SymbolicR

Find non-linear formulas that fits your input data. You can systematically explore and memoize the possible formulas and its cross-validation performance, in an incremental fashon. Three main interoperable search functions are available:

random.search performs a random exploration,
genetic.search employs a genetic optimization algorithm
comb.search allows the user to provide a data.frame of formulas to be tested

After installation, see tutorials with

browseVignettes('symbolicr')

Installation

You can install the development version of symbolicr like so:

devtools::install_github('cosbi-research/symbolicr', build_vignettes = TRUE)

Genetic search

This is a minimum viable example on how to use genetic search to find the non-linear relationship:

library(symbolicr)
set.seed(1)

x1<-runif(100, min=2, max=67)
x2<-runif(100, min=0.01, max=0.1)

y <- log10(x1^2*x2) + rnorm(100, 0, 0.001)

X <- data.frame(x1=x1, x2=x2)

results <- genetic.search(
  X, y, 
  n.squares=2, 
  max.formula.len = 1, 
  N=2,
  K=10,
  best.vars.l = list(
   c('log.x1')
  ),
  transformations = list(
   "log"=function(rdf, x, stats){ log(x) },
   "inv"=function(rdf, x, stats){ 1/x }
  ),
  keepBest=T,
  cv.norm=F
)
#> [1] "## Total number of single terms: 27"
#> [1] "Iteration: 1 Mean/Max fitness:-9.80e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 2 Mean/Max fitness:-9.80e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 3 Mean/Max fitness:-9.80e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 4 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 5 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 6 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 7 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 8 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 9 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 10 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 11 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 12 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 13 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 14 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 15 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 16 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 17 Mean/Max fitness:-9.60e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 18 Mean/Max fitness:-9.40e+07 / 1.66e+00 Best: log.x1"
#> [1] "Iteration: 19 Mean/Max fitness:-9.00e+07 / 1.69e+00 Best: log.x1"
#> [1] "Iteration: 20 Mean/Max fitness:-9.60e+07 / 1.69e+00 Best: log.x1"
#> [1] "Iteration: 21 Mean/Max fitness:-9.00e+07 / 1.69e+00 Best: log.x1"
#> [1] "Iteration: 22 Mean/Max fitness:-9.40e+07 / 1.69e+00 Best: log.x1"
#> [1] "Iteration: 23 Mean/Max fitness:-9.40e+07 / 1.69e+00 Best: log.mul.x1.x1"
#> [1] "Iteration: 24 Mean/Max fitness:-9.60e+07 / 1.69e+00 Best: log.mul.x1.x1"
#> [1] "Iteration: 25 Mean/Max fitness:-9.60e+07 / 1.69e+00 Best: log.mul.x1.x1"
#> [1] "Iteration: 26 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 27 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 28 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 29 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 30 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 31 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 32 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 33 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 34 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 35 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 36 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 37 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 38 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 39 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 40 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 41 Mean/Max fitness:-9.20e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 42 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 43 Mean/Max fitness:-9.20e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 44 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 45 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 46 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 47 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 48 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 49 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 50 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 51 Mean/Max fitness:-9.20e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 52 Mean/Max fitness:-9.20e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 53 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 54 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 55 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 56 Mean/Max fitness:-9.20e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 57 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 58 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 59 Mean/Max fitness:-9.20e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 60 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 61 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 62 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 63 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 64 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 65 Mean/Max fitness:-9.40e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 66 Mean/Max fitness:-9.60e+07 / 1.71e+00 Best: log.x1"
#> [1] "Iteration: 67 Mean/Max fitness:-9.40e+07 / 6.22e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 68 Mean/Max fitness:-9.40e+07 / 6.22e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 69 Mean/Max fitness:-9.40e+07 / 6.22e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 70 Mean/Max fitness:-9.20e+07 / 6.22e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 71 Mean/Max fitness:-9.20e+07 / 6.22e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 72 Mean/Max fitness:-9.20e+07 / 6.22e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 73 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 74 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 75 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 76 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 77 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 78 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 79 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 80 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 81 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 82 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 83 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 84 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 85 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 86 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 87 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 88 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 89 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 90 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 91 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 92 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 93 Mean/Max fitness:-9.60e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 94 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 95 Mean/Max fitness:-9.20e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 96 Mean/Max fitness:-9.20e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 97 Mean/Max fitness:-9.20e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 98 Mean/Max fitness:-9.40e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 99 Mean/Max fitness:-9.00e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"
#> [1] "Iteration: 100 Mean/Max fitness:-9.20e+07 / 6.23e+00 Best: log.mul.x1.mul.x1.x2"

We found the correct non-linear formula starting from an initial guess! We can now get the best formula

results$best
#> [1] "log.mul.x1.mul.x1.x2"

And all the formula the genetic algorithm found to be best at each one of the 100 evolution iterations (last 5 shown for brevity)

results$best.iter[seq(length(results$best.iter)-5,length(results$best.iter))]
#> [[1]]
#> [1] "log.mul.x1.mul.x1.x2"
#> 
#> [[2]]
#> [1] "log.mul.x1.mul.x1.x2"
#> 
#> [[3]]
#> [1] "log.mul.x1.mul.x1.x2"
#> 
#> [[4]]
#> [1] "log.mul.x1.mul.x1.x2"
#> 
#> [[5]]
#> [1] "log.mul.x1.mul.x1.x2"
#> 
#> [[6]]
#> [1] "log.mul.x1.mul.x1.x2"

Note that cv.norm=FALSE means data is used as-is. Before running this example we checked for the non-negativeness of x1^2*x2. If you would like to normalize data to avoid scaling issues just use cv.norm=TRUE but in this case, to avoid computing the log of a negative value, we use this updated transformation function

# NB: function applied to standardized X values!
   #     they can be negative
log.std <- function(x, stats){ log(0.1 + abs(stats$min) + x) }

The stats object contains - min: the minimum of the column values - absmin: the minimum of the absolute values of the columns - absmax: the maximum of the absolute values of the columns - projzero: -mean/sd of the columns, that is the position of the zero in the original, non-normalized space.

