Harold C. Fritts 5703 North Lady Lane Tucson, Arizona 85704, USA Internet:hfritts@ltrr.arizona.edu Phone: (520) 887 7291 FAX: (520) 887 7291 March 7, 1999 Dear PRECON owners: At the end of this note is a description of the Bootstrapped Response Function These disks contain a copy of the revised 32 bit PRECON,Version 5.17b now in a file called PRECON32.EXE. Version B corrects Runtime errors in A when processing climatic files that are shorter than the chronology file by one or two years. In addition, I have not included the kalman.exe file By Ed Cook in recent versions as few people seem to be using it. If you understand it and wish to run the Kalman filter you can download it from my web site http://www.LTRR.arizona.edu/ people/hal/HAL1.html. DOSXMSF.EXE must be in the same directory as PRECON32.EXE. The 32 bit version of the program runs about 6 times faster than the older PRECON16.EXE versions. In addition, you will be able to call up Graph-in-the-Box anywhere in the program with an Alt-G and will not have to use the Menu Item 9. To run under Win95 you can leave the bat file in the Precon directory, make a shortcut to PRECON32.BAT and place it on the desktop. Left click on the shortcut and select properties. Go to screen tab and check the full screen button. Click on OK. Then, enter PRECON32.BAT to run the program. This version allows much longer path statements and has fixed some bugs. You will have to make new INI files as the paths are now up to 60 characters long. If you have any problems with this version or any of the versions, please contact me promptly giving me the details so that I can correct the problem ar advise you what the difficulty may be. Backup any old copies if you have them. If you are a first user and have not made a \PRECON directory, then make a directory C:\PRECON and log onto the upgrade disks and enter INSTALL or INSTALL followed by a RETURN. All files will be transferred to that directory and extracted. You may need to put the PRECON32.BAT files into a directory in the path and the PRWIN32.BAT files in thy PRECON directory for running in DOS. You may need to edit the path statement in these bat files to operate on your system if the PRECON directory is not on the C: drive and/or you are using a path other than C:\PRECON\. Edit the BAT files as ASCII documents. If you use WINDOWS 3.1 to run PRECON call PRWIN32.BAT rather than PRECON.BAT. However, don't remove these files from your disk as they are called by PRECON32.BAT. Be sure to remove your old PRECON.EXE and PRECON.BAT fileS as they could conflict with PRECON.BAT. If you exit with a Control C or Break key, be sure to remove Graph-in-the-Box from memory by running GBKILL from the Precon directory. If you are using your own chronology and climatic files, I recommend that you loadthem in another directory and enter "PRECON32.BAT" from that directory. All output files will be placed in that directory or any other directory from which you start the program. For best results, do not put the PRECON directory in the DOS path but be sure to copy PRECON32.BAT to a directory in your path. You can then run the program in any directory as long as the original program EXE files are in the PRECON directory. This new version of PRECON 5.17a corrects some bugs INCLUDING A BAD ONE THAT APPEARED IN VERSION 5.0 RELEASED IN 6/96 and a less important bug that created additional meaningless plots in the time- series plots, B) allows assignment of longer paths and file names, C) enables the analysis of earlywood and latewood data arranging them as the correct prior growth variables when lags are included in the analysis,and D) allows a more complete analysis of data from the Southern Hemisphere as well as a few other conveniences. Now the maximum number of iterations for the response is 99999 if you ever care to run that many. I recommend using 30 iterations for exploritory work. Then use more from 100 to 1000 depending upon what statistical guidelines you wish to follow. Joel Guiot who wrote the response subroutine says there is no significant change after 50 iterations although some in the literature recommend 1000. I found that when I compared results between 100 and 1000 replications there was sometimes improvement in that more variables were significant. An update of the new Graph-in-the-box Analytic, Version 1.01, copyright by New England Software, is included in disks that were purchased. If this program is used with this new update, the Months in the response function plots will be labeled with letters rather than numbers. Also it includes more printer drivers. If you enter E or L in the earlywood or latewood option, then the program assumes that the first file read is either earlywood or latewood and will prompt you for a second file that will be the latewood of the previous year or earlywood of the current year. In the analysis of the response function, the three prior growth values for earlywood will be the late- wood and earlywood of the preceding year and latewood two years previously. For latewood the prior growth values will be the earlywood of the current year and the latewood and early- wood of the preceding year. If you choose the option for Southern Hemisphere data, the climatic data are read, moved backward in time by 6 months, and rewritten in a file with the .t_t or .p_p extent. This is the same as beginning the calendar year in July instead of January so Month 1 in the Southern Hemisphere year will be July, month 2, August and so forth. The program simply displaces the climatic data so you can include values for the years prior to growth comparable to the Northern Hemisphere. The year of growth follows the convention for Southern Hemisphere chronology data in that the year assigned to the ring-width index is the year in which growth begins. There is no longer any need to change the date to run PRECON and you can include more months extending the coverage to the end of the growing season. Those of you who have data using these two options, please try the options out and let me know if you encounter any problems. Just as in the earlier version of PRECONK (2.1 - 5.1) the program can generate input for the KALMAN FILTER PROGRAM written by Ed Cook, which I have revised to use input from PRECONK. Read PRECONK.TXT for instructions and current information. If for some reason it is necessary to reinstall PRECON, delete the old files and the PRECON Directory, install the data from the original disk. I have corrected as many of the problems as I could find and added the suggestions I have received for the program. It should be much easier to import files and to run the program than version 1.0 as it can learn the input parameters needed to run your appli- cations. For those who encounter difficulties in the future, please do me a great favor by sending as much information as you can on the problem, including the *.out file generated by the problem, and I will try to correct the code, recompile it and send you a new free updated version as promptly as possible. The bootstrapped response function. A response function is a statistical