- 1 Model
- 2 Load Data
- 3 Data Exploration
- 4 Initial Model - Single Line-of-Business, Single Insurer (SISLOB)
- 5 Altenative Models
- 6 References and SessionInfo
1 Model
Loss curves are a standard actuarial technique for helping insurance companies assess the amount of reserve capital they need to keep on hand to cover claims from a line of business. Claims made and reported for a given accounting period are tracked seperately over time. This enables the use of historical patterns of claim development to predict expected total claims for newer policies.
In insurance, depending on the types of risks, it can take many years for an insurer to learn the amount of liability incurred on policies written during any particular year. So, at a particular point in time after the policy is written some claims may not reported or known about by then, or some claims are still working through the legal system so the final amount due is not determined.
Total claim amounts from a simple accounting period are laid out in a single row of a table, each column showing the total claim amount after that period of time. Subsequent accounting periods have less development, so the data takes a triangular shape - hence the term ‘loss triangles’. Using previous patterns, data in the upper part of the triangle is used to predict values in the unknown lower triangle, giving the insurer a probabilistic forecast of the ultimate claim amounts to be paid for all business written.
The ChainLadder package provides functionality to generate and use these loss triangles.
In this case study, we take a related but different approach: we model the growth of the losses in each accounting period as an increasing function of time, and use the model to estimate the parameters which determine the shape and form of this growth. We also use the sampler to estimate the values of the “ultimate loss ratio”, i.e. the ratio of the total claims on an accounting period to the total premium received to write those policies. We treat each accounting period as a cohort.
1.1 Overview
We will work with two different functional forms for the growth behaviour of the loss curves: a ‘Weibull’ model and a ‘loglogistic’ model:
\[\begin{align*} g(t \, ; \, \theta, \omega) &= \frac{t^\omega}{t^\omega + \theta^\omega} & (\text{Weibull}) \\ g(t \, ; \, \theta, \omega) &= 1 - \exp\left(-\left(\frac{t}{\theta}\right)^\omega\right) & (\text{Log-logistic}) \end{align*}\]2 Load Data
We load the Schedule P loss data from casact.org.
### File was downloaded from http://www.casact.org/research/reserve_data/ppauto_pos.csv
data_files <- dir("data/", pattern = "\\.csv", full.names = TRUE)
data_cols <- cols(GRCODE = col_character())
rawdata_tbl <- data_files %>%
map(read_claim_datafile, col_type = data_cols) %>%
bind_rows
glimpse(rawdata_tbl)## Observations: 77,900
## Variables: 14
## $ grcode <chr> "266", "266", "266", "266", "266", "266", "266", "2...
## $ grname <chr> "Public Underwriters Grp", "Public Underwriters Grp...
## $ accidentyear <int> 1988, 1988, 1988, 1988, 1988, 1988, 1988, 1988, 198...
## $ developmentyear <int> 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 199...
## $ developmentlag <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7,...
## $ incurloss <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 24, 21, 24, 25, 2...
## $ cumpaidloss <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 20, 21, 23, 24, 24...
## $ bulkloss <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, ...
## $ earnedpremdir <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 25, 25, 25, 25, 2...
## $ earnedpremceded <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ earnedpremnet <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 25, 25, 25, 25, 2...
## $ single <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ postedreserve97 <int> 932, 932, 932, 932, 932, 932, 932, 932, 932, 932, 9...
## $ lob <chr> "comauto", "comauto", "comauto", "comauto", "comaut...claimdata_tbl <- rawdata_tbl %>%
mutate(acc_year = as.character(accidentyear)
,dev_year = developmentyear
,dev_lag = developmentlag
,premium = earnedpremdir
,cum_loss = cumpaidloss
,loss_ratio = cum_loss / premium) %>%
select(grcode, grname, lob, acc_year, dev_year, dev_lag, premium, cum_loss, loss_ratio)With the data in the format we will use in this analysis, we take a look at it in tabular form:
print(claimdata_tbl)## # A tibble: 77,900 x 9
## grcode grname lob acc_year dev_year dev_lag premium
## <chr> <chr> <chr> <chr> <int> <int> <int>
## 1 266 Public Underwriters Grp comauto 1988 1988 1 0
## 2 266 Public Underwriters Grp comauto 1988 1989 2 0
## 3 266 Public Underwriters Grp comauto 1988 1990 3 0
## 4 266 Public Underwriters Grp comauto 1988 1991 4 0
## 5 266 Public Underwriters Grp comauto 1988 1992 5 0
## 6 266 Public Underwriters Grp comauto 1988 1993 6 0
## 7 266 Public Underwriters Grp comauto 1988 1994 7 0
## 8 266 Public Underwriters Grp comauto 1988 1995 8 0
## 9 266 Public Underwriters Grp comauto 1988 1996 9 0
## 10 266 Public Underwriters Grp comauto 1988 1997 10 0
## # ... with 77,890 more rows, and 2 more variables: cum_loss <int>,
## # loss_ratio <dbl>3 Data Exploration
In terms of modeling, we first confine ourselves to a single line of business ‘ppauto’ and ensure the data we work with is a snapshot in time. We remove all data timestamped after 1997 and use the remaining data as our modelling dataset.
