Is fitting for several dependent variables possible

[jranke]

From the documentation, it is not clear to me how one would go about fitting functions that return several dependent variables for the same independent variable, as occurs e.g. when fitting systems of ordinary differential equations. A simple example would be

x1' = - k1 * x1 x2' = k1 * x1 - k2 x2

I could provide a test case with data if you should be interested.

[sczesla]

Hi, sounds like an interesting question, but I am not sure whether I fully understand what you want to do. Below, I provide an example of how I would approach the problem of estimating k1,2. Here, I set up a custom model and combined x1' and x2' into one "data" array; I am not sure whether this really is what you are going at, however. Indeed, I would be interested in a test case. Cheers, Stefan

from PyAstronomy import funcFit as fuf import matplotlib.pylab as plt import numpy as np

k1 = 0.764 k2 = 1.96

x1 = np.linspace(0., 1., 100) x2 = np.linspace(-1., 1., 100)

y = -k1_x1 z = k1_x1 - k2*x2

yerr = np.ones(len(x1)) * 0.01 y += np.random.normal(0.0, 0.01, len(x1)) z += np.random.normal(0.0, 0.01, len(x1))

class XY(fuf.OneDFit):

def init(self): fuf.OneDFit.init(self, ["k1", "k2"])

def evaluate(self, x): y = -self["k1"]_x[0] z = self["k1"]_x[0] - self["k2"]*x[1] return np.concatenate((y,z))

xy = XY() xy["k1"] = 1.0 xy["k2"] = 1.0 xy.thaw(["k1", "k2"])

xy.fit((x1,x2), np.concatenate((y,z)), yerr=np.concatenate((yerr,yerr)))

xy.parameterSummary()

[jranke]

Hi,

sorry, I did not see your comment until today, when I came back from holiday and read the message that the issue was closed...

Your code above does not take into account that y should be a derivative of x1 (x1') and z should be a derivative of x2 (x2'). Therefore, within the optimisation routine, we need to solve the system of ODEs specified by x1' and x2'.

Here a simple test case evaluated with the R package mkin (I was looking into alternatives e.g. written in python when I opened this issue):

This is just a very simple vignette showing how to fit a degradation model for a parent compound with one transformation product using mkin. After loading the library we look a the data. We have observed concentrations in the column named value at the times specified in column time for the two observed variables named parent and m1.

library("mkin")
print(FOCUS_2006_D)

##      name time  value
## 1  parent    0  99.46
## 2  parent    0 102.04
## 3  parent    1  93.50
## 4  parent    1  92.50
## 5  parent    3  63.23
## 6  parent    3  68.99
## 7  parent    7  52.32
## 8  parent    7  55.13
## 9  parent   14  27.27
## 10 parent   14  26.64
## 11 parent   21  11.50
## 12 parent   21  11.64
## 13 parent   35   2.85
## 14 parent   35   2.91
## 15 parent   50   0.69
## 16 parent   50   0.63
## 17 parent   75   0.05
## 18 parent   75   0.06
## 19 parent  100     NA
## 20 parent  100     NA
## 21 parent  120     NA
## 22 parent  120     NA
## 23     m1    0   0.00
## 24     m1    0   0.00
## 25     m1    1   4.84
## 26     m1    1   5.64
## 27     m1    3  12.91
## 28     m1    3  12.96
## 29     m1    7  22.97
## 30     m1    7  24.47
## 31     m1   14  41.69
## 32     m1   14  33.21
## 33     m1   21  44.37
## 34     m1   21  46.44
## 35     m1   35  41.22
## 36     m1   35  37.95
## 37     m1   50  41.19
## 38     m1   50  40.01
## 39     m1   75  40.09
## 40     m1   75  33.85
## 41     m1  100  31.04
## 42     m1  100  33.13
## 43     m1  120  25.15
## 44     m1  120  33.31

Next we specify the degradation model: The parent compound degrades with simple first-order kinetics (SFO) to one metabolite named m1, which also degrades with SFO kinetics.

The call to mkinmod returns a degradation model. The differential equations represented in R code can be found in the character vector $diffs of the mkinmod object. If the ccSolve package is installed and functional, the differential equation model will be compiled from auto-generated C code.

SFO_SFO <- mkinmod(parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"))

## Compiling differential equation model from auto-generated C code...

print(SFO_SFO$diffs)

##                                                       parent 
## "d_parent = - k_parent_sink * parent - k_parent_m1 * parent" 
##                                                           m1 
##             "d_m1 = + k_parent_m1 * parent - k_m1_sink * m1"

We do the fitting without progress report (quiet = TRUE).

fit <- mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE)

A comprehensive report of the results is obtained using the summary method for mkinfit objects.

summary(fit)

