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Editing: _entropy.cpython-311.pyc
� d�c8 � � � d Z ddlmZ ddlZddlZddlmZ ddlm Z m Z ddgZ d d!d�Z ddddd�d"d�Z d� Zd � Zd � Zd � Zd � ZdS )#z5 Created on Fri Apr 2 09:06:05 2021 @author: matth � )� annotationsN)�special)�Optional�Union�entropy�differential_entropy�pk�np.typing.ArrayLike�qk� Optional[np.typing.ArrayLike]�base�Optional[float]�axis�int�return� Union[np.number, np.ndarray]c �� � |�|dk rt d� � �t j | � � } d| z t j | |d�� � z } |�t j | � � }n^t j |� � }t j | |� � \ } }d|z t j ||d�� � z }t j | |� � }t j ||�� � }|�|t j |� � z }|S )a� Calculate the Shannon entropy/relative entropy of given distribution(s). If only probabilities `pk` are given, the Shannon entropy is calculated as ``H = -sum(pk * log(pk))``. If `qk` is not None, then compute the relative entropy ``D = sum(pk * log(pk / qk))``. This quantity is also known as the Kullback-Leibler divergence. This routine will normalize `pk` and `qk` if they don't sum to 1. Parameters ---------- pk : array_like Defines the (discrete) distribution. Along each axis-slice of ``pk``, element ``i`` is the (possibly unnormalized) probability of event ``i``. qk : array_like, optional Sequence against which the relative entropy is computed. Should be in the same format as `pk`. base : float, optional The logarithmic base to use, defaults to ``e`` (natural logarithm). axis : int, optional The axis along which the entropy is calculated. Default is 0. Returns ------- S : {float, array_like} The calculated entropy. Notes ----- Informally, the Shannon entropy quantifies the expected uncertainty inherent in the possible outcomes of a discrete random variable. For example, if messages consisting of sequences of symbols from a set are to be encoded and transmitted over a noiseless channel, then the Shannon entropy ``H(pk)`` gives a tight lower bound for the average number of units of information needed per symbol if the symbols occur with frequencies governed by the discrete distribution `pk` [1]_. The choice of base determines the choice of units; e.g., ``e`` for nats, ``2`` for bits, etc. The relative entropy, ``D(pk|qk)``, quantifies the increase in the average number of units of information needed per symbol if the encoding is optimized for the probability distribution `qk` instead of the true distribution `pk`. Informally, the relative entropy quantifies the expected excess in surprise experienced if one believes the true distribution is `qk` when it is actually `pk`. A related quantity, the cross entropy ``CE(pk, qk)``, satisfies the equation ``CE(pk, qk) = H(pk) + D(pk|qk)`` and can also be calculated with the formula ``CE = -sum(pk * log(qk))``. It gives the average number of units of information needed per symbol if an encoding is optimized for the probability distribution `qk` when the true distribution is `pk`. It is not computed directly by `entropy`, but it can be computed using two calls to the function (see Examples). See [2]_ for more information. References ---------- .. [1] Shannon, C.E. (1948), A Mathematical Theory of Communication. Bell System Technical Journal, 27: 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x .. [2] Thomas M. Cover and Joy A. Thomas. 2006. Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, USA. Examples -------- The outcome of a fair coin is the most uncertain: >>> import numpy as np >>> from scipy.stats import entropy >>> base = 2 # work in units of bits >>> pk = np.array([1/2, 1/2]) # fair coin >>> H = entropy(pk, base=base) >>> H 1.0 >>> H == -np.sum(pk * np.log(pk)) / np.log(base) True The outcome of a biased coin is less uncertain: >>> qk = np.array([9/10, 1/10]) # biased coin >>> entropy(qk, base=base) 0.46899559358928117 The relative entropy between the fair coin and biased coin is calculated as: >>> D = entropy(pk, qk, base=base) >>> D 0.7369655941662062 >>> D == np.sum(pk * np.log(pk/qk)) / np.log(base) True The cross entropy can be calculated as the sum of the entropy and relative entropy`: >>> CE = entropy(pk, base=base) + entropy(pk, qk, base=base) >>> CE 1.736965594166206 >>> CE == -np.sum(pk * np.log(qk)) / np.log(base) True Nr �+`base` must be a positive number or `None`.g �?T�r �keepdims�r ) � ValueError�np�asarray�sumr �entr�broadcast_arrays�rel_entr�log)r r r r �vec�Ss �6/usr/lib/python3/dist-packages/scipy/stats/_entropy.pyr r s� � �d ��D�A�I�I��F�G�G�G� ��B� � �B� �R��"�&��$��6�6�6� 6�B� �z��l�2� � ��� �Z��^�^���$�R��,�,���B� ��V�b�f�R�d�T�:�:�:� :��� �r�2�&�&�� ��s�� � � �A� �� �R�V�D�\�\��� �H� �auto)� window_lengthr r �method�valuesr% � Optional[int]r& �strc � � t j | � � } t j | |d� � } | j d }|�)t j t j |� � dz � � }dd|z cxk r|k sn t d|� d|� d�� � �|�|dk rt d � � �t j | d� � � }t t t t t d �}|� � � }||vr!d t |� � � �}t |� � �|d k r|dk rd}n |dk rd}nd} || ||� � } |�| t j |� � z } | S )aV Given a sample of a distribution, estimate the differential entropy. Several estimation methods are available using the `method` parameter. By default, a method is selected based the size of the sample. Parameters ---------- values : sequence Sample from a continuous distribution. window_length : int, optional Window length for computing Vasicek estimate. Must be an integer between 1 and half of the sample size. If ``None`` (the default), it uses the heuristic value .. math:: \left \lfloor \sqrt{n} + 0.5 \right \rfloor where :math:`n` is the sample size. This heuristic was originally proposed in [2]_ and has become common in the literature. base : float, optional The logarithmic base to use, defaults to ``e`` (natural logarithm). axis : int, optional The axis along which the differential entropy is calculated. Default is 0. method : {'vasicek', 'van es', 'ebrahimi', 'correa', 'auto'}, optional The method used to estimate the differential entropy from the sample. Default is ``'auto'``. See Notes for more information. Returns ------- entropy : float The calculated differential entropy. Notes ----- This function will converge to the true differential entropy in the limit .. math:: n \to \infty, \quad m \to \infty, \quad \frac{m}{n} \to 0 The optimal choice of ``window_length`` for a given sample size depends on the (unknown) distribution. Typically, the smoother the density of the distribution, the larger the optimal value of ``window_length`` [1]_. The following options are available for the `method` parameter. * ``'vasicek'`` uses the estimator presented in [1]_. This is one of the first and most influential estimators of differential entropy. * ``'van