**5. Learning effects**

#### **5.1. Introduction**

So, with the exception of the USA, there is no indication from the limited available data of non-random sampling or of countries having different overall accident rates. The USA data indicate that here either sampling is not random or the accident rate is higher than in the rest of the world. The present data do not allow us to determine which of these alternatives is the

There have been several studies on reactor safety in the past. The first was the reactor safety study or Rasmussen report published in 1975 by the US Nuclear Regulatory Commission as report WASH-1400 or NUREG75/014 [11]. Five years later the German reactors were analysed in the Deutsche Risikostudie Kernkraftwerke [12]. In 1990 *Severe Accident Risks: An Assessment for Five U.S. Nuclear Power Plants* [13] was published. While the first two studies analysed typical reactors in their respective countries, the last one investigated five specified

'The Reactor Safety Study carefully examined the various paths leading to core melt. Using methods developed in recent years for estimating the likelihood of such accidents, a probability of occurrence was determined for each core melt accident identified. These probabilities were combined to obtain the total probability of melting the core. The value obtained was about one in 20,000 per reactor per year. With 100 reactors operating, as is anticipated for the U.S. by about 1980, this means that the chance for one such accident is one in 200 per year' [11].

The results of NUREG 1150 [14] can be found in Tables 3.2, 4.2, 5.2, 6.2 and 7.2 for the reactors Surry, Peach Bottom, Sequoyah, Grand Gulf and Zion, respectively. The German data are in

Based on the list and information of Sovacool, the following accidents are not included in the present study of severe accidents: Chalk River (1952) showed no core meltdown; Windscale (1957) was a military reactor only used for weapon production; Simi Valley (1959) was an experimental reactor; Monroe (1966) was an experimental reactor; and Lucens (1969) was an experimental reactor and probably showed no core meltdown. In Fukushima three of the six reactors at the site suffered severe destruction with INES ratings of 5–7. This threefold accident is counted as one because all three were triggered by the same cause, the tsunami with

.

and 3.40 × 10−4

accidents

more likely explanation and further studies are needed.

What are the results of these studies? WASH-1400 states:

So, the probability for a core melt accident per reactor year is 5 × 10<sup>−</sup><sup>5</sup>

the Deutsche Risikostudie [12]. The mean values vary between 4 × 10<sup>−</sup><sup>6</sup>

**4. Statistics of severe nuclear accidents**

142 Statistics - Growing Data Sets and Growing Demand for Statistics

**4.1. Results of previous PRA calculations**

reactors.

per reactor-year.

**4.2. Empirical analysis**

subsequent earthquake.

Experience and learning from operating power reactors and from analysing incidents and accidents are important for further reducing accident rates. Increasing operational experience should result in decreasing accident rates. This can be tested empirically by comparing accident rates with the amount of operational experience. In a simple approach, operational experience can be measured by the cumulative number of reactor-years up to a given date.

The small number of core melt accidents makes it difficult to detect any learning effect. Therefore, for this analysis we also included minor accidents and incidents. The two different datasets from *The Guardian* with 35 accidents and from Sovacool with 99 accidents were analysed independently. *The Guardian* data were grouped according to INES levels, and here all incidents of level 2 and above were included. One of the criteria for a level 2 incident is a 'significant contamination within a facility into an area not expected by design'. So, these incidents must be avoided by all means. From Sovacool's data all accidents related to nuclear power generation were included. Some of the basic results given below are summarised in [1], but the analysis here is more detailed.

## **5.2. Preliminary analysis**

In order to analyse the rather low number of accidents, the total number of accidents, which is the cumulative number of accidents that had happened until a given year, was compared to the total number of reactor-years until that year, which is the cumulative reactor-years. Thus, the accident rate is

the accident rate is

$$\text{accident rate} = \frac{\text{cumulative number of accidents}}{\text{cumulative reactor years}}.\tag{3}$$

Let *nt* be the number of reactors that are operational in year *t*, coded as *<sup>t</sup>* <sup>=</sup> 1, …,*T*. For *<sup>r</sup>* <sup>=</sup> 1, …,*nt*

*t* or approximately the probability of at least one accident at the reactor in year *t*. Assuming independence of the *Ytr* over the reactors' operational time *t*, the total number of accidents *Yt* <sup>=</sup>

ber of accidents in year *t*. If we further assume that the reactors have the same probability of

sure of nuclear operational experience in year *t* and postulate that the expected number of

**Figure 2.** Accident rate = cumulative accidents/cumulative reactor-years on a log scale vs. cumulative reactor years, each

data point representing 1 year. The lines are 95% pointwise confidence limits. *Source*: *The Guardian*.

