Model

Model composition is the process of defining a probabilistic model as a collection of model components, which are ultimate fitted to a dataset via a non-linear search.

This cookbook provides an overview of basic model composition tools.

Contents:

If first describes how to use the af.Model object to define models with a single model component from single Python classes, with the following sections:

• Python Class Template: The template of a model component written as a Python class.

• Model Composition (Model): Creating a model via af.Model().

• Priors (Model): How the default priors of a model are set and how to customize them.

• Instances (Model): Creating an instance of a model via input parameters.

• Model Customization (Model): Customizing a model (e.g. fixing parameters or linking them to one another).

• Json Output (Model): Output a model in human readable text via a .json file and loading it back again.

It then describes how to use the af.Collection object to define models with many model components from multiple Python classes, with the following sections:

• Model Composition (Collection): Creating a model via af.Collection().

• Priors (Collection): How the default priors of a collection are set and how to customize them.

• Instances (Collection): Create an instance of a collection via input parameters.

• Model Customization (Collection): Customize a collection (e.g. fixing parameters or linking them to one another).

• Json Output (Collection): Output a collection in human readable text via a .json file and loading it back again.

• Extensible Models (Collection): Using collections to extend models with new model components, including the use of Python dictionaries and lists.

Python Class Template

A model component is written as a Python class using the following format:

• The name of the class is the name of the model component, in this case, “Gaussian”.

• The input arguments of the constructor are the parameters of the mode (here centre, normalization and sigma).

• The default values of the input arguments tell PyAutoFit whether a parameter is a single-valued float or a multi-valued tuple.

We define a 1D Gaussian model component to illustrate model composition in PyAutoFit.

class Gaussian:
    def __init__(
        self,
        centre=30.0,  # <- **PyAutoFit** recognises these constructor arguments
        normalization=1.0,  # <- are the Gaussian``s model parameters.
        sigma=5.0,
    ):
        self.centre = centre
        self.normalization = normalization
        self.sigma = sigma

Model Composition (Model)

We can instantiate a Python class as a model component using af.Model().

model = af.Model(Gaussian)

The model has 3 free parameters, corresponding to the 3 parameters defined above (centre, normalization and sigma).

Each parameter has a prior associated with it, meaning they are fitted for if the model is passed to a non-linear search.

print(f"Model Total Free Parameters = {model.total_free_parameters}")

If we print the info attribute of the model we get information on all of the parameters and their priors.

print(model.info)

This gives the following output:

Total Free Parameters = 3

model                         Gaussian (N=3)

centre                        UniformPrior [1], lower_limit = 0.0, upper_limit = 100.0
normalization                 LogUniformPrior [2], lower_limit = 1e-06, upper_limit = 1000000.0
sigma                         UniformPrior [3], lower_limit = 0.0, upper_limit = 25.0

Priors (Model)

The model has a set of default priors, which have been loaded from a config file in the PyAutoFit workspace.

The config cookbook describes how to setup config files in order to produce custom priors, which means that you do not need to manually specify priors in your Python code every time you compose a model.

If you do not setup config files, all priors must be manually specified before you fit the model, as shown below.

model = af.Model(Gaussian)
model.centre = af.UniformPrior(lower_limit=0.0, upper_limit=100.0)
model.normalization = af.LogUniformPrior(lower_limit=1e-4, upper_limit=1e4)
model.sigma = af.GaussianPrior(mean=0.0, sigma=1.0, lower_limit=0.0, upper_limit=1e5)

Instances (Model)

Instances of the model components above (created via af.Model) can be created, where an input vector of parameters is mapped to create an instance of the Python class of the model.

We first need to know the order of parameters in the model, so we know how to define the input vector. This information is contained in the models paths attribute:

print(model.paths)

The paths appear as follows:

[('centre',), ('normalization',), ('sigma',)]

We create an instance of the Gaussian class via the model where centre=30.0, normalization=2.0 and sigma=3.0.

instance = model.instance_from_vector(vector=[30.0, 2.0, 3.0])

print("Model Instance: \n")
print(instance)

print("Instance Parameters \n")
print("centre = ", instance.centre)
print("normalization = ", instance.normalization)
print("sigma = ", instance.sigma)

This gives the following output:

Model Instance:
<__main__.Gaussian object at 0x7f6f11d437c0>

Instance Parameters

centre =  30.0
normalization =  2.0
sigma =  3.0

We can create an instance by inputting unit values (e.g. between 0.0 and 1.0) which are mapped to the input values via the priors.

