Next Kölner R User Meeting: Friday, 4 December 2015
The 16th Cologne R user group meeting is scheduled for this Friday, 4 December 2015 and we have great line up with three talks followed by networking drinks.
- Monitoring process change using Bayesian methods (Mick Cooney)
A common business problem is to evaluate the effect of a change of process, and this talk will discuss a straightforward approach to this using conjugate priors.
- Editing R files with DataJoy (Dietmar Janetzko)
Brief introduction to the online collaborative data analysis platform DataJoy.
- Emoticons decoder for social media sentiment analysis in R (Jessica Peterka-Bonetta)
While R seems to be able to display some emoticons properly, many other times it doesn’t, making any further analysis impossible unless you get rid of them. With a little hack, I decoded these emoticons and put them all in a dictionary for further use.
Drinks and Networking
The event will be followed by drinks (Kölsch!) and networking opportunities.For further details visit our KölnRUG Meetup site. Please sign up if you would like to come along. Notes from past meetings are available here.
The organisers, Kirill Pomogajko and Markus Gesmann, gratefully acknowledge the sponsorship of Revolution Analytics, who support the Cologne R user group as part of their Matrix programme.
Kölner R Users
,
KölnR
,
News
,
R
Notes from Warsaw R meetup
I had the great pleasure time to attend the Warsaw R meetup last Thursday. The organisers Olga Mierzwa and Przemyslaw Biecek had put together an event with a focus on R in Insurance (btw, there is a conference with the same name), discussing examples of pricing and reserving in general and life insurance.
As an example, Emilia talked about the claims reserving process, which in her view should be embedded in the full life cycle of insurance, namely product development, claims, risk and performance management. Following this thought, she presented an idea for claims reserving that models the life of a claim from not incurred and not reported (NINR), to incurred but not reported (IBNR), reported but not settled (RBNS) and finally paid.
Although life and non-life companies are usually separated for regulatory reasons, they both share the fundamental challenge of predicting future cash flows. An example where the two industries meet is product liability.
Over the last century, technology has changed our environment fundamentally, more so than ever before. Yet, we still don't know which long-term impact some of the new technologies and products will have on our life expectancy. Some will prolong our lives, others may make us ill.
A classic example is asbestos, initial regarded as a miracle mineral, as it was impossible to set on fire, abundant, cheap to mine, and easy to manufacture. Not surprisingly it was widely used until it was linked to causing cancer. Over the last 35 years, the non-life insurance industry has paid well in excess of hundred billion dollars in compensations.
Experience vs. Data
I kicked off with some observations of the challenges in insurance pricing. Accidents are thankfully rare events, that's why we buy insurance. Hence, there is often not a lot of claims data available for pricing. Combining the information from historical data and experts with domain knowledge can provide a rich basis for the assessment of risk. I presented some examples using Bayesian analysis to understand the probability of an event occurring. Regular readers of my blog will recognise the examples from earlier posts. You find my slides on GitHub. |
| Download slides |
Non-life insurance in R
Emilia Kalarus from Triple A shared some of her experience of using R in non-life insurance companies. She focused on the challenges in working across teams, with different systems, data sets and mentalities.As an example, Emilia talked about the claims reserving process, which in her view should be embedded in the full life cycle of insurance, namely product development, claims, risk and performance management. Following this thought, she presented an idea for claims reserving that models the life of a claim from not incurred and not reported (NINR), to incurred but not reported (IBNR), reported but not settled (RBNS) and finally paid.
Stochastic mortality modelling
The final talk was given by Adam Wróbel from the life insurer Nationale Nederlanden, discussing stochastic mortality modelling. Adam's talk on analysing mortality made me realise that life and non-life insurance companies may be much closer to each other than I thought.Although life and non-life companies are usually separated for regulatory reasons, they both share the fundamental challenge of predicting future cash flows. An example where the two industries meet is product liability.
Over the last century, technology has changed our environment fundamentally, more so than ever before. Yet, we still don't know which long-term impact some of the new technologies and products will have on our life expectancy. Some will prolong our lives, others may make us ill.
A classic example is asbestos, initial regarded as a miracle mineral, as it was impossible to set on fire, abundant, cheap to mine, and easy to manufacture. Not surprisingly it was widely used until it was linked to causing cancer. Over the last 35 years, the non-life insurance industry has paid well in excess of hundred billion dollars in compensations.
Slides and Code
The slides and R code of the presentations are hosted on the Warsaw R GitHub page.
Insurance
,
R
,
R in Insurance
Hierarchical Loss Reserving with Stan
I continue with the growth curve model for loss reserving from last week's post. Today, following the ideas of James Guszcza [2] I will add an hierarchical component to the model, by treating the ultimate loss cost of an accident year as a random effect. Initially, I will use the
Last week's model assumed that cumulative claims payment could be described by a growth curve. I used the Weibull curve and will do so here again, but others should be considered as well, e.g. the log-logistic cumulative distribution function for long tail business, see [1].
