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- Uncertainties and Distributions (`astropy.uncertainty`)

# Uncertainties and Distributions (`astropy.uncertainty`)

## Introduction

Note

This subpackage is still in development.

`astropy` provides a `Distribution` object to represent statistical distributions in a form that acts as a drop-in replacement for a `Quantity` object or a regular `numpy.ndarray`. Used in this manner, `Distribution` provides uncertainty propagation at the cost of additional computation. It can also more generally represent sampled distributions for Monte Carlo calculation techniques, for instance.

The core object for this feature is the `Distribution`. Currently, all such distributions are Monte Carlo sampled. While this means each distribution may take more memory, it allows arbitrarily complex operations to be performed on distributions while maintaining their correlation structure. Some specific well-behaved distributions (e.g., the normal distribution) have analytic forms which may eventually enable a more compact and efficient representation. In the future, these may provide a coherent uncertainty propagation mechanism to work with `NDData`. However, this is not currently implemented. Hence, details of storing uncertainties for `NDData` objects can be found in the N-Dimensional Datasets (astropy.nddata) section.

## Getting Started

To demonstrate a basic use case for distributions, consider the problem of uncertainty propagation of normal distributions. Assume there are two measurements you wish to add, each with normal uncertainties. We start with some initial imports and setup:

    >>> import numpy as np
    >>> from astropy import units as u
    >>> from astropy import uncertainty as unc
    >>> rng = np.random.default_rng(12345)  # ensures reproducible example numbers

Now we create two `Distribution` objects to represent our distributions:

    >>> a = unc.normal(1*u.kpc, std=30*u.pc, n_samples=10000)
    >>> b = unc.normal(2*u.kpc, std=40*u.pc, n_samples=10000)

For normal distributions, the centers should add as expected, and the standard deviations add in quadrature. We can check these results (to the limits of our Monte Carlo sampling) trivially with `Distribution` arithmetic and attributes:

    >>> c = a + b
    >>> c
    <QuantityDistribution [...] kpc with n_samples=10000>
    >>> c.pdf_mean()
    <Quantity 2.99970555 kpc>
    >>> c.pdf_std().to(u.pc)
    <Quantity 50.07120457 pc>

Indeed these are close to the expectations. While this may seem unnecessary for the basic Gaussian case, for more complex distributions or arithmetic operations where error analysis becomes untenable, `Distribution` still powers through:

    >>> d = unc.uniform(center=3*u.kpc, width=800*u.pc, n_samples=10000)
    >>> e = unc.Distribution(((rng.beta(2,5, 10000)-(2/7))/2 + 3)*u.kpc)
    >>> f = (c * d * e) ** (1/3)
    >>> f.pdf_mean()
    <Quantity 2.99760998 kpc>
    >>> f.pdf_std()
    <Quantity 0.08308941 kpc>
    >>> from matplotlib import pyplot as plt
    >>> from astropy.visualization import quantity_support
    >>> with quantity_support():
    ...     plt.hist(f.distribution, bins=50)

(png, svg, pdf)

## Using `astropy.uncertainty`

### Creating Distributions

The most direct way to create a distribution is to use an array or `Quantity` that carries the samples in the last dimension:

    >>> import numpy as np
    >>> from astropy import units as u
    >>> from astropy import uncertainty as unc
    >>> rng = np.random.default_rng(123456)  # ensures "random" numbers match examples below
    >>> unc.Distribution(rng.poisson(12, (1000)))
    NdarrayDistribution([..., 12,...]) with n_samples=1000
    >>> pq = rng.poisson([1, 5, 30, 400], (1000, 4)).T * u.ct # note the transpose, required to get the sampling on the *last* axis
    >>> distr = unc.Distribution(pq)
    >>> distr
    <QuantityDistribution [[...],
               [...],
               [...],
               [...]] ct with n_samples=1000>

Note the distinction for these two distributions: the first is built from an array and therefore does not have `Quantity` attributes like `unit`, while the latter does have these attributes. This is reflected in how they interact with other objects, for example, the `NdarrayDistribution` will not combine with `Quantity` objects containing units.

