Statistical Characteristics of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 18

For the positively skewed data series, the GM is usually fairly close to the
median of the series. In fact, the GM is an unbiased estimate of the median
when the logarithms of the datasets are symmetric (Helsel and Hirsch, 2002).
This is because the median and mean logarithms are equal. When transformed
back to original units, the GM continues to be an estimate for the median, but
is not an estimate for the mean.
In mathematics, the harmonic mean (sometimes also called 'sub-contrary
mean') is one of several kinds of averages. Typically, it is appropriate for
situations when the average of rates is desired (Shahin et al., 1993). The
harmonic mean (HM) of positive real numbers of a time series, x 1 , x 2 , ..., x n > 0
is defined as:

n n

Ç

HM = (8)

11 1

n

1

...

xx x

x

1 2 n

i

i 1

The harmonic mean is related to the arithmetic and geometric means.
Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of
reciprocals. For all positive datasets containing at least one pair of non-equal
values, the harmonic mean is always the least of the three means, while the
arithmetic mean is always the greatest of the three and the geometric mean is
always in between. Since the harmonic mean of a list of numbers tends
strongly toward the least elements of the list, it tends (compared to the arithmetic
mean) to mitigate the impact of large outliers and aggravate the impact of
small ones.
Moreover, compromises between the median and mean can be made by
trimming off several of the lowest and highest observations in the time series,
and then calculating the mean of remaining data. Unlike the mean, such an
estimate of location is not influenced by the most extreme (or abnormal) tails
of the sample. Nevertheless, unlike the median, it allows the magnitudes of
most data values to affect the location estimate (Helsel and Hirsch, 2002).
This estimate of location is known as 'trimmed mean' because it is computed
after trimming away a desirable percentage of the data. The most common
trimming is to remove 25% of the data on each tail—the resulting mean of the
central 50% of data is commonly called 'trimmed mean', but it is more
precisely 25 percent trimmed mean. A zero percent trimmed mean results in
the arithmetic mean itself, whereas trimming all but 1 or 2 mid data points
produces the median. Percentage of trimming should be explicitly stated when
'trimmed mean' is used. The trimmed mean is a robust estimator of location
because it is not strongly influenced by outliers, and works well for a wide
variety of distributional shapes such as normal, lognormal, etc. (Helsel and
Hirsch, 2002). It may be considered as a weighted mean, where data beyond
the cutoff 'window' are given a weight of zero, and those within the window
a weight of one (Fig. 2.3).