The growth curve describes the proportion of claims paid up to a given development period compared to the ultimate claims cost at the end of time, hence often called development pattern. Cumulative distribution functions are often considered, as they increase monotonously from 0 to 100%. Multiplying the development pattern with the expected ultimate loss cost gives me then the expected cumulative paid to date value.
However, what I'd like to do is the opposite, I know the cumulative claims position to date and wish to estimate the ultimate claims cost instead. If the claims process is fairly stable over the years and say, once a claim has been notified the payment process is quite similar from year to year and claim to claim, then a growth curve model is not unreasonable. Yet, the number and the size of the yearly claims will be random, e.g. if a windstorm, fire, etc occurs or not. Hence, a random effect for the ultimate loss cost across accident years sounds very convincing to me.
Here is James' model as described in [2]:
\[
\begin{align}
CL_{AY, dev} & \sim \mathcal{N}(\mu_{AY, dev}, \sigma^2_{dev}) \\
\mu_{AY,dev} & = Ult_{AY} \cdot G(dev|\omega, \theta)\\
\sigma_{dev} & = \sigma \sqrt{\mu_{dev}}\\
Ult_{AY} & \sim \mathcal{N}(\mu_{ult}, \sigma^2_{ult})\\
G(dev|\omega, \theta) & = 1 - \exp\left(-\left(\frac{dev}{\theta}\right)^\omega\right)
\end{align}
\]The cumulative losses \(CL_{AY, dev}\) for a given accident year \(AY\) and development period \(dev\) follow a Normal distribution with parameters \(\mu_{AY, dev}\) and \(\sigma_{dev}\).
The mean itself is modelled as the product of an accident year specific ultimate loss cost \(Ult_{AY}\) and a development period specific parametric growth curve \(G(dev | \omega, \theta)\). The variance is believed to increase in proportion with the mean. Finally, the ultimate loss cost is modelled with a Normal distribution as well.
Assuming a Gaussian distribution of losses doesn't sound quite intuitive to me, as loss are often skewed to the right, but I shall continue with this assumption here to make a comparison with [2] possible.
Using the example data set given in the paper I can reproduce the result in R with
The fit looks pretty good, with only 5 parameters. See James' paper for a more detailed discussion.
Let's move this model into Stan. Here is my attempt, which builds on last week's pooled model. With the generated quantities code block I go beyond the scope of the original paper, as I try to estimate the full posterior predictive distribution as well.
The 'trick' is the line
The notation
Let's run the model:
The estimated parameters look very similar to the
Let's take a look at the parameter traceplot and the densities of the estimated ultimate loss costs by origin year.
This looks all not too bad. The trace plots don't show any particular patterns, apart from \(\sigma_{ult}\), which shows a little skewness.
The
The model seems to work rather well, even with the Gaussian distribution assumptions. Yet, it has still only 5 parameters. Note, this model doesn't need an additional artificial tail factor either.
James Guszcza contributed to a follow-up paper with Y. Zhank and V. Dukic [3] that extends the model described in [2]. It deals with skewness in loss data sets and the autoregressive nature of the errors in a cumulative time series.
Frank Schmid offers a more complex Bayesian analysis of claims reserving in [4], while Jake Morris highlights the similarities between a compartmental model used in drug research and loss reserving [5].
Finally, Glenn Meyers published a monograph on Stochastic Loss Reserving Using Bayesian MCMC Models earlier this year [7] that is worth taking a look at.
[2] James Guszcza. Hierarchical Growth Curve Models for Loss Reserving, 2008, CAS Fall Forum, pp. 146–173.
[3] Y. Zhang, V. Dukic, and James Guszcza. A Bayesian non-linear model for forecasting insurance loss payments. 2012. Journal of the Royal Statistical Society: Series A (Statistics in Society), 175: 637–656. doi: 10.1111/j.1467-985X.2011.01002.x
[4] Frank A. Schmid. Robust Loss Development Using MCMC. Available at SSRN. See also http://lossdev.r-forge.r-project.org/
[5] Jake Morris. Compartmental reserving in R. 2015. R in Insurance Conference.
[6] Stan Development Team. Stan: A C++ Library for Probability and Sampling, Version 2.8.0. 2015. http://mc-stan.org/.
[7] Glenn Meyers. Stochastic Loss Reserving Using Bayesian MCMC Models. Issue 1 of CAS Monograph Series. 2015.
nlme R package, just as James did in his paper, and then move on to Stan/RStan [6], which will allow me to estimate the full distribution of future claims payments.Last week's model assumed that cumulative claims payment could be described by a growth curve. I used the Weibull curve and will do so here again, but others should be considered as well, e.g. the log-logistic cumulative distribution function for long tail business, see [1].
However, what I'd like to do is the opposite, I know the cumulative claims position to date and wish to estimate the ultimate claims cost instead. If the claims process is fairly stable over the years and say, once a claim has been notified the payment process is quite similar from year to year and claim to claim, then a growth curve model is not unreasonable. Yet, the number and the size of the yearly claims will be random, e.g. if a windstorm, fire, etc occurs or not. Hence, a random effect for the ultimate loss cost across accident years sounds very convincing to me.
