# Space, Observable and Range¶

Inside zfit, Space defines the domain of objects by specifying the observables/axes and maybe also the limits. Any model and data needs to be specified in a certain domain, which is usually done using the obs argument. It is crucial that the axis used by the observable of the data and the model match, and this matching is handle by the Space class.

obs = zfit.Space("x")
model = zfit.pdf.Gauss(obs=obs, ...)
data = zfit.Data.from_numpy(obs=obs, ...)

## Definitions¶

Space: an n-dimensional definition of a domain (either by using one or more observables or axes), with or without limits.

Note

compared to RooFit, a space is **not* the equivalent of an observable but rather corresponds to an object combining a set of observables (which of course can be of size 1). Furthermore, there is a strong distinction in zfit between a Space (or observables) and a Parameter, both conceptually and in terms of implementation and usage.*

Observable: a string defining the axes; a named axes.

(for advanced usage only, can be skipped on first read) Axis: integer defining the axes internally of a model. There is always a mapping of observables <-> axes once inside a model.

Limit The range on a certain axis. Typically defines an interval. In fact, there are two times of limits:
• rectangular: This type is the usual limit as e.g. (-2, 5) for a simple, 1 dimensional interval. It is rectangular. This can either be given as limits of a Space or as rect_limits.

• functional: In order to define arbitrary limits, a function can be used that receives a tensor-like object x and returns True on every position that is inside the limits, False for every value outside. When a functional limit is given, rectangular limits that contain the functional limit as a subset must be defined as well.

Since every object has a well defined domain, it is possible to combine them in an unambiguous way. While not enforced, a space should usually be created with limits that define the default space of an object. This correspond for example to the default normalization range norm_range or sampling range.

lower1, upper1 = [0, 1], [2, 3]
lower2, upper2 = [-4, 1], [10, 3]
obs1 = zfit.Space(['x', 'y'], limits=(lower1, upper2))
obs2 = zfit.Space(['z', 'y'], limits=(lower2, upper2))

model1 = zfit.pdf.Gauss(obs=obs1, ...)
model2 = zfit.pdf.Gauss(obs=obs2, ...)

# creating a composite pdf
product = model1 * model2
# OR, equivalently
product = zfit.pdf.ProductPDF([model1, model2])

assert obs1 * obs2 = product.space

The product is now defined in the space with observables [‘x’, ‘y’, ‘z’]. Any Data object to be combined with product has to be specified in the same space.

# create the space
combined_obs = obs1 * obs2

data = zfit.Data.from_numpy(obs=combined_obs, ...)

Now we have a Data object that is defined in the same domain as product and can be used to build a loss function.

## Limits¶

In many places, just defining the observables is not enough and an interval, specified by its limits, is required. Examples are a normalization range, the limits of an integration or sampling in a certain region.

Simple, 1-dimensional limits can be specified as follows. Operations like addition (creating a space with two intervals) or combination (increase the dimensionality) are also possible.

simple_limit1 = zfit.Space(obs='obs1', limits=(-5, 1))
simple_limit2 = zfit.Space(obs='obs1', limits=(3, 7.5))

In this case, added_limits is now a zfit.Space with observable ‘obs1’ defined in the intervals (-5, 1) and (3, 7.5). This can be useful, e.g., when fitting in two regions. An example of the product of different zfit.Space instances has been shown before as combined_obs.

### Functional limits¶

Limits can be defined by a function that returns whether a value is inside the boundaries or not and rectangular limits (note that specifying rect_limit does not enforce them, the function itself has to take care of that).

This example specifies the bounds between (-4, 0.5) with the limit_fn (which, in this simple case, could be better achieved by directly specifying them as rectangular limits).

def limit_fn(x):
x = z.unstack_x(x)
inside_lower = tf.greater_equal(x, -4)
inside_upper = tf.less_equal(x, 0.5)
inside = tf.logical_and(inside_lower, inside_upper)
return inside

space = zfit.Space(obs='obs1', limits=limit_fn, rect_limits=(-5, 1))

### Combining limits¶

To define simple, 1-dimensional limits, a tuple with two numbers or a functional limit in 1 dimension is enough. For anything more complicated, the operators product * or addition + respectively their functional API zfit.dimension.combine_spaces() and zfit.dimension.add_spaces() can be used.

A working code example of Space handling is provided in spaces.py in examples.

### Using the limits¶

To use the limits of any object, the methods :py:meth~zfit.Space.inside (to test if values are inside or outside of the boundaries) and :py:meth~zfit.Space.filter can be used.

The rectangular limits can also direclty be accessed by rect_limits, rect_lower or rect_upper. The returned shape is of (n_events, n_obs), for the lower respectively upper limit (rect_limits is a tuple of (lower, upper)). This should be used with caution and only if the rectangular limits are desired.