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<h1>Functions to transform and backtransform kinetic parameters for fitting</h1>
</div>
<p>The transformations are intended to map parameters that should only take
on restricted values to the full scale of real numbers. For kinetic rate
constants and other paramters that can only take on positive values, a
simple log transformation is used. For compositional parameters, such as
the formations fractions that should always sum up to 1 and can not be
negative, the <code><a href='ilr.html'>ilr</a></code> transformation is used.</p>
<p>The transformation of sets of formation fractions is fragile, as it supposes
the same ordering of the components in forward and backward transformation.
This is no problem for the internal use in <code><a href='mkinfit.html'>mkinfit</a></code>.</p>
<pre><span class='fu'>transform_odeparms</span>(<span class='no'>parms</span>, <span class='no'>mkinmod</span>,
<span class='kw'>transform_rates</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>transform_fractions</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>backtransform_odeparms</span>(<span class='no'>transparms</span>, <span class='no'>mkinmod</span>,
<span class='kw'>transform_rates</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>transform_fractions</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</pre>
<h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a> Arguments</h2>
<dl class="dl-horizontal">
<dt>parms</dt>
<dd>
Parameters of kinetic models as used in the differential equations.
</dd>
<dt>transparms</dt>
<dd>
Transformed parameters of kinetic models as used in the fitting procedure.
</dd>
<dt>mkinmod</dt>
<dd>
The kinetic model of class <code><a href='mkinmod.html'>mkinmod</a></code>, containing the names of
the model variables that are needed for grouping the formation fractions
before <code><a href='ilr.html'>ilr</a></code> transformation, the parameter names and
the information if the pathway to sink is included in the model.
</dd>
<dt>transform_rates</dt>
<dd>
Boolean specifying if kinetic rate constants should be transformed in the
model specification used in the fitting for better compliance with the
assumption of normal distribution of the estimator. If TRUE, also
alpha and beta parameters of the FOMC model are log-transformed, as well
as k1 and k2 rate constants for the DFOP and HS models and the break point tb
of the HS model.
</dd>
<dt>transform_fractions</dt>
<dd>
Boolean specifying if formation fractions constants should be transformed in the
model specification used in the fitting for better compliance with the
assumption of normal distribution of the estimator. The default (TRUE) is
to do transformations. The g parameter of the DFOP and HS models are also
transformed, as they can also be seen as compositional data. The
transformation used for these transformations is the <code><a href='ilr.html'>ilr</a></code>
transformation.
</dd>
</dl>
<h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
<p>A vector of transformed or backtransformed parameters with the same names
as the original parameters.</p>
<h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
<pre class="examples"><div class='input'><span class='no'>SFO_SFO</span> <span class='kw'><-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(
<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"m1"</span>, <span class='kw'>sink</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>),
<span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>))</div><div class='output co'>#> <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'><span class='co'># Fit the model to the FOCUS example dataset D using defaults</span>
<span class='no'>fit</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>summary</span>(<span class='no'>fit</span>, <span class='kw'>data</span><span class='kw'>=</span><span class='fl'>FALSE</span>) <span class='co'># See transformed and backtransformed parameters</span></div><div class='output co'>#> mkin version: 0.9.44.9000
#> R version: 3.3.2
#> Date of fit: Fri Nov 18 22:53:30 2016
#> Date of summary: Fri Nov 18 22:53:30 2016
#>
#> Equations:
#> d_parent/dt = - k_parent_sink * parent - k_parent_m1 * parent
#> d_m1/dt = + k_parent_m1 * parent - k_m1_sink * m1
#>
#> Model predictions using solution type deSolve
#>
#> Fitted with method Port using 153 model solutions performed in 0.618 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 with symmetric confidence intervals:
#> Estimate Std. Error Lower Upper
#> parent_0 99.600 1.61400 96.330 102.900
#> log_k_parent_sink -3.038 0.07826 -3.197 -2.879
#> log_k_parent_m1 -2.980 0.04124 -3.064 -2.897
#> log_k_m1_sink -5.248 0.13610 -5.523 -4.972
#>
#> 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:
#> Confidence intervals for internally transformed parameters are asymmetric.
