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# Cube Statistics

## Collapsing Entire Data Dimensions

In the Subsetting a Cube section we saw how to extract a subset of a cube in order to reduce either its dimensionality or its resolution. Instead of simply extracting a sub-region of the data, we can produce statistical functions of the data values across a particular dimension, such as a ‘mean over time’ or ‘minimum over latitude’.

For instance, suppose we have a cube:

```>>> import iris
>>> filename = iris.sample_data_path('uk_hires.pp')
>>> print(cube)
air_potential_temperature / (K)     (time: 3; model_level_number: 7; grid_latitude: 204; grid_longitude: 187)
Dimension coordinates:
time                             x                      -                 -                    -
model_level_number               -                      x                 -                    -
grid_latitude                    -                      -                 x                    -
grid_longitude                   -                      -                 -                    x
Auxiliary coordinates:
forecast_period                  x                      -                 -                    -
level_height                     -                      x                 -                    -
sigma                            -                      x                 -                    -
surface_altitude                 -                      -                 x                    x
Derived coordinates:
altitude                         -                      x                 x                    x
Scalar coordinates:
forecast_reference_time     2009-11-19 04:00:00
Attributes:
STASH                       m01s00i004
source                      Data from Met Office Unified Model
um_version                  7.3
```

In this case we have a 4 dimensional cube; to mean the vertical (z) dimension down to a single valued extent we can pass the coordinate name and the aggregation definition to the `Cube.collapsed()` method:

```>>> import iris.analysis
>>> vertical_mean = cube.collapsed('model_level_number', iris.analysis.MEAN)
>>> print(vertical_mean)
air_potential_temperature / (K)     (time: 3; grid_latitude: 204; grid_longitude: 187)
Dimension coordinates:
time                             x                 -                    -
grid_latitude                    -                 x                    -
grid_longitude                   -                 -                    x
Auxiliary coordinates:
forecast_period                  x                 -                    -
surface_altitude                 -                 x                    x
Derived coordinates:
altitude                         -                 x                    x
Scalar coordinates:
forecast_reference_time     2009-11-19 04:00:00
level_height                696.6666 m, bound=(0.0, 1393.3333) m
model_level_number          10, bound=(1, 19)
sigma                       0.92292976, bound=(0.8458596, 1.0)
Cell methods:
mean                        model_level_number
Attributes:
STASH                       m01s00i004
source                      Data from Met Office Unified Model
um_version                  7.3
```

Similarly other analysis operators such as `MAX`, `MIN` and `STD_DEV` can be used instead of `MEAN`, see `iris.analysis` for a full list of currently supported operators.

For an example of using this functionality, the Hovmoller Diagram of Monthly Surface Temperature example found in the gallery takes a zonal mean of an `XYT` cube by using the `collapsed` method with `latitude` and `iris.analysis.MEAN` as arguments.

### Area Averaging

Some operators support additional keywords to the `cube.collapsed` method. For example, `iris.analysis.MEAN` supports a weights keyword which can be combined with `iris.analysis.cartography.area_weights()` to calculate an area average.

Let’s use the same data as was loaded in the previous example. Since `grid_latitude` and `grid_longitude` were both point coordinates we must guess bound positions for them in order to calculate the area of the grid boxes:

```import iris.analysis.cartography
cube.coord('grid_latitude').guess_bounds()
cube.coord('grid_longitude').guess_bounds()
grid_areas = iris.analysis.cartography.area_weights(cube)
```

These areas can now be passed to the `collapsed` method as weights:

```>>> new_cube = cube.collapsed(['grid_longitude', 'grid_latitude'], iris.analysis.MEAN, weights=grid_areas)
>>> print(new_cube)
air_potential_temperature / (K)     (time: 3; model_level_number: 7)
Dimension coordinates:
time                             x                      -
model_level_number               -                      x
Auxiliary coordinates:
forecast_period                  x                      -
level_height                     -                      x
sigma                            -                      x
Derived coordinates:
altitude                         -                      x
Scalar coordinates:
forecast_reference_time     2009-11-19 04:00:00
grid_latitude               1.5145501 degrees, bound=(0.14430022, 2.8848) degrees
grid_longitude              358.74948 degrees, bound=(357.494, 360.00497) degrees
surface_altitude            399.625 m, bound=(-14.0, 813.25) m
Cell methods:
mean                        grid_longitude, grid_latitude
Attributes:
STASH                       m01s00i004
source                      Data from Met Office Unified Model
um_version                  7.3
```

Several examples of area averaging exist in the gallery which may be of interest, including an example on taking a global area-weighted mean.

