def polfit_residuals_with_sigma_rejection(
        x, y, deg, times_sigma_reject,
        color='b', size=75,
        xlim=None, ylim=None,
        xlabel=None, ylabel=None, title=None,
        use_r=None,
        geometry=(0,0,640,480),
        debugplot=0):
    """Polynomial fit with iterative rejection of points.

    This function makes use of function polfit_residuals for display
    purposes.

    Parameters
    ----------
    x : 1d numpy array, float
        X coordinates of the data being fitted.
    y : 1d numpy array, float
        Y coordinates of the data being fitted.
    deg : int
        Degree of the fitting polynomial.
    times_sigma_reject : float or None
        Number of times the standard deviation to reject points
        iteratively. If None, the fit does not reject any point.
    color : single character or 1d numpy array of characters
        Color for all the symbols (single character) or for each
        individual symbol (array of color names with the same length as
        'x' or 'y'). If 'color' is a single character, the rejected
        points are displayed in red color, whereas when 'color' is an
        array of color names, rejected points are displayed with the
        color provided in this array.
    size : int
        Marker size for all the symbols (single character) or for each
        individual symbol (array of integers with the same length as
        'x' or 'y').
    xlim : tuple (floats)
        Plot limits in the X axis.
    ylim : tuple (floats)
        Plot limits in the Y axis.
    xlabel : string
        Character string for label in X axis.
    ylabel : string
        Character string for label in y axis.
    title : string
        Character string for graph title.
    use_r : bool
        If True, the function computes several fits, using R, to
        polynomials of degree deg, deg+1 and deg+2 (when possible).
    geometry : tuple (4 integers) or None
        x, y, dx, dy values employed to set the window geometry.
    debugplot : int
        Determines whether intermediate computations and/or plots
        are displayed. The valid codes are defined in
        numina.array.display.pause_debugplot.

    Return
    ------
    poly : instance of Polynomial (numpy)
        Result from the polynomial fit using numpy Polynomial. Only
        points not flagged as rejected are employed in the fit.
    yres : 1d numpy array, float
        Residuals from polynomial fit. Note that the residuals are
        computed for all the points, including the rejected ones. In
        this way the dimension of this array is the same as the
        dimensions of the input 'x' and 'y' arrays.
    reject : 1d numpy array, bool
        Boolean array indicating rejected points.

    """

    # protections
    if type(x) is not np.ndarray:
        raise ValueError("x=" + str(x) + " must be a numpy.ndarray")
    elif x.ndim != 1:
        raise ValueError("x.ndim=" + str(x.ndim) + " must be 1")
    if type(y) is not np.ndarray:
        raise ValueError("y=" + str(y) + " must be a numpy.ndarray")
    elif y.ndim != 1:
        raise ValueError("y.ndim=" + str(y.ndim) + " must be 1")
    npoints = x.size
    if npoints != y.size:
        raise ValueError("x.size != y.size")
    if type(deg) not in [np.int, np.int64]:
        raise ValueError("deg=" + str(deg) +
                         " is not a valid integer")
    if deg >= npoints:
        raise ValueError("Polynomial degree=" + str(deg) +
                         " can't be fitted with npoints=" + str(npoints))

    # initialize boolean rejection array
    reject = np.zeros(npoints, dtype=np.bool)

    # if there is no room to remove any point, compute a fit without
    # rejection
    if deg == npoints - 1:
        poly, yres = polfit_residuals(x=x, y=y, deg=deg, reject=None,
                                      color=color, size=size,
                                      xlim=xlim, ylim=ylim,
                                      xlabel=xlabel, ylabel=ylabel,
                                      title=title,
                                      use_r=use_r,
                                      geometry=geometry,
                                      debugplot=debugplot)
        return poly, yres, reject