Type ?dataset.min.maxs in your R console for further informations.

Random search

This is a minimum viable example on how to use random search to test multiple non-linear relationships, and get a summary of the performances in a data.frame:

random.results <- random.search(
  X, y, 
  n.squares=2, 
  formula.len = 1, 
  N=2,
  K=10,
  transformations = list(
   "log"=function(rdf, x, stats){ log(x) },
   "inv"=function(rdf, x, stats){ 1/x }
  ),
  cv.norm=F
)
#> [1] "Regression on mul.x2.mul.x2.x2"
#> [1] "Regression on log.mul.x1.x2"
#> [1] "Regression on inv.mul.x1.mul.x2.x2"
#> [1] "Regression on mul.x1.x1"
#> [1] "Regression on mul.x1.x2"
#> [1] "Regression on x2"
#> [1] "Regression on x1"
#> [1] "Regression on log.x2"
#> [1] "Regression on mul.x1.mul.x1.x1"
#> [1] "Regression on log.mul.x2.x2"
#> [1] "Regression on inv.mul.x2.x2"
#> [1] "Regression on inv.x2"
#> [1] "Regression on inv.mul.x1.mul.x1.x2"
#> [1] "Regression on log.mul.x1.mul.x2.x2"
#> [1] "Regression on inv.mul.x2.mul.x2.x2"
#> [1] "Regression on inv.mul.x1.mul.x1.x1"
#> [1] "Regression on inv.x1"
#> [1] "Regression on log.mul.x1.mul.x1.x1"
#> [1] "Regression on mul.x2.x2"
#> [1] "Regression on log.mul.x1.mul.x1.x2"
#> [1] "Regression on inv.mul.x1.x2"
#> [1] "Regression on log.mul.x2.mul.x2.x2"
#> [1] "Regression on log.mul.x1.x1"
#> [1] "Regression on inv.mul.x1.x1"
#> [1] "Regression on mul.x1.mul.x1.x2"
#> [1] "Regression on log.x1"
#> [1] "Regression on mul.x1.mul.x2.x2"

You can then inspect results in the resulting data.frame:

random.results[order(random.results$base.r.squared, decreasing = T), ][seq(5), ]
#>         base.pe  base.cor base.r.squared base.max.pe  base.iqr.pe
#> 20 0.0004285218 0.9999988      0.9999976 0.004748987 0.0005285826
#> 2  0.0725826920 0.9536868      0.9095013 1.433554884 0.1219353333
#> 26 0.1038707364 0.9276918      0.8606065 1.093821498 0.0978448963
#> 23 0.1005819365 0.9272320      0.8597596 1.111784453 0.0969222114
#> 18 0.1039579306 0.9261947      0.8578210 1.085202757 0.0888076193
#>    base.max.cooksd base.max.cooksd.name                 vars n.squares
#> 20       0.4515147                   47 log.mul.x1.mul.x1.x2         2
#> 2        0.2559093                   27        log.mul.x1.x2         2
#> 26       0.1554620                   47               log.x1         2
#> 23       0.1542245                   47        log.mul.x1.x1         2
#> 18       0.1567286                   47 log.mul.x1.mul.x1.x1         2
#>    formula.len
#> 20           1
#> 2            1
#> 26           1
#> 23           1
#> 18           1

Combinatorial Search

Search over all non-duplicated combinations of terms taken from an user-supplied list of formulas.

# test the three formula x1, x2 and log(x2)
comb.search(
  X, y, 
  combinations=data.frame(t1=c('x1','x2','log.x2')), 
  N=2,
  K=10,
  transformations = list(
   "log"=function(rdf, x, stats){ log(x) },
   "exp"=function(rdf, x, stats){ exp(x) }
  ),
  cv.norm=F
)
#> [1] "Regression on x1"
#> [1] "Regression on x2"
#> [1] "Regression on log.x2"
#>     base.pe  base.cor base.r.squared base.max.pe base.iqr.pe base.max.cooksd
#> 1 0.1302693 0.8630977     0.74487976    3.080926   0.1708999       0.3215164
#> 2 0.2173972 0.3203810     0.09942148    7.524692   0.1991448       0.1019341
#> 3 0.2215673 0.2952262     0.08528910    7.786758   0.1758801       0.1168211
#>   base.max.cooksd.name   vars n.squares formula.len
#> 1                   47     x1         0           1
#> 2                   47     x2         0           1
#> 3                   47 log.x2         0           1

Name		Name	Last commit message	Last commit date
Latest commit History 69 Commits
R		R
man		man
vignettes		vignettes
.Rbuildignore		.Rbuildignore
.gitignore		.gitignore
DESCRIPTION		DESCRIPTION
LICENSE		LICENSE
NAMESPACE		NAMESPACE
README.Rmd		README.Rmd
README.md		README.md
symbolicr.Rproj		symbolicr.Rproj

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