# Source code for multipletau._multipletau

    #!/usr/bin/python
    # -*- coding: utf-8 -*-
    """
    A multiple-τ algorithm for Python 2.7 and 3.x.

    Copyright (c) 2014 Paul Müller

    Redistribution and use in source and binary forms, with or without
    modification, are permitted provided that the following conditions are
    met:

      1. Redistributions of source code must retain the above copyright
         notice, this list of conditions and the following disclaimer.

      2. Redistributions in binary form must reproduce the above copyright
         notice, this list of conditions and the following disclaimer in
         the documentation and/or other materials provided with the
         distribution.

      3. Neither the name of multipletau nor the names of its contributors
         may be used to endorse or promote products derived from this
         software without specific prior written permission.

    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
    A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INFRAE OR
    CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
    EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
    PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
    PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
    LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
    NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
    SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
    """

    from __future__ import division

    import numpy as np
    import warnings

    __all__ = ["autocorrelate", "correlate", "correlate_numpy"]

    def autocorrelate(a, m=16, deltat=1, normalize=False,
                      copy=True, dtype=None):
        """
        Autocorrelation of a 1-dimensional sequence on a log2-scale.

        This computes the correlation according to
        :py:func:`numpy.correlate` for positive :math:`k`  on a base 2
        logarithmic scale.

            :func:`numpy.correlate(a, a, mode="full")[len(a)-1:]`

            :math:`z_k = \Sigma_n a_n a_{n+k}`

        Note that only the correlation in the positive direction is
        computed.

        Parameters
        ----------
        a : array_like
            input sequence of real numbers
        m : even integer
            defines the number of points on one level, must be an
            even integer
        deltat : float
            distance between bins
        normalize : bool
            normalize the result to the square of the average input
            signal and the factor `M-k`.
        copy : bool
            copy input array, set to False to save memory
        dtype : dtype, optional
            The type of the returned array and of the accumulator in
            which the elements are summed.  By default, the dtype of
            `a` is used.

        Returns
        -------
        autocorrelation : ndarray
            Nx2 array containing lag time and autocorrelation

        Notes
        -----
        The algorithm computes the correlation with the convention of the
        curve decaying to zero.

        For experiments like e.g. fluorescence correlation spectroscopy,
        the signal can be normalized to `M-k` by invoking:

               normalize = True

        For emulating the numpy.correlate behavior on a logarithmic
        scale (default behavior) use:

               normalize = False

        Examples
        --------
        >>> from numpy import dtype
        >>> from multipletau import autocorrelate
        >>> autocorrelate(range(42), m=2, dtype=dtype(float))
        array([[  1.00000000e+00,   2.29600000e+04],
               [  2.00000000e+00,   2.21000000e+04],
               [  4.00000000e+00,   2.03775000e+04],
               [  8.00000000e+00,   1.50612000e+04]])

        """
        traceavg = np.average(a)
        if normalize and traceavg == 0:
            raise ZeroDivisionError("Normalization not possible. The " +
                                    "average of the input *binned_array* " +
                                    "is zero.")

        trace = np.array(a, dtype=dtype, copy=copy)
        dtype = trace.dtype

        if dtype.kind in ["b", "i", "u"]:
            warnings.warn("Converting input data type ({}) to float.".
                          format(dtype))
            dtype = np.dtype(float)
            trace = np.array(a, dtype=dtype, copy=copy)

        # Complex data
        if dtype.kind == "c":
            raise NotImplementedError(
                "Please use `multipletau.correlate` for complex data.")

        # Check parameters
        if np.around(m / 2) != m / 2:
            mold = 1 * m
            m = np.int((np.around(m / 2) + 1) * 2)
            warnings.warn("Invalid value of m={}. Using m={} instead"
                          .format(mold, m))
        else:
            m = np.int(m)

        N = N0 = len(trace)

        # Find out the length of the correlation function.
        # The integer k defines how many times we can average over
        # two neighboring array elements in order to obtain an array of
        # length just larger than m.
        k = np.int(np.floor(np.log2(N / m)))

        # In the base2 multiple-tau scheme, the length of the correlation
        # array is (only taking into account values that are computed from
        # traces that are just larger than m):
        lenG = np.int(np.floor(m + k * m // 2))

        G = np.zeros((lenG, 2), dtype=dtype)

        normstat = np.zeros(lenG, dtype=dtype)
        normnump = np.zeros(lenG, dtype=dtype)

