Overview

The Statistics tab of a chart enables you to perform several types of statistical analyses for the graphed data. For simplicity, they can be categorised into two groups:

univariate statistics for a single time series, and
bivariate statistics for multiple time series via custom charts.

For all types of analyses, you can export results (to *.csv) or view them in Excel.

Note: Statistics are affected by the chart type (selected in the Chart tab) and any transforms applied to result(s). See Data Manipulation for more details.

Single result

If a single time series is selected from the left-hand tree menu of the Results Manager window, the statistics tab will show univariate statistics (Figure 1). Univariate statistics provide information on a single result and are intended to summarise and reveal patterns in that result.

For statistics for a single result, you can:

Choose the period type (Table 1) used to calculate the statistics from the drop-down menu;
Change the Water Year Start from the default date (the default is set in Project Options);
View the result used to calculate the statistics under Data;
View the statistics themselves.

Multiple results

For multiple results in a custom chart, the statistics tab shows both univariate comparison (Figure 2) and bivariate statistics (Figure 3) for two of the results in the chart:

Univariate comparison statistics are the same as univariate statistics for a single result, except that by default the statistics are only calculated from data that overlaps between the two results (see Table 2). To calculate univariate statistics for all data in each result, disable Calculate using overlapping data only.
Bivariate statistics compare two results for the purpose of determining empirical relationships between them.

For statistics of multiple results, you can also:

Toggle on or off two Statistics Options for how the two results are compared (Table 3);
View and change the result used as the X Data (reference) and the Y Data for statistic calculations (in Figure 2, this is Downstream Flow Volume); and
Select to view either Univariate Comparison or Bivariate statistics for the two results.

Table 1. Basic statistical analysis, period types

Period type	Description
Total	Provides statistics for the entire run. This is the default period.
Yearly summary	Provides annual statistics, with one row for each year in the run.
Monthly summary	Provides monthly statistics by combining the data for each month, regardless of the year. There are 12 rows, one for each calendar month. For example, the January row displays statistics calculated from data for every January for all years in the run.
MonthxYear summary	Provides monthly statistics by month and year, with one row for each month-year pair in the run. For example, December 1999, January 2000, February 2000 ... etc.

Table 2. Multiple results, parameters

Parameter	Description	Default
Calculate using overlapping data only	Sets the start and end dates to the latest common start date and earliest common end date. Sets common nulls (nulls in one series are set to null in the second, and vice-versa).	Enabled
Set all N/A to 0	Converts any missing value to 0.	Disabled

Figure 1. Univariate statistics, single result.

Figure 2. Univariate statistics, multiple results, yearly summary

Naming conventions of difference, double mass and scatter statistics

If you have a custom chart with a chart type of difference, double mass or scatter, then the following naming conventions are used in the statistics tab (Figure 3):

Each result in the custom chart is given a letter. This is shown in a Legend in the bottom left of the statistics tab.
The series used as a reference in the chart is always given letter A. To change the reference series, go to the Chart view then select Chart Settings » Charts and modify Chart Type Reference Series.
Under X Data (Reference) and Y (Data), the results are listed using the letters to represent each result.

For example, in Figure 3, A - B corresponds to the difference between Crab Creek's Downstream flow volume (result A) and Fish Creek's Downstream Flow Volume (result B).

Figure 3. Statistics, Difference Chart Type

Univariate statistics

You can view univariate statistics for a single result (as shown in Figure 1) or for each individual result when multiple results are combined (Figure 2). A brief description of each univariate statistic is given in Table 1, for more detailed information see Univariate Statistics SRG.

Table 1. Univariate Statistics

Statistic	Definition	Example for [-9999, 0, 1, 3, 5, 9, 9]
Minimum	Minimum value in the time series.	0
Maximum	Maximum value in the time series.	9
Number of Values	The number of values in the time series, not including nulls.	6
Number of Nulls	The number of nulls, either missing values or values entered as -9999. These values are ignored in all other univariate statistics.	1
Total	The sum of all values in the time series.	27
Mean	The sum of all values in the time series divided by the number of values,	5
Median	The middle value in the sorted list of all values in a time series. For n values, the middle value is . When n is even, the median is the mean of the two middle values.	4
Standard Deviation	How widely values in the time series vary from the mean. See Standard Deviation.	3.89 (to 2 decimal places)
Skew	The skewness of the distribution of values in the time series. See Skew.	0.23 (to 2 decimal places).

Bivariate statistics

When two or more results are in a custom chart, a set of bivariate statistics is automatically generated and can be viewed on the Bivariate Statistics tab (Figure 4). A brief description of each bivariate statistic is given in Table 4, where:

NSE (Nash-Sutcliffe Efficiency) measures the relative magnitude of the model error variance compared to the measured data variance. It can be applied at any time step size (eg. daily, monthly). See Bivariate Statistics SRG - Nash-Sutcliffe Efficiency
The NSE of Flow Duration measures the NSE of the modelled and observed flow duration curves. It can be applied at any time step size (eg. daily, monthly). See Bivariate Statistics SRG - Flow Duration.
Relative Bias measures the magnitude of the model errors relative to the magnitude of the observations. See Bivariate Statistics SRG - Relative Bias.
Bias Penalty is log transformation of the absolute value of the relative bias. In Source, the bias penalty is always used in combination with the NSE and is not available on its own. It is designed to be used in model calibration to penalise biased solutions. See Bivariate Statistics SRG - Bias Penalty.
Kling-Gupta Efficiency (KGE) measures several aspects of model performance such as correlation, bias and variance. It can also be applied to any time step.
Trotter is used to ensure capturing of both high and low flow aspects of a hydrograph and also to obtain minimal volumetric bias. Trotter is a composite bivariate statistics involving the KGE. Split Trotter and Split Trotter Weighted are other modified Trotter objective functions.