Once we have fits and predictions, we use the later timestamped data as a way to validate the model.
use_grcode <- c(43,353,388,620)
carrier_full_tbl <- claimdata_tbl %>%
filter(lob == 'ppauto')
carrier_snapshot_tbl <- carrier_full_tbl %>%
filter(grcode %in% use_grcode
,dev_year < 1998)We are looking at four insurers with the GRCODEs above. Before we proceed with any analysis, we first plot the data, grouping the loss curves by accounting year and faceting by carrier.
ggplot(carrier_snapshot_tbl) +
geom_line(aes(x = dev_lag, y = loss_ratio, colour = as.character(acc_year))
,size = 0.3) +
expand_limits(y = c(0,1)) +
facet_wrap(~grcode) +
xlab('Development Time') +
ylab('Loss Ratio') +
ggtitle('Snapshot of Loss Curves for 10 Years of Loss Development'
,subtitle = 'Private Passenger Auto Insurance for Single Organisation') +
guides(colour = guide_legend(title = 'Cohort Year'))
We look at the chain ladder of the data, rather than looking at the loss ratios we just look at the dollar amounts of the losses.
snapshot_tbl <- carrier_snapshot_tbl %>%
filter(grcode %in% use_grcode[1])
snapshot_tbl %>%
select(acc_year, dev_lag, premium, cum_loss) %>%
spread(dev_lag, cum_loss) %>%
print## # A tibble: 10 x 12
## acc_year premium `1` `2` `3` `4` `5` `6` `7` `8` `9` `10`
## * <chr> <int> <int> <int> <int> <int> <int> <int> <int> <int> <int> <int>
## 1 1988 957 133 333 431 570 615 615 615 614 614 614
## 2 1989 3695 934 1746 2365 2579 2763 2966 2940 2978 2978 NA
## 3 1990 6138 2030 4864 6880 8087 8595 8743 8763 8762 NA NA
## 4 1991 17533 4537 11527 15123 16656 17321 18076 18308 NA NA NA
## 5 1992 29341 7564 16061 22465 25204 26517 27124 NA NA NA NA
## 6 1993 37194 8343 19900 26732 30079 31249 NA NA NA NA NA
## 7 1994 46095 12565 26922 33867 38338 NA NA NA NA NA NA
## 8 1995 51512 13437 26012 31677 NA NA NA NA NA NA NA
## 9 1996 52481 12604 23446 NA NA NA NA NA NA NA NA
## 10 1997 56978 12292 NA NA NA NA NA NA NA NA NAIn the above ‘triangle’, we see the cumulative amounts of ‘incurred losses’ for each accounting year. 1988 was the first year and so has ten years of claims development by 1998. Similarily, 1989 has nine years of development and so on. Incurred claims come in two forms: closed claims that the insurer has paid out and will have no further changes, or open claims known to the insurer but not fully settled and paid out yet.
As claims develop, we see that the total claims is an approximately-monotonically increasing function of time, providing the motivation to model this pattern as a growth curve.
The premium column details the total premium received by the insurer for the policies written in that accounting year. Recall that the ratio of total claims paid to total premium received is the ‘Loss Ratio’ (LR).
For this insurer we see that the premium collected in each account year increases significantly over time, suggesting that the size of this line of business grew as time went on.