#> t-test (unrealistically) based on the assumption of normal distribution
#> for estimators of untransformed parameters.
#> Estimate t value Pr(>t) Lower Upper
#> parent_0 99.600000 61.720 2.024e-38 96.330000 1.029e+02
#> k_parent_sink 0.047920 12.780 3.050e-15 0.040890 5.616e-02
#> k_parent_m1 0.050780 24.250 3.407e-24 0.046700 5.521e-02
#> k_m1_sink 0.005261 7.349 5.758e-09 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</div><div class='input'>
<span class='no'>fit.2</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>transform_rates</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>summary</span>(<span class='no'>fit.2</span>, <span class='kw'>data</span><span class='kw'>=</span><span class='fl'>FALSE</span>)</div><div class='output co'>#> mkin version: 0.9.44.9000
#> R version: 3.3.2
#> Date of fit: Fri Nov 18 22:53:31 2016
#> Date of summary: Fri Nov 18 22:53:31 2016
#>
#> Equations:
#> d_parent/dt = - k_parent_sink * parent - k_parent_m1 * parent
#> d_m1/dt = + k_parent_m1 * parent - k_m1_sink * m1
#>
#> Model predictions using solution type deSolve
#>
#> Fitted with method Port using 327 model solutions performed in 1.335 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_0 100.7500 -Inf Inf
#> k_parent_sink 0.1000 0 Inf
#> k_parent_m1 0.1001 0 Inf
#> k_m1_sink 0.1002 0 Inf
#>
#> Fixed parameter values:
#> value type
#> m1_0 0 state
#>
#> Optimised, transformed parameters with symmetric confidence intervals:
#> Estimate Std. Error Lower Upper
#> parent_0 99.600000 1.6140000 96.330000 1.029e+02
#> k_parent_sink 0.047920 0.0037500 0.040310 5.553e-02
#> k_parent_m1 0.050780 0.0020940 0.046530 5.502e-02
#> k_m1_sink 0.005261 0.0007159 0.003809 6.713e-03
#>
#> Parameter correlation:
#> parent_0 k_parent_sink k_parent_m1 k_m1_sink
#> parent_0 1.00000 0.6075 -0.06625 -0.1701
#> k_parent_sink 0.60752 1.0000 -0.08740 -0.6253
#> k_parent_m1 -0.06625 -0.0874 1.00000 0.4716
#> k_m1_sink -0.17006 -0.6253 0.47164 1.0000
#>
#> Residual standard error: 3.211 on 36 degrees of freedom
#>
#> Backtransformed parameters:
#> Confidence intervals for internally transformed parameters are asymmetric.
#> t-test (unrealistically) based on the assumption of normal distribution
#> for estimators of untransformed parameters.
#> Estimate t value Pr(>t) Lower Upper
#> parent_0 99.600000 61.720 2.024e-38 96.330000 1.029e+02
#> k_parent_sink 0.047920 12.780 3.050e-15 0.040310 5.553e-02
#> k_parent_m1 0.050780 24.250 3.407e-24 0.046530 5.502e-02
#> k_m1_sink 0.005261 7.349 5.758e-09 0.003809 6.713e-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</div><div class='input'>
<span class='no'>initials</span> <span class='kw'><-</span> <span class='no'>fit</span>$<span class='no'>start</span>$<span class='no'>value</span>
<span class='fu'>names</span>(<span class='no'>initials</span>) <span class='kw'><-</span> <span class='fu'>rownames</span>(<span class='no'>fit</span>$<span class='no'>start</span>)
<span class='no'>transformed</span> <span class='kw'><-</span> <span class='no'>fit</span>$<span class='no'>start_transformed</span>$<span class='no'>value</span>