For more detailed information on bivariate statistics, see Bivariate Statistics SRG.

Parameter selection for bivariate statistics is the same as for a univariate comparison. If you select any period type other than Total, or you have disabled Calculate using overlapping data only and your start and/or end dates do not match, a subset of the statistics are available:

Pearson's Correlation (r);
NSE Daily;
Volume Bias (%);
Values used; and
Values not used.

Table 4. Bivariate Statistics

Statistic	Definition	Scientific Reference Guide entry	Range
Values Used	The number of time steps for which there are complete data pairs ie. both the X data and Y data time series have values. These pairs are used to calculate the bivariate statistics. Time steps where either series has missing values are not used.		0 to +∞
Values not used from X	The number of time steps in the X Data series that are not used in bivariate statistics calculations because either the X data or the Y data have missing values for those time steps.		0 to +∞
Values not used for Y	The number of time steps in the Y Data series that are not used in bivariate statistics calculations because either the X data or the Y data have missing values for those time steps.		0 to +∞
Pearson's Correlation (r)	Pearson's correlation coefficient measures the linear correlation between two variables. Pearson's correlation coefficient is symmetric, meaning that the value will be the same regardless of which time series is defined as X data (reference) and which as the Y data.	Pearson's Correlation Coefficient	-1 to 1
Volume Bias (%)	Relative bias expressed as a percentage.	Relative Bias	-100 to 100
Minimise Absolute Bias	The absolute value of the relative bias.	Relative Bias	0 to +∞
NSE Daily	The NSE for using a daily time step.	Nash-Sutcliffe Efficiency	-∞ to 1
NSE Daily & Bias Penalty	The difference between NSE daily and the Bias Penalty.	NSE Daily & Bias Penalty	-∞ to 1
NSE Daily & Flow Duration	Combines the NSE Daily and Flow Duration using a user-defined weighting factor.	NSE Daily & Flow Duration	-∞ to 1
NSE Daily & log Flow Duration	Combines the NSE Daily and log Flow Duration using a user-defined weighting factor. Log flow duration is the NSE of flow duration of the logarithm of data, calculated using a daily time step.	NSE Daily & log Flow Duration	-∞ to 1
NSE Log Daily	The NSE of the logarithm of data, using a daily time step.	NSE of Log Data	-∞ to 1
NSE Log Daily & Bias Penalty	The difference between NSE Log daily and the Bias Penalty.	NSE Log Daily & Bias Penalty	-∞ to 1
NSE Monthly	The NSE using a monthly time step	Nash-Sutcliffe Efficiency	-∞ to 1
NSE Monthly & Bias Penalty	The difference between NSE monthly and the Bias Penalty.	NSE Monthly & Bias Penalty.	-∞ to 1
Square-root Daily, Exceedance and Bias	Combines three terms, the sum of errors on power transformed flow, the same sum on sorted flow values, and the relative simulation bias.	SDEB	0 to +∞
Kling-Gupta Efficiency (KGE)	Combines linear correlation, measure of flow variability and the bias.	KGE	-∞ to 1
Trotter	Combines KGE and Bias Penalty to emphasise both high and low flows	Trotter	-∞ to 1
Split Trotter	Explicitly considers different subperiods of the calibration period	Split Trotter	-∞ to 1
Split Trotter Weighted	Aims to place additional emphasis on model fitting to drier years	Split Trotter Weighted	-∞ to 1

Figure 4. Bivariate statistics, Total period

MUSIC volumetric ratio statistic

In MUSIC Results Manager, is an additional bivariate statistic, Volumetric Ratio, which can be used to calculate the volumetric ratio between two flows, for example, pre and post development outflow. For each flow, the user chooses a threshold, and the area between the curve and the threshold is calculated. The ratio between these two areas is the Volumetric Ratio:

where x_i is the i^th value of the flow series x, x_t is the user-defined threshold value for x, y_j is the j^th value of the flow series y, and y_t is the user-defined threshold for y.

This statistic can be used to calculate several useful parameters such as Volumetric runoff coefficients and Stream Erosion Index (SEI). To calculate SEI, it is necessary to know the flow threshold (critical flow) below which no erosion is expected to occur within a waterway. This threshold can be represented (EarthTech, 2005) as a percentage of the pre‐development two-year ARI peak flow at the location in question. The percentage depends on the stream bed material and usually varies between 10 – 50%. The pre-development two-year ARI peak discharge can be estimated using flood frequency analysis or the rational method as described in Australian Rainfall and Runoff (Pilgrim, D.H., 2001). Flux Files containing the pre- and post-development outflows can be generated from MUSIC model run. SEI can be calculated by creating a custom chart in Results Manager where X data is the post-development outflow, Y data is the pre-development outflow, and both the X data and Y data threshold values are manually set to the value of critical flow using the appropriated fields next to the volumetric ratio.

Although Results Manager can calculate a volumetric ratio for any two data series, currently this statistic supports data expressed as either volume (eg. ML) or rate (eg. m³/s).

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