, is some function *e*

no learning if and only if the function *β* is identically zero, in which case *e*(*Nt*

to be the cumulative number of reactor-years at year *t*. We use *Nt*

*<sup>t</sup>* = *e*(*Nt*

be the cumulative number of accidents up to time *t*. Assuming independence

 be the number of accidents at reactor *r* in year *t*. It is assumed that accidents at a given reactor in any given year occur independently. Then, accidents at that reactor over a 1-year period will occur according to a (possibly nonhomogeneous) Poisson process, so that *Ytr*

), where *λ<sup>t</sup>* <sup>=</sup> <sup>∑</sup>*<sup>r</sup>*=1

. Any variation across reactors will lead to extra-Poisson variation, which

) of *Nt*

*<sup>N</sup> <sup>β</sup>*(*x*)*dx*), where *β*(*N*) is the (instantaneous) rate of learning when the

), where *Λ<sup>t</sup>* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1

= *Nt*

) = *α*. If, however, there is learning, then *β* > 0 and *e*(*N*) will be a

is the expected number of accidents at reactor *r* in year

*nt <sup>λ</sup>tr*

is the expected number of accidents per reactor

let *Ytr*

∑*<sup>r</sup>*=1 *nt Ytr*

be distributed as Poisson (*λtr*) where *λtr*

failure in any given year, then *λtr* <sup>=</sup> *<sup>e</sup>*

*<sup>t</sup> nu*

accidents per reactor per year, *e*

*<sup>t</sup> Yu*

we may write *e*(*N*) <sup>=</sup> *<sup>α</sup>* exp(−<sup>∫</sup>

cumulative accident rate *E*(*Xt*

*e t*

can be assessed following model fitting.

number of reactor-years has reached *N*.

in year *t* and *λ<sup>t</sup>* <sup>=</sup> *nt*

Define *Nt* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1

Now, let *Xt* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1

of the *Yt* ′

in year *t* will be distributed as Poisson (*λ<sup>t</sup>*

*t* , where *e t*

*t*

*<sup>s</sup>* over time, *Xt* will be distributed as Poisson (*Λ<sup>t</sup>*

expected number of accidents per reactor per year, and *Λ<sup>t</sup>*

/*Nt*

0

,

145

will

as a mea-

*<sup>t</sup> nu <sup>e</sup>*(*Nu*). There is

) = *α*, the constant

is the expected total num-

Severe Nuclear Accidents and Learning Effects http://dx.doi.org/10.5772/intechopen.76637

. Without any loss of generality,

*α*. It follows that the expected

*<sup>t</sup> <sup>λ</sup><sup>u</sup>* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1

Without any learning effect, the increase in accidents per reactor-year should be the same for every reactor-year; so, this accident rate should remain constant. A learning effect would decrease the accident rate.

We start by investigating *The Guardian* data. As discussed in Section 2, after excluding some accidents from the study, the final number of nuclear power-related incidents or accidents with level 2 and above is 16. The accident rate calculated from these data is plotted against the cumulative reactor-years in **Figure 2**. In order to present the data more clearly, the accident rate is displayed in a logarithmic scale. Every point represents the data of 1 year. The lines are 95% pointwise confidence intervals obtained from Poisson statistics.

A decreasing trend in this plot would indicate the presence of a learning effect. As can be readily seen, the first accident in 1957 resulted in a relatively high accident rate of about 0.05 per reactor-year. The following years saw no (publicly known) accident so the observed rate decreases drastically. Such a decreasing behaviour would be expected if an initial learning effect exists. However, after around 500 reactor-years, the plot appears to stabilise, with the accident rate varying around a constant value of about 1 in 1000 reactor-years. The plot does not indicate a learning effect. We investigate this further using a more detailed statistical analysis in Section 4.2.

Next, the Sovacool data is considered. As discussed in Section 2, after excluding some accidents from the study, the final number of nuclear power-related incidents or accidents with level 2 and above is 99. **Figure 3** is a plot of the log accident rate against cumulative reactoryears for these data, along with 95% pointwise confidence limits.

The slight decreasing trend in the latter portion of the graph along with the confidence limits suggests the possible presence of a small learning effect, with a larger effect apparent in the early years. We investigate this further using a more detailed statistical analysis in Section 5.3.

### **5.3. Formal statistical analysis**

In order to investigate the possibility of a learning effect more formally, we constructed a suitable statistical model. The notation and assumptions below, summarised in the supplementary online material for [1], are common to the analyses of both *The Guardian* and the Sovacool data.

