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- minimum: Xmin = min(X1, X2, X3, ..., XN )
- maximum: Xmax = max(X1, X2, X3, ..., XN )
- arithmetic mean:
- median: middle value of the ranked series Xi
- mode: value of X which occurs with the greatest frequency; i.e. the middle value of the class with the greatest frequency. If classes have equal greatest frequency then the middle value of the class with the lowest class levels will be indicated as mode.
- standard deviation:
- variance = sxσx*sxσx
- skewness:
- kurtosis:
!05 - Time Series Analysis^image009.gif! - quantiles and deciles: the
- the quantiles and deciles are computed with the function:
where k = the rank in sorted data array.
If k is not an integer X{~}k~ is interpolated between the two closest values
- the quantiles and deciles are computed with the function:
- empirical frequency distribution: graphical presentation of number of data per class; number of classes and minimum and maximum class levels are input
- empirical cumulative frequency distribution: cumulative representation of frequencies per class. The relative cumulative frequencies are computed by:
As can be seen, the relative cumulative frequency is plotted with the Chegodayev function
For the mean and the variance also the 95% confidence intervals will be computed. The Student-t distribution is to be applied and the percentage points tn,
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α /2 and tn,1-
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α /2 .are computed, where n = N - 1 is the number of degrees of freedom. The confidence limits for the mean then read:
Given an estimate of the sample variance the true variance s{~}Y~ 2 will be contained within the following confidence interval with a probability of 100(1-
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α) %:
The values for c
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2n,
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α/2 and c2^ n,1-
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α/2 are read from the tables of the Chi-square distribution for given aand n.
The 95% confidence limits for the median, 25% quantile and 75% quantile are computed with:
Selection of
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Data
Data for statistical analysis is be read from the hymosdatabase.
Series
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Codes
Series can be selected by clicking the series in the 'series codes' list box. Only one series may be selected at a time.
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- actual values
- annual minimum values, and
- annual maximum values.
In case of actual values a threshold selection menu is displayed from which no threshold, minimum, maximum, both, peaks over threshold or peaks under threshold can be selected.
The annual minimum/maximum values may be the minimum/maximum of full years or of a part of years, like seasonally or monthly extremes.
Computation Period
The computation period can be set to 'Full years' or 'Part of years'. When 'Full years' is selected all data values of the complete year will be taken into the analysis, make sure the start date and end date of the processing period are form the first of January to the first of January. When 'Part of years' is selected a start date and end date for the sub-period must be entered. When data for only the month of March must be selected, the start date of the sub-period must be set to "01-03" and the end date of the sub-period to "01-04".
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Note that the time series that are investigated should not contain missing values!
POT
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Method in HYMOS
The basic statistics and fitting distribution functions of HYMOS can use the POT method for selecting data. The Basic Statistics method also has an option of selecting data by a threshold. How these data selection options work will be explained here.
In case of actual values a threshold selection menu is displayed from which no threshold, minimum, maximum, both, Peaks over Threshold or Peaks under Threshold can be selected. If peaks over threshold is selected also a value for the 'Horizon' can be entered (a period in time interval units of the original series, e.g. days, which is used to skip lower peaks within that period before or after a peak). Default value = 1 (no horizon). A maximum of 1000 values/peaks of a maximum of 50000 input values will enter analysis.
Data Selection for
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Tthreshold Method
The selection of data for threshold method is very straightforward:
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