Here is James' model as described in [2]:
\[
\begin{align}
CL_{AY, dev} & \sim \mathcal{N}(\mu_{AY, dev}, \sigma^2_{dev}) \\
\mu_{AY,dev} & = Ult_{AY} \cdot G(dev|\omega, \theta)\\
\sigma_{dev} & = \sigma \sqrt{\mu_{dev}}\\
Ult_{AY} & \sim \mathcal{N}(\mu_{ult}, \sigma^2_{ult})\\
G(dev|\omega, \theta) & = 1 - \exp\left(-\left(\frac{dev}{\theta}\right)^\omega\right)
\end{align}
\]The cumulative losses \(CL_{AY, dev}\) for a given accident year \(AY\) and development period \(dev\) follow a Normal distribution with parameters \(\mu_{AY, dev}\) and \(\sigma_{dev}\).
The mean itself is modelled as the product of an accident year specific ultimate loss cost \(Ult_{AY}\) and a development period specific parametric growth curve \(G(dev | \omega, \theta)\). The variance is believed to increase in proportion with the mean. Finally, the ultimate loss cost is modelled with a Normal distribution as well.
Assuming a Gaussian distribution of losses doesn't sound quite intuitive to me, as loss are often skewed to the right, but I shall continue with this assumption here to make a comparison with [2] possible.
Using the example data set given in the paper I can reproduce the result in R with
nlme:Let's move this model into Stan. Here is my attempt, which builds on last week's pooled model. With the generated quantities code block I go beyond the scope of the original paper, as I try to estimate the full posterior predictive distribution as well.
The 'trick' is the line
mu[i] <- ult[origin[i]] * weibull_cdf(dev[i], omega, theta); where I have an accident year (here labelled origin) specific ultimate loss. The notation
ult[origin[i]] illustrates the hierarchical nature in Stan's language nicely.Let's run the model:
The estimated parameters look very similar to the
nlme output above.Let's take a look at the parameter traceplot and the densities of the estimated ultimate loss costs by origin year.
The
generated quantities code block in Stan allows me to get also the predictive distribution beyond the current data range. Here I forecast claims up to development year 12 and plot the predictions, including the 95% credibility interval of the posterior predictive distribution with the observations. Conclusions
The Bayesian approach sounds to me a lot more natural than many classical techniques around the chain-ladder methods. Thanks to Stan, I can get the full posterior distributions on both, the parameters and predictive distribution. I find communicating credibility intervals much easier than talking about the parameter, process and mean squared error.James Guszcza contributed to a follow-up paper with Y. Zhank and V. Dukic [3] that extends the model described in [2]. It deals with skewness in loss data sets and the autoregressive nature of the errors in a cumulative time series.
Frank Schmid offers a more complex Bayesian analysis of claims reserving in [4], while Jake Morris highlights the similarities between a compartmental model used in drug research and loss reserving [5].
Finally, Glenn Meyers published a monograph on Stochastic Loss Reserving Using Bayesian MCMC Models earlier this year [7] that is worth taking a look at.
References
[1] David R. Clark. LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach. Casualty Actuarial Society, 2003. CAS Fall Forum.[2] James Guszcza. Hierarchical Growth Curve Models for Loss Reserving, 2008, CAS Fall Forum, pp. 146–173.
[3] Y. Zhang, V. Dukic, and James Guszcza. A Bayesian non-linear model for forecasting insurance loss payments. 2012. Journal of the Royal Statistical Society: Series A (Statistics in Society), 175: 637–656. doi: 10.1111/j.1467-985X.2011.01002.x
[4] Frank A. Schmid. Robust Loss Development Using MCMC. Available at SSRN. See also http://lossdev.r-forge.r-project.org/
[5] Jake Morris. Compartmental reserving in R. 2015. R in Insurance Conference.
[6] Stan Development Team. Stan: A C++ Library for Probability and Sampling, Version 2.8.0. 2015. http://mc-stan.org/.
[7] Glenn Meyers. Stochastic Loss Reserving Using Bayesian MCMC Models. Issue 1 of CAS Monograph Series. 2015.