The inputs of 0.5 below are mapped as follows:

• centre: goes to 0.5 because this is the midpoint of a UniformPrior with lower_limit=0.0 and upper_limit=1.0.

• normalization goes to 1.0 because this is the midpoint of the LogUniformPrior’ with lower_limit=1e-4 and upper_limit=1e4 corresponding to log10 space.

• sigma: goes to 0.0 because this is the mean of the GaussianPrior.

instance = model.instance_from_unit_vector(unit_vector=[0.5, 0.5, 0.5])

print("Model Instance:\n")
print(instance)

print("\nInstance Parameters \n")
print("centre = ", instance.centre)
print("normalization = ", instance.normalization)
print("sigma = ", instance.sigma)

This gives the following output:

Model Instance:
<__main__.Gaussian object at 0x7f6f11d43f70>

Instance Parameters

centre =  50.0
normalization =  1.0
sigma =  0.0

We can create instances of the Gaussian using the median value of the prior of every parameter.

instance = model.instance_from_prior_medians()

print("Instance Parameters \n")
print("centre = ", instance.centre)
print("normalization = ", instance.normalization)
print("sigma = ", instance.sigma)

This gives the following output:

Instance Parameters

centre =  50.0
normalization =  1.0
sigma =  0.0

We can create a random instance, where the random values are unit values drawn between 0.0 and 1.0.

This means the parameter values of this instance are randomly drawn from the priors.

model = af.Model(Gaussian)
instance = model.random_instance()

Model Customization (Model)

We can fix a free parameter to a specific value (reducing the dimensionality of parameter space by 1):

model = af.Model(Gaussian)
model.centre = 0.0

We can link two parameters together such they always assume the same value (reducing the dimensionality of parameter space by 1):

model.centre = model.normalization

Offsets between linked parameters or with certain values are possible:

model.centre = model.normalization + model.sigma

Assertions remove regions of parameter space (but do not reduce the dimensionality of parameter space):

model.add_assertion(model.sigma > 5.0)
model.add_assertion(model.centre > model.normalization)

The customized model can be inspected by printing its info attribute.

print(model.info)

This gives the following output:

Total Free Parameters = 2

model                            Gaussian (N=2)
    centre                       SumPrior (N=2)

centre
    self                         LogUniformPrior [14], lower_limit = 1e-06, upper_limit = 1000000.0
    other                        UniformPrior [15], lower_limit = 0.0, upper_limit = 25.0
normalization                    LogUniformPrior [14], lower_limit = 1e-06, upper_limit = 1000000.0
sigma                            UniformPrior [15], lower_limit = 0.0, upper_limit = 25.0

The overwriting of priors shown above can be achieved via the following alternative API:

model = af.Model(
    Gaussian,
    centre=af.UniformPrior(lower_limit=0.0, upper_limit=1.0),
    normalization=af.LogUniformPrior(lower_limit=1e-4, upper_limit=1e4),
    sigma=af.GaussianPrior(mean=0.0, sigma=1.0),
)

This API can also be used for fixing a parameter to a certain value:

model = af.Model(Gaussian, centre=0.0)

Json Outputs (Model)

A model has a dict attribute, which expresses all information about the model as a Python dictionary.

By printing this dictionary we can therefore get a concise summary of the model.

model = af.Model(Gaussian)

print(model.dict())

This gives the following output:

{
    'class_path': '__main__.Gaussian', 'type': 'model',
    'centre': {'lower_limit': 0.0, 'upper_limit': 100.0, 'type': 'Uniform'},
    'normalization': {'lower_limit': 1e-06, 'upper_limit': 1000000.0, 'type': 'LogUniform'},
    'sigma': {'lower_limit': 0.0, 'upper_limit': 25.0, 'type': 'Uniform'}
}

The dictionary representation printed above can be saved to hard disk as a .json file.