        # We use the fluctuation of the signal around the mean
        if normalize:
            trace -= traceavg
        if N < 2 * m:
            # Otherwise the following for-loop will fail:
            raise ValueError("len(binned_array) must be larger than 2m.")
        # Calculate autocorrelation function for first m bins
        # Discrete convolution of m elements
        for n in range(1, m + 1):
            G[n - 1, 0] = deltat * n
            # This is the computationally intensive step
            G[n - 1, 1] = np.sum(trace[:N - n] * trace[n:], dtype=dtype)
            normstat[n - 1] = N - n
            normnump[n - 1] = N
        # Now that we calculated the first m elements of G, let us
        # go on with the next m/2 elements.
        # Check if len(trace) is even:
        if N % 2 == 1:
            N -= 1
        # Add up every second element
        trace = (trace[:N:2] + trace[1:N + 1:2]) / 2
        N /= 2
        N = np.int(N)
        # Start iteration for each m/2 values
        for step in range(1, k + 1):
            # Get the next m/2 values via correlation of the trace
            for n in range(1, np.int(m // 2) + 1):
                idx = np.int(m + n - 1 + (step - 1) * m // 2)
                if len(trace[:N - (n + m // 2)]) == 0:
                    # This is a shortcut that stops the iteration once the
                    # length of the trace is too small to compute a corre-
                    # lation. The actual length of the correlation function
                    # does not only depend on k - We also must be able to
                    # perform the sum with repect to k for all elements.
                    # For small N, the sum over zero elements would be
                    # computed here.
                    #
                    # One could make this for loop go up to maxval, where
                    #   maxval1 = int(m/2)
                    #   maxval2 = int(N-m/2-1)
                    #   maxval = min(maxval1, maxval2)
                    # However, we then would also need to find out which
                    # element in G is the last element...
                    G = G[:idx - 1]
                    normstat = normstat[:idx - 1]
                    normnump = normnump[:idx - 1]
                    # Note that this break only breaks out of the current
                    # for loop. However, we are already in the last loop
                    # of the step-for-loop. That is because we calculated
                    # k in advance.
                    break
                else:
                    G[idx, 0] = deltat * (n + m // 2) * 2**step
                    # This is the computationally intensive step
                    G[idx, 1] = np.sum(trace[:N - (n + m // 2)] *
                                       trace[(n + m // 2):], dtype=dtype)
                    normstat[idx] = N - (n + m // 2)
                    normnump[idx] = N
            # Check if len(trace) is even:
            if N % 2 == 1:
                N -= 1
            # Add up every second element
            trace = (trace[:N:2] + trace[1:N + 1:2]) / 2
            N /= 2
            N = np.int(N)

        if normalize:
            G[:, 1] /= traceavg**2 * normstat
        else:
            G[:, 1] *= N0 / normnump

        return G

    def correlate(a, v, m=16, deltat=1, normalize=False,
                  copy=True, dtype=None):
        """
        Cross-correlation of two 1-dimensional sequences
        on a log2-scale.

        This computes the cross-correlation according to
        :py:func:`numpy.correlate` for positive :math:`k`  on a base 2
        logarithmic scale.

            numpy.correlate(a, v, mode="full")[len(a)-1:]

            :math:`z_k = \Sigma_n a_n v_{n+k}`

        Note that only the correlation
        in the positive direction is computed.

        Parameters
        ----------
        a, v : array_like
            input sequences with equal length
        m : even integer
            defines the number of points on one level, must be an
            even integer
        deltat : float
            distance between bins
        normalize : bool
            normalize the result to the square of the average input
            signal and the factor `M-k`.
        copy : bool
            copy input array, set to False to save memory
        dtype : dtype, optional
            The type of the returned array and of the accumulator in
            which the elements are summed.  By default, the dtype of
            `a` is used.

        Returns
        -------
        crosscorrelation : ndarray
            Nx2 array containing lag time and cross-correlation

        Notes
        -----
        The algorithm computes the correlation with the convention of the
        curve decaying to zero.

        For experiments like e.g. fluorescence correlation spectroscopy,
        the signal can be normalized to `M-k` by invoking:

               normalize = True

        For emulating the numpy.correlate behavior on a logarithmic
        scale (default behavior) use:

               normalize = False

        Examples
        --------
        >>> from numpy import dtype
        >>> from multipletau import correlate
        >>> correlate(range(42), range(1,43), m=2, dtype=dtype(float))
        array([[  1.00000000e+00,   2.38210000e+04],
               [  2.00000000e+00,   2.29600000e+04],
               [  4.00000000e+00,   2.12325000e+04],
               [  8.00000000e+00,   1.58508000e+04]])

        """
        # See `autocorrelation` for better documented code.
        traceavg1 = np.average(v)
        traceavg2 = np.average(a)
        if normalize and traceavg1 * traceavg2 == 0:
            raise ZeroDivisionError("Normalization not possible. The " +
                                    "average of the input *binned_array* " +
                                    "is zero.")

        trace1 = np.array(v, dtype=dtype, copy=copy)
        dtype = trace1.dtype

        if dtype.kind in ["b", "i", "u"]:
            warnings.warn(
                "Converting input data type ({}) to float.".format(dtype))
            dtype = np.dtype(float)
            trace1 = np.array(v, dtype=dtype, copy=copy)

        # Prevent traces from overwriting each other
        if a is v:
            # Force copying trace 2
            copy = True

        trace2 = np.array(a, dtype=dtype, copy=copy)

        # Complex data
        if dtype.kind == "c":
            trace1.imag *= -1

        # Check parameters
        if np.around(m / 2) != m / 2:
            mold = 1 * m
            m = np.int((np.around(m / 2) + 1) * 2)
            warnings.warn("Invalid value of m={}. Using m={} instead"
                          .format(mold, m))
        else:
            m = np.int(m)

        if len(a) != len(v):
            raise ValueError("Input arrays must be of equal length.")