Next, we look at loss ratios in a similar fashion:
snapshot_tbl %>%
select(acc_year, dev_lag, premium, loss_ratio) %>%
spread(dev_lag, loss_ratio) %>%
print.data.frame(digits = 2)## acc_year premium 1 2 3 4 5 6 7 8 9 10
## 1 1988 957 0.14 0.35 0.45 0.60 0.64 0.64 0.64 0.64 0.64 0.64
## 2 1989 3695 0.25 0.47 0.64 0.70 0.75 0.80 0.80 0.81 0.81 NA
## 3 1990 6138 0.33 0.79 1.12 1.32 1.40 1.42 1.43 1.43 NA NA
## 4 1991 17533 0.26 0.66 0.86 0.95 0.99 1.03 1.04 NA NA NA
## 5 1992 29341 0.26 0.55 0.77 0.86 0.90 0.92 NA NA NA NA
## 6 1993 37194 0.22 0.54 0.72 0.81 0.84 NA NA NA NA NA
## 7 1994 46095 0.27 0.58 0.73 0.83 NA NA NA NA NA NA
## 8 1995 51512 0.26 0.50 0.61 NA NA NA NA NA NA NA
## 9 1996 52481 0.24 0.45 NA NA NA NA NA NA NA NA
## 10 1997 56978 0.22 NA NA NA NA NA NA NA NA NA3.1 Loss Ratio Ladders
We are working with the loss ratio, so we recreate the chain ladder format but look at loss ratios instead of dollar losses.
ggplot(snapshot_tbl) +
geom_line(aes(x = dev_lag, y = loss_ratio, colour = acc_year)
,size = 0.3) +
expand_limits(y = 0) +
xlab('Development Time') +
ylab('Loss Ratio') +
ggtitle("Loss Ratio Curves by Development Time") +
guides(colour = guide_legend(title = 'Cohort Year'))
4 Initial Model - Single Line-of-Business, Single Insurer (SISLOB)
For our first model, we wish to keep things simple and so restrict the model to considering a single line-of-buiness for a single insurer. Thus, our data is in the form of a single triangle, and the problem confines itself to modelling a single triangle, giving us a simple starting place for improving and extending the model.
The basic concept is to model the growth of losses in a cohort of policies as a function of time. As mentioned, we use two different growth functions, the Weibull and the Log-Logistic. Both functions have two parameters which we label \(\theta\) and \(\omega\).
As each cohort year has different volumes of business, we scale the losses by the total premium received for that cohort, allowing us to more directly compare the cohorts. The total losses is then given by
\[ \text{Total Loss}(t) = \text{Premium} \times \text{Final Loss Ratio} \times \text{GF}(t) \]
Each accounting year, \(Y\), in the cohort gets its own value for the Final Loss Ratio, \(\text{LR}_{Y}\), each cumulated loss value in the data can be modelled as
\[ \text{Loss}(Y, t) \sim \text{Normal}(\mu(Y, t), \, \sigma_Y) \]
where we have
\[\begin{eqnarray*} \mu(Y, t) &=& \text{Premium}(Y) \times \text{LR}(Y) \times \text{GF}(t) \\ \text{GF}(t) &=& \text{growth function of } t \\ \sigma_Y &=& \text{Premium}(Y) \times \sigma \\ \text{LR}_Y &\sim& \text{Lognormal}(\mu_{\text{LR}}, \sigma_{\text{LR}}) \\ \mu_{\text{LR}} &\sim& \text{Normal}(0, 0.5) \end{eqnarray*}\]All other parameters in the model, (\(\sigma_{\text{LR}}\), \(\omega\), \(\theta\), \(\sigma\)), have lognormal priors to constrain them to be positive.
The priors on the hyper-parameters are weakly infomative - chosen to cover the feasiable range of parameter values with a lot of uncertainty.
By setting the model up in this way, Stan can fit for both the shape of the growth curve - as this is determined by \(\theta\) and \(\omega\) - and the loss ratios for each cohort simultaneously.
4.1 Weibull vs Log-logistic
We have no prior preference for using either the Weibull or the Log-logistic function to model the growth of the losses.
Visuals are important in this case so we first look at how the two functions differ in value for a given set of parameters.
There are references that the Weibull function tends to result in heavier losses in the model, but we have no empirical evidence for such a claim. It is true that for any given set of values for \((\omega, \theta)\) the log-logistic function plateaus at smaller levels of \(t\), but this seems no guarantee to me: the model could just choose different values for the parameters to counteract this.
