# Computation¶

xframe is actually more than a multi-dimensional data frame library; like xtensor, it is an expression engine that allows numerical computation on any object implementing the variable or the frame interfaces.

## Expressions¶

Assume `x`, `y` and `z` are variables with compatible coordinates (we’ll come back to that later), the return type of an expression such as `x + exp(y) * sin(z)` is not a variable. The result is a variable expression which offers the same interface as `xvariable` but does not hold any value. Such expressions can be plugged into others to build more cmoplex expressions:

```auto f = x + exp(y) * sin(z);
auto f2 = x + exp(w) * cos(f);
```

The expression engines avoids the evaluation of intermediate results and their storage in temporary variables, so you can achieve the same performance as if you had written a loop. Such a loop is quite more complicated than an array loop since labels and dimension names are involved in the assignment mechanism.

Since a variable expression provides the same API as `xvariable`, all the indexing and selection operations are available:

```auto v = (x + exp(y) * sin(z)).select({{"city", "Paris"}, {"group", "a"}});
auto view = select(x + exp(y) * sin(z), {{"city", xf::keep("Paris")}, {"group", xf::drop({"b", "d", "h"})}});
```

## Lazy evaluation¶

An expression such as `x + exp(y) * sin(z)` does not hold the result. Values are only computed upon access or when the expression is assigned to a variable. this allows to operate symbolically on very large data sets and only compute the result for the indices of interest:

```// Assuming x and y are variables each holding ~1M values
auto f = cos(x) + sin(y);

double first_res = f.locate("a", "London");
double second_res = f.locate("b", "Pekin");
// Only two values have been computed
```

That means if you use the same expression in two assign statements, the computation of the expression will be done twice. Depending on the complexity of the cmoputation and the size of the data, it might be convenient to store the result of the expression in a temporary variable:

```variable_type tmp = cos(x) + sin(y);
variable_type res1 = tmp + 2 * exp(z);
variable_type res2 = tmp - 3 * exp(w);
```

Like xarray, broadcasting in xframe is done according to the dimension names rather than their positions. This way, you do not need to transpose variable or insert dimensions of length 1 to to get array operations to work, as commonly done in xtensor with `xt::reshape` or `xt::newaxis`.

This can be illustrated with the following examples. First, consider two one-dimensional variable aligned along different dimensions:

```auto v1 = variable_type(data_type({1., 2.}), {{"x", xf::axis(1, 3)}});
auto v2 = variable_type(data_type({3., 7.}), {{"y", xf::axis(2, 5)}});
```

We can apply mathematical operations to these variables, their dimension are expanded automatically:

```variable_type res = v1 + v2;
std::cout << res << std::endl;
// Output:
// {{ 4.  8.}
//  { 5.  9.}}
// Coordinates:
// x: (1, 3, )
// y: (2, 5, )
```

Contrary to xarray , dimensions are not reordered to the order in which they first appeared:

```auto v3 = variable_type(data_type({{1., 2.}, {3., 4.}}),
{{"y", xf::axis({2, 5})}, {"x", xf::axis({1, 4})}});
variable_type res2 = v1 + v3;
std::cout << res2 << std::endl;
// Output:
// {{ 2.  4. }
//  { 4.  6. }}
// Coordinates:
// y: (2, 5, )
// x: (1, 3, )
```

This allows many optimizations in the assignment mechanism.

## Automatic alignment¶

xframe enforces alignment between coordinate labels on objects involved in operations. The default result of an operation is by the intersection of the coordinate labels:

```auto v4 = variable_type(data_type({1., 2., 3.}), {{"x", xf::axis({1, 3, 5})}});
auto v5 = variable_type(data_type({4., 7., 12.}), {{"x", xf::axis({1, 5, 7})}});

variable_type res3 = v4 + v5;
std::cout << res3 << std::endl;
// Output:
// { 5.  10. }
// Coordinates:
// x: (1, 5,)
```

Operations are slower when variables are not aligned, so it might be useful to explicitly align variables involved in loops of performance critical code.

## Operators and functions¶

xframe provides all the basic operators and mathematical functions:

• arithmetic operators: `+`, `-`, `*`, `/`

• logical operators: `&&`, `||`, `!`

• comparison operators: `==`, `!=`, `<`, `<=`, `>`, `>=`

• basic functions: `abs`, `remainder`, `fma`, …

• exponential functions: `exp`, `expm1`, `log`, `log1p`, …

• power functions: `pow`, `sqrt`, `cbrt`, …

• trigonometric functions: `sin`, `cos`, `tan`, …

• hyperbolic functions: `sinh`, `cosh`, `tanh`, …

• error and gamma functions: `erf`, `erfc`, `tgamma`, `lgamma`, …

Actually, any function operating on `xtensor` expressions can work with `xvariable` expressions without any additional work; the only constraint is to accept and return expressions:

```template <class E1, class E2>
inline auto distance(const xexpression<E1>& e1, const xexpression<E2>& e2)
{
const E1& de1 = e1.derived_cast();
const E2& de2 = e2.derived_cast();
//
return sqrt(de1 * de1 + de2 * de2);
}
```

This function can work with both tnesor and variable expressions, performing broadcasting according to the rules of xtensor or xframe depending on its argument type:

```xt::xarray<double> a1 = {1., 2. };
xt::xarray<double> a2 = {{1., 3.}, {4., 7.}};

// Broadcasting is applied according to xtensor rules,
// that is by dimension order
xt::xarray<double> ares = distance(a1, a2);

// Broadcasting is applied according to xframe rules,
// that is by dimension name
variable_type vres = distance(v1, v3);
```

## Missing values¶

Contrary to pandas or xarray, xframe does not use particular values for representing missing values. Instead, it makes use of the dedicated type `xtl::xoptional` which gathers the value and a flag to specify whether the value is missing or not:

```data_type d = {1., 2., 3. };
d(1).has_value() = false;

auto v = variable_type(d, {{"x", xf::axis({1, 3, 4})}});
std::cout << v << std::endl;
// Output:
// { 1.  N/A  3. }
// Coordinates:
// x: (1, 3, 4,)
```