# NDData Arithmetic¶

## Introduction¶

NDDataRef implements the following arithmetic operations:

## Using Basic Arithmetic Methods¶

Using the standard arithmetic methods requires that the first operand is an NDDataRef instance:

>>> from astropy.nddata import NDDataRef
>>> from astropy.wcs import WCS
>>> import numpy as np
>>> ndd1 = NDDataRef([1, 2, 3, 4])


While the requirement for the second operand is that it must be convertible to the first operand. It can be a number:

>>> ndd1.add(3)
NDDataRef([4, 5, 6, 7])


Or a list:

>>> ndd1.subtract([1,1,1,1])
NDDataRef([0, 1, 2, 3])

>>> ndd1.multiply(np.arange(4, 8))
NDDataRef([ 4, 10, 18, 28])
>>> ndd1.divide(np.arange(1,13).reshape(3,4))  # a 3 x 4 numpy array
NDDataRef([[1.        , 1.        , 1.        , 1.        ],
[0.2       , 0.33333333, 0.42857143, 0.5       ],
[0.11111111, 0.2       , 0.27272727, 0.33333333]])


Here, broadcasting takes care of the different dimensions. Several other classes are also possible.

## Using Arithmetic Classmethods¶

Here both operands do not need to be NDDataRef-like:

>>> NDDataRef.add(1, 3)
NDDataRef(4)


To wrap the result of an arithmetic operation between two Quantities:

>>> import astropy.units as u
>>> ndd = NDDataRef.multiply([1,2] * u.m, [10, 20] * u.cm)
>>> ndd
NDDataRef([10., 40.])
>>> ndd.unit
Unit("cm m")


Or take the inverse of an NDDataRef object:

>>> NDDataRef.divide(1, ndd1)
NDDataRef([1.        , 0.5       , 0.33333333, 0.25      ])


### Possible Operands¶

The possible types of input for operands are:

• Scalars of any type

• Lists containing numbers (or nested lists)

• numpy arrays

• numpy masked arrays

• astropy quantities

• Other nddata classes or subclasses

The normal Python operators +, -, etc. are not implemented because the methods provide several options on how to proceed with the additional attributes.

### Data and Unit¶

For data and unit there are no parameters. Every arithmetic operation lets the astropy.units.Quantity-framework evaluate the result or fail and abort the operation.

Adding two NDData objects with the same unit works:

>>> ndd1 = NDDataRef([1,2,3,4,5], unit='m')
>>> ndd2 = NDDataRef([100,150,200,50,500], unit='m')

>>> ndd.data
array([101., 152., 203.,  54., 505.])
>>> ndd.unit
Unit("m")


Adding two NDData objects with compatible units also works:

>>> ndd1 = NDDataRef(ndd1, unit='pc')
INFO: overwriting NDData's current unit with specified unit. [astropy.nddata.nddata]
>>> ndd2 = NDDataRef(ndd2, unit='lyr')
INFO: overwriting NDData's current unit with specified unit. [astropy.nddata.nddata]

>>> ndd = ndd1.subtract(ndd2)
>>> ndd.data
array([ -29.66013938,  -43.99020907,  -58.32027876,  -11.33006969,
-148.30069689])
>>> ndd.unit
Unit("pc")


This will keep by default the unit of the first operand. However, units will not be decomposed during division:

>>> ndd = ndd2.divide(ndd1)
>>> ndd.data
array([100.        ,  75.        ,  66.66666667,  12.5       , 100.        ])
>>> ndd.unit
Unit("lyr / pc")


The handle_mask parameter for the arithmetic operations implements what the resulting mask will be. There are several options.

• None, the result will have no mask:

>>> ndd1 = NDDataRef(1, mask=True)
True

• "first_found" or "ff", the result will have the mask of the first operand or if that is None, the mask of the second operand:

>>> ndd1 = NDDataRef(1, mask=True)
True
>>> ndd3 = NDDataRef(1)
False

• A function (or an arbitrary callable) that takes at least two arguments. For example, numpy.logical_or is the default:

>>> ndd1 = NDDataRef(1, mask=np.array([True, False, True, False]))
>>> ndd2 = NDDataRef(1, mask=np.array([True, False, False, True]))
array([ True, False,  True,  True]...)