<span class='fu'>names</span>(<span class='no'>transformed</span>) <span class='kw'><-</span> <span class='fu'>rownames</span>(<span class='no'>fit</span>$<span class='no'>start_transformed</span>)
<span class='fu'>transform_odeparms</span>(<span class='no'>initials</span>, <span class='no'>SFO_SFO</span>)</div><div class='output co'>#> parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink
#> 100.750000 -2.302585 -2.301586 -2.300587 </div><div class='input'><span class='fu'>backtransform_odeparms</span>(<span class='no'>transformed</span>, <span class='no'>SFO_SFO</span>)</div><div class='output co'>#> parent_0 k_parent_sink k_parent_m1 k_m1_sink
#> 100.7500 0.1000 0.1001 0.1002 </div><div class='input'>
<span class='co'># The case of formation fractions</span>
<span class='no'>SFO_SFO.ff</span> <span class='kw'><-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(
<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"m1"</span>, <span class='kw'>sink</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>),
<span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>),
<span class='kw'>use_of_ff</span> <span class='kw'>=</span> <span class='st'>"max"</span>)</div><div class='output co'>#> <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'>
<span class='no'>fit.ff</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO.ff</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>summary</span>(<span class='no'>fit.ff</span>, <span class='kw'>data</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)</div><div class='output co'>#> mkin version: 0.9.44.9000
#> R version: 3.3.2
#> Date of fit: Fri Nov 18 22:53:32 2016
#> Date of summary: Fri Nov 18 22:53:32 2016
#>
#> Equations:
#> d_parent/dt = - k_parent * parent
#> d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1
#>
#> Model predictions using solution type deSolve
#>
#> Fitted with method Port using 185 model solutions performed in 0.749 s
#>
#> Weighting: none
#>
#> Starting values for parameters to be optimised:
#> value type
#> parent_0 100.7500 state
#> k_parent 0.1000 deparm
#> k_m1 0.1001 deparm
#> f_parent_to_m1 0.5000 deparm
#>
#> Starting values for the transformed parameters actually optimised:
#> value lower upper
#> parent_0 100.750000 -Inf Inf
#> log_k_parent -2.302585 -Inf Inf
#> log_k_m1 -2.301586 -Inf Inf
#> f_parent_ilr_1 0.000000 -Inf Inf
#>
#> Fixed parameter values:
#> value type
#> m1_0 0 state
#>
#> Optimised, transformed parameters with symmetric confidence intervals:
#> Estimate Std. Error Lower Upper
#> parent_0 99.60000 1.61400 96.3300 102.9000
#> log_k_parent -2.31600 0.04187 -2.4010 -2.2310
#> log_k_m1 -5.24800 0.13610 -5.5230 -4.9720
#> f_parent_ilr_1 0.04096 0.06477 -0.0904 0.1723
#>
#> Parameter correlation:
#> parent_0 log_k_parent log_k_m1 f_parent_ilr_1
#> parent_0 1.0000 0.5178 -0.1701 -0.5489
#> log_k_parent 0.5178 1.0000 -0.3285 -0.5451
#> log_k_m1 -0.1701 -0.3285 1.0000 0.7466
#> f_parent_ilr_1 -0.5489 -0.5451 0.7466 1.0000
#>
#> Residual standard error: 3.211 on 36 degrees of freedom
#>
#> Backtransformed parameters:
#> Confidence intervals for internally transformed parameters are asymmetric.
#> t-test (unrealistically) based on the assumption of normal distribution
#> for estimators of untransformed parameters.