Session Info
R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.1 (El Capitan)
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ChainLadder_0.2.3 rstan_2.8.0 ggplot2_1.0.1 Rcpp_0.12.1
[5] lattice_0.20-33
loaded via a namespace (and not attached):
[1] nloptr_1.0.4 plyr_1.8.3 tools_3.2.2
[4] digest_0.6.8 lme4_1.1-10 statmod_1.4.21
[7] gtable_0.1.2 nlme_3.1-122 mgcv_1.8-8
[10] Matrix_1.2-2 parallel_3.2.2 biglm_0.9-1
[13] SparseM_1.7 proto_0.3-10 coda_0.18-1
[16] gridExtra_2.0.0 stringr_1.0.0 MatrixModels_0.4-1
[19] lmtest_0.9-34 stats4_3.2.2 grid_3.2.2
[22] nnet_7.3-11 tweedie_2.2.1 inline_0.3.14
[25] cplm_0.7-4 minqa_1.2.4 actuar_1.1-10
[28] reshape2_1.4.1 car_2.1-0 magrittr_1.5
[31] scales_0.3.0 codetools_0.2-14 MASS_7.3-44
[34] splines_3.2.2 rsconnect_0.3.79 systemfit_1.1-18
[37] pbkrtest_0.4-2 colorspace_1.2-6 quantreg_5.19
[40] labeling_0.3 sandwich_2.3-4 stringi_1.0-1
[43] munsell_0.4.2 zoo_1.7-12
Insurance
,
R
,
reserving
,
Stan
,
stochastic reserving
Loss Developments via Growth Curves and Stan
Last week I posted a biological example of fitting a non-linear growth curve with Stan/RStan. Today, I want to apply a similar approach to insurance data using ideas by David Clark [1] and James Guszcza [2].
Instead of predicting the growth of dugongs (sea cows), I would like to predict the growth of cumulative insurance loss payments over time, originated from different origin years. Loss payments of younger accident years are just like a new generation of dugongs, they will be small in size initially, grow as they get older, until the losses are fully settled.
Here is my example data set:
Following the articles cited above I will assume that the growth can be explained by a Weibull curve, with two parameters \(\theta\) (scale) and \(\omega\) (shape):
\[
G(dev|\omega, \theta) = 1 - \exp\left(-\left(\frac{dev}{\theta}\right)^\omega\right)
\]Inspired by the classical over-dispersed Poisson GLM in actuarial science, Guszcza [2] assumes a power variance structure for the process error as well. For the purpose of this post I will assume that claims cost follow a Normal distribution, to make a comparison with a the classical least square regression easier. With a prior estimate of the ultimate claims cost the cumulative loss can be modelled as:
\[
\begin{align}
CL_{AY, dev} & \sim \mathcal{N}(\mu_{dev}, \sigma^2_{dev}) \\
\mu_{dev} & = Ult \cdot G(dev|\omega, \theta)\\
\sigma_{dev} & = \sigma \sqrt{\mu_{dev}}
\end{align}
\]Perhaps, the above formula suggests a hierarchical modelling approach, with different ultimate loss costs by accident year. Indeed, this is the focus of [2] and I will endeavour to reproduce the results with Stan in my next post, but today I will stick to a pooled model that assumes a constant ultimate loss across all accident years, i.e. \(Ult_{AY} = Ult \;\forall\; AY\).
To prepare my analysis I read in the data as a long data frame instead of the matrix structure above. Additionally, I compose another data frame that I will use later to predict payments two years beyond the first 10 years. Furthermore, to align the output with [2] I relabelled the development periods from years to months, so that year 1 becomes month 6, year 2 becomes month 18, etc. The accident years run from 1991 to 2000, while the variable
To get a reference point for Stan I start with a non-linear least square regression:
The output above doesn't look unreasonable, apart from the accident years 1991, 1992 and 1998. The output of
The Stan code below is very similar to last week. Again, I am interested here in the posterior distributions, hence I add a block to generate quantities from those. Note, Stan comes with a build-in function for the cumulative Weibull distribution
I store the Stan code in a separate text file, which I read into R to compile the model. The compilation takes a little time. The sampling itself is done in a few seconds.
Let's take a look at the output:
The parameters are a little different to the output of
Finally, I can plot the 90% credible intervals of the posterior distributions.
The 90% prediction credible interval captures most of the data and although this model might not be suitable for reserving individual accident years, it could provide an initial starting point for further investigations. Additionally, thanks to the Bayesian model I have access to the full distribution, not just point estimators and standard errors.
My next post will continue with this data set and the ideas of James Guszcza by allowing the ultimate loss cost to vary by accident year, treating it as a random effect. Here is a teaser of what the output will look like:
[2] Guszcza, J. Hierarchical Growth Curve Models for Loss Reserving, 2008, CAS Fall Forum,
pp. 146–173.
Instead of predicting the growth of dugongs (sea cows), I would like to predict the growth of cumulative insurance loss payments over time, originated from different origin years. Loss payments of younger accident years are just like a new generation of dugongs, they will be small in size initially, grow as they get older, until the losses are fully settled.
Here is my example data set:
\[
G(dev|\omega, \theta) = 1 - \exp\left(-\left(\frac{dev}{\theta}\right)^\omega\right)
\]Inspired by the classical over-dispersed Poisson GLM in actuarial science, Guszcza [2] assumes a power variance structure for the process error as well. For the purpose of this post I will assume that claims cost follow a Normal distribution, to make a comparison with a the classical least square regression easier. With a prior estimate of the ultimate claims cost the cumulative loss can be modelled as:
\[
\begin{align}
CL_{AY, dev} & \sim \mathcal{N}(\mu_{dev}, \sigma^2_{dev}) \\
\mu_{dev} & = Ult \cdot G(dev|\omega, \theta)\\
\sigma_{dev} & = \sigma \sqrt{\mu_{dev}}
\end{align}
\]Perhaps, the above formula suggests a hierarchical modelling approach, with different ultimate loss costs by accident year. Indeed, this is the focus of [2] and I will endeavour to reproduce the results with Stan in my next post, but today I will stick to a pooled model that assumes a constant ultimate loss across all accident years, i.e. \(Ult_{AY} = Ult \;\forall\; AY\).
