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# Source code for safe.common.interpolation1d

    """**Module for 1D interpolation**

    This module:

    * provides piecewise constant (nearest neighbour) and bilinear interpolation
    * is fast (based on numpy vector operations)
    * depends only on numpy
    * guarantees that interpolated values never exceed the two nearest neighbours
    * handles missing values in domain sensibly using NaN
    * is unit tested with a range of common and corner cases

    See interpolation2d.py for documentation of the mathematical derivation used.
    """

    import numpy
    # pylint: disable=W0105

    def interpolate1d(x, z, points, mode='linear', bounds_error=False):
        """Fundamental 2D interpolation routine

        Args:
            * x: 1D array of x-coordinates on which to interpolate
            * z: 1D array of values for each x
            * points: 1D array of coordinates where interpolated values are sought
            * mode: Determines the interpolation order. Options are:
                * 'constant' - piecewise constant nearest neighbour interpolation
                * 'linear' - bilinear interpolation using the two nearest \
                  neighbours (default)
            * bounds_error: Boolean flag. If True (default) an exception will
                            be raised when interpolated values are requested
                            outside the domain of the input data. If False, nan
                            is returned for those values
        Returns:
            * 1D array with same length as points with interpolated values

        Note:
            Input coordinates x are assumed to be monotonically increasing,
            but need not be equidistantly spaced.

            z is assumed to have dimension M where M = len(x).
        """

        # Input checks
        x, z, xi = check_inputs(x, z, points, mode, bounds_error)

        # Identify elements that are outside interpolation domain or NaN
        outside = (xi < x[0]) + (xi > x[-1])
        outside += numpy.isnan(xi)

        inside = -outside
        xi = xi[inside]

        # Find upper neighbours for each interpolation point
        idx = numpy.searchsorted(x, xi, side='left')

        # Internal check (index == 0 is OK)
        msg = ('Interpolation point outside domain. This should never happen. '
               'Please email Ole.Moller.Nielsen@gmail.com')
        if len(idx) > 0:
            if not max(idx) < len(x):
                raise RuntimeError(msg)

        # Get the two neighbours for each interpolation point
        x0 = x[idx - 1]
        x1 = x[idx]

        z0 = z[idx - 1]
        z1 = z[idx]

        # Coefficient for weighting between lower and upper bounds
        alpha = (xi - x0) / (x1 - x0)

        if mode == 'linear':
            # Bilinear interpolation formula
            dx = z1 - z0
            zeta = z0 + alpha * dx
        else:
            # Piecewise constant (as verified in input_check)

            # Set up masks for the quadrants
            left = alpha < 0.5

            # Initialise result array with all elements set to right neighbour
            zeta = z1

            # Then set the left neigbours
            zeta[left] = z0[left]

        # Self test
        if len(zeta) > 0:
            mzeta = numpy.nanmax(zeta)
            mz = numpy.nanmax(z)
            msg = ('Internal check failed. Max interpolated value %.15f '
                   'exceeds max grid value %.15f ' % (mzeta, mz))
            if not(numpy.isnan(mzeta) or numpy.isnan(mz)):
                if not mzeta <= mz:
                    raise RuntimeError(msg)

        # Populate result with interpolated values for points inside domain
        # and NaN for values outside
        r = numpy.zeros(len(points))
        r[inside] = zeta
        r[outside] = numpy.nan

        return r

    def check_inputs(x, z, points, mode, bounds_error):
        """Check inputs for interpolate1d function
        """

        msg = 'Only mode "linear" and "constant" are implemented. I got %s' % mode
        if mode not in ['linear', 'constant']:
            raise RuntimeError(msg)

        x = numpy.array(x)

        msg = ('Input vector x must be monotoneously increasing. I got '
               'min(x) == %.15f, but x[0] == %.15f' % (min(x), x[0]))
        if not min(x) == x[0]:
            raise RuntimeError(msg)

        msg = ('Input vector x must be monotoneously increasing. I got '
               'max(x) == %.15f, but x[-1] == %.15f' % (max(x), x[-1]))
        if not max(x) == x[-1]:
            raise RuntimeError(msg)

        msg = 'Z must be a 1d numpy array'
        try:
            z = numpy.array(z)
        except Exception, e:
            msg += '%s: %s' % (msg, e)
            raise Exception(msg)

        if not len(z.shape) == 1:
            raise RuntimeError(msg)

        m = len(z)
        Nx = len(x)
        msg = ('Input array z must have same length as x (%i).'
               'However, Z has length %i.' % (Nx, m))
        if not Nx == m:
            raise RuntimeError(msg)

        # Get interpolation points
        points = numpy.array(points)
        msg = 'Interpolation points must be a 1d array'
        if not len(points.shape) == 1:
            raise RuntimeError(msg)
        xi = points[:]

        if bounds_error:
            msg = ('Interpolation point %f was less than the smallest value in '
                   'domain %f and bounds_error was requested.' % (xi[0], x[0]))
            if xi[0] < x[0]:
                raise Exception(msg)

            msg = ('Interpolation point %f was greater than the largest value in '
                   'domain %f and bounds_error was requested.' % (xi[-1], x[-1]))
            if xi[-1] > x[-1]:
                raise Exception(msg)

        return x, z, xi

    # Mathematical derivation of the interpolation formula used
    """
    Bilinear interpolation is based on the standard 1D linear interpolation
    formula:

    Given points (x0, y0) and (x1, x0) and a value of x where x0 <= x <= x1,
    the linearly interpolated value y at x is given as

    alpha*(y1-y0) + y0

    or

    alpha*y1 + (1-alpha)*y0                (1)

    where alpha = (x-x0)/(x1-x0)           (1a)

    Piecewise constant interpolation can be implemented using the same coefficients
    (1a) that are used for bilinear interpolation as they are a measure of
    the relative distance to the left and lower neigbours. A value of 0 will pick
    the lower bound whereas a value of 1 will pick the higher bound.
    Hence z can be assigned to its nearest neigbour as follows

        | z0   alpha < 0.5    # lower value
    z = |
        | z1   alpha >= 0.5   # higher value

    """
    # pylint: enable=W0105

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