Let *nt* be the number of reactors that are operational in year *t*, coded as *<sup>t</sup>* <sup>=</sup> 1, …,*T*. For *<sup>r</sup>* <sup>=</sup> 1, …,*nt* , let *Ytr* be the number of accidents at reactor *r* in year *t*. It is assumed that accidents at a given reactor in any given year occur independently. Then, accidents at that reactor over a 1-year period will occur according to a (possibly nonhomogeneous) Poisson process, so that *Ytr* will be distributed as Poisson (*λtr*) where *λtr* is the expected number of accidents at reactor *r* in year *t* or approximately the probability of at least one accident at the reactor in year *t*. Assuming independence of the *Ytr* over the reactors' operational time *t*, the total number of accidents *Yt* <sup>=</sup> ∑*<sup>r</sup>*=1 *nt Ytr* in year *t* will be distributed as Poisson (*λ<sup>t</sup>* ), where *λ<sup>t</sup>* <sup>=</sup> <sup>∑</sup>*<sup>r</sup>*=1 *nt <sup>λ</sup>tr* is the expected total number of accidents in year *t*. If we further assume that the reactors have the same probability of failure in any given year, then *λtr* <sup>=</sup> *<sup>e</sup> t* , where *e t* is the expected number of accidents per reactor in year *t* and *λ<sup>t</sup>* <sup>=</sup> *nt e t* . Any variation across reactors will lead to extra-Poisson variation, which can be assessed following model fitting.

**5.2. Preliminary analysis**

144 Statistics - Growing Data Sets and Growing Demand for Statistics

decrease the accident rate.

analysis in Section 4.2.

**5.3. Formal statistical analysis**

Sovacool data.

the accident rate is

In order to analyse the rather low number of accidents, the total number of accidents, which is the cumulative number of accidents that had happened until a given year, was compared to the total number of reactor-years until that year, which is the cumulative reactor-years. Thus,

accident rate <sup>=</sup> cumulative number of accidents \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ cumulative reactor years . (3)

Without any learning effect, the increase in accidents per reactor-year should be the same for every reactor-year; so, this accident rate should remain constant. A learning effect would

We start by investigating *The Guardian* data. As discussed in Section 2, after excluding some accidents from the study, the final number of nuclear power-related incidents or accidents with level 2 and above is 16. The accident rate calculated from these data is plotted against the cumulative reactor-years in **Figure 2**. In order to present the data more clearly, the accident rate is displayed in a logarithmic scale. Every point represents the data of 1 year. The lines are

A decreasing trend in this plot would indicate the presence of a learning effect. As can be readily seen, the first accident in 1957 resulted in a relatively high accident rate of about 0.05 per reactor-year. The following years saw no (publicly known) accident so the observed rate decreases drastically. Such a decreasing behaviour would be expected if an initial learning effect exists. However, after around 500 reactor-years, the plot appears to stabilise, with the accident rate varying around a constant value of about 1 in 1000 reactor-years. The plot does not indicate a learning effect. We investigate this further using a more detailed statistical

Next, the Sovacool data is considered. As discussed in Section 2, after excluding some accidents from the study, the final number of nuclear power-related incidents or accidents with level 2 and above is 99. **Figure 3** is a plot of the log accident rate against cumulative reactor-

The slight decreasing trend in the latter portion of the graph along with the confidence limits suggests the possible presence of a small learning effect, with a larger effect apparent in the early years. We investigate this further using a more detailed statistical analysis in Section 5.3.

In order to investigate the possibility of a learning effect more formally, we constructed a suitable statistical model. The notation and assumptions below, summarised in the supplementary online material for [1], are common to the analyses of both *The Guardian* and the

95% pointwise confidence intervals obtained from Poisson statistics.

years for these data, along with 95% pointwise confidence limits.

Define *Nt* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1 *<sup>t</sup> nu* to be the cumulative number of reactor-years at year *t*. We use *Nt* as a measure of nuclear operational experience in year *t* and postulate that the expected number of accidents per reactor per year, *e t* , is some function *e <sup>t</sup>* = *e*(*Nt* ) of *Nt* . Without any loss of generality, we may write *e*(*N*) <sup>=</sup> *<sup>α</sup>* exp(−<sup>∫</sup> 0 *<sup>N</sup> <sup>β</sup>*(*x*)*dx*), where *β*(*N*) is the (instantaneous) rate of learning when the number of reactor-years has reached *N*.