es'`` uses the bias-corrected estimator presented in [3]_, which is not only consistent but, under some conditions, asymptotically normal. * ``'ebrahimi'`` uses an estimator presented in [4]_, which was shown in simulation to have smaller bias and mean squared error than the Vasicek estimator. * ``'correa'`` uses the estimator presented in [5]_ based on local linear regression. In a simulation study, it had consistently smaller mean square error than the Vasiceck estimator, but it is more expensive to compute. * ``'auto'`` selects the method automatically (default). Currently, this selects ``'van es'`` for very small samples (<10), ``'ebrahimi'`` for moderate sample sizes (11-1000), and ``'vasicek'`` for larger samples, but this behavior is subject to change in future versions. All estimators are implemented as described in [6]_. References ---------- .. [1] Vasicek, O. (1976). A test for normality based on sample entropy. Journal of the Royal Statistical Society: Series B (Methodological), 38(1), 54-59. .. [2] Crzcgorzewski, P., & Wirczorkowski, R. (1999). Entropy-based goodness-of-fit test for exponentiality. Communications in Statistics-Theory and Methods, 28(5), 1183-1202. .. [3] Van Es, B. (1992). Estimating functionals related to a density by a class of statistics based on spacings. Scandinavian Journal of Statistics, 61-72. .. [4] Ebrahimi, N., Pflughoeft, K., & Soofi, E. S. (1994). Two measures of sample entropy. Statistics & Probability Letters, 20(3), 225-234. .. [5] Correa, J. C. (1995). A new estimator of entropy. Communications in Statistics-Theory and Methods, 24(10), 2439-2449. .. [6] Noughabi, H. A. (2015). Entropy Estimation Using Numerical Methods. Annals of Data Science, 2(2), 231-241. https://link.springer.com/article/10.1007/s40745-015-0045-9 Examples -------- >>> import numpy as np >>> from scipy.stats import differential_entropy, norm Entropy of a standard normal distribution: >>> rng = np.random.default_rng() >>> values = rng.standard_normal(100) >>> differential_entropy(values) 1.3407817436640392 Compare with the true entropy: >>> float(norm.entropy()) 1.4189385332046727 For several sample sizes between 5 and 1000, compare the accuracy of the ``'vasicek'``, ``'van es'``, and ``'ebrahimi'`` methods. Specifically, compare the root mean squared error (over 1000 trials) between the estimate and the true differential entropy of the distribution. >>> from scipy import stats >>> import matplotlib.pyplot as plt >>> >>> >>> def rmse(res, expected): ... '''Root mean squared error''' ... return np.sqrt(np.mean((res - expected)**2)) >>> >>> >>> a, b = np.log10(5), np.log10(1000) >>> ns = np.round(np.logspace(a, b, 10)).astype(int) >>> reps = 1000 # number of repetitions for each sample size >>> expected = stats.expon.entropy() >>> >>> method_errors = {'vasicek': [], 'van es': [], 'ebrahimi': []} >>> for method in method_errors: ... for n in ns: ... rvs = stats.expon.rvs(size=(reps, n), random_state=rng) ... res = stats.differential_entropy(rvs, method=method, axis=-1) ... error = rmse(res, expected) ... method_errors[method].append(error) >>> >>> for method, errors in method_errors.items(): ... plt.loglog(ns, errors, label=method) >>> >>> plt.legend() >>> plt.xlabel('sample size') >>> plt.ylabel('RMSE (1000 trials)') >>> plt.title('Entropy Estimator Error (Exponential Distribution)') ���Ng �?