This defaults to "first_found" in case only one mask is not None:

>>> ndd1 = NDDataRef(1)
>>> ndd2 = NDDataRef(1, mask=np.array([True, False, False, True]))
array([ True, False, False,  True]...)


Custom functions are also possible:

>>> def take_alternating_values(mask1, mask2, start=0):
...     return result


This function is nonsense, but we can still see how it performs:

>>> ndd1 = NDDataRef(1, mask=np.array([True, False, True, False]))
>>> ndd2 = NDDataRef(1, mask=np.array([True, False, False, True]))
array([ True, False,  True,  True]...)


Additional parameters can be given by prefixing them with mask_ (which will be stripped before passing it to the function):

>>> ndd1.add(ndd2, handle_mask=take_alternating_values, mask_start=1).mask
array([False, False, False, False]...)
array([False, False,  True,  True]...)


### Meta¶

The handle_meta parameter for the arithmetic operations implements what the resulting meta will be. The options are the same as for the mask:

• If None the resulting meta will be an empty collections.OrderedDict.

>>> ndd1 = NDDataRef(1, meta={'object': 'sun'})
>>> ndd2 = NDDataRef(1, meta={'object': 'moon'})
OrderedDict()


For meta this is the default so you do not need to pass it in this case:

>>> ndd1.add(ndd2).meta
OrderedDict()

• If "first_found" or "ff", the resulting meta will be the meta of the first operand or if that contains no keys, the meta of the second operand is taken.

>>> ndd1 = NDDataRef(1, meta={'object': 'sun'})
>>> ndd2 = NDDataRef(1, meta={'object': 'moon'})
{'object': 'sun'}

• If it is a callable it must take at least two arguments. Both meta attributes will be passed to this function (even if one or both of them are empty) and the callable evaluates the result’s meta. For example, a function that merges these two:

>>> # It's expected with arithmetics that the result is not a reference,
>>> # so we need to copy
>>> from copy import deepcopy

>>> def combine_meta(meta1, meta2):
...     if not meta1:
...         return deepcopy(meta2)
...     elif not meta2:
...         return deepcopy(meta1)
...     else:
...         meta_final = deepcopy(meta1)
...         meta_final.update(meta2)
...         return meta_final

>>> ndd1 = NDDataRef(1, meta={'time': 'today'})
>>> ndd2 = NDDataRef(1, meta={'object': 'moon'})
>>> ndd1.subtract(ndd2, handle_meta=combine_meta).meta
{'object': 'moon', 'time': 'today'}


Here again additional arguments for the function can be passed in using the prefix meta_ (which will be stripped away before passing it to this function). See the description for the mask-attribute for further details.

#### World Coordinate System (WCS)¶

The compare_wcs argument will determine what the result’s wcs will be or if the operation should be forbidden. The possible values are identical to mask and meta:

• If None the resulting wcs will be an empty None.

>>> ndd1 = NDDataRef(1, wcs=None)
>>> ndd2 = NDDataRef(1, wcs=WCS())
True

• If "first_found" or "ff" the resulting wcs will be the wcs of the first operand or if that is None, the meta of the second operand is taken.

>>> wcs = WCS()
>>> ndd1 = NDDataRef(1, wcs=wcs)
>>> ndd2 = NDDataRef(1, wcs=None)
True

• If it is a callable it must take at least two arguments. Both wcs attributes will be passed to this function (even if one or both of them are None) and the callable should return True if these wcs are identical (enough) to allow the arithmetic operation or False if the arithmetic operation should be aborted with a ValueError. If True the wcs are identical and the first one is used for the result:

>>> def compare_wcs_scalar(wcs1, wcs2, allowed_deviation=0.1):
...     if wcs1 is None and wcs2 is None:
...         return True  # both have no WCS so they are identical
...     if wcs1 is None or wcs2 is None:
...         return False  # one has WCS, the other doesn't not possible
...     else:
...         # Consider wcs close if centers are close enough
...         return all(abs(wcs1.wcs.crpix - wcs2.wcs.crpix) < allowed_deviation)

>>> ndd1 = NDDataRef(1, wcs=None)
>>> ndd2 = NDDataRef(1, wcs=None)
>>> ndd1.subtract(ndd2, compare_wcs=compare_wcs_scalar).wcs


Additional arguments can be passed in prefixing them with wcs_ (this prefix will be stripped away before passing it to the function):

>>> ndd1 = NDDataRef(1, wcs=WCS())
>>> ndd1.wcs.wcs.crpix = [1, 1]
>>> ndd2 = NDDataRef(1, wcs=WCS())
>>> ndd1.subtract(ndd2, compare_wcs=compare_wcs_scalar, wcs_allowed_deviation=2).wcs.wcs.crpix
array([1., 1.])