This means we can save any PyAutoFit model to hard-disk in a human readable format.

Checkout the file autofit_workspace/*/cookbooks/jsons/model.json to see the model written as a .json.

model_path = path.join("scripts", "cookbooks", "jsons")

os.makedirs(model_path, exist_ok=True)

model_file = path.join(model_path, "model.json")

with open(model_file, "w+") as f:
    json.dump(model.dict(), f, indent=4)

We can load the model from its .json file, meaning that one can easily save a model to hard disk and load it elsewhere.

model = af.Model.from_json(file=model_file)

Model Composition (Collection)

To illustrate Collection objects we define a second model component, representing a Exponential profile.

class Exponential:
    def __init__(
        self,
        centre=0.0,  # <- PyAutoFit recognises these constructor arguments are the model
        normalization=0.1,  # <- parameters of the Exponential.
        rate=0.01,
    ):
        self.centre = centre
        self.normalization = normalization
        self.rate = rate

To instantiate multiple Python classes into a combined model component we combine the af.Collection() and af.Model() objects.

By passing the key word arguments gaussian and exponential below, these are used as the names of the attributes of instances created using this model (which is illustrated clearly below).

model = af.Collection(gaussian=af.Model(Gaussian), exponential=af.Model(Exponential))

We can check the model has a total_free_parameters of 6, meaning the 3 parameters defined above (centre, normalization, sigma and rate) for both the Gaussian and Exponential classes all have priors associated with them .

This also means each parameter is fitted for if we fitted the model to data via a non-linear search.

print(f"Model Total Free Parameters = {model.total_free_parameters}")

Printing the info attribute of the model gives us information on all of the parameters.

print(model.info)

This gives the following output:

Total Free Parameters = 6

model                         Collection (N=6)
    gaussian                  Gaussian (N=3)
    exponential               Exponential (N=3)

gaussian
    centre                    UniformPrior [39], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [40], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [41], lower_limit = 0.0, upper_limit = 25.0
exponential
    centre                    UniformPrior [42], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [43], lower_limit = 1e-06, upper_limit = 1000000.0
    rate                      UniformPrior [44], lower_limit = 0.0, upper_limit = 1.0

Priors (Collection)

The model has a set of default priors, which have been loaded from a config file in the PyAutoFit workspace.

The configs cookbook describes how to setup config files in order to produce custom priors, which means that you do not need to manually specify priors in your Python code every time you compose a model.

If you do not setup config files, all priors must be manually specified before you fit the model, as shown below.

model.gaussian.centre = af.UniformPrior(lower_limit=0.0, upper_limit=100.0)
model.gaussian.normalization = af.UniformPrior(lower_limit=0.0, upper_limit=1e2)
model.gaussian.sigma = af.UniformPrior(lower_limit=0.0, upper_limit=30.0)
model.exponential.centre = af.UniformPrior(lower_limit=0.0, upper_limit=100.0)
model.exponential.normalization = af.UniformPrior(lower_limit=0.0, upper_limit=1e2)
model.exponential.rate = af.UniformPrior(lower_limit=0.0, upper_limit=10.0)

When creating a model via a Collection, there is no need to actually pass the python classes as an af.Model() because PyAutoFit implicitly assumes they are to be created as a Model().

This enables more concise code, whereby the following code:

model = af.Collection(gaussian=af.Model(Gaussian), exponential=af.Model(Exponential))

Can instead be written as:

model = af.Collection(gaussian=Gaussian, exponential=Exponential)

Instances (Collection)

We can create an instance of collection containing both the Gaussian and Exponential classes using this model.

Below, we create an instance where:

• The Gaussian class has centre=30.0, normalization=2.0 and sigma=3.0.

