# mages' blog

## ChainLadder 0.2.2 is out with improved glmReserve function

We released version 0.2.2 of 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  NA
The rows present different origin periods in which accidents occurred and the columns along each row show the incremental reported claims over the years (the data itself is stored in a cumulative form).

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 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")
##    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

Let's visualise the data:
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")
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:
# Poisson model
mdl.pois <- glm(inc.value ~ originf + devf, data=na.omit(claims),
# 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
summary(MackChainLadder(UKMotor, est.sigma = "Mack"))$Totals[4,] ## [1] 28655.77 Ok, this worked. However, both of these models make actually fairly strong assumptions. The Poisson model by its very nature will only produce whole numbers, and although payments could be regarded as whole numbers, say in pence or cents, it does feel a little odd to me. Similarly, modelling the year on year developments via a weighted linear regression through the origin, as in the case of the chain-ladder model, sounds not intuitive either. 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: • 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 0.05961042 Perfect, I get the same results, plus further information about the model. Setting 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:
##     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

From the model I note that the dispersion parameter $$\phi$$ was estimated as 10.788 and the index parameter $$p$$ as 1.01.

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:

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        

## Notes from the Kölner R meeting, 18 September 2015

Last Friday the Cologne R user group came together for the 15th time. Since its inception over three years ago the group evolved from a small gathering in a pub into an active data science community, covering wider topics than just R. Still, R is the link and clue between the different interests. Last Friday's agenda was a good example of this, with three talks touching on workflow management, web development and risk analysis.

### R in a big data pipeline

Yuki Katoh had travelled all the way from Berlin to present on how to embed R with luigi into a heterogeneous workflow of different applications. This is especially useful when R needs to be integrated with hadoop/hdfs based technologies, such as Spark and Hive. Luigi is not unlike Make, which Kirill presented at our last meeting in June. In a configuration file Yuki specified the various workflow steps and dependencies between the jobs.

Kicking off the luigi script starts the workflow, and luigid server allows Yuki to monitor the various parts of the dependency graph visually. Thus, he can see the progress of his workflow in real time and identify quickly, when and where a sub process fails. As Yuki pointed out, this becomes critical in production systems, where failures need to be known and fixed quickly, unlike when ones carries out an explorative analysis in a development/research environment. See also Yuki's blog post for further details.

### Shiny + Shinyjs

Shiny is a very popular R package that allows users to develop interactive browser applications. Paul Viefers introduced us to the extension shinyjs, a package written by Dean Attali. The name suggests already that the package provides additional JavaScript functionality. Indeed, it does, but without the need to learn JavaScript, as those functions are wrapped into R.

Paul showed us an example of a shinyapp that depending on the user plotted a different graph. Behind the scene his script would either hide or shows those plots, conditioned on the user. With only a few lines in R it allowed him to develop a user specific application. To achieve this he created a login screen that checks for user name and password. In his example he had hard coded the login credentials, instead of using a secure connection via a professional shiny server instance. However this was sufficient for his purpose, where he tests how students react to different economic scenarios in a lab environment at university.

### Experience vs. Data

The last talk of the meeting had a more statistical focus with examples from insurance. I repeated my talk from the LondonR user group meeting in June. One of the challenge in insurance is that despite of having many customers , insurance companies will have little claims data per customer to assess risks.

I presented some Bayesian ideas to analyse risks with little data. I used the wonderful "Hit and run accident" example from Daniel Kahneman's book Thinking, fast and slow to explain Bayes' formula, introduced Bayesian belief networks for a claims analysis and discussed the challenge of predicting events when they haven't happened yet (also in Stan). Along the way I mentioned a few ideas on communicating risk, which I learned from David Spiegelhalter earlier this year.

### Next Kölner R meeting

The next meeting will be scheduled in December. Details will be published on our Meetup site. Thanks again to Revolution Analytics/Microsoft for their sponsorship.

Please get in touch, if you would like to present at the next meeting.

## Next Kölner R User Meeting: Friday, 18 September 2015

The 15th Cologne R user group meeting is scheduled for this Friday, 18 September 2015 and we have a full agenda with three talks followed by networking drinks.

