Statistical Characteristics of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 27

2.6.1 Reliability

The 'reliability' of a system is defined as the number of data in a satisfactory
state divided by the total number of data in the time series. Assuming
satisfactory values in the hydrologic time series x n containing n values are
those equal to or greater than some threshold x T , the reliability of the system
can be expressed as:
Reliability ( x ) =[Number of time periods t when x t x T ]/ n

(23)

The reliability of the original time series shown in blue colour (Fig. 2.7)
is 0.7, which suggests that it failed three times in ten. The reliability of the
rotated time series shown in red colour (Fig. 2.7) is also 0.7, indicating three
times failure in ten. In general, a more reliable system is better than a less
reliable system, but it is not always true. The reliability measure does not tell
anything about how fast a system recovers and returns to a satisfactory value,
nor does it indicate how bad an unsatisfactory value might be if it occurs. It
may be fine that a system that fails relatively often, but by insignificant
amounts and for short durations, will be much preferable to the system whose
reliability is much higher but when a failure does occur, it is likely to be much
more severe. 'Resilience' and 'vulnerability' measures, which are discussed
below, can quantify these system characteristics.

2.6.2 Resilience

The 'resilience' of a system is defined as the probability that if a system is in
an unsatisfactory state, the next state will be satisfactory. In other words, it is
the probability of having a satisfactory value in time period t - 1, given an
unsatisfactory value in any time period t . It can be expressed as:

Number of times a satisfactory value
follows an unsatisfactory value
Number of times an unsatisfactory
value occurred

Ë Û

Ì Ü

Í Ý

Resilience ( x ) = (24)

Ë Û

Ì Ü

Í Ý

Note that 'resilience' cannot be defined if no unsatisfactory values occur
in the time series. For the original time series shown in blue colour (Fig. 2.7),
the resilience is 2/2 = 1, again assuming the value of 300 mm or less is
considered a failure. We cannot judge the resilience of the blue time series on
the basis of the last failure in period 10 because we do not have an observation
in period 11. For the rotated time series shown in red colour (Fig. 2.7), the
resilience is 1/3 = 0.33.

2.6.3 Vulnerability
The term 'vulnerability' is a measure of the extent of the differences between
the threshold value and the unsatisfactory time series values. Obviously, this
is a probabilistic measure. Some use expected values, some use maximum
observed values, and others may assign a probability of exceedance to their