To prepare my analysis I read in the data as a long data frame instead of the matrix structure above. Additionally, I compose another data frame that I will use later to predict payments two years beyond the first 10 years. Furthermore, to align the output with [2] I relabelled the development periods from years to months, so that year 1 becomes month 6, year 2 becomes month 18, etc. The accident years run from 1991 to 2000, while the variable
origin maps those years from 1 to 10.To get a reference point for Stan I start with a non-linear least square regression:
gnls gives me an opportunity to provide my prior distributions with good starting points. I will use an inverse Gamma as a prior for \(\sigma\), constrained Normals for the parameters \(\theta, \omega\) and \(Ult\) as well. If you have better ideas, then please get in touch.The Stan code below is very similar to last week. Again, I am interested here in the posterior distributions, hence I add a block to generate quantities from those. Note, Stan comes with a build-in function for the cumulative Weibull distribution
weibull_cdf.I store the Stan code in a separate text file, which I read into R to compile the model. The compilation takes a little time. The sampling itself is done in a few seconds.
Let's take a look at the output:
gnls, but well within the standard error. From the plots I notice that the samples for the ultimate loss as well as for \(\theta\) are a little skewed to the right. Well, assuming cumulative losses to follow a Normal distribution was a bit a of stretch to start with. Still, the samples seem to converge.Finally, I can plot the 90% credible intervals of the posterior distributions.
My next post will continue with this data set and the ideas of James Guszcza by allowing the ultimate loss cost to vary by accident year, treating it as a random effect. Here is a teaser of what the output will look like:
References
[1] David R. Clark. LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach. Casualty Actuarial Society, 2003. CAS Fall Forum.[2] Guszcza, J. Hierarchical Growth Curve Models for Loss Reserving, 2008, CAS Fall Forum,
pp. 146–173.
Session Info
R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.1 (El Capitan)
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods
[7] base
other attached packages:
[1] rstan_2.8.0 ggplot2_1.0.1 Rcpp_0.12.1
[4] nlme_3.1-122 lattice_0.20-33 ChainLadder_0.2.3
loaded via a namespace (and not attached):
[1] nloptr_1.0.4 plyr_1.8.3 tools_3.2.2
[4] digest_0.6.8 lme4_1.1-10 statmod_1.4.21
[7] gtable_0.1.2 mgcv_1.8-8 Matrix_1.2-2
[10] parallel_3.2.2 biglm_0.9-1 SparseM_1.7
[13] proto_0.3-10 gridExtra_2.0.0 coda_0.18-1
[16] stringr_1.0.0 MatrixModels_0.4-1 stats4_3.2.2
[19] lmtest_0.9-34 grid_3.2.2 nnet_7.3-11
[22] inline_0.3.14 tweedie_2.2.1 cplm_0.7-4
[25] minqa_1.2.4 reshape2_1.4.1 car_2.1-0
[28] actuar_1.1-10 magrittr_1.5 codetools_0.2-14
[31] scales_0.3.0 MASS_7.3-44 splines_3.2.2
[34] systemfit_1.1-18 rsconnect_0.3.79 pbkrtest_0.4-2
[37] colorspace_1.2-6 labeling_0.3 quantreg_5.19
[40] sandwich_2.3-4 stringi_1.0-1 munsell_0.4.2
[43] zoo_1.7-12Non-linear growth curves with Stan
I suppose the go to tool for fitting non-linear models in R is
The original example itself is taken from OpenBUGS. The data describes the length and age measurements for 27 captured dugongs (sea cows). Carlin and Gelfand (1991) model the data using a nonlinear growth curve with no inflection point and an asymptote as \(x_i\) tends to infinity:
\[
Y_i \sim \mathcal{N}(\mu_i, \sigma^2),\; i = 1,\dots,27\\
\mu_i = \alpha - \beta \lambda^{x_i},\; \alpha,\, \beta > 0;\, 0 < \lambda < 1 \] Fitting the curve with
Writing the model in Stan requires a few more lines, but gives me also the opportunity to generate output from the posterior distributions.