• R in big data pipeline with luigi (Yuki Katoh)
R in big data pipeline: Put your awesome R codes into production. Learn how to build solid big data pipeline around it.
• shinyjs (Paul Viefers)
Using JavaScript in shiny, without knowing JavaScript
• Experience vs Data (Markus Gesmann)
How to asses risks with small data sets. Bayesian ideas, belief networks and a small simulation example in Stan/rstan.
Please note: Our venue changed! We have outgrown the seminar room at the Institute of Sociology and moved to Startplatz, a start-up incubator venue: Im Mediapark, 550670 Köln

### Drinks and Networking

The event will be followed by drinks (Kölsch!) and networking opportunities.

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.

## Bayesian regression models using Stan in R

It seems the summer is coming to end in London, so I shall take a final look at my ice cream data that I have been playing around with to predict sales statistics based on temperature for the last couple of weeks [1], [2], [3].

Here I will use the new brms (GitHub, CRAN) package by Paul-Christian Bürkner to derive the 95% prediction credible interval for the four models I introduced in my first post about generalised linear models. Additionally, I am interested to predict how much ice cream I should hold in stock for a hot day at 35ºC, such that I only run out of ice cream with a probability of 2.5%.

### Stan models with brms

Like in my previous post about the log-transformed linear model with Stan, I will use Bayesian regression models to estimate the 95% prediction credible interval from the posterior predictive distribution.

Thanks to brms this will take less than a minute of coding, because brm allows me to specify my models in the usual formula syntax and I can leave it to the package functions to create and execute the Stan files.

Let's start. Here is the data again:

My models are written down in very much the same way as with glm. Only the binomial model requires a slightly different syntax. Here I use the default priors and link functions:

Last week I wrote the Stan model for the log-transformed linear model myself. Here is the output of brm. The estimated parameters are quite similar, apart from $$\sigma$$:

I access the underlying Stan model via log.lin.mod\$model and note that the prior of $$\sigma$$ is modelled via a Cauchy distribution, unlike the inverse Gamma I used last week. I believe that explains my small difference in $$\sigma$$.

To review the model I start by plotting the trace and density plots for the MCMC samples.

### Prediction credible interval

The predict function gives me access to the posterior predictive statistics, including the 95% prediction credible interval.

Combining the outputs of all four models into one data frame gives me then the opportunity to compare the prediction credible intervals of the four models in one chart.

There are no big news in respect of the four models, but for the fact that here I can look at the posterior prediction credible intervals, rather then the theoretical distributions two weeks ago. The over-prediction of the log-transformed linear model is apparent again.

### How much stock should I hold on a hot day?

Running out of stock on a hot summer's day would be unfortunate, because those are days when sells will be highest. But how much stock should I hold?

Well, if I set the probability of selling out at 2.5%, then I will have enough ice cream to sell with 97.5% certainty. To estimate those statistics I have to calculate the 97.5% percentile of the posterior predictive samples.

Ok, I have four models and four answers ranging from 761 to 2494. The highest number is more than 3 times the lowest number!

I had set the market size at 800 in my binomial model, so I am not surprised by its answer of 761. Also, I noted earlier that the log-normal distribution is skewed to the right, so that explains the high prediction of 2494. The Poisson model, like the log-transform linear model, has the implicit exponential growth assumption. Its mean forecast is well over 1000 as I pointed out earlier and hence the 97.5% prediction of 1510 is to be expected.

### Conclusions

How much ice cream should I hold in stock? Well, if I believe strongly in my assumption of a market size of 800, then I should stick with the output of the binomial model, or perhaps hold 800 just in case.

Another aspect to consider would be the cost of holding unsold stock. Ice cream can usually be stored for some time, but the situation would be quite different for freshly baked bread or bunches of flowers that have to be sold quickly.

### Session Info

Note, I used the developer version of brms from GitHub.
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
[6] methods   base

other attached packages:
[1] lattice_0.20-33 brms_0.4.1.9000 ggplot2_1.0.1
[4] rstan_2.7.0-1   inline_0.3.14   Rcpp_0.12.0

loaded via a namespace (and not attached):
[1] codetools_0.2-14 digest_0.6.8     MASS_7.3-43
[4] grid_3.2.2       plyr_1.8.3       gtable_0.1.2
[7] stats4_3.2.2     magrittr_1.5     scales_0.2.5
[10] stringi_0.5-5    reshape2_1.4.1   proto_0.3-10
[13] tools_3.2.2      stringr_1.0.0    munsell_0.4.2
[16] parallel_3.2.2   colorspace_1.2-6