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Binning

Functions:

Name Description
doane

Doane's rule defines the bin count as
freedman_diaconis

The Freedman-Diaconis rule defines the bin width as
rice

Rice's rule defines the bin count as
scott

Scott's rule defines the bin width as
sqrt

The square root rule defines the bin count as
sturges

Sturges' rule defines the bin count as

doane(x)

Doane's rule defines the bin count as

\[ k = 1 + \log_{2}(n) + \log_{2}\left(1 + \frac{|g_{1}|}{\sigma_{g_{1}}}\right) \]

where

\[ \sigma_{g_{1}} = \sqrt{\frac{6(n-2)}{(n+1)(n+3)}} \]

Parameters:

Name Type Description Default
x ArrayLike
required

Returns:

Type Description
int

Bin count

Raises:

Type Description
ValueError

If \(n < 2\)
Source code in python/rapidstats/bin.py 
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def doane(x: ArrayLike) -> int:
    r"""Doane's rule defines the bin count as

    \[
        k = 1 + \log_{2}(n) + \log_{2}\left(1 + \frac{|g_{1}|}{\sigma_{g_{1}}}\right)
    \]

    where

    \[
        \sigma_{g_{1}} = \sqrt{\frac{6(n-2)}{(n+1)(n+3)}}
    \]

    Parameters
    ----------
    x : ArrayLike

    Returns
    -------
    int
        Bin count

    Raises
    ------
    ValueError
        If $n < 2$
    """
    x = pl.Series(x)
    x_len = x.len()

    if x_len <= 2:
        raise ValueError("Doane's rule requires at least 3 observations")

    g1 = abs(x.skew())
    sg1 = math.sqrt(6.0 * (x_len - 2) / ((x_len + 1.0) * (x_len + 3)))

    return int(1 + math.log2(x_len) + math.log2(1 + (g1 / sg1)))

freedman_diaconis(x)

The Freedman-Diaconis rule defines the bin width as

\[ h = 2\frac{IQR(x)}{\sqrt[3]{n}} \]

where \(x\) is the input array and \(n\) is the length of \(x\).

The bin width is converted to a bin count via

\[ k = \lceil \frac{\max{x} - \min{x}}{h} \rceil \]

Parameters:

Name Type Description Default
x ArrayLike
required

Returns:

Type Description
int

Bin count
Source code in python/rapidstats/bin.py 
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def freedman_diaconis(x: ArrayLike) -> int:
    r"""The Freedman-Diaconis rule defines the bin width as

    \[
        h = 2\frac{IQR(x)}{\sqrt[3]{n}}
    \]

    where $x$ is the input array and $n$ is the length of $x$.

    The bin width is converted to a bin count via

    \[
        k = \lceil \frac{\max{x} - \min{x}}{h} \rceil
    \]

    Parameters
    ----------
    x : ArrayLike

    Returns
    -------
    int
        Bin count
    """
    x = pl.Series(x)

    iqr = x.quantile(0.75, interpolation="linear") - x.quantile(
        0.25, interpolation="linear"
    )

    bin_width = 2.0 * iqr * x.len() ** (-1.0 / 3.0)

    return _bin_width_to_count(x, bin_width)

rice(x)

Rice's rule defines the bin count as

\[ k = \lceil 2n^{\frac{1}{3}} \rceil \]

Parameters:

Name Type Description Default
x ArrayLike
required

Returns:

Type Description
int

Bin count
Source code in python/rapidstats/bin.py 
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def rice(x: ArrayLike) -> int:
    r"""Rice's rule defines the bin count as

    \[
        k = \lceil 2n^{\frac{1}{3}} \rceil
    \]

    Parameters
    ----------
    x : ArrayLike

    Returns
    -------
    int
        Bin count
    """
    return math.ceil(2 * (len(x) ** (1 / 3)))

scott(x)

Scott's rule defines the bin width as

\[ h = 3.49\sigma n^{-\frac{1}{3}} \]

The bin count is given by

\[ k = \lceil \frac{\max{x} - \min{x}}{h} \rceil \]

Parameters:

Name Type Description Default
x ArrayLike
required

Returns:

Type Description
int

Bin count
Source code in python/rapidstats/bin.py 
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def scott(x: ArrayLike) -> int:
    r"""Scott's rule defines the bin width as

    \[
        h = 3.49\sigma n^{-\frac{1}{3}}
    \]

    The bin count is given by

    \[
        k = \lceil \frac{\max{x} - \min{x}}{h} \rceil
    \]

    Parameters
    ----------
    x : ArrayLike

    Returns
    -------
    int
        Bin count
    """
    x = pl.Series(x)

    bin_width = (3.49 * x.std()) / (len(x) ** (1 / 3))

    return _bin_width_to_count(x, bin_width)

sqrt(x)

The square root rule defines the bin count as

\[ k = \lceil\sqrt{n}\rceil \]

Parameters:

Name Type Description Default
x ArrayLike
required

Returns:

Type Description
int

Bin count
Source code in python/rapidstats/bin.py 
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def sqrt(x: ArrayLike) -> int:
    r"""The square root rule defines the bin count as

    \[
        k = \lceil\sqrt{n}\rceil
    \]

    Parameters
    ----------
    x : ArrayLike

    Returns
    -------
    int
        Bin count
    """
    return math.ceil(math.sqrt(len(x)))

sturges(x)

Sturges' rule defines the bin count as

\[ k = \lceil 1 + \log_{2}(n) \rceil \]

Parameters:

Name Type Description Default
x ArrayLike
required

Returns:

Type Description
int

Bin count
Source code in python/rapidstats/bin.py 
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def sturges(x: ArrayLike) -> int:
    r"""Sturges' rule defines the bin count as

    \[
        k = \lceil 1 + \log_{2}(n) \rceil
    \]

    Parameters
    ----------
    x : ArrayLike

    Returns
    -------
    int
        Bin count
    """
    return math.ceil(math.log2(len(x))) + 1