The predicted parameters and errors are very much the same as in the least square output of
nls of the stats package. In this post I will show an alternative approach with Stan/RStan, as illustrated in the example, Dugongs: "nonlinear growth curve", that is part of Stan's documentation. The original example itself is taken from OpenBUGS. The data describes the length and age measurements for 27 captured dugongs (sea cows). Carlin and Gelfand (1991) model the data using a nonlinear growth curve with no inflection point and an asymptote as \(x_i\) tends to infinity:
\[
Y_i \sim \mathcal{N}(\mu_i, \sigma^2),\; i = 1,\dots,27\\
\mu_i = \alpha - \beta \lambda^{x_i},\; \alpha,\, \beta > 0;\, 0 < \lambda < 1 \] Fitting the curve with
nls gives the following results:Writing the model in Stan requires a few more lines, but gives me also the opportunity to generate output from the posterior distributions.
The predicted parameters and errors are very much the same as in the least square output of
nls, but with the Stan output I can also review the 90% credible intervals.Session Info
R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.1 (El Capitan)
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets
[6] methods base
other attached packages:
[1] rstan_2.8.0 ggplot2_1.0.1.9003 Rcpp_0.12.1
loaded via a namespace (and not attached):
[1] colorspace_1.2-6 scales_0.3.0 plyr_1.8.3
[4] parallel_3.2.2 tools_3.2.2 inline_0.3.14
[7] gtable_0.1.2 gridExtra_2.0.0 codetools_0.2-14
[10] grid_3.2.2 stats4_3.2.2 munsell_0.4.2
growth model
,
non-linear
,
R
,
Stan
R in Insurance 2016
Following the successful 3rd R in Insurance conference in Amsterdam this year, we will return to London next year. We will be back at Cass Business School, 11 July 2016.
The event will focus again on the use of R in insurance, bringing together experts from industry and academia with a diverse background of disciplines, such as actuarial science, catastrophe modelling, finance, statistics and computer science.
We are delighted to announce or keynote speakers already: Dan Murphy and Mario V. Wüthrich.
More details about the event will be published in due course here. For now, save the date in your diary.
PS: The picture above is a cluster analysis of the Lloyd's building in R.
ChainLadder 0.2.2 is out with improved glmReserve function
We released version 0.2.2 of
Ok, what does this all mean? I will run through a couple of examples and look behind the scene of
Like most other functions in ChainLadder,
The data at hand is often presented in form of a claims triangle, such as the following example data set from the ChainLadder package:
Suppose all claims will be reported within 7 years, then I'd like to know how much money should be set aside for the origin years 2008 to 2013 for claims that have incurred but not been reported (IBNR) yet. Or, to put it differently, I have to predict the
First, let's reformat the data as it would be stored in a database, that is in a long format of incremental claims over the years (I add the years also as factors, which I will use later):
The left plot of the chart above shows the cumulative claims payment over time, while the right plot shows the log-transformed incremental claims development for each origin/accident year.
One of the oldest methods to predict future claims development is called chain-ladder, which can be regarded as a weighted linear regression through the origin of cumulative claims over the development periods. Multiplying those development factors to the latest available cumulative position allows me to predict future claims in an iterative fashion.
It is well know in actuarial science that a Poisson GLM produces the same forecasts as the chain-ladder model.
Let's check:
There is another aspect to highlight with the Poisson model; its variance is equal to the mean. Yet, in real data I often observe that the variance increases in proportion to the mean. Well, this can be remedied by using an over-dispersed quasi-Poisson model.
I think a more natural approach would be to assume a compound distribution that models the frequency and severity of claims separately, e.g. Poisson frequency and Gamma severity.
Here the Tweedie distributions comes into play.
Tweedie distributions are a subset of what are called Exponential Dispersion Models. EDMs are two parameter distributions from the linear exponential family that also have a dispersion parameter \(\Phi\).
Furthermore, the variance is a power function of the mean, i.e. \(\mbox{Var}(X)=\Phi \, E[X]^p\).
The canonical link function for a Tweedie distribution in a GLM is the power link \(\mu^q\) with \(q=1-p\). Note, \(q=0\) is interpreted as \(\log(\mu)\).
Thus, let \(\mu_i = E(y_i)\) be the expectation of the ith response. Then I have the following model.
\[
y \sim \mbox{Tweedie}(q, p)\\
E(y) = \mu^q = Xb \\
\mbox{Var}(y) = \Phi \mu^p
\]The variance power \(p\) characterises the distribution of the responses \(y\). The following are some special cases:
Finally, I get back to the
With
In my first example I set use the parameters \(p=1, q=0\), which should return the results of the Poisson model.
Setting the argument
Not surprisingly the estimated reserves are similar to the Poisson model, but with a smaller predicted standard error.
Intuitively the modelling approach makes a lot more sense, but I end up with one parameter for each origin and development period, hence there is a danger of over-parametrisation.
Looking at the plots above again I note that many origin periods have a very similar development. Perhaps a hierarchical model would be more appropriate?