If you are using WCS objects, a very handy function to use might be:

>>> def wcs_compare(wcs1, wcs2, *args, **kwargs):
...     return wcs1.wcs.compare(wcs2.wcs, *args, **kwargs)


See astropy.wcs.Wcsprm.compare() for the arguments this comparison allows.

### Uncertainty¶

The propagate_uncertainties argument can be used to turn the propagation of uncertainties on or off.

• If None the result will have no uncertainty:

>>> from astropy.nddata import StdDevUncertainty
>>> ndd1 = NDDataRef(1, uncertainty=StdDevUncertainty(0))
>>> ndd2 = NDDataRef(1, uncertainty=StdDevUncertainty(1))
True

• If False the result will have the first found uncertainty.

Note

Setting propagate_uncertainties=False is generally not recommended.

• If True both uncertainties must be NDUncertainty subclasses that implement propagation. This is possible for StdDevUncertainty:

>>> ndd1 = NDDataRef(1, uncertainty=StdDevUncertainty([10]))
>>> ndd2 = NDDataRef(1, uncertainty=StdDevUncertainty([10]))
StdDevUncertainty([14.14213562])


### Uncertainty with Correlation¶

If propagate_uncertainties is True you can also give an argument for uncertainty_correlation. StdDevUncertainty cannot keep track of its correlations by itself, but it can evaluate the correct resulting uncertainty if the correct correlation is given.

The default (0) represents uncorrelated while 1 means correlated and -1 anti-correlated. If given a numpy.ndarray it should represent the element-wise correlation coefficient.

### Examples¶

Without correlation, subtracting an NDDataRef instance from itself results in a non-zero uncertainty:

>>> ndd1 = NDDataRef(1, uncertainty=StdDevUncertainty([10]))
>>> ndd1.subtract(ndd1, propagate_uncertainties=True).uncertainty
StdDevUncertainty([14.14213562])


Given a correlation of 1 (because they clearly correlate) gives the correct uncertainty of 0:

>>> ndd1 = NDDataRef(1, uncertainty=StdDevUncertainty([10]))
>>> ndd1.subtract(ndd1, propagate_uncertainties=True,
...               uncertainty_correlation=1).uncertainty
StdDevUncertainty([0.])


Which would be consistent with the equivalent operation ndd1 * 0:

>>> ndd1.multiply(0, propagate_uncertainties=True).uncertainty
StdDevUncertainty([0.])


Warning

The user needs to calculate or know the appropriate value or array manually and pass it to uncertainty_correlation. The implementation follows general first order error propagation formulas. See, for example: Wikipedia.

You can also give element-wise correlations:

>>> ndd1 = NDDataRef([1,1,1,1], uncertainty=StdDevUncertainty([1,1,1,1]))
>>> ndd2 = NDDataRef([2,2,2,2], uncertainty=StdDevUncertainty([2,2,2,2]))
StdDevUncertainty([3.        , 2.64575131, 2.23606798, 1.        ])


The correlation np.array([1, 0.5, 0, -1]) would indicate that the first element is fully correlated and the second element partially correlates, while the third element is uncorrelated, and the fourth is anti-correlated.

### Uncertainty with Unit¶

StdDevUncertainty implements correct error propagation even if the unit of the data differs from the unit of the uncertainty:

>>> ndd1 = NDDataRef([10], unit='m', uncertainty=StdDevUncertainty([10], unit='cm'))
>>> ndd2 = NDDataRef([20], unit='m', uncertainty=StdDevUncertainty([10]))
>>> ndd1.subtract(ndd2, propagate_uncertainties=True).uncertainty
StdDevUncertainty([10.00049999])


But it needs to be convertible to the unit for the data.