## Partially Reducing Data Dimensions

Instead of completely collapsing a dimension, other methods can be applied to reduce or filter the number of data points of a particular dimension.

### Aggregation of Grouped Data

The `Cube.aggregated_by` operation combines data for all points with the same value of a given coordinate. To do this, you need a coordinate whose points take on only a limited set of different values – the number of these then determines the size of the reduced dimension. The `iris.coord_categorisation` module can be used to make such ‘categorical’ coordinates out of ordinary ones: The most common use is to aggregate data over regular time intervals, such as by calendar month or day of the week.

For example, let’s create two new coordinates on the cube to represent the climatological seasons and the season year respectively:

```import iris
import iris.coord_categorisation

filename = iris.sample_data_path('ostia_monthly.nc')

```

Note

The ‘season year’ is not the same as year number, because (e.g.) the months Dec11, Jan12 + Feb12 all belong to ‘DJF-12’. See `iris.coord_categorisation.add_season_year()`.

Printing this cube now shows that two extra coordinates exist on the cube:

```>>> print(cube)
surface_temperature / (K)           (time: 54; latitude: 18; longitude: 432)
Dimension coordinates:
time                             x             -              -
latitude                         -             x              -
longitude                        -             -              x
Auxiliary coordinates:
clim_season                      x             -              -
forecast_reference_time          x             -              -
season_year                      x             -              -
Scalar coordinates:
forecast_period             0 hours
Cell methods:
mean                        month, year
Attributes:
Conventions                 CF-1.5
STASH                       m01s00i024
```

These two coordinates can now be used to aggregate by season and climate-year:

```>>> annual_seasonal_mean = cube.aggregated_by(
...     ['clim_season', 'season_year'],
...     iris.analysis.MEAN)
>>> print(repr(annual_seasonal_mean))
<iris 'Cube' of surface_temperature / (K) (time: 19; latitude: 18; longitude: 432)>
```

The primary change in the cube is that the cube’s data has been reduced in the ‘time’ dimension by aggregation (taking means, in this case). This has collected together all data points with the same values of season and season-year. The results are now indexed by the 19 different possible values of season and season-year in a new, reduced ‘time’ dimension.

We can see this by printing the first 10 values of season+year from the original cube: These points are individual months, so adjacent ones are often in the same season:

```>>> for season, year in zip(cube.coord('clim_season')[:10].points,
...                         cube.coord('season_year')[:10].points):
...     print(season + ' ' + str(year))
mam 2006
mam 2006
jja 2006
jja 2006
jja 2006
son 2006
son 2006
son 2006
djf 2007
djf 2007
```

Compare this with the first 10 values of the new cube’s coordinates: All the points now have distinct season+year values:

```>>> for season, year in zip(
...         annual_seasonal_mean.coord('clim_season')[:10].points,
...         annual_seasonal_mean.coord('season_year')[:10].points):
...     print(season + ' ' + str(year))
mam 2006
jja 2006
son 2006
djf 2007
mam 2007
jja 2007
son 2007
djf 2008
mam 2008
jja 2008
```

Because the original data started in April 2006 we have some incomplete seasons (e.g. there were only two months worth of data for ‘mam-2006’). In this case we can fix this by removing all of the resultant ‘times’ which do not cover a three month period (note: judged here as > 3*28 days):

```>>> tdelta_3mth = datetime.timedelta(hours=3*28*24.0)
>>> spans_three_months = lambda t: (t.bound - t.bound) > tdelta_3mth
>>> three_months_bound = iris.Constraint(time=spans_three_months)
>>> full_season_means = annual_seasonal_mean.extract(three_months_bound)
>>> full_season_means
<iris 'Cube' of surface_temperature / (K) (time: 17; latitude: 18; longitude: 432)>
```

The final result now represents the seasonal mean temperature for 17 seasons from jja-2006 to jja-2010:

```>>> for season, year in zip(full_season_means.coord('clim_season').points,
...                         full_season_means.coord('season_year').points):
...     print(season + ' ' + str(year))
jja 2006
son 2006
djf 2007
mam 2007
jja 2007
son 2007
djf 2008
mam 2008
jja 2008
son 2008
djf 2009
mam 2009
jja 2009
son 2009
djf 2010
mam 2010
jja 2010
```