For more details on
ChainLadder a few weeks ago. This version adds back the functionality to estimate the index parameter for the compound Poisson model in glmReserve using the cplm package by Wayne Zhang. Ok, what does this all mean? I will run through a couple of examples and look behind the scene of
glmReserve. However, the clue is in the title, glmReserve is a function that uses a generalised linear model to estimate future claims, assuming claims follow a Tweedie distribution. I should actually talk about a family of distributions that is known as Tweedie, named by Bent Jørgensen after Maurice Tweedie. Joe Rickert published a nice post about the Tweedie distribution last year.Like most other functions in ChainLadder,
glmReserve purpose is to predict future insurance claims based on historical data.The data at hand is often presented in form of a claims triangle, such as the following example data set from the ChainLadder package:
library(ChainLadder)
cum2incr(UKMotor)
## dev
## origin 1 2 3 4 5 6 7
## 2007 3511 3215 2266 1712 1059 587 340
## 2008 4001 3702 2278 1180 956 629 NA
## 2009 4355 3932 1946 1522 1238 NA NA
## 2010 4295 3455 2023 1320 NA NA NA
## 2011 4150 3747 2320 NA NA NA NA
## 2012 5102 4548 NA NA NA NA NA
## 2013 6283 NA NA NA NA NA NASuppose all claims will be reported within 7 years, then I'd like to know how much money should be set aside for the origin years 2008 to 2013 for claims that have incurred but not been reported (IBNR) yet. Or, to put it differently, I have to predict the
NA fields in the bottom right hand triangle.First, let's reformat the data as it would be stored in a database, that is in a long format of incremental claims over the years (I add the years also as factors, which I will use later):
claims <- as.data.frame(cum2incr(UKMotor)) # convert into long format
library(data.table)
claims <- data.table(claims)
claims <- claims[ , ':='(cal=origin+dev-1, # calendar period
originf=factor(origin),
devf=factor(dev))]
claims <- claims[order(dev), cum.value:=cumsum(value), by=origin]
setnames(claims, "value", "inc.value")
head(claims)
## origin dev inc.value cal originf devf cum.value
## 1: 2007 1 3511 2007 2007 1 3511
## 2: 2008 1 4001 2008 2008 1 4001
## 3: 2009 1 4355 2009 2009 1 4355
## 4: 2010 1 4295 2010 2010 1 4295
## 5: 2011 1 4150 2011 2011 1 4150
## 6: 2012 1 5102 2012 2012 1 5102
library(lattice)
xyplot(cum.value/1000 + log(inc.value) ~ dev , groups=origin, data=claims,
t="b", par.settings = simpleTheme(pch = 16),
auto.key = list(space="right",
title="Origin\nperiod", cex.title=1,
points=FALSE, lines=TRUE, type="b"),
xlab="Development period", ylab="Amount",
main="Claims development by origin period",
scales="free")One of the oldest methods to predict future claims development is called chain-ladder, which can be regarded as a weighted linear regression through the origin of cumulative claims over the development periods. Multiplying those development factors to the latest available cumulative position allows me to predict future claims in an iterative fashion.
It is well know in actuarial science that a Poisson GLM produces the same forecasts as the chain-ladder model.
Let's check:
# Poisson model
mdl.pois <- glm(inc.value ~ originf + devf, data=na.omit(claims),
family=poisson(link = "log"))
# predict claims
claims <- claims[, ':='(
pred.inc.value=predict(mdl.pois,
.SD[, list(originf, devf, inc.value)],
type="response")), by=list(originf, devf)]
# sum of future payments
claims[cal>max(origin)][, sum(pred.inc.value)]
## [1] 28655.77
# Chain-ladder forecast
summary(MackChainLadder(UKMotor, est.sigma = "Mack"))$Totals[4,]
## [1] 28655.77There is another aspect to highlight with the Poisson model; its variance is equal to the mean. Yet, in real data I often observe that the variance increases in proportion to the mean. Well, this can be remedied by using an over-dispersed quasi-Poisson model.
I think a more natural approach would be to assume a compound distribution that models the frequency and severity of claims separately, e.g. Poisson frequency and Gamma severity.
Here the Tweedie distributions comes into play.
Tweedie distributions are a subset of what are called Exponential Dispersion Models. EDMs are two parameter distributions from the linear exponential family that also have a dispersion parameter \(\Phi\).
Furthermore, the variance is a power function of the mean, i.e. \(\mbox{Var}(X)=\Phi \, E[X]^p\).
The canonical link function for a Tweedie distribution in a GLM is the power link \(\mu^q\) with \(q=1-p\). Note, \(q=0\) is interpreted as \(\log(\mu)\).
Thus, let \(\mu_i = E(y_i)\) be the expectation of the ith response. Then I have the following model.
\[
y \sim \mbox{Tweedie}(q, p)\\
E(y) = \mu^q = Xb \\
\mbox{Var}(y) = \Phi \mu^p
\]The variance power \(p\) characterises the distribution of the responses \(y\). The following are some special cases:
- Normal distribution, p = 0
- Poisson distribution, p = 1
- Compound Poisson-Gamma distribution, 1 < p < 2
- Gamma distribution, p = 2
- Inverse-Gaussian, p = 3
- Stable, with support on the positive real numbers, p > 2
Finally, I get back to the
glmReserve function, which Wayne Zhang, the other author of the cplm package, contributed to the ChainLadder package.With
glmReserve I can model a claims triangle using the Tweedie distribution family.In my first example I set use the parameters \(p=1, q=0\), which should return the results of the Poisson model.
