**4. Data analysis**

• Draw a mean residual life plot using (26)

*Natural Hazards - Impacts, Adjustments and Resilience*

*3.3.2 Parameter stability plot*

*ζ*, but with scale parameter of:

such that:

interval to select this point.

significant statistical change.

*<sup>r</sup>* <sup>¼</sup> <sup>P</sup>*<sup>r</sup>*

a series:

**282**

where *U* <sup>∗</sup>

*3.3.3 Gertensgarbe and Werner plot*

(This follows from theory 1.3).

Let us re-parametrise the scale parameter *σr*:

• Choose a threshold above which the plot is approximately linear. We use

Assuming that the exceedances, (*z* � *v*), over a threshold *v* follow a GPD (*ζ*, *σv*), the exceedances will still follow a GPD for any higher threshold *r*>*v*, with the same

> *σ<sup>r</sup>* � *ζ<sup>r</sup>* ¼ *σ<sup>v</sup>* � *ζv <sup>σ</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>σ</sup><sup>v</sup>* � *<sup>ζ</sup><sup>v</sup>*

The parameter *σ* <sup>∗</sup> now only depends on a sufficiently high threshold *v*. The parameter stability plot involves plotting the GPD parameter estimates for a range of values *v*. The threshold is chosen to be the point where the shape and the modified scale parameters become stable (that is, the parameter estimates is constant above the threshold at which the GPD becomes valid). We use confidence

The test was proposed by Gertensgarbe and Werner (1989) and is used to select a threshold by detecting the starting point of the extreme region. The idea behind the test is that the behaviour of a series of differences that correspond to the extreme observations is different from the one corresponding to the non-extreme observations. So, given a series of differences Δ*<sup>r</sup>* ¼ *z*ð Þ*<sup>r</sup>* � *z*ð Þ *<sup>r</sup>*þ<sup>1</sup> , *i* ¼ 2, 3, ⋯, *n*, of the order statistics, *z*ð Þ<sup>1</sup> ≤*z*ð Þ<sup>2</sup> ≤⋯≤*z*ð Þ *<sup>n</sup>* , the starting point of the extreme region, and hence the threshold, will be the point at which the series of differences exhibit a

To detect this point, we apply a sequential version of the Mann-Kendall test to check the null hypothesis that there is no change in the series of differences. Define

> *<sup>r</sup>* � *r r*ð Þ �<sup>1</sup> <sup>4</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *i r*ð Þ �1 ð Þ *i*þ5 72

*<sup>j</sup>*¼<sup>1</sup>*<sup>n</sup> <sup>j</sup>* and *<sup>n</sup> <sup>j</sup>* is the number of values in the the series of

<sup>q</sup> (30)

*Ur* <sup>¼</sup> *<sup>U</sup>* <sup>∗</sup>

differences Δ1, ⋯, Δ *<sup>j</sup>* less than Δ *<sup>j</sup>*. We also define another series, *Up*, by applying the same procedure to the series of differences from the end to start,Δ*n*, ⋯, Δ1,

*σ<sup>r</sup>* ¼ *σ<sup>v</sup>* þ *ζ*ð Þ *r* � *v* (27)

*<sup>σ</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>σ</sup><sup>r</sup>* � *<sup>ζ</sup><sup>r</sup>* (28)

*σ<sup>r</sup>* ¼ *σ<sup>v</sup>* þ *ζr* � *ζv* (29)

confidence interval to determine whether the plot looks linear.

We use data for all the natural disasters that has been recorded in Kenya in the period 1964 2018. The data is obtained from CRED database, which is the currently the most comprehensive database for natural events. The data was also crossreferenced with that from other sources including UN-agencies and NDMU. The impact of natural disasters is quantified in terms of the total number of people affected on an annual basis, which we deemed to be more reliable than the the total damage in monetary terms. The total number of people affected includes those who were injured, died, left homeless or affected in any other way by natural disasters. A total of 112 events have been recorded over that period of time with a resulting damage of approximately 62 million people affected.

Descriptive statistics for both the annual occurrence and the impacts are provided in **Table 1**. The minimum number of disasters and the resulting impact is zero, which corresponds to those years where no natural disasters occurred. A total of 22 years recorded no natural disaster events. The average annual number of natural disaster occurrences in Kenya is two, and about 1*:*3 million people are affected every year. The maximum number of natural disaster occurrences observed in any year is 9 and the worst disaster recorded in any year affected approximately 23, 331, 469 people. The mean is greater than the median for both variables, indicating that the data is right-skewed. We can also observe that the spread of the impacts is large as suggested by both the standard deviation and the inter-quantile range (about 3*:*5 million and 252718 respectively), as opposed to that of the occurrence.

We are interested in the distribution of the number of occurences and the corresponding impact of natural disasters in Kenya in the period of study. **Figure 1** shows the distribution of the number of natural disaster occurrences in the last 55 years. The number of natural disasters between 1964 and the late 1990s was fairly low, with no year experiencing more than two events. We can then observe a sharp


**Table 1.** *Descriptive Statistics.*

**Figure 4.**

**Figure 5.** *Mean Excess Plot.*

**285**

**Figure 3.**

*Scatter plot of Annual Impact.*

*On Modelling Extreme Damages from Natural Disasters in Kenya*

*DOI: http://dx.doi.org/10.5772/intechopen.94578*

*Scatter plot of Annual Occurrence against Annual Impact.*

#### **Figure 1.**

*Distribution of Natural Disasters in Kenya in the period: 1964–2018.*

increase in the annual occurrences in the last 20 years. During this period, all the years have experienced at least two events.

A similar trend is observed for the impacts of the natural disasters, as shown in **Figure 1b**. Very few people were being affected in the years between 1964 and 1990. Then from 1990, the severity of the natural disasters increased and a lot more people were being affected.