• The Exponential class has centre=60.0, normalization=4.0 and rate=1.0``.

instance = model.instance_from_vector(vector=[30.0, 2.0, 3.0, 60.0, 4.0, 1.0])

Because we passed the key word arguments gaussian and exponential above, these are the names of the attributes of instances created using this model (e.g. this is why we write instance.gaussian):

print("Model Instance: \n")
print(instance)

print("Instance Parameters \n")
print("centre (Gaussian) = ", instance.gaussian.centre)
print("normalization (Gaussian)  = ", instance.gaussian.normalization)
print("sigma (Gaussian)  = ", instance.gaussian.sigma)
print("centre (Exponential) = ", instance.exponential.centre)
print("normalization (Exponential) = ", instance.exponential.normalization)
print("rate (Exponential) = ", instance.exponential.rate)

This gives the following output:

Model Instance:
<autofit.mapper.model.ModelInstance object at 0x7f6f11b73580>

Instance Parameters

centre (Gaussian) =  30.0
normalization (Gaussian)  =  2.0
sigma (Gaussian)  =  3.0
centre (Exponential) =  60.0
normalization (Exponential) = 4.0
rate (Exponential) =  1.0

Alternatively, the instance’s variables can also be accessed as a list, whereby instead of using attribute names (e.g. gaussian_0) we input the list index.

Note that the order of the instance model components is determined from the order the components are input into the Collection.

For example, for the line af.Collection(gaussian=gaussian, exponential=exponential), the first entry in the list is the gaussian because it is the first input to the Collection.

print("centre (Gaussian) = ", instance[0].centre)
print("normalization (Gaussian)  = ", instance[0].normalization)
print("sigma (Gaussian)  = ", instance[0].sigma)
print("centre (Gaussian) = ", instance[1].centre)
print("normalization (Gaussian) = ", instance[1].normalization)
print("rate (Exponential) = ", instance[1].rate)

This gives the following output:

centre (Gaussian) =  30.0
normalization (Gaussian)  =  2.0
sigma (Gaussian)  =  3.0
centre (Exponential) =  60.0
normalization (Exponential) = 4.0
rate (Exponential) =  1.0

Model Customization (Collection)

By setting up each Model first the model can be customized using either of the API’s shown above:

gaussian = af.Model(Gaussian)
gaussian.normalization = 1.0
gaussian.sigma = af.GaussianPrior(mean=0.0, sigma=1.0)

exponential = af.Model(Exponential)
exponential.centre = 50.0
exponential.add_assertion(exponential.rate > 5.0)

model = af.Collection(gaussian=gaussian, exponential=exponential)

print(model.info)

This gives the following output:

Total Free Parameters = 4

model Collection (N=4)

gaussian Gaussian (N=2) exponential Exponential (N=2)

gaussian

centre UniformPrior [71], lower_limit = 0.0, upper_limit = 100.0 normalization 1.0 sigma GaussianPrior [70], mean = 0.0, sigma = 1.0

exponential

centre 50.0 normalization LogUniformPrior [72], lower_limit = 1e-06, upper_limit = 1000000.0 rate UniformPrior [73], lower_limit = 0.0, upper_limit = 1.0

Below is an alternative API that can be used to create the same model as above.

Which API is used is up to the user and which they find most intuitive.

gaussian = af.Model(
    Gaussian, normalization=1.0, sigma=af.GaussianPrior(mean=0.0, sigma=1.0)
)
exponential = af.Model(Exponential, centre=50.0)
exponential.add_assertion(exponential.rate > 5.0)

model = af.Collection(gaussian=gaussian, exponential=exponential)

print(model.info)

This gives the following output:

Total Free Parameters = 4

model                         Collection (N=4)
    gaussian                  Gaussian (N=2)
    exponential               Exponential (N=2)

gaussian
    centre                    UniformPrior [63], lower_limit = 0.0, upper_limit = 100.0
    normalization             1.0
    sigma                     GaussianPrior [66], mean = 0.0, sigma = 1.0
exponential
    centre                    50.0
    normalization             LogUniformPrior [68], lower_limit = 1e-06, upper_limit = 1000000.0
    rate                      UniformPrior [69], lower_limit = 0.0, upper_limit = 1.0