(m1 <- glmReserve(UKMotor, var.power = 1, link.power = 0))
## Latest Dev.To.Date Ultimate IBNR S.E CV
## 2008 12746 0.9732000 13097 351 125.8106 0.35843464
## 2009 12993 0.9260210 14031 1038 205.0826 0.19757473
## 2010 11093 0.8443446 13138 2045 278.8519 0.13635790
## 2011 10217 0.7360951 13880 3663 386.7919 0.10559429
## 2012 9650 0.5739948 16812 7162 605.2741 0.08451188
## 2013 6283 0.3038201 20680 14397 1158.1250 0.08044210
## total 62982 0.6872913 91638 28656 1708.1963 0.05961042Setting the argument
var.power=NULL will estimate \(p\) in the interval \((1,2)\) using the cplm package.(m2 <- glmReserve(UKMotor, var.power=NULL))
## Latest Dev.To.Date Ultimate IBNR S.E CV
## 2008 12746 0.9732000 13097 351 110.0539 0.31354394
## 2009 12993 0.9260870 14030 1037 176.9361 0.17062307
## 2010 11093 0.8444089 13137 2044 238.5318 0.11669851
## 2011 10217 0.7360951 13880 3663 335.6824 0.09164138
## 2012 9650 0.5739948 16812 7162 543.6472 0.07590718
## 2013 6283 0.3038201 20680 14397 1098.7988 0.07632138
## total 62982 0.6873063 91636 28654 1622.4616 0.05662252
m2$model
##
## Call:
## cpglm(formula = value ~ factor(origin) + factor(dev), link = link.power,
## data = ldaFit, offset = offset)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -6.7901 -1.6969 0.0346 1.6087 8.4465
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.25763 0.04954 166.680 < 2e-16 ***
## factor(origin)2008 0.03098 0.05874 0.527 0.605588
## factor(origin)2009 0.09999 0.05886 1.699 0.110018
## factor(origin)2010 0.03413 0.06172 0.553 0.588369
## factor(origin)2011 0.08933 0.06365 1.403 0.180876
## factor(origin)2012 0.28091 0.06564 4.279 0.000659 ***
## factor(origin)2013 0.48797 0.07702 6.336 1.34e-05 ***
## factor(dev)2 -0.11740 0.04264 -2.753 0.014790 *
## factor(dev)3 -0.62829 0.05446 -11.538 7.38e-09 ***
## factor(dev)4 -1.03168 0.06957 -14.830 2.28e-10 ***
## factor(dev)5 -1.31346 0.08857 -14.829 2.28e-10 ***
## factor(dev)6 -1.86307 0.13826 -13.475 8.73e-10 ***
## factor(dev)7 -2.42868 0.25468 -9.536 9.30e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Estimated dispersion parameter: 10.788
## Estimated index parameter: 1.01
##
## Residual deviance: 299.27 on 15 degrees of freedom
## AIC: 389.18
##
## Number of Fisher Scoring iterations: 4
Not surprisingly the estimated reserves are similar to the Poisson model, but with a smaller predicted standard error.
Intuitively the modelling approach makes a lot more sense, but I end up with one parameter for each origin and development period, hence there is a danger of over-parametrisation.
Looking at the plots above again I note that many origin periods have a very similar development. Perhaps a hierarchical model would be more appropriate?
For more details on
glmReserve see the help file and package vignette.Session Info
R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.5 (Yosemite)
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] data.table_1.9.6 ChainLadder_0.2.2
loaded via a namespace (and not attached):
[1] Rcpp_0.12.1 nloptr_1.0.4 plyr_1.8.3
[4] tools_3.2.2 digest_0.6.8 lme4_1.1-9
[7] statmod_1.4.21 gtable_0.1.2 nlme_3.1-121
[10] lattice_0.20-33 mgcv_1.8-7 Matrix_1.2-2
[13] parallel_3.2.2 biglm_0.9-1 SparseM_1.7
[16] proto_0.3-10 coda_0.17-1 stringr_1.0.0
[19] MatrixModels_0.4-1 stats4_3.2.2 lmtest_0.9-34
[22] grid_3.2.2 nnet_7.3-10 tweedie_2.2.1
[25] cplm_0.7-4 minqa_1.2.4 ggplot2_1.0.1
[28] reshape2_1.4.1 car_2.1-0 actuar_1.1-10
[31] magrittr_1.5 scales_0.3.0 MASS_7.3-44
[34] splines_3.2.2 systemfit_1.1-18 pbkrtest_0.4-2
[37] colorspace_1.2-6 quantreg_5.19 sandwich_2.3-4
[40] stringi_0.5-5 munsell_0.4.2 chron_2.3-47
[43] zoo_1.7-12 