Now, let *Xt* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1 *<sup>t</sup> Yu* be the cumulative number of accidents up to time *t*. Assuming independence of the *Yt* ′ *<sup>s</sup>* over time, *Xt* will be distributed as Poisson (*Λ<sup>t</sup>* ), where *Λ<sup>t</sup>* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1 *<sup>t</sup> <sup>λ</sup><sup>u</sup>* <sup>=</sup> <sup>∑</sup>*<sup>u</sup>*=1 *<sup>t</sup> nu <sup>e</sup>*(*Nu*). There is no learning if and only if the function *β* is identically zero, in which case *e*(*Nt* ) = *α*, the constant expected number of accidents per reactor per year, and *Λ<sup>t</sup>* = *Nt α*. It follows that the expected cumulative accident rate *E*(*Xt* /*Nt* ) = *α*. If, however, there is learning, then *β* > 0 and *e*(*N*) will be a

**Figure 2.** Accident rate = cumulative accidents/cumulative reactor-years on a log scale vs. cumulative reactor years, each data point representing 1 year. The lines are 95% pointwise confidence limits. *Source*: *The Guardian*.

decreasing function of *N*, so that a plot of *Xt* /*Nt* against *Nt* will exhibit a decreasing trend, as illustrated in **Figures 2** and **3**.

*Guardian* data, the GLM results based on the years 1958–2011 produce a negative estimate of

to be zero, then the estimated probability throughout this period is 0.0010, which is the same

Finally, consideration of only the more recent data from 1970 onwards produces a positive

from 0.0011 to 0.0010 over this period. However, again the result is not statistically signifi-

throughout this period is again 0.0010. So, overall, there is no evidence from these data of any

The larger size of the Sovacool dataset allows us to elaborate the model to investigate the possibility of a learning effect more formally. To this end we choose a suitable formulation for the function *e*(*N*). A change-point model could be used, but we preferred to use a smooth alternative that does no presuppose the existence of a sudden change in the accident rate. A commonly used functional form that models different rates of change at the early and late

Here, *β* is the ultimate rate of learning relevant in the later years. The initial rate of learning *β*<sup>I</sup>

relevant for the early years, can be obtained as a function of all the parameters in the model.

In particular, the initial rate is *<sup>β</sup><sup>I</sup>* <sup>=</sup> *<sup>β</sup>* <sup>+</sup> *<sup>η</sup>*/(1 <sup>+</sup> *<sup>e</sup>*<sup>−</sup>). If the change from the initial to the final rate is quite pronounced, then it can be shown that this model will approximate to a change-point model, with the change-point at *N* = *ϕ*. We can now set up the likelihood function *L*(*θ*), where *θ* = (*γ*, *β*, *ϕ*, *η*) and *γ* = log α, and carry out a likelihood analysis [16]. Starting values for the computation may be obtained from graphical inspection and/or by fitting a generalised linear

Another hypothesis of interest is that there is a constant rate of learning throughout the entire

parameters, along with the *p*-values for the indicated null hypotheses, are exhibited in **Table 1**. We see that there is some evidence of a learning effect over the latter portion of the data, formally verifying what seems to be indicated in **Figure 3**. Moreover, the rate of learning is

:*β<sup>I</sup>* <sup>=</sup> *<sup>β</sup>*. The maximum likelihood estimates and standard errors for various

A convenient parameterisation of this function is *e*(*N*) <sup>=</sup> *<sup>e</sup>* <sup>−</sup>*<sup>N</sup>*{1 <sup>+</sup> *<sup>e</sup>* <sup>−</sup>*η*(*N*−*<sup>ϕ</sup>*)}, where *<sup>η</sup>* <sup>=</sup> *<sup>β</sup>*<sup>0</sup>

/*α*)}/*η*. With this parameterisation the instantaneous learning rate is

model to the data after 1962, using the Poisson family with a log link function.

for *β*, indicating an increasing accident rate. However, the associated standard error

is large, and so again this value of *β* is far from statistically significant. If *β* is taken

for *β*, which would give rise to a very slight decrease in the accident rate

. If *β* is taken to be zero, then the estimated probability

Severe Nuclear Accidents and Learning Effects http://dx.doi.org/10.5772/intechopen.76637

*<sup>N</sup>* + *e* <sup>−</sup>*<sup>N</sup>*. (4)

<sup>1</sup> <sup>+</sup> *<sup>e</sup> <sup>η</sup>*(*N*−*ϕ*). (5)

:*β* = 0, which corresponds to no learning in the later years.