For commonly used distributions, helper functions exist to make creating them more convenient. The examples below demonstrate several equivalent ways to create a normal/Gaussian distribution:

    >>> center = [1, 5, 30, 400]
    >>> n_distr = unc.normal(center*u.kpc, std=[0.2, 1.5, 4, 1]*u.kpc, n_samples=1000)
    >>> n_distr = unc.normal(center*u.kpc, var=[0.04, 2.25, 16, 1]*u.kpc**2, n_samples=1000)
    >>> n_distr = unc.normal(center*u.kpc, ivar=[25, 0.44444444, 0.625, 1]*u.kpc**-2, n_samples=1000)
    >>> n_distr.distribution.shape
    (4, 1000)
    >>> unc.normal(center*u.kpc, std=[0.2, 1.5, 4, 1]*u.kpc, n_samples=100).distribution.shape
    (4, 100)
    >>> unc.normal(center*u.kpc, std=[0.2, 1.5, 4, 1]*u.kpc, n_samples=20000).distribution.shape
    (4, 20000)

Additionally, Poisson and uniform `Distribution` creation functions exist:

    >>> unc.poisson(center*u.count, n_samples=1000)
    <QuantityDistribution [[...],
                 [...],
                 [...],
                 [...]] ct with n_samples=1000>
    >>> uwidth = [10, 20, 10, 55]*u.pc
    >>> unc.uniform(center=center*u.kpc, width=uwidth, n_samples=1000)
    <QuantityDistribution [[...],
                 [...],
                 [...],
                 [...]] kpc with n_samples=1000>
    >>> unc.uniform(lower=center*u.kpc - uwidth/2,  upper=center*u.kpc + uwidth/2, n_samples=1000)
    <QuantityDistribution [[...],
                 [...],
                 [...],
                 [...]] kpc with n_samples=1000>

Users are free to create their own distribution classes following similar patterns.

### Using Distributions

This object now acts much like a `Quantity` or `numpy.ndarray` for all but the non-sampled dimension, but with additional statistical operations that work on the sampled distributions:

    >>> distr.shape
    (4,)
    >>> distr.size
    4
    >>> distr.unit
    Unit("ct")
    >>> distr.n_samples
    1000
    >>> distr.pdf_mean()
    <Quantity [  1.034,   5.026,  29.994, 400.365] ct>
    >>> distr.pdf_std()
    <Quantity [ 1.04539179,  2.19484031,  5.47776998, 19.87022333] ct>
    >>> distr.pdf_var()
    <Quantity [  1.092844,   4.817324,  30.005964, 394.825775] ct2>
    >>> distr.pdf_median()
    <Quantity [   1.,   5.,  30., 400.] ct>
    >>> distr.pdf_mad()  # Median absolute deviation
    <Quantity [ 1.,  1.,  4., 13.] ct>
    >>> distr.pdf_smad()  # Median absolute deviation, rescaled to match std for normal
    <Quantity [ 1.48260222,  1.48260222,  5.93040887, 19.27382884] ct>
    >>> distr.pdf_percentiles([10, 50, 90])
    <Quantity [[  0. ,   2. ,  23. , 375. ],
               [  1. ,   5. ,  30. , 400. ],
               [  2. ,   8. ,  37. , 426.1]] ct>
    >>> distr.pdf_percentiles([.1, .5, .9]*u.dimensionless_unscaled)
    <Quantity [[  0. ,   2. ,  23. , 375. ],
              [  1. ,   5. ,  30. , 400. ],
              [  2. ,   8. ,  37. , 426.1]] ct>

If need be, the underlying array can then be accessed from the `distribution` attribute:

    >>> distr.distribution
    <Quantity [[  2.,   2.,   0., ...,   1.,   0.,   1.],
               [  3.,   2.,   8., ...,   8.,   3.,   3.],
               [ 31.,  30.,  32., ...,  20.,  34.,  31.],
               [354., 373., 384., ..., 410., 404., 395.]] ct>
    >>> distr.distribution.shape
    (4, 1000)

A `Quantity` distribution interacts naturally with non-`Distribution` `Quantity` objects, assuming the `Quantity` is a Dirac delta distribution:

    >>> distr_in_kpc = distr * u.kpc/u.count  # for the sake of round numbers in examples
    >>> distrplus = distr_in_kpc + [2000,0,0,500]*u.pc
    >>> distrplus.pdf_median()
    <Quantity [   3. ,   5. ,  30. , 400.5] kpc>
    >>> distrplus.pdf_var()
    <Quantity [  1.092844,   4.817324,  30.005964, 394.825775] kpc2>

It also operates as expected with other distributions (but see below for a discussion of covariances):

    >>> means = [2000, 0, 0, 500]
    >>> sigmas = [1000, .01, 3000, 10]
    >>> another_distr = unc.Distribution((rng.normal(means, sigmas, (1000,4))).T * u.pc)
    >>> combined_distr = distr_in_kpc + another_distr
    >>> combined_distr.pdf_median()
    <Quantity [  2.81374275,   4.99999631,  29.7150889 , 400.49576691] kpc>
    >>> combined_distr.pdf_var()
    <Quantity [  2.15512118,   4.817324  ,  39.0614616 , 394.82969655] kpc2>

### Covariance in Distributions and Discrete Sampling Effects

One of the main applications for distributions is uncertainty propagation, which critically requires proper treatment of covariance. This comes naturally in the Monte Carlo sampling approach used by the `Distribution` class, as long as proper care is taken with sampling error.