After creating the model as a Collection we can customize it afterwards:

model = af.Collection(gaussian=Gaussian, exponential=Exponential)

model.gaussian.normalization = 1.0
model.gaussian.sigma = af.GaussianPrior(mean=0.0, sigma=1.0)

model.exponential.centre = 50.0
model.exponential.add_assertion(exponential.rate > 5.0)

print(model.info)

This gives the following output:

Total Free Parameters = 4

model                         Collection (N=4)
    gaussian                  Gaussian (N=2)
    exponential               Exponential (N=2)

gaussian
    centre                    UniformPrior [71], lower_limit = 0.0, upper_limit = 100.0
    normalization             1.0
    sigma                     GaussianPrior [70], mean = 0.0, sigma = 1.0
exponential
    centre                    50.0
    normalization             LogUniformPrior [72], lower_limit = 1e-06, upper_limit = 1000000.0
    rate                      UniformPrior [73], lower_limit = 0.0, upper_limit = 1.0

JSon Outputs (Collection)

A Collection has a dict attribute, which express all information about the model as a Python dictionary.

By printing this dictionary we can therefore get a concise summary of the model.

model = af.Model(Gaussian)

print(model.dict())

This gives the following output:

{
    'type': 'collection',
    'gaussian': {
        'class_path': '__main__.Gaussian', 'type': 'model',
        'centre': {'lower_limit': 0.0, 'upper_limit': 100.0, 'type': 'Uniform'},
        'normalization': 1.0, 'sigma': {'lower_limit': -inf, 'upper_limit': inf, 'type': 'Gaussian', 'mean': 0.0, 'sigma': 1.0}},
    'exponential': {
        'class_path': '__main__.Exponential', 'type': 'model',
        'centre': 50.0,
        'normalization': {'lower_limit': 1e-06, 'upper_limit': 1000000.0, 'type': 'LogUniform'},
        'rate': {'lower_limit': 0.0, 'upper_limit': 1.0, 'type': 'Uniform'}}
}

Python dictionaries can easily be saved to hard disk as a .json file.

This means we can save any PyAutoFit model to hard-disk.

Checkout the file autofit_workspace/*/model/jsons/model.json to see the model written as a .json.

model_path = path.join("scripts", "model", "jsons")

os.makedirs(model_path, exist_ok=True)

model_file = path.join(model_path, "collection.json")

with open(model_file, "w+") as f:
    json.dump(model.dict(), f, indent=4)

We can load the model from its .json file, meaning that one can easily save a model to hard disk and load it elsewhere.

model = af.Model.from_json(file=model_file)

print(f"\n Model via Json Prior Count = {model.prior_count}")

Extensible Models (Collection)

There is no limit to the number of components we can use to set up a model via a Collection.

model = af.Collection(
    gaussian_0=Gaussian,
    gaussian_1=Gaussian,
    exponential_0=Exponential,
    exponential_1=Exponential,
    exponential_2=Exponential,
)

print(model.info)

This gives the following output:

Total Free Parameters = 15

model                         Collection (N=15)
    gaussian_0                Gaussian (N=3)
    gaussian_1                Gaussian (N=3)
    exponential_0             Exponential (N=3)
    exponential_1             Exponential (N=3)
    exponential_2             Exponential (N=3)

gaussian_0
    centre                    UniformPrior [91], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [92], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [93], lower_limit = 0.0, upper_limit = 25.0
gaussian_1
    centre                    UniformPrior [94], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [95], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [96], lower_limit = 0.0, upper_limit = 25.0
exponential_0
    centre                    UniformPrior [97], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [98], lower_limit = 1e-06, upper_limit = 1000000.0
    rate                      UniformPrior [99], lower_limit = 0.0, upper_limit = 1.0
exponential_1
    centre                    UniformPrior [100], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [101], lower_limit = 1e-06, upper_limit = 1000000.0
    rate                      UniformPrior [102], lower_limit = 0.0, upper_limit = 1.0
exponential_2
    centre                    UniformPrior [103], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [104], lower_limit = 1e-06, upper_limit = 1000000.0
    rate                      UniformPrior [105], lower_limit = 0.0, upper_limit = 1.0
Total Free Parameters = 6