,

147

− *β* and *ϕ*=

−8.61 × 10<sup>−</sup><sup>5</sup>

of 5.7 × 10<sup>−</sup><sup>5</sup>

{log(*α*<sup>0</sup>

estimate of 7.29 × 10<sup>−</sup><sup>6</sup>

as the result based on the complete dataset.

learning effect, at least beyond the initial few years of operation.

portions of a series is the biexponential function, given by

*<sup>β</sup>*(*N*) <sup>=</sup> *<sup>β</sup>* <sup>+</sup> *<sup>η</sup>* \_\_\_\_\_\_\_

The main hypothesis of interest is *H*<sup>0</sup>

period, that is, *H*<sup>1</sup>

*e*(*N*) = *α*<sup>0</sup> *e* <sup>−</sup>*β*<sup>0</sup>

cant, with a standard error of 6.0 × 10<sup>−</sup><sup>5</sup>

*5.3.2. Analysis of the Sovacool data*

#### *5.3.1. Analysis of The Guardian data*

For *The Guardian* data, we took *β*(*N*) = *β*, so that there is either no learning or a constant rate of learning. In this case the expected number of accidents per reactor per year *e t* (*Nt* ) = *α* exp(−*β Nt* ), an exponentially decreasing function of the number of reactor-years. Since log*λ<sup>t</sup>* <sup>=</sup> log*nt* + log *α* − *β Nt* , the model is a generalised linear model (GLM) with Poisson family and log link function [15]. The analysis was implemented in the programming language R.

**Figure 2** suggests the absence of any learning effect, but to investigate this formally, we set up and tested the null hypothesis *<sup>H</sup>* <sup>0</sup> :*β* = 0. Based on the dataset from 1956 to 2011, a likelihood analysis produced a positive estimate of 1.58 × 10<sup>−</sup><sup>5</sup> for *β*, but with a standard error of 5.5 × 10<sup>−</sup><sup>5</sup> , this is far from being statistically significant (with a *p*-value of 0.78). If all the estimated values were the true values of the parameters, then the probability of a severe accident per reactor-year would reduce from 0.0012 to 0.0009 over the period. If, however, *β* is taken to be zero, then the estimated probability of a severe accident throughout the period is 0.0010.

Given the erratic behaviour in the early years, with just one accident in 1957 followed by a run of zero accidents over the next 19 years, it is important to investigate the sensitivity of the results to the early data. For the somewhat more informative data discussed in the next section with Sovacool's data, we will proceed more formally by elaborating the model to take into account the possibly different learning behaviour in the early years. In the case of *The* 

**Figure 3.** Accident rate = cumulative accidents/cumulative reactor-years on a log scale vs. cumulative reactor-years, each data point representing 1 year. The lines are 95% pointwise confidence limits. *Source*: Sovacool.

*Guardian* data, the GLM results based on the years 1958–2011 produce a negative estimate of −8.61 × 10<sup>−</sup><sup>5</sup> for *β*, indicating an increasing accident rate. However, the associated standard error of 5.7 × 10<sup>−</sup><sup>5</sup> is large, and so again this value of *β* is far from statistically significant. If *β* is taken to be zero, then the estimated probability throughout this period is 0.0010, which is the same as the result based on the complete dataset.

Finally, consideration of only the more recent data from 1970 onwards produces a positive estimate of 7.29 × 10<sup>−</sup><sup>6</sup> for *β*, which would give rise to a very slight decrease in the accident rate from 0.0011 to 0.0010 over this period. However, again the result is not statistically significant, with a standard error of 6.0 × 10<sup>−</sup><sup>5</sup> . If *β* is taken to be zero, then the estimated probability throughout this period is again 0.0010. So, overall, there is no evidence from these data of any learning effect, at least beyond the initial few years of operation.

#### *5.3.2. Analysis of the Sovacool data*

decreasing function of *N*, so that a plot of *Xt*

146 Statistics - Growing Data Sets and Growing Demand for Statistics

set up and tested the null hypothesis *<sup>H</sup>* <sup>0</sup>

trated in **Figures 2** and **3**.

*Nt*

of 5.5 × 10<sup>−</sup><sup>5</sup>

period is 0.0010.