� zWindow length (z7) must be positive and less than half the sample size (z).r r r )�vasicek�van es�correa�ebrahimir$ z`method` must be one of r$ � r. i� r0 r- )r r �moveaxis�shape�math�floor�sqrtr �sort�_vasicek_entropy�_van_es_entropy�_correa_entropy�_ebrahimi_entropy�lower�setr ) r' r% r r r& �n� sorted_data�methods�message�ress r" r r � s� � �` �Z�� � �F� �[���r� *� *�F�� �R��A�� � � �4�9�Q�<�<�#�#5�6�6� � ��M�!� %� %� %� %�A� %� %� %� %�� 0�m� 0� 0�*+� 0� 0� 0� � � � ��D�A�I�I��F�G�G�G��'�&�r�*�*�*�K�*�(�(�,�'� )� )�G� �\�\�^�^�F� �W� � �;�S��\�\�;�;����!�!�!� ���� ��7�7� �F�F� �$�Y�Y� �F�F� �F� �'�&�/�+�}� 5� 5�C� �� �r�v�d�|�|��� �Jr# c �� � t j | j � � }||d< t j | ddgf |� � }t j | ddgf |� � }t j || |fd�� � S )z9Pad the data for computing the rolling window difference.r+ .r r )r �arrayr3 � broadcast_to� concatenate)�X�mr3 �Xl�Xrs r" �_pad_along_last_axisrK Q so � � �H�Q�W� � �E��E�"�I� � ��3���8��e� ,� ,�B� � ��3���9� �u� -� -�B� �>�2�q�"�+�B� /� /� /�/r# c �� � | j d }t | |� � } | dd|z d�f | ddd|z �f z }t j |d|z z |z � � }t j |d�� � S )z:Compute the Vasicek estimator as described in [6] Eq. 1.3.r+ .r, N���r )r3 rK r r �mean)rG rH r> � differences�logss r" r8 r8 [ sx � � ��� �A� �Q��"�"�A��C��Q����K�.�1�S�)�B��F�)�^�#4�4�K� �6�!�Q�q�S�'�K�'� (� (�D� �7�4�b� !� !� !�!r# c � � | j d }| d|d�f | dd| �f z }d||z z t j t j |dz |z |z � � d�� � z }t j ||dz � � }|t j d|z � � z t j |� � z t j |dz � � z S )z1Compute the van Es estimator as described in [6].r+ .N� r )r3 r r r �arange)rG rH r> � difference�term1�ks r" r9 r9 d s� � � ��� �A��3����7��a��S�q�b�S��k�)�J� �q��s�G�b�f�R�V�Q�q�S�!�G�j�$8� 9� 9��C�C�C� C�E� � �!�Q�q�S���A� �2�6�!�A�#�;�;� ������ *�R�V�A�a�C�[�[� 8�8r# c � � | j d }t | |� � } | dd|z d�f | ddd|z �f z }t j d|dz � � � t � � }t j |� � dz }d|||k dz |z z |||k < d|||||z dz k z |z z ||||z dz k < t j ||z ||z z � � }t j |d�� � S )z3Compute the Ebrahimi estimator as described in [6].r+ .r, NrM rR r ) r3 rK r rS �astype�float� ones_liker rN )rG rH r> rO �i�cirP s r" r; r; o s � � ��� �A� �Q��"�"�A��C��Q����K�.�1�S�)�B��F�)�^�#4�4�K� � �!�Q�q�S��� � ��'�'�A� � �a� � �� �B��a��Q��i�!�m�Q�&�&�B�q�A�v�J��a�!�A��1��Q��J�-� /�� 2�2�B�q�A��E�A�I�~�� �6�!�k�/�R�!�V�,� -� -�D� �7�4�b� !� !� !�!r# c �� � | j d }t | |� � } t j d|dz � � }t j | |dz � � dd�df }||z }||z dz }t j | d|f dd�� � }| d|f |z }t j ||z d�� � } |t j |d z d�� � z } t j t j | | z � � d�� � S ) z1Compute the Correa estimator as described in [6].r+ rR N.rM Tr r r, )r3 rK r rS rN r r ) rG rH r> r[ �dj�j�j0�XibarrT �num�dens r" r: r: � s� � � ��� �A� �Q��"�"�A� � �!�Q�q�S���A� ��A�2�q��s� � �A�A�A�t�G� $�B� �B��A� �Q����B� �G�A�c�2�g�J�R�$� 7� 7� 7�E��3��7��e�#�J� �&��B� �R� (� (� (�C� �B�F�:�q�=�r� *� *� *� *�C� �G�B�F�3�s�7�O�O�"� -� -� -� -�-r# )NNr ) r r r r r r r r r r ) r' r r% r( r r r r r& r) r r )�__doc__� __future__r r4 �numpyr �scipyr �typingr r �__all__r r rK r8 r9 r; r: � r# r" �<module>rk s ��� � #� "� "� "� "� "� � � � � � � � � � � � � � � "� "� "� "� "� "� "� "� �,� -�� 15�$(��A � A � A � A � A �N $(� ��� y� y� y� y� y� y�x0� 0� 0�"� "� "�9� 9� 9�"� "� "�".� .� .� .� .r#
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