    # main loop to reject points iteratively
    loop_to_reject_points = True
    poly = None
    yres = None
    while loop_to_reject_points:
        if abs(debugplot) in [21, 22]:
            poly, yres = polfit_residuals(x=x, y=y, deg=deg, reject=reject,
                                          color=color, size=size,
                                          xlim=xlim, ylim=ylim,
                                          xlabel=xlabel, ylabel=ylabel,
                                          title=title,
                                          use_r=use_r,
                                          geometry=geometry,
                                          debugplot=debugplot)
        else:
            poly, yres = polfit_residuals(x=x, y=y, deg=deg, reject=reject)
        # check that there is room to remove a point with the current
        # polynomial degree
        npoints_effective = npoints - np.sum(reject)
        if deg < npoints_effective - 1:
            # determine robuts standard deviation, excluding points
            # already rejected
            # --- method 1 ---
            # yres_fitted = yres[np.logical_not(reject)]
            # q25, q75 = np.percentile(yres_fitted, q=[25.0, 75.0])
            # rms = 0.7413 * (q75 - q25)
            # --- method 2 ---
            yres_fitted = np.abs(yres[np.logical_not(reject)])
            rms = np.median(yres_fitted)
            if abs(debugplot) >= 10:
                print("--> robust rms:", rms)
            # reject fitted point exceeding the threshold with the
            # largest deviation (note: with this method only one point
            # is removed in each iteration of the loop; this allows the
            # recomputation of the polynomial fit which, sometimes,
            # transforms deviant points into good ones)
            index_to_remove = []
            for i in range(npoints):
                if not reject[i]:
                    if np.abs(yres[i]) > times_sigma_reject * rms:
                        index_to_remove.append(i)
                        if abs(debugplot) >= 10:
                            print('--> suspicious point #', i + 1)
            if len(index_to_remove) == 0:
                if abs(debugplot) >= 10:
                    print('==> no need to remove any point')
                loop_to_reject_points = False
            else:
                imax = np.argmax(np.abs(yres[index_to_remove]))
                reject[index_to_remove[imax]] = True
                if abs(debugplot) >= 10:
                    print('==> removing point #', index_to_remove[imax] + 1)
        else:
            loop_to_reject_points = False

    # plot final fit in case it has not been already shown
    if abs(debugplot) % 10 != 0:
        if abs(debugplot) not in [21, 22]:
            poly, yres = polfit_residuals(x=x, y=y, deg=deg, reject=reject,
                                          color=color, size=size,
                                          xlim=xlim, ylim=ylim,
                                          xlabel=xlabel, ylabel=ylabel,
                                          title=title,
                                          use_r=use_r,
                                          geometry=geometry,
                                          debugplot=debugplot)
        else:
            if abs(debugplot) >= 10:
                print(' ')

    # return result
    return poly, yres, reject
Polynomial fit with iterative rejection of points.

    This function makes use of function polfit_residuals for display
    purposes.

    Parameters
    ----------
    x : 1d numpy array, float
        X coordinates of the data being fitted.
    y : 1d numpy array, float
        Y coordinates of the data being fitted.
    deg : int
        Degree of the fitting polynomial.
    times_sigma_reject : float or None
        Number of times the standard deviation to reject points
        iteratively. If None, the fit does not reject any point.
    color : single character or 1d numpy array of characters
        Color for all the symbols (single character) or for each
        individual symbol (array of color names with the same length as
        'x' or 'y'). If 'color' is a single character, the rejected
        points are displayed in red color, whereas when 'color' is an
        array of color names, rejected points are displayed with the
        color provided in this array.
    size : int
        Marker size for all the symbols (single character) or for each
        individual symbol (array of integers with the same length as
        'x' or 'y').
    xlim : tuple (floats)
        Plot limits in the X axis.
    ylim : tuple (floats)
        Plot limits in the Y axis.
    xlabel : string
        Character string for label in X axis.
    ylabel : string
        Character string for label in y axis.
    title : string
        Character string for graph title.
    use_r : bool
        If True, the function computes several fits, using R, to
        polynomials of degree deg, deg+1 and deg+2 (when possible).
    geometry : tuple (4 integers) or None
        x, y, dx, dy values employed to set the window geometry.
    debugplot