*5.3.1. Analysis of The Guardian data*

/*Nt*

learning. In this case the expected number of accidents per reactor per year *e*

[15]. The analysis was implemented in the programming language R.

likelihood analysis produced a positive estimate of 1.58 × 10<sup>−</sup><sup>5</sup>

an exponentially decreasing function of the number of reactor-years. Since log*λ<sup>t</sup>* <sup>=</sup> log*nt*

against *Nt*

For *The Guardian* data, we took *β*(*N*) = *β*, so that there is either no learning or a constant rate of

, the model is a generalised linear model (GLM) with Poisson family and log link function

**Figure 2** suggests the absence of any learning effect, but to investigate this formally, we

estimated values were the true values of the parameters, then the probability of a severe accident per reactor-year would reduce from 0.0012 to 0.0009 over the period. If, however, *β* is taken to be zero, then the estimated probability of a severe accident throughout the

Given the erratic behaviour in the early years, with just one accident in 1957 followed by a run of zero accidents over the next 19 years, it is important to investigate the sensitivity of the results to the early data. For the somewhat more informative data discussed in the next section with Sovacool's data, we will proceed more formally by elaborating the model to take into account the possibly different learning behaviour in the early years. In the case of *The* 

**Figure 3.** Accident rate = cumulative accidents/cumulative reactor-years on a log scale vs. cumulative reactor-years, each

data point representing 1 year. The lines are 95% pointwise confidence limits. *Source*: Sovacool.

, this is far from being statistically significant (with a *p*-value of 0.78). If all the

will exhibit a decreasing trend, as illus-

*t* (*Nt*

for *β*, but with a standard error

:*β* = 0. Based on the dataset from 1956 to 2011, a

) = *α* exp(−*β Nt*

+ log *α* − *β*

),

The larger size of the Sovacool dataset allows us to elaborate the model to investigate the possibility of a learning effect more formally. To this end we choose a suitable formulation for the function *e*(*N*). A change-point model could be used, but we preferred to use a smooth alternative that does no presuppose the existence of a sudden change in the accident rate. A commonly used functional form that models different rates of change at the early and late portions of a series is the biexponential function, given by

$$
\mathcal{e}(\mathcal{N}) = \alpha\_0 e^{-\mathfrak{q}\mathcal{N}} + a e^{-\mathfrak{q}\mathcal{N}}.\tag{4}
$$

Here, *β* is the ultimate rate of learning relevant in the later years. The initial rate of learning *β*<sup>I</sup> , relevant for the early years, can be obtained as a function of all the parameters in the model.

A convenient parameterisation of this function is *e*(*N*) <sup>=</sup> *<sup>e</sup>* <sup>−</sup>*<sup>N</sup>*{1 <sup>+</sup> *<sup>e</sup>* <sup>−</sup>*η*(*N*−*<sup>ϕ</sup>*)}, where *<sup>η</sup>* <sup>=</sup> *<sup>β</sup>*<sup>0</sup> − *β* and *ϕ*= {log(*α*<sup>0</sup> /*α*)}/*η*. With this parameterisation the instantaneous learning rate is

$$
\beta(\text{N}) = \beta + \frac{\eta}{1 + e^{\eta(\text{N} + \eta)}}.\tag{5}
$$

In particular, the initial rate is *<sup>β</sup><sup>I</sup>* <sup>=</sup> *<sup>β</sup>* <sup>+</sup> *<sup>η</sup>*/(1 <sup>+</sup> *<sup>e</sup>*<sup>−</sup>). If the change from the initial to the final rate is quite pronounced, then it can be shown that this model will approximate to a change-point model, with the change-point at *N* = *ϕ*. We can now set up the likelihood function *L*(*θ*), where *θ* = (*γ*, *β*, *ϕ*, *η*) and *γ* = log α, and carry out a likelihood analysis [16]. Starting values for the computation may be obtained from graphical inspection and/or by fitting a generalised linear model to the data after 1962, using the Poisson family with a log link function.

The main hypothesis of interest is *H*<sup>0</sup> :*β* = 0, which corresponds to no learning in the later years. Another hypothesis of interest is that there is a constant rate of learning throughout the entire period, that is, *H*<sup>1</sup> :*β<sup>I</sup>* <sup>=</sup> *<sup>β</sup>*. The maximum likelihood estimates and standard errors for various parameters, along with the *p*-values for the indicated null hypotheses, are exhibited in **Table 1**.

We see that there is some evidence of a learning effect over the latter portion of the data, formally verifying what seems to be indicated in **Figure 3**. Moreover, the rate of learning is fairly constant throughout the period from around 1962 to 2010, as can be seen from **Figures 4** and **5**. In these figures the observed accident rate *Yt* /*nt* per reactor in year *t* is plotted against year, in contrast to **Figures 3** and **4** in which the cumulative accident rate is plotted against cumulative reactor-years. The superimposed lines in **Figures 4** and **5** are the estimated theoretical annual accident rates *e*(*Nt* ) obtained from the biexponential Poisson model. **Figure 5** is the same as in **Figure 4**, except that omitting the data before 1964 allows for a higher resolution of the *y* axis.