## mkin version:    0.9.36 
## R version:       3.2.0 
## Date of fit:     Fri Jun  5 13:29:10 2015 
## Date of summary: Fri Jun  5 13:29:10 2015 
## 
## Equations:
## d_parent = - k_parent_sink * parent - k_parent_m1 * parent
## d_m1 = + k_parent_m1 * parent - k_m1_sink * m1
## 
## Model predictions using solution type deSolve 
## 
## Fitted with method Port using 153 model solutions performed in 0.592 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##                  value   type
## parent_0      100.7500  state
## k_parent_sink   0.1000 deparm
## k_parent_m1     0.1001 deparm
## k_m1_sink       0.1002 deparm
## 
## Starting values for the transformed parameters actually optimised:
##                        value lower upper
## parent_0          100.750000  -Inf   Inf
## log_k_parent_sink  -2.302585  -Inf   Inf
## log_k_parent_m1    -2.301586  -Inf   Inf
## log_k_m1_sink      -2.300587  -Inf   Inf
## 
## Fixed parameter values:
##      value  type
## m1_0     0 state
## 
## Optimised, transformed parameters:
##                   Estimate Std. Error  Lower   Upper t value  Pr(>|t|)
## parent_0            99.600    1.61400 96.330 102.900   61.72 4.048e-38
## log_k_parent_sink   -3.038    0.07826 -3.197  -2.879  -38.82 5.601e-31
## log_k_parent_m1     -2.980    0.04124 -3.064  -2.897  -72.27 1.446e-40
## log_k_m1_sink       -5.248    0.13610 -5.523  -4.972  -38.56 7.087e-31
##                      Pr(>t)
## parent_0          2.024e-38
## log_k_parent_sink 2.800e-31
## log_k_parent_m1   7.228e-41
## log_k_m1_sink     3.543e-31
## 
## Parameter correlation:
##                   parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink
## parent_0           1.00000            0.6075        -0.06625       -0.1701
## log_k_parent_sink  0.60752            1.0000        -0.08740       -0.6253
## log_k_parent_m1   -0.06625           -0.0874         1.00000        0.4716
## log_k_m1_sink     -0.17006           -0.6253         0.47163        1.0000
## 
## Residual standard error: 3.211 on 36 degrees of freedom
## 
## Backtransformed parameters:
##                Estimate     Lower     Upper
## parent_0      99.600000 96.330000 1.029e+02
## k_parent_sink  0.047920  0.040890 5.616e-02
## k_parent_m1    0.050780  0.046700 5.521e-02
## k_m1_sink      0.005261  0.003992 6.933e-03
## 
## Chi2 error levels in percent:
##          err.min n.optim df
## All data   6.398       4 15
## parent     6.827       3  6
## m1         4.490       1  9
## 
## Resulting formation fractions:
##                 ff
## parent_sink 0.4855
## parent_m1   0.5145
## m1_sink     1.0000
## 
## Estimated disappearance times:
##           DT50   DT90
## parent   7.023  23.33
## m1     131.761 437.70
## 
## Data:
##  time variable observed predicted   residual
##     0   parent    99.46 9.960e+01 -1.385e-01
##     0   parent   102.04 9.960e+01  2.442e+00
##     1   parent    93.50 9.024e+01  3.262e+00
##     1   parent    92.50 9.024e+01  2.262e+00
##     3   parent    63.23 7.407e+01 -1.084e+01
##     3   parent    68.99 7.407e+01 -5.083e+00
##     7   parent    52.32 4.991e+01  2.408e+00
##     7   parent    55.13 4.991e+01  5.218e+00
##    14   parent    27.27 2.501e+01  2.257e+00
##    14   parent    26.64 2.501e+01  1.627e+00
##    21   parent    11.50 1.253e+01 -1.035e+00
##    21   parent    11.64 1.253e+01 -8.946e-01
##    35   parent     2.85 3.148e+00 -2.979e-01
##    35   parent     2.91 3.148e+00 -2.379e-01
##    50   parent     0.69 7.162e-01 -2.624e-02
##    50   parent     0.63 7.162e-01 -8.624e-02
##    75   parent     0.05 6.074e-02 -1.074e-02
##    75   parent     0.06 6.074e-02 -7.382e-04
##   100   parent       NA 5.151e-03         NA
##   100   parent       NA 5.151e-03         NA
##   120   parent       NA 7.155e-04         NA
##   120   parent       NA 7.155e-04         NA
##     0       m1     0.00 0.000e+00  0.000e+00
##     0       m1     0.00 0.000e+00  0.000e+00
##     1       m1     4.84 4.803e+00  3.704e-02
##     1       m1     5.64 4.803e+00  8.370e-01
##     3       m1    12.91 1.302e+01 -1.140e-01
##     3       m1    12.96 1.302e+01 -6.400e-02
##     7       m1    22.97 2.504e+01 -2.075e+00
##     7       m1    24.47 2.504e+01 -5.748e-01
##    14       m1    41.69 3.669e+01  5.000e+00
##    14       m1    33.21 3.669e+01 -3.480e+00
##    21       m1    44.37 4.165e+01  2.717e+00
##    21       m1    46.44 4.165e+01  4.787e+00
##    35       m1    41.22 4.331e+01 -2.093e+00
##    35       m1    37.95 4.331e+01 -5.363e+00
##    50       m1    41.19 4.122e+01 -2.831e-02
##    50       m1    40.01 4.122e+01 -1.208e+00
##    75       m1    40.09 3.645e+01  3.643e+00
##    75       m1    33.85 3.645e+01 -2.597e+00
##   100       m1    31.04 3.198e+01 -9.416e-01
##   100       m1    33.13 3.198e+01  1.148e+00
##   120       m1    25.15 2.879e+01 -3.640e+00
##   120       m1    33.31 2.879e+01  4.520e+00