To test for this, let us treat the Log-logistic function as the ‘true’ value and then try to fit a new set of parameters \((\omega, \theta)\) to see how well a different set of parameters can match another.
ll_vals <- loglogistic_tbl$value
new_param_func <- function(x) {
omega <- x[1]
theta <- x[2]
new_vals <- weibull_func(t_seq, omega, theta)
tot_ss <- sum((new_vals - ll_vals)^2)
return(tot_ss)
}
optim_params <- optim(c(1, 1), new_param_func)
fittedweibull_tbl <- tibble(
label = 'Weibull (fitted)'
,t = t_seq
,value = weibull_func(t_seq
,optim_params$par[1]
,optim_params$par[2])
)
plot_tbl <- bind_rows(weibull_tbl
,loglogistic_tbl
,fittedweibull_tbl)
ggplot(plot_tbl) +
geom_line(aes(x = t, y = value, colour = label)) +
xlab(expression(t)) +
ylab(expression("Functional Forms for Growth/Development Factors")) +
ggtitle("Comparison Plot for Weibull and Log-Logistic Curves") 
We see that the Weibull growth function comes quite close to a log-logistic function so the choice should be a matter of preference in the main.
That said, we shall try both and see if we see much difference between the two.
4.2 Configure Data
We want to only use the data at a given snapshot, so we choose all data current to 1998. Thus, we have 10 years of development for our first 1988 cohort, and one less for each subsequent year. Our final cohort for 1997 has only a single year of development
modeldata_tbl <- claimdata_tbl %>%
filter(lob == 'ppauto'
,grcode == use_grcode[1])
usedata_tbl <- modeldata_tbl %>%
filter(dev_year < 1998)
cohort_maxtime <- usedata_tbl %>%
group_by(acc_year) %>%
summarise(maxtime = max(dev_lag)) %>%
arrange(acc_year) %>%
pull(maxtime)
cohort_premium <- usedata_tbl %>%
group_by(acc_year) %>%
summarise(premium = unique(premium)) %>%
pull(premium)
t_values <- usedata_tbl %>%
select(dev_lag) %>%
arrange(dev_lag) %>%
unique %>%
pull(dev_lag)
standata_lst <- list(
growthmodel_id = 1 # Use weibull rather than loglogistic
,n_data = usedata_tbl %>% nrow
,n_time = usedata_tbl %>% select(dev_lag) %>% unique %>% nrow
,n_cohort = usedata_tbl %>% select(acc_year) %>% unique %>% nrow
,cohort_id = get_character_index(usedata_tbl$acc_year)
,cohort_maxtime = cohort_maxtime
,t_value = t_values
,t_idx = get_character_index(usedata_tbl$dev_lag)
,premium = cohort_premium
,loss = usedata_tbl$cum_loss
)The full Stan file is shown below:
stan_file <- "losscurves_sislob.stan"
cat(read_lines(stan_file), sep = "\n")functions {
real growth_factor_weibull(real t, real omega, real theta) {
return 1 - exp(-(t/theta)^omega);
}
real growth_factor_loglogistic(real t, real omega, real theta) {
real pow_t_omega = t^omega;
return pow_t_omega / (pow_t_omega + theta^omega);
}
}
data {
int<lower=0,upper=1> growthmodel_id;
int n_data;
int n_time;
int n_cohort;
int cohort_id[n_data];
int t_idx[n_data];
int cohort_maxtime[n_cohort];
vector<lower=0>[n_time] t_value;
vector[n_cohort] premium;
vector[n_data] loss;
}
parameters {
real<lower=0> omega;
real<lower=0> theta;
vector<lower=0>[n_cohort] LR;
real mu_LR;
real<lower=0> sd_LR;
real<lower=0> loss_sd;
}
transformed parameters {
vector[n_time] gf;
vector[n_data] lm;
for(i in 1:n_time) {
gf[i] = growthmodel_id == 1 ?