Although the data indicate a possible nonconstant learning effect over the period, with a larger effect at the beginning of the period up to about 1962, we see from **Table 1** that this is not statistically significant, owing to the highly variable nature of the early data when there were relatively few reactors and only two accidents. If the initial and final rates of learning do differ, then the best estimate of *ϕ*, the effective change-point in terms of the number of reactor years, is 43.10, which corresponds to the year 1961. This estimate is highly variable; however, a 90% confidence interval for *ϕ*, constructed from the profile likelihood of log *ϕ*, gave values of *ϕ* between 3 and 221, which roughly correspond to the years 1957 and1966, respectively. These change-point results are unreliable, however, and more reliable estimates are obtained later in this section.

The high variability in the change-point contributes to the high degree of error in the estimate

As a diagnostic for the model, one may calculate the standardised response residuals *rt* <sup>=</sup> (*yt* <sup>−</sup> *<sup>λ</sup>*̂

and the estimated model values *λ*̂

these showed no unusual pattern. Moreover, the observed standard deviation of these residuals

one. Specifically, if we suppose that there is a positive but constant variation over reactors, so that

We further carried out a Bayesian analysis of these data. We used a noninformative prior of the form π(*θ*) ∝ 1/*α*. A higher-order asymptotic approximation was computed, using the method in [17]. This was supplemented by the Monte Carlo method described in that paper. The results of the latter analysis, which may be considered to be exact having negligible simu-

sis, providing evidence of a learning effect over the latter portion of the data. The credible

learning, although this difference may be very small. If the initial and final rates of learning do differ, then the Bayes estimate of the change-point *ϕ* is 39.37, which corresponds to the year 1961, as in the likelihood analysis. However, the exact Bayesian 90% credible interval

**Parameter Estimate Standard error Null hypothesis** *p***-Value** *γ* −4.690 0.194 — *β* × 10<sup>−</sup><sup>5</sup> 5.362 2.476 *β* = 0 0.029 *β<sup>I</sup>* − *β* 0.053 0.056 *β<sup>I</sup>* = *β* 0.218

**Table 1.** Likelihood results from the biexponential Poisson model for the Sovacool data.

− *β* provides some evidence of a difference between the initial and final rates of

lation error, are given in **Table 2**. These are very similar to the asymptotic results.

, then the theoretical variance of the *t*th residual at the true parameter values will be

. Thus, the observed residuals would exhibit extra-Poisson variability, which does not

*t*

 as seen in **Table 1**. However, whether or not there is a change in the rate of learning over the period, the estimated probability of an accident or incident at a reactor in 1 year falls from

> *<sup>t</sup>*)/√ \_\_ *λ*̂ *t*

149

. When plotted against the year,

Severe Nuclear Accidents and Learning Effects http://dx.doi.org/10.5772/intechopen.76637

is constant over reactors was a reasonable

is consistent with the likelihood analy-

of *β<sup>I</sup>*

var(*λtr*) <sup>=</sup> *<sup>σ</sup>*<sup>2</sup>

interval for *<sup>β</sup> <sup>I</sup>*

1 + *e*(*Nt* ) *σ*<sup>2</sup>

0.010 in 1963 to 0.004 in 2010.

from the observed values *yt*

appear to be the case here.

of *Yt*

was 0.982, indicating that our initial assumption that *λtr*

We see that the Bayesian credible interval for *β* × 10<sup>−</sup><sup>5</sup>

*ϕ* 43.10 33.42

**Figure 4.** Observed and theoretical annual accident rate per year.

**Figure 5.** Same as in **Figure 4** for the years 1964–2010.

Although the data indicate a possible nonconstant learning effect over the period, with a larger effect at the beginning of the period up to about 1962, we see from **Table 1** that this is not statistically significant, owing to the highly variable nature of the early data when there were relatively few reactors and only two accidents. If the initial and final rates of learning do differ, then the best estimate of *ϕ*, the effective change-point in terms of the number of reactor years, is 43.10, which corresponds to the year 1961. This estimate is highly variable; however, a 90% confidence interval for *ϕ*, constructed from the profile likelihood of log *ϕ*, gave values of *ϕ* between 3 and 221, which roughly correspond to the years 1957 and1966, respectively. These change-point results are unreliable, however, and more reliable estimates are obtained later in this section.

fairly constant throughout the period from around 1962 to 2010, as can be seen from **Figures 4**

year, in contrast to **Figures 3** and **4** in which the cumulative accident rate is plotted against cumulative reactor-years. The superimposed lines in **Figures 4** and **5** are the estimated theo-

the same as in **Figure 4**, except that omitting the data before 1964 allows for a higher resolu-

/*nt*

) obtained from the biexponential Poisson model. **Figure 5** is

per reactor in year *t* is plotted against

and **5**. In these figures the observed accident rate *Yt*

148 Statistics - Growing Data Sets and Growing Demand for Statistics

**Figure 4.** Observed and theoretical annual accident rate per year.

**Figure 5.** Same as in **Figure 4** for the years 1964–2010.

retical annual accident rates *e*(*Nt*

tion of the *y* axis.