#> Estimate t value Pr(>t) Lower Upper
#> parent_0 99.600000 61.720 2.024e-38 96.330000 1.029e+02
#> k_parent 0.098700 23.880 5.701e-24 0.090660 1.074e-01
#> k_m1 0.005261 7.349 5.758e-09 0.003992 6.933e-03
#> f_parent_to_m1 0.514500 22.490 4.374e-23 0.468100 5.606e-01
#>
#> Chi2 error levels in percent:
#> err.min n.optim df
#> All data 6.398 4 15
#> parent 6.459 2 7
#> m1 4.690 2 8
#>
#> Resulting formation fractions:
#> ff
#> parent_m1 0.5145
#> parent_sink 0.4855
#>
#> Estimated disappearance times:
#> DT50 DT90
#> parent 7.023 23.33
#> m1 131.761 437.70</div><div class='input'><span class='no'>initials</span> <span class='kw'><-</span> <span class='fu'>c</span>(<span class='st'>"f_parent_to_m1"</span> <span class='kw'>=</span> <span class='fl'>0.5</span>)
<span class='no'>transformed</span> <span class='kw'><-</span> <span class='fu'>transform_odeparms</span>(<span class='no'>initials</span>, <span class='no'>SFO_SFO.ff</span>)
<span class='fu'>backtransform_odeparms</span>(<span class='no'>transformed</span>, <span class='no'>SFO_SFO.ff</span>)</div><div class='output co'>#> f_parent_to_m1
#> 0.5 </div><div class='input'>
<span class='co'># And without sink</span>
<span class='no'>SFO_SFO.ff.2</span> <span class='kw'><-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(
<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"m1"</span>, <span class='kw'>sink</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>),
<span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>),
<span class='kw'>use_of_ff</span> <span class='kw'>=</span> <span class='st'>"max"</span>)</div><div class='output co'>#> <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'>
<span class='no'>fit.ff.2</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO.ff.2</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>summary</span>(<span class='no'>fit.ff.2</span>, <span class='kw'>data</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)</div><div class='output co'>#> mkin version: 0.9.44.9000
#> R version: 3.3.2
#> Date of fit: Fri Nov 18 22:53:33 2016
#> Date of summary: Fri Nov 18 22:53:33 2016
#>
#> Equations:
#> d_parent/dt = - k_parent * parent
#> d_m1/dt = + k_parent * parent - k_m1 * m1
#>
#> Model predictions using solution type deSolve
#>
#> Fitted with method Port using 104 model solutions performed in 0.426 s
#>
#> Weighting: none
#>
#> Starting values for parameters to be optimised:
#> value type
#> parent_0 100.7500 state
#> k_parent 0.1000 deparm
#> k_m1 0.1001 deparm
#>
#> Starting values for the transformed parameters actually optimised:
#> value lower upper
#> parent_0 100.750000 -Inf Inf
#> log_k_parent -2.302585 -Inf Inf
#> log_k_m1 -2.301586 -Inf Inf
#>
#> Fixed parameter values:
#> value type
#> m1_0 0 state
#>
#> Optimised, transformed parameters with symmetric confidence intervals:
#> Estimate Std. Error Lower Upper
#> parent_0 84.790 2.96500 78.78 90.800
#> log_k_parent -2.756 0.08088 -2.92 -2.593
#> log_k_m1 -4.214 0.11150 -4.44 -3.988
#>
#> Parameter correlation:
#> parent_0 log_k_parent log_k_m1
#> parent_0 1.0000 0.11059 0.46156
#> log_k_parent 0.1106 1.00000 0.06274
#> log_k_m1 0.4616 0.06274 1.00000
#>
#> Residual standard error: 8.333 on 37 degrees of freedom
#>
#> Backtransformed parameters:
#> Confidence intervals for internally transformed parameters are asymmetric.
#> t-test (unrealistically) based on the assumption of normal distribution
#> for estimators of untransformed parameters.
#> Estimate t value Pr(>t) Lower Upper
#> parent_0 84.79000 28.600 3.939e-27 78.78000 90.80000
#> k_parent 0.06352 12.360 5.237e-15 0.05392 0.07483
#> k_m1 0.01478 8.966 4.114e-11 0.01179 0.01853
#>
#> Chi2 error levels in percent:
#> err.min n.optim df
#> All data 19.66 3 16
#> parent 17.56 2 7
#> m1 18.71 1 9
#>
#> Estimated disappearance times:
#> DT50 DT90
#> parent 10.91 36.25
#> m1 46.89 155.75</div><div class='input'>
</div></pre>
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<h2>Contents</h2>
<ul class="nav nav-pills nav-stacked">
<li><a href="#arguments">Arguments</a></li>
<li><a href="#value">Value</a></li>
<li><a href="#examples">Examples</a></li>
</ul>
<h2>Author</h2>
Johannes Ranke
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