Initial simplex for non-linear application of nelder-mead [closed] - r

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I am using a modified Weibull CDF to predict the height of trees in a stand at a given age: Y = b0 * {1.0 + EXP[-b1 * (X**b2)]}. Where Y = height of dominant trees; X = stand age; b0 is the asymptote parameter to be predicted; b1 is the rate parameter to be predicted; and b2 is the shape parameter to be predicted. Each of b0, b1, b2 are linear functions of one or more bio-geo-climatic and/or physiographic variables. I have a data set of N ~ 2000 stands. I get an initial fit with a higher tolerance, and then sample the data with replacement (i.e. bootstrap) and re-fit at a lower tolerance. With ~200 iterations, I generate a distribution of parameter estimates from which I can identify the least-significant element across b0, b1, b2. Eliminate (or add), and repeat.
I have a working version of this process in R using OPTIM, where the minimizing function is evaluating Z = the sum of squares (SSQ) rather than the Y values directly. My problem is computer time: the initial fit requires about 1 day, and the 200 bootstraps require an additional 2-3 days depending. The 40 or so additions/reductions in variables have been spinning around continuously on a server since August 2022. So I am attempting to port this into a FORTRAN 90 .dll called from R.
Here is what my data look like:
`
Y <- c(50.1, 80.3, 60.4, ... , 75.5, 90.2), len(Y) = 2000
X <- c(21, 38, 27, ... , 34, 37), len(X) = 2000
b0 <- f(P1, P2, P3, P4) len(b0) = 4
b1 <- f(P3, P5, P7, P8, P12) len(b1) = 5
b2 <- f(P6, P2, P8, P9, P10) len(b2) = 5
`
where P is the set of bio-geo-climatic and physiographic variables, with values at each X. Note that some of the same predictors are repeated across the parameters, but since they act on different parts of the equation (asymp,rate,shape), their sign and magnitude will vary, and are therefore treated as separate variables. My data matrix has 2000 rows by (len(b0) + len(b1) + len(b2) + 3) columns, one for each predictor in each parameter, plus an intercept term for each of b0,b1,b2. The number of columns may vary over time as I'm adding/subtracting terms to/from each. so my fitting data is a matrix with 2000 rows and a column structure that looks like this:
(icpt0, P1, P2, P3, P4, icpt1, P3, P5, P7, P8, P12, icpt, P6, P2, P8, P9, P10), cols = 17, rows = 2000
When evaluating the function I grab the appropriate columns to get the parameters:
Y.hat = (icpt0 + P1 + P2 + P3 + P4) * {1.0 + EXP[-(icpt1 + P3 + P5 + P7 + P8 + P12) * (X**(icpt2 + P6 + P2 + P8 + P9 + P10))]}
residual = Y(X) - Y.hat(X)
squares = residual**2
sum squares across D to get SSQ
Z(i) = SSQ (this is what I'm actually minimizing, not Y across D)
I need help constructing the initial coefficient simplex S = (cols+1) x cols vertices to pass to the FORTRAN implementation of Nelder-Mead. Within R and OPTIM, I only need to pass a single point (b0,b1,b2), where the estimates are first applied only to the intercepts. I'm not sure how to construct the appropriate unit vectors to build a robust initial matrix.
if my initial point esimate is b0 = 200.0, b1 = 0.001, b2 = 1.5, then my first row of vertices S(1) looks like:
icpt0
P1
P2
P3
P4
icpt1
P3
P5
P7
P8
P12
icpt
P6
P2
P8
P9
P10
200.0
0.0
0.0
0.0
0.0
0.001
0.0
0.0
0.0
0.0
0.0
1.5
0.0
0.0
0.0
0.0
0.0
0.0
4.0
0.0
0.0
0.0
0.001
0.0
0.0
0.0
0.0
0.0
1.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.7
0.0
0.0
0.001
0.0
0.0
0.0
0.0
0.0
1.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
12.0
0.0
0.001
0.0
0.0
0.0
0.0
0.0
1.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2.3
0.001
0.0
0.0
0.0
0.0
0.0
1.5
0.0
0.0
0.0
0.0
0.0
If the average of P1 across D is 50, then S(2) looks like row 2 with the expected parameter = 200/50. I repeated this process for each row, so that I have a diagonal of positive coefficient values across S, but zeroes or intercept terms otherwise. I see that I could input 0.00025 for all zeroes, and mess with positive values by 0.05, but I'm not sure what that really changes.
My FORTRAN .dll appears to work, however the results do no match outputs from R/OPTIM version (using same data/predictors); The FORTRAN results yield Z values that are at least an order of magnitude larger than R, and my re-starts don't improve anything. I figure that R/OPTIM is constructing a different version of S than above. What would a better initial simplex S look like? what would the unit vectors look like? I'm struggling since my initial points are (b0,b1,b2), but each are linear functions, so I don't know how to construct unit vectors to get a full array of positive values, which I suspect is what's needed.