The high variability in the change-point contributes to the high degree of error in the estimate of *β<sup>I</sup>* as seen in **Table 1**. However, whether or not there is a change in the rate of learning over the period, the estimated probability of an accident or incident at a reactor in 1 year falls from 0.010 in 1963 to 0.004 in 2010.

As a diagnostic for the model, one may calculate the standardised response residuals *rt* <sup>=</sup> (*yt* <sup>−</sup> *<sup>λ</sup>*̂ *<sup>t</sup>*)/√ \_\_ *λ*̂ *t* from the observed values *yt* of *Yt* and the estimated model values *λ*̂ *t* . When plotted against the year, these showed no unusual pattern. Moreover, the observed standard deviation of these residuals was 0.982, indicating that our initial assumption that *λtr* is constant over reactors was a reasonable one. Specifically, if we suppose that there is a positive but constant variation over reactors, so that var(*λtr*) <sup>=</sup> *<sup>σ</sup>*<sup>2</sup> , then the theoretical variance of the *t*th residual at the true parameter values will be 1 + *e*(*Nt* ) *σ*<sup>2</sup> . Thus, the observed residuals would exhibit extra-Poisson variability, which does not appear to be the case here.

We further carried out a Bayesian analysis of these data. We used a noninformative prior of the form π(*θ*) ∝ 1/*α*. A higher-order asymptotic approximation was computed, using the method in [17]. This was supplemented by the Monte Carlo method described in that paper. The results of the latter analysis, which may be considered to be exact having negligible simulation error, are given in **Table 2**. These are very similar to the asymptotic results.

We see that the Bayesian credible interval for *β* × 10<sup>−</sup><sup>5</sup> is consistent with the likelihood analysis, providing evidence of a learning effect over the latter portion of the data. The credible interval for *<sup>β</sup> <sup>I</sup>* − *β* provides some evidence of a difference between the initial and final rates of learning, although this difference may be very small. If the initial and final rates of learning do differ, then the Bayes estimate of the change-point *ϕ* is 39.37, which corresponds to the year 1961, as in the likelihood analysis. However, the exact Bayesian 90% credible interval


**Table 1.** Likelihood results from the biexponential Poisson model for the Sovacool data.


significant events (including safety system malfunctions and unplanned and immediate

Severe Nuclear Accidents and Learning Effects http://dx.doi.org/10.5772/intechopen.76637 151

Our work shows the possibility of studying learning effects within the nuclear industry. But more detailed results require more analysis and more information from reactor operators and regulators. But this is difficult on an international scale because of the restrictive information

**Country Total energy produced Reactor years Accidents**

Argentina 225.63 66.5150685 1 Armenia 90.14 44.1945205 0 Belgium 1422.8 247.421918 0 Brazil 228.57 41.2219178 0 Bulgaria 518.07 151.057534 0 Canada 2907.4 692.589041 1 China 1155.36 125.219178 0 Czechoslovakia 540.62 122.561644 1 East Germany 143.21 79.9534247 1 Finland 719.34 131.328767 0 France 12137.14 1813.29863 10 Germany 4741.16 701.482192 2 Hungary 404.27 106.027397 1 India 459.94 356.736986 7 Iran 12.46 0.3260274 0 Italy 89.78 80.569863 0 Japan 7310.61 1558.76986 7 Lithuania 241.58 43.3972603 0 Mexico 211.94 39.8794521 0 Netherlands 152.9 66.9726027 0 Pakistan 53.56 52.5863014 0 Romania 143.54 19.8849315 0 Russia 4286.29 947.312329 2 Slovakia 433.25 140.19726 0 Slovenia 160.83 30.2657534 1 South Africa 351.43 54.2109589 0

reactor shutdowns)' [18].

policy of the IAEA.

**A. Appendix**

**Table 2.** Bayes results from the biexponential Poisson model for the Sovacool data.

is tighter than the approximate confidence interval produced earlier and corresponds to the years 1959–1963.

Whether or not there is a change in the rate of learning over the period, the estimated probabilities of an accident or incident at a reactor in 1963 and 2010 are identical to those obtained earlier from the likelihood analysis.