        N = N0 = len(trace1)
        # Find out the length of the correlation function.
        # The integer k defines how many times we can average over
        # two neighboring array elements in order to obtain an array of
        # length just larger than m.
        k = np.int(np.floor(np.log2(N / m)))

        # In the base2 multiple-tau scheme, the length of the correlation
        # array is (only taking into account values that are computed from
        # traces that are just larger than m):
        lenG = np.int(np.floor(m + k * m // 2))

        G = np.zeros((lenG, 2), dtype=dtype)
        normstat = np.zeros(lenG, dtype=dtype)
        normnump = np.zeros(lenG, dtype=dtype)

        # We use the fluctuation of the signal around the mean
        if normalize:
            trace1 -= traceavg1
            trace2 -= traceavg2
        if N < 2 * m:
            # Otherwise the following for-loop will fail:
            raise ValueError("len(binned_array) must be larger than 2m.")
        # Calculate autocorrelation function for first m bins
        for n in range(1, m + 1):
            G[n - 1, 0] = deltat * n
            G[n - 1, 1] = np.sum(trace1[:N - n] * trace2[n:])
            normstat[n - 1] = N - n
            normnump[n - 1] = N
        # Check if len(trace) is even:
        if N % 2 == 1:
            N -= 1
        # Add up every second element
        trace1 = (trace1[:N:2] + trace1[1:N + 1:2]) / 2
        trace2 = (trace2[:N:2] + trace2[1:N + 1:2]) / 2
        N /= 2
        N = np.int(N)

        for step in range(1, k + 1):
            # Get the next m/2 values of the trace
            for n in range(1, np.int(m // 2) + 1):
                idx = np.int(m + n - 1 + (step - 1) * m // 2)
                if len(trace1[:N - (n + m // 2)]) == 0:
                    # Abort
                    G = G[:idx - 1]
                    normstat = normstat[:idx - 1]
                    normnump = normnump[:idx - 1]
                    break
                else:
                    G[idx, 0] = deltat * (n + m // 2) * 2**step
                    G[idx, 1] = np.sum(
                        trace1[:N - (n + m // 2)] * trace2[(n + m // 2):])
                    normstat[idx] = N - (n + m // 2)
                    normnump[idx] = N

            # Check if len(trace) is even:
            if N % 2 == 1:
                N -= 1
            # Add up every second element
            trace1 = (trace1[:N:2] + trace1[1:N + 1:2]) / 2
            trace2 = (trace2[:N:2] + trace2[1:N + 1:2]) / 2
            N /= 2
            N = np.int(N)

        if normalize:
            G[:, 1] /= traceavg1 * traceavg2 * normstat
        else:
            G[:, 1] *= N0 / normnump

        return G

    def correlate_numpy(a, v, deltat=1, normalize=False,
                        dtype=None, copy=True):
        """
        Convenience function that wraps around numpy.correlate and
        returns the data as multipletau.correlate does.

        Parameters
        ----------
        a, v : array_like
            input sequences
        deltat : float
            distance between bins
        normalize : bool
            normalize the result to the square of the average input
            signal and the factor (M-k). The resulting curve follows
            the convention of decaying to zero for large lag times.
        copy : bool
            copy input array, set to False to save memory
        dtype : dtype, optional
            The type of the returned array and of the accumulator in
            which the elements are summed.  By default, the dtype of
            `a` is used.

        Returns
        -------
        crosscorrelation : ndarray
            Nx2 array containing lag time and cross-correlation
        """

        avg = np.average(a)
        vvg = np.average(v)

        if dtype is None:
            dtype = a.dtype

        if len(a) != len(v):
            raise ValueError("Arrays must be of same length.")

        ab = np.array(a, dtype=dtype, copy=copy)
        vb = np.array(v, dtype=dtype, copy=copy)

        Gd = np.correlate(ab - avg, vb - vvg, mode="full")[len(ab) - 1:]

        if normalize:
            N = len(Gd)
            m = N - np.arange(N)
            Gd /= m * avg * vvg
        G = np.zeros((len(Gd), 2))
        G[:, 1] = Gd
        G[:, 0] = np.arange(len(Gd)) * deltat
        return G