model                         Collection (N=6)
    gaussian_0                Gaussian (N=3)
    gaussian_1                Gaussian (N=3)

gaussian_0
    centre                    UniformPrior [106], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [107], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [108], lower_limit = 0.0, upper_limit = 25.0
gaussian_1
    centre                    UniformPrior [109], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [110], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [111], lower_limit = 0.0, upper_limit = 25.0
Total Free Parameters = 6

model                         Collection (N=6)
    gaussian_0                Gaussian (N=3)
    gaussian_1                Gaussian (N=3)

gaussian_0
    centre                    UniformPrior [112], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [113], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [114], lower_limit = 0.0, upper_limit = 25.0
gaussian_1
    centre                    UniformPrior [115], lower_limit = 0.0, upper_limit = 100.0
    normalization             LogUniformPrior [116], lower_limit = 1e-06, upper_limit = 1000000.0
    sigma                     UniformPrior [117], lower_limit = 0.0, upper_limit = 25.0

A model can be created via af.Collection() where a dictionary of af.Model() objects are passed to it.

The two models created below are identical- one uses the API detailed above whereas the second uses a dictionary.

model = af.Collection(gaussian_0=Gaussian, gaussian_1=Gaussian)

model_dict = {"gaussian_0": Gaussian, "gaussian_1": Gaussian}
model = af.Collection(**model_dict)

The keys of the dictionary passed to the model (e.g. gaussian_0 and gaussian_1 above) are used to create the names of the attributes of instances of the model.

instance = model.instance_from_vector(vector=[1.0, 2.0, 3.0, 4.0, 5.0, 6.0])

print("Model Instance: \n")
print(instance)

print("Instance Parameters \n")
print("centre (Gaussian) = ", instance.gaussian_0.centre)
print("normalization (Gaussian)  = ", instance.gaussian_0.normalization)
print("sigma (Gaussian)  = ", instance.gaussian_0.sigma)
print("centre (Gaussian) = ", instance.gaussian_1.centre)
print("normalization (Gaussian) = ", instance.gaussian_1.normalization)
print("sigma (Gaussian) = ", instance.gaussian_1.sigma)

This gives the following output:

Model Instance:
<autofit.mapper.model.ModelInstance object at 0x7f10a40f3a60>

Instance Parameters:

centre (Gaussian) =  1.0
normalization (Gaussian)  =  2.0
sigma (Gaussian)  =  3.0
centre (Gaussian) =  4.0
normalization (Gaussian) =  5.0
sigma (Gaussian) =  6.0

A list of model components can also be passed to an af.Collection to create a model:

model = af.Collection([Gaussian, Gaussian])

print(model.info)

When a list is used, there is no string with which to name the model components (e.g. we do not input gaussian_0 and gaussian_1 anywhere.

The instance therefore can only be accessed via list indexing.

instance = model.instance_from_vector(vector=[1.0, 2.0, 3.0, 4.0, 5.0, 6.0])

print("Model Instance: \n")
print(instance)

print("Instance Parameters \n")
print("centre (Gaussian) = ", instance[0].centre)
print("normalization (Gaussian)  = ", instance[0].normalization)
print("sigma (Gaussian)  = ", instance[0].sigma)
print("centre (Gaussian) = ", instance[1].centre)
print("normalization (Gaussian) = ", instance[1].normalization)
print("sigma (Gaussian) = ", instance[1].sigma)

This gives the following output:

Model Instance:
<autofit.mapper.model.ModelInstance object at 0x7f10a40f3a60>

Instance Parameters:

centre (Gaussian) =  1.0
normalization (Gaussian)  =  2.0
sigma (Gaussian)  =  3.0
centre (Gaussian) =  4.0
normalization (Gaussian) =  5.0
sigma (Gaussian) =  6.0

Wrap Up

This cookbook shows how to compose models consisting of multiple components using the af.Model() and af.Collection() object.

Advanced model composition uses multi-level models, which compose models from hierarchies of Python classes. This is described in the multi-level model cookbook.