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Extraction Frequency

Extraction Frequency

compute_frequency_features(x)

Returns the spectral centroid (mean), variance, skew, and kurtosis of the absolute fourier transform spectrum.

Parameters:

Name Type Description Default
x array

the time series on which the spectral features are calculated

 required

Returns:

Type Description
array

An array containig the spectral centroid, variance, skew, and kurtosis of the absolute fourier transform spectrum.
Source code in ceruleo/transformation/features/extraction_frequency.py 
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@jit(debug=True, nopython=True, error_model='numpy')
def compute_frequency_features(x: np.array) -> np.array:
    """
    Returns the spectral centroid (mean), variance, skew, and kurtosis of the absolute fourier transform spectrum.

    Parameters:
        x: the time series on which the spectral features are calculated

    Returns:
        An array containig the spectral centroid, variance, skew, and kurtosis of the absolute fourier transform spectrum.
    """

    with objmode(fft_abs='float64[:]'):
        fft_abs = np.abs(fft.rfft(x))

    ps = fft_abs**2

    data = np.zeros(8, dtype=np.float32)
    data[0] = get_centroid(fft_abs)
    data[1] = get_variance(fft_abs)
    data[2] = get_skew(fft_abs)
    data[3] = get_kurtosis(fft_abs)
    data[4] = get_centroid(ps)
    data[5] = get_variance(ps)
    data[6] = get_skew(ps)
    data[7] = get_kurtosis(ps)
    return data

get_centroid(y)

Compute the centroid of the input distribution (aka distribution mean, first moment)

Parameters:

Name Type Description Default
y array

the discrete distribution from which one wants to calculate the centroid

 required

Returns:

Type Description
float32

The centroid of distribution y
Source code in ceruleo/transformation/features/extraction_frequency.py 
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@jit(nopython=True, error_model='numpy')
def get_centroid(y: np.array) -> np.float32:
    """
    Compute the centroid of the input distribution (aka distribution mean, first moment)

    Parameters:
        y: the discrete distribution from which one wants to calculate the centroid

    Returns:
        The centroid of distribution y
    """
    return get_moment(y, 1)

get_kurtosis(y)

Calculates the kurtosis as the fourth standardized moment. (Ref: https://en.wikipedia.org/wiki/Kurtosis#Pearson_moments)

Parameters:

Name Type Description Default
y array

the discrete distribution from which one wants to calculate the kurtosis

 required

Returns:

Type Description
float32

The kurtosis of distribution y
Source code in ceruleo/transformation/features/extraction_frequency.py 
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@jit(nopython=True, error_model='numpy')
def get_kurtosis(y: np.array) -> np.float32:
    """
    Calculates the kurtosis as the fourth standardized moment. (Ref: https://en.wikipedia.org/wiki/Kurtosis#Pearson_moments)

    Parameters:
        y: the discrete distribution from which one wants to calculate the kurtosis

    Returns:
        The kurtosis of distribution y
    """

    variance = get_variance(y)
    # In the limit of a dirac delta, kurtosis should be 3 and variance 0.  However, in the discrete limit,
    # the kurtosis blows up as variance --> 0, hence return nan when variance is smaller than a resolution of 0.5:
    if variance < 0.5:
        return np.nan
    else:
        return (
            get_moment(y, 4) - 4 * get_centroid(y) * get_moment(y, 3)
            + 6 * get_moment(y, 2) * get_centroid(y)**2 -
            3 * get_centroid(y)
        ) / get_variance(y)**2

get_moment(y, moment)

Compute the (non centered) moment of the input distribution

Parameters:

Name Type Description Default
y array

the discrete distribution from which one wants to calculate the moment

 required
moment int

the moment one wants to calculate (choose 1,2,3, ... )

 required

Returns:

Type Description
float32

The moment requested
Source code in ceruleo/transformation/features/extraction_frequency.py 
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@jit(nopython=True, error_model='numpy')
def get_moment(y: np.array, moment: int) -> np.float32:
    """
    Compute the (non centered) moment of the input distribution 

    Parameters:
        y: the discrete distribution from which one wants to calculate the moment
        moment: the moment one wants to calculate (choose 1,2,3, ... )

    Returns:
        The moment requested
    """
    return y.dot(np.arange(len(y), dtype=np.float64)**moment) / y.sum()

get_skew(y)

Calculates the skew as the third standardized moment. (Ref: https://en.wikipedia.org/wiki/Skewness#Definition)

Parameters:

Name Type Description Default
y array

the discrete distribution from which one wants to calculate the skew

 required

Returns:

Type Description
float32

The skew of distribution y
Source code in ceruleo/transformation/features/extraction_frequency.py 
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@jit(nopython=True, error_model='numpy')
def get_skew(y: np.array) -> np.float32:
    """
    Calculates the skew as the third standardized moment. (Ref: https://en.wikipedia.org/wiki/Skewness#Definition)

    Parameters:
        y: the discrete distribution from which one wants to calculate the skew

    Returns:
        The skew of distribution y
    """

    variance = get_variance(y)
    # In the limit of a dirac delta, skew should be 0 and variance 0.  However, in the discrete limit,
    # the skew blows up as variance --> 0, hence return nan when variance is smaller than a resolution of 0.5:
    if variance < 0.5:
        return np.nan
    else:
        return (
            get_moment(y, 3) - 3 * get_centroid(y) *
            variance - get_centroid(y)**3
        ) / get_variance(y)**(1.5)

get_variance(y)

Compute the variance of the input distribution (aka distribution second central moment)

Parameters:

Name Type Description Default
y array

the discrete distribution from which one wants to calculate the variance

 required

Returns:

Type Description
float32

The variance of distribution y
Source code in ceruleo/transformation/features/extraction_frequency.py 
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@jit(nopython=True, error_model='numpy')
def get_variance(y: np.array) -> np.float32:
    """
    Compute the variance of the input distribution (aka distribution second central moment)

    Parameters:
        y: the discrete distribution from which one wants to calculate the variance

    Returns:
        The variance of distribution y
    """
    # Here we are implementing the formula var(X) = E[X**2] (second moment) - E[X]**2 (first moment squared)
    return get_moment(y, 2) - get_centroid(y) ** 2