# Introducing ‘propagate’

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With this post, I want to introduce the new ‘propagate’ package on CRAN.

It has one single purpose: propagation of uncertainties (“error propagation”). There is already one package on CRAN available for this task, named ‘metRology’ (http://cran.r-project.org/web/packages/metRology/index.html).

‘propagate’ has some additional functionality that some may find useful. The most important functions are:

* `propagate`

: A general function for the calculation of uncertainty propagation by first-/**second-order** Taylor expansion and Monte Carlo simulation including covariances. Input data can be any symbolic/numeric differentiable expression and data based on replicates, summaries (mean & s.d.) or sampled from a distribution. Uncertainty propagation is based completely on matrix calculus accounting for full covariance structure. Monte Carlo simulation is conducted using multivariate normal or t-distributions with covariance structure. The **second-order** Taylor approximation is the new aspect, because it is not based on the assumption of linearity around but uses a second-order polynomial to account for nonlinearities, making heavy use of numerical or symbolical Hessian matrices. Interestingly, the second-order approximation gives results quite similar to the MC simulations!

* `plot.propagate`

: Graphing error propagation with the histograms of the MC simulations and MC/Taylor-based confidence intervals.

* `predictNLS`

: The `propagate`

function is used to calculate the propagated error to the fitted values of a nonlinear model of type `nls`

or `nlsLM`

. Please refer to my post here: http://rmazing.wordpress.com/2013/08/26/predictnls-part-2-taylor-approximation-confidence-intervals-for-nls-models/.

* `makeGrad, makeHess, numGrad, numHess`

are functions to create symbolical or numerical gradient and Hessian matrices from an `expression`

containing first/second-order partial derivatives. These can then be evaluated in an environment with `evalDerivs`

.

* `fitDistr`

: This function fits 21 different continuous distributions by (weighted) NLS to the histogram or kernel density of the Monte Carlo simulation results as obtained by propagate or any other vector containing large-scale observations. Finally, the fits are sorted by ascending AIC.

* random samplers for 15 continuous distributions under one hood, some of them previously unavailable:

Skewed-normal distribution, Generalized normal distributionm, Scaled and shifted t-distribution, Gumbel distribution, Johnson SU distribution, Johnson SB distribution, 3P Weibull distribution, 4P Beta distribution, Triangular distribution, Trapezoidal distribution, Curvilinear Trapezoidal distribution, Generalized trapezoidal distribution, Laplacian distribution, Arcsine distribution, von Mises distribution.

Most of them sample from the inverse cumulative distribution function, but 11, 12 and 15 use a vectorized version of “Rejection Sampling” giving roughly 100000 random numbers/s.

An example (without covariance for simplicity):

:

```
>DAT <- data.frame(a = c(5, 0.1), b = c(10, 0.1), x = c(1, 0.1))
>EXPR <- expression(a^b*x)
>res <- propagate(EXPR, DAT)
Results from error propagation:
Mean.1 Mean.2 sd.1 sd.2 2.5% 97.5%
9765625 10067885 2690477 2739850 4677411 15414333
Results from Monte Carlo simulation:
Mean sd Median MAD 2.5% 97.5%
10072640 2826027 9713207 2657217 5635222 16594123
```

The plot reveals the resulting distribution obtained from Monte Carlo simulation:

```
>plot(res)
```

Seems like a skewed distributions. We can now use `fitDistr`

to find out which comes closest:

> fitDistr(res$resSIM)

Fitting Normal distribution...Done.

Fitting Skewed-normal distribution...Done.

Fitting Generalized normal distribution...Done.

Fitting Log-normal distribution...Done.

Fitting Scaled/shifted t- distribution...Done.

Fitting Logistic distribution...Done.

Fitting Uniform distribution...Done.

Fitting Triangular distribution...Done.

Fitting Trapezoidal distribution...Done.

Fitting Curvilinear Trapezoidal distribution...Done.

Fitting Generalized Trapezoidal distribution...Done.

Fitting Gamma distribution...Done.

Fitting Cauchy distribution...Done.

Fitting Laplace distribution...Done.

Fitting Gumbel distribution...Done.

Fitting Johnson SU distribution...........10.........20.........30.........40.........50

.........60.........70.........80.Done.

Fitting Johnson SB distribution...........10.........20.........30.........40.........50

.........60.........70.........80.Done.

Fitting 3P Weibull distribution...........10.........20.......Done.

Fitting 4P Beta distribution...Done.

Fitting Arcsine distribution...Done.

Fitting von Mises distribution...Done.

$aic

Distribution AIC

4 Log-normal -4917.823

16 Johnson SU -4861.960

15 Gumbel -4595.917

19 4P Beta -4509.716

12 Gamma -4469.780

9 Trapezoidal -4340.195

1 Normal -4284.706

5 Scaled/shifted t- -4283.070

6 Logistic -4266.171

3 Generalized normal -4264.102

14 Laplace -4144.870

13 Cauchy -4099.405

2 Skewed-normal -4060.936

11 Generalized Trapezoidal -4032.484

10 Curvilinear Trapezoidal -3996.495

8 Triangular -3970.993

7 Uniform -3933.513

20 Arcsine -3793.793

18 3P Weibull -3783.041

21 von Mises -3715.034

17 Johnson SB -3711.034

```
```Log-normal wins, which makes perfect sense after using an exponentiation function...

Have fun with the package. Comments welcome!

Cheers,

Andrej

Filed under: General, R Internals Tagged: confidence interval, first-order, fitting, Monte Carlo, nls, nonlinear, predict, second-order, Taylor approximation
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