growth_factor_weibull (t_value[i], omega, theta) :
growth_factor_loglogistic(t_value[i], omega, theta);
}
for (i in 1:n_data) {
lm[i] = LR[cohort_id[i]] * premium[cohort_id[i]] * gf[t_idx[i]];
}
}
model {
mu_LR ~ normal(0, 0.5);
sd_LR ~ lognormal(0, 0.5);
LR ~ lognormal(mu_LR, sd_LR);
loss_sd ~ lognormal(0, 0.7);
omega ~ lognormal(0, 0.5);
theta ~ lognormal(0, 0.5);
loss ~ normal(lm, (loss_sd * premium)[cohort_id]);
}
generated quantities {
vector<lower=0>[n_time] loss_sample[n_cohort];
vector<lower=0>[n_time] loss_prediction[n_cohort];
vector<lower=0>[n_time] step_ratio[n_cohort];
real mu_LR_exp;
real<lower=0> ppc_minLR;
real<lower=0> ppc_maxLR;
real<lower=0> ppc_EFC;
for(i in 1:n_cohort) {
step_ratio[i] = rep_vector(1, n_time);
loss_sample[i] = LR[i] * premium[i] * gf;
}
mu_LR_exp = exp(mu_LR);
for(i in 1:n_data) {
loss_prediction[cohort_id[i], t_idx[i]] = loss[i];
}
for(i in 1:n_cohort) {
for(j in 2:n_time) {
step_ratio[i, j] = gf[t_idx[j]] / gf[t_idx[j-1]];
}
}
for(i in 1:n_cohort) {
for(j in (cohort_maxtime[i]+1):n_time) {
loss_prediction[i,j] = loss_prediction[i,j-1] * step_ratio[i,j];
}
}
// Create PPC distributions for the max/min of LR
ppc_minLR = min(LR);
ppc_maxLR = max(LR);
// Create total reserve PPC
ppc_EFC = 0;
for(i in 1:n_cohort) {
ppc_EFC = ppc_EFC + loss_prediction[i, n_time] - loss_prediction[i, cohort_maxtime[i]];
}
}There are a few points of note about this Stan model worth highlighting here.
4.2.1 Local functions
To avoid having near-duplicate versions of the Stan model, we define local functions to calculate both the Weibull and Log-logistic functions. In general, Weibull models are said to produce fits with heavier losses, as fact we shall test.
4.2.2 Loss Ratios and Ensuring Positivity
To ensure positivity on the loss ratios, we use the lognormal distribution for the LR variables. We also use them for the standard deviations, but we may alter this approach and try half-Cauchy distributions as an alternative.
4.2.3 Prior for mu_LR
Because we use lognormals for the underlying losses, we want the prior for the mean of the distribution to take both positive and negative values, and thus use a normal distribution for \(\mu_{\text{LR}}\), with mean 0 and std dev 0.5.
4.2.4 The generated quantities block
We use the generated quantities block to facilitate both posterior predictive checks and to also make loss reserving projections across all the cohorts. The use and analysis of the output of this block is discussed in a later section.
4.3 Fitting the Stan Model
We now proceed with fitting the stan model and examing the output.
model_sislob_stanmodel <- stan_model(stan_file)model_sislob_stanfit <- sampling(
object = model_sislob_stanmodel
,data = standata_lst
,iter = 500
,chains = 8
,seed = stan_seed
)The Stan sample contains no divergent transitions, a good start.
4.4 Sampler Diagnostic Plots
It is always worth checking convergence of the model by checking the \(\hat{R}\) and ensuring it is less than about 1.1
# Plot of convergence statistics
model_sislob_draws <- extract(model_sislob_stanfit, permuted = FALSE, inc_warmup = TRUE)
model_sislob_monitor_tbl <- as.data.frame(monitor(model_sislob_draws, print = FALSE))
model_sislob_monitor_tbl <- model_sislob_monitor_tbl %>%
mutate(variable = rownames(model_sislob_monitor_tbl)
,parameter = gsub("\\[.*]", "", variable)
)
ggplot(model_sislob_monitor_tbl) +
aes(x = parameter, y = Rhat, color = parameter) +
geom_jitter(height = 0, width = 0.2, show.legend = FALSE) +
geom_hline(aes(yintercept = 1), size = 0.5) +
ylab(expression(hat(italic(R)))) +
ggtitle("R-hat Plots for Sampler Parameters") +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5))
The \(\hat{R}\) values for the parameter appear to be in or around 1. A positive sign, but not sufficient for convergence.
We check the size of n_eff for each of the variables:
ggplot(model_sislob_monitor_tbl) +
aes(x = parameter, y = n_eff, color = parameter) +
geom_jitter(height = 0, width = 0.1, show.legend = FALSE) +
expand_limits(y = 0) +
xlab("Parameter") +
ylab(paste0("Effective Sample Count (n_eff)")) +
ggtitle("N_eff Plots for Sampler Parameters") +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5))
The lowest sample size for the parameters is around 800, about 40% of the/ maximum. This is reasonable, though the higher is better.
Traceplots for the parameters are a useful diagnostic. The large parameter count makes the plots messy, so we break them up into groups. First we look at omega, theta and LR.
traceplot(model_sislob_stanfit, pars = c("omega", "theta", "LR")) +
ggtitle("Traceplot of omega, theta and LR") +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5))
We have 8 chains, and the plots show signs the chains have mixed well, with no indications of difficult exploration of the posterior.
