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- NDData Arithmetic

# NDData Arithmetic

## Introduction

`NDDataRef` implements the following arithmetic operations:

- Addition: `add()`

- Subtraction: `subtract()`

- Multiplication: `multiply()`

- Division: `divide()`

## 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])

Or a `numpy.ndarray`:

    >>> 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.], unit='cm m')
    >>> 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

## Advanced Options

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 = ndd1.add(ndd2)
    >>> 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")

### Mask

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)
      >>> ndd2 = NDDataRef(1, mask=False)
      >>> ndd1.add(ndd2, handle_mask=None).mask is None
      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)
      >>> ndd2 = NDDataRef(1, mask=False)
      >>> ndd1.add(ndd2, handle_mask="first_found").mask
      True
      >>> ndd3 = NDDataRef(1)
      >>> ndd3.add(ndd2, handle_mask="first_found").mask
      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]))
      >>> ndd1.add(ndd2).mask
      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]))
      >>> ndd1.add(ndd2).mask
      array([ True, False, False,  True]...)

  Custom functions are also possible:

      >>> def take_alternating_values(mask1, mask2, start=0):
      ...     result = np.zeros(mask1.shape, dtype=np.bool_)
      ...     result[start::2] = mask1[start::2]
      ...     result[start+1::2] = mask2[start+1::2]
      ...     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]))
      >>> ndd1.add(ndd2, handle_mask=take_alternating_values).mask
      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]...)
      >>> ndd1.add(ndd2, handle_mask=take_alternating_values, mask_start=2).mask
      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'})
      >>> ndd1.add(ndd2, handle_meta=None).meta
      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'})
      >>> ndd1.add(ndd2, handle_meta='ff').meta
      {'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())
      >>> ndd1.add(ndd2, compare_wcs=None).wcs is None
      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)
      >>> str(ndd1.add(ndd2, compare_wcs='ff').wcs) == str(wcs)
      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))
      >>> ndd1.add(ndd2, propagate_uncertainties=None).uncertainty is None
      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]))
      >>> ndd1.add(ndd2, propagate_uncertainties=True).uncertainty
      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]))
    >>> ndd1.add(ndd2,uncertainty_correlation=np.array([1,0.5,0,-1])).uncertainty
    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.