To start with a basic example, two un-correlated distributions should produce an un-correlated joint distribution plot:

    >>> import numpy as np
    >>> from astropy import units as u
    >>> from astropy import uncertainty as unc
    >>> from matplotlib import pyplot as plt
    >>> n1 = unc.normal(center=0., std=1, n_samples=10000)
    >>> n2 = unc.normal(center=0., std=2, n_samples=10000)
    >>> plt.scatter(n1.distribution, n2.distribution, s=2, lw=0, alpha=.5)
    >>> plt.xlim(-4, 4)
    >>> plt.ylim(-4, 4)

(png, svg, pdf)

Indeed, the distributions are independent. If we instead construct a covariant pair of Gaussians, it is immediately apparent:

    >>> rng = np.random.default_rng(357)
    >>> ncov = rng.multivariate_normal([0, 0], [[1, .5], [.5, 2]], size=10000)
    >>> n1 = unc.Distribution(ncov[:, 0])
    >>> n2 = unc.Distribution(ncov[:, 1])
    >>> plt.scatter(n1.distribution, n2.distribution, s=2, lw=0, alpha=.5)
    >>> plt.xlim(-4, 4)
    >>> plt.ylim(-4, 4)

(png, svg, pdf)

Most importantly, the proper correlated structure is preserved or generated as expected by appropriate arithmetic operations. For example, ratios of uncorrelated normal distribution gain covariances if the axes are not independent, as in this simulation of iron, hydrogen, and oxygen abundances in a hypothetical collection of stars:

    >>> fe_abund = unc.normal(center=-2, std=.25, n_samples=10000)
    >>> o_abund = unc.normal(center=-6., std=.5, n_samples=10000)
    >>> h_abund = unc.normal(center=-0.7, std=.1, n_samples=10000)
    >>> feh = fe_abund - h_abund
    >>> ofe = o_abund - fe_abund
    >>> plt.scatter(ofe.distribution, feh.distribution, s=2, lw=0, alpha=.5)
    >>> plt.xlabel('[Fe/H]')
    >>> plt.ylabel('[O/Fe]')

(png, svg, pdf)

This demonstrates that the correlations naturally arise from the variables, but there is no need to explicitly account for it: the sampling process naturally recovers correlations that are present.

An important note of warning, however, is that the covariance is only preserved if the sampling axes are exactly matched sample by sample. If they are not, all covariance information is (silently) lost:

    >>> n2_wrong = unc.Distribution(ncov[::-1, 1])  #reverse the sampling axis order
    >>> plt.scatter(n1.distribution, n2_wrong.distribution, s=2, lw=0, alpha=.5)
    >>> plt.xlim(-4, 4)
    >>> plt.ylim(-4, 4)

(png, svg, pdf)

Moreover, an insufficiently sampled distribution may give poor estimates or hide correlations. The example below is the same as the covariant Gaussian example above, but with 200x fewer samples:

    >>> ncov = rng.multivariate_normal([0, 0], [[1, .5], [.5, 2]], size=50)
    >>> n1 = unc.Distribution(ncov[:, 0])
    >>> n2 = unc.Distribution(ncov[:, 1])
    >>> plt.scatter(n1.distribution, n2.distribution, s=5, lw=0)
    >>> plt.xlim(-4, 4)
    >>> plt.ylim(-4, 4)
    >>> np.cov(n1.distribution, n2.distribution)
    array([[0.95534365, 0.35220031],
           [0.35220031, 1.99511743]])

(png, svg, pdf)

The covariance structure is much less apparent by eye, and this is reflected in significant discrepancies between the input and output covariance matrix. In general this is an intrinsic trade-off using sampled distributions: a smaller number of samples is computationally more efficient, but leads to larger uncertainties in any of the relevant quantities. These tend to be of order \(\sqrt{n_{\rm samples}}\) in any derived quantity, but that depends on the complexity of the distribution in question.

## Reference/API

### astropy.uncertainty Package

This sub-package contains classes and functions for creating distributions that work similar to `Quantity` or array objects, but can propagate uncertainties.

#### Functions

`normal`(center, *[, std, var, ivar, cls]) | Create a Gaussian/normal distribution.
`poisson`(center, n_samples[, cls]) | Create a Poisson distribution.
`uniform`(*[, lower, upper, center, width, cls]) | Create a Uniform distriution from the lower and upper bounds.

#### Classes

`Distribution`(samples) | A scalar value or array values with associated uncertainty distribution.

#### Class Inheritance Diagram