Now we look at the traces for gf and loss_sd.
traceplot(model_sislob_stanfit, pars = c("gf", "loss_sd")) +
ggtitle("Traceplot of gf and loss_sd") +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5))
4.4.1 Using bayesplot Functionality
The packages bayesplot provides some simple diagnostic plots for fits, so we look at those, restricting ourselves to omega, theta, LR and loss_sd.
stanmodel_pars <- c('omega','theta','LR','mu_LR','sd_LR','loss_sd')We have some simple lineplots for \(\hat{R}\), and this should convey similar information to our previous plots.
model_sislob_stanfit %>%
rhat(pars = stanmodel_pars) %>%
mcmc_rhat(.) +
yaxis_text() +
ggtitle("Parameter Plot of R-hat Statistic")
Related to this is \(n_{\text{eff}}\):
model_sislob_stanfit %>%
neff_ratio(pars = stanmodel_pars) %>%
mcmc_neff(.) +
yaxis_text() +
ggtitle("Parameter Plot of Effective Sample Size")
The lowest sample size is just under 50% of the full count, which is good.
A major benefit of using Hamiltonian Monte Carlo and the NUTS sampler is the presence of powerful diagnostic tools for convergence. One useful plot is the energy diagnostic.
If the two histograms broadly overlap, the sampler is doing a good job of exploring the posterior probability space.
model_sislob_stanfit %>%
nuts_params %>%
mcmc_nuts_energy(binwidth = 1) +
facet_wrap(~Chain, ncol = 2) +
ggtitle("Energy Diagnostic Plots Across Chains")
The diagnostics are not flagging any potential issues with the sample.
Finally we check the parameter traceplots for convergence, looking for indications of a lack of mixing.
model_sislob_stanfit %>%
as.matrix %>%
mcmc_trace(regex_pars = c('theta','omega','LR\\[','mu_LR','sd_LR','loss_sd')) +
ggtitle("Parameter Traceplots")
These plots are similar to the traceplots from before, and show no causes for concern.
4.5 Assessing the Fit
Having convinced ourselves that the samples have converged, we proceed to checking the quality of the fit. We first look at the 50% credibility intervals of the parameters
param_root <- c("omega", "theta", "LR", "mu_LR_exp", "gf", "loss_sd")
use_vars <- model_sislob_monitor_tbl %>%
filter(parameter %in% param_root) %>%
pull(variable)
plotdata_tbl <- model_sislob_monitor_tbl %>%
filter(variable %in% use_vars) %>%
select(mean, `25%`, `50%`, `75%`) %>%
mutate(variable = factor(use_vars, levels = use_vars))
ggplot(plotdata_tbl) +
geom_point(aes(x = variable, y = mean)) +
geom_errorbar(aes(x = variable, ymin = `25%`, ymax = `75%`), width = 0) +
expand_limits(y = 0) +
theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
xlab("Parameter") +
ggtitle("Posterior Credibility Intervals for Sampler Parameters")
The bayesplot package also provides some functionality for plotting the parameters values.
weibull_param_plot <- model_sislob_stanfit %>%
extract(inc_warmup = FALSE, permuted = FALSE) %>%
mcmc_intervals(regex_pars = c('omega','theta','LR\\[', 'mu_LR','sd_LR','loss_sd')) +
expand_limits(x = c(-0.5, 2.5)) +
ggtitle("Posterior Credibility Intervals for Sampler Parameters")
weibull_param_plot %>% plot
Now that we have our fit we can start looking at some plots for it. First we look at some very simple sanity-check plots. We look at the full development of an accounting year and see how well our model fits the pattern observed from the data.
fitted_curves_tbl <- extract(model_sislob_stanfit)$loss_sample[,1,] %>%
as_data_frame() %>%
mutate(iter = 1:n()) %>%
gather("timelbl", "value", -iter) %>%
mutate(time = gsub("V", "", timelbl) %>% as.numeric())
ggplot(snapshot_tbl %>% filter(acc_year == 1988)) +
geom_line (aes(x = time, y = value, group = iter)
,data = fitted_curves_tbl, alpha = 0.01) +
geom_line (aes(x = dev_lag, y = cum_loss), colour = 'red') +
geom_point(aes(x = dev_lag, y = cum_loss), colour = 'blue') +
expand_limits(y = 0) +
scale_y_continuous(labels = dollar) +
xlab("Time") +
ylab("Loss") +
ggtitle("Plot of 1988 Year Loss Development Against Posterior Distribution")