representation of the important climatic factors as well as prior growth factors that control ring width index growth. The first attempt at creating a response function used a multiple regression equation built in a stepwise fashion (Fritts 1976). However that form of the response function provided a limited measure of the effects of climate. Only the coefficients that provided a significant retrodiction of climate were obtained. There is no reason to believe that the actual tree response to climate turns off and then back on for only those months and variables that had significant regression coefficients. Many factors influence the number of significant coefficients including the length of the common period used between climate and the ring-width index, the number of variables entering the regression, collinearity in the data set, distance from the climatic station, differences in the response, the variance of response from the different trees included in the chronology data set and measurement error. The basic procedure was changed to using a multiple regression after extracting the principal components of the correlation matrix of the climatic and prior growth variables (Fritts 1976). The most important eigenvectors were extracted from the array of climatic and prior growth variables using Equation 7.18 (Fritts 1976). The normalized climatic and prior growth variables were multiplied by the eigenvectors to obtain what is called the principal components, factor scores or eigenvector amplitudes (Equation 7.9 Fritts 1976). These principal components are orthogonal modes of climate and prior growth values and contain the same information that was within the original correlation matrix, but in a rotated form so that they are uncorrelated (orthogonal) with one another. This procedure transforms what was originally a highly intercorrelated set of variables to an uncorrelated set of principal components. Some of the eigenvectors and principal components explain too little variance and these were excluded from the analysis using the PVP criterion of Guiot (1985, 1991). These principal components with significant eigenvectors were used as predictors of the tree-ring width indices in a standard, not stepwise, multiple regression analysis. The regression coefficients using the principal components cannot be interpreted easily so it was necessary to transform them back to the original monthly climatic and prior growth variables. This was accomplished by simply multiplying the regression weights by the eigenvectors that met the PVP criterion. This rerotated set of regression coefficients are the coefficients of the response function. The multiple correlation squared RSQ measures how much of the chronology variance was actually predicted by the coefficients of climate and prior growth. By solving the equation first for only climate and then for prior growth one can obtain an estimate of the variance reduced by climate and then the variance reduced by prior growth. Since the variables are correlated, sometimes the variance due to climate and prior growth do not add up to the total variance. One problem with this calculation is that the significance of the coefficients is overestimated because of the correlation between the climatic and prior growth data sets (Cropper 1985). Thus we had to add the bootstrap procedure which estimates the errors in the results by replicating the calculations m times by randomly drawing n sets of data with replacement, simulating the n years of data (including both climate, prior growth and the ring-width index data). This added a great deal more calculation but obtained unbiased estimates of the regression coefficients and their errors used to test significance. The boot strap procedure uses the basic calculations described above and adds the following steps. Instead of calculating the response function once, it is calculated m times (with the default values m = 50). In addition for each of these 50 calculations we added another calculation which is the independent check or verification. The procedure begins by sampling from the n years common between the climatic data and chronology indices n times with replacement. When we do this there are still n sets used in the computation but these are not the same ones since we use random sampling. A number of observations are drawn two or more times and some are not drawn at all. The procedure keeps tract of which values are drawn and which ones are not drawn. For each step we proceed by sampling the n sets as stated before, rotating the climatic data, calculating the eigenvectors, calculating the principal components, multiple regression on the set of principle components of climate and prior growth and then rerotating the coefficients back to the estimated response function weights. The pooled multiple correlation Rd is calculated with its standard error derived from all replicates. (On the first replication there is no error because there is only one replication. Because the data were drawn randomly with replacement, a set of observations remain that were not drawn for calculating the particular response function. These observations are now applied to the response function to obtain an independent estimate. This is a check on how stable this response function is when it is applied to the population in general. A new set of n observations is drawn from this independent set; and these are multiplied by the response coefficients and a new multiple regression coefficient, Ri, is calculated. If the response is a perfect and unbiased estimate of the population as a whole Ri = Rd. In reality, this can never happen and Ri is some value below that of Rd. The drop in variance between Rd and Ri is a measure of the reliability of the response function. This procedure is repeated m times and the mth response function pooled from all replications is used to estimate the actual, estimated and residual time series. The statistics from each set of calculations are pooled and used to calculate the standard errors of Rd and Ri. At the end of the calculations these pooled data are also used to calculate the standard errors of each response function coefficient. All coefficients with absolute values exceeding two standard errors are marked as significant. You can see that you left out the whole part of the response that involved principal component regression and the translation back to coefficients of the original climatic and prior growth data. Since the number of eigenvectors is reduced to those that are significant a normal multiple regression, not the stepwise was used. Cropper, J. P. 1985. Tree-ring response functions: An evaluation by means of simulations. Ph.D. dissertation, The University of Arizona, Tucson. University Microfilms International, Ann Arbor. Fritts, Harold C. 1976. Tree rings and climate. Academic Press, London pp 340-370. Guiot, J. 1991. The bootstrapped response function. Tree-Ring Bulletin, 51:39-40. Guiot, J. 1985. The extrapolation of recent climatological series with spectral canonical regression. Journal of Climatology 5:325-335.