National Convention Held at Richmond, VA, 25-28 October 1992.
- T. Honer and F. Hegyi, “Forest Inventory—Growth and Yield in Canada: Past, Present and Future,” The Forestry Chronicle, Vol. 66, 1990, pp. 112-117.
- H. Schreuder, T. Gregoire and G. Wood, “Sampling Methods for Multiresource Forest Inventory,” John Wiley and Sons, New York, 1993.
- R. Sepp?l?, “Forest Inventories and the Development of Sampling Methods,” Silva Fennica, Vol. 3, 1985, pp. 218-219.
- D. Van Hooser, N. Cost and H. Lund, “The History of Forest Survey Program in the United States,” In: G. Preto and B. Koch, Eds., Forest Resource Inventory and Monitoring and Remote Sensing Technology, Proceedings of the IUFRO Centennial Meeting in Berlin, 31 August-4 September 1992, Japan Society of Forest Planning Press, Tokyo University of Agriculture and Technology, Saiwaicho, Fuchu, Tokyo, Japan, 1992, pp. 19-27.
- S. Rao, “Optimization Theory and Application,” Wiley Eastern Ltd., New Delhi, 1984.
- W. Cochran, “Sampling Techniques—3rd Edition,” Wiley, New York, 1977.
- P. De Vries, “Sampling for Forest Inventory,” Springer, Berlin, 1986. doi:10.1007/978-3-642-71581-5
- K. Matis, “Sampling of Natural Resources,” Pigasos Publications, Thessaloniki, Greece, 2004.
- J. Deville and C. S?rndal, “Calibration Estimators in Survey Sampling,” Journal of the American Statistical Association, Vol. 87, No. 418, 1992, pp. 376-382. doi:10.2307/2290268