# Missing values¶

xtensor handles missing values and provides specialized container types for an optimized support of missing values.

## Optional expressions¶

Support of missing values in xtensor is primarily provided through the xoptional value type and the xtensor_optional and xarray_optional containers. In the following example, we instantiate a 2-D tensor with a missing value:

xtensor_optional<double, 2> m
{{ 1.0 ,       2.0         },
{ 3.0 , missing<double>() }};


This code is semantically equivalent to

xtensor<xoptional<double>, 2> m
{{ 1.0 ,       2.0         },
{ 3.0 , missing<double>() }};


The xtensor_optional container is optimized to handle missing values. Internally, instead of holding a single container of optional values, it holds an array of double and a boolean container where each value occupies a single bit instead of sizeof(bool) bytes.

The xtensor_optional::reference typedef, which is the return type of operator() is a reference proxy which can be used as an lvalue for assigning new values in the array. It happens to be an instance of xoptional<T, B> where T and B are actually the reference types of the underlying storage for values and boolean flags.

This technique enables performance improvements in mathematical operations over boolean arrays including SIMD optimizations, and reduces the memory footprint of optional arrays. It should be transparent to the user.

## Operating on missing values¶

Arithmetic operators and mathematical universal functions are overloaded for optional values so that they can be operated upon in the same way as regular scalars.

xtensor_optional<double, 2> a
{{ 1.0 ,       2.0         },
{ 3.0 , missing<double>() }};

xtensor<double, 1> b
{ 1.0, 2.0 };

// b is broadcasted to match the shape of a
std::cout << a + b << std::endl;


outputs:

{{  2,   4},
{  4, N/A}}


## Optional assemblies¶

The classes xoptional_assembly and xoptional_assembly_adaptor provide containers and adaptors holding missing values that are optimized for element-wise operations. Contrary to xtensor_optional and xarray_optional, the optional assemblies hold two expressions, one holding the values, the other holding the mask for the missing values. The difference between xoptional_assembly and xoptional_assembly_adaptor is that the first one is the owner of the two expressions while the last one holds a reference on at least one of the two expressions.

xarray<double> v
{{ 1.0, 2.0 },
{ 3.0, 4.0 }};

xarray<bool> hv
{{ true, true  },
{ true, false }};

xoptional_assembly<xarray<double>, xarray<bool>> assembly(v, hv);
std::cout << assembly << std::endl;


outputs:

{{ 1,  2 },
{ 3, N/A}}


## Handling expressions with missing values¶

Functions has_value(E&& e) and value(E&& e) return expressions corresponding to the underlying value and flag of optional elements. When e is an lvalue, value(E&& e) and has_value(E&& e) are lvalues too.

xtensor_optional<double, 2> a
{{ 1.0 ,       2.0         },
{ 3.0 , missing<double>() }};

xtensor<bool, 2> b = has_value(a);

std::cout << b << std::endl;


outputs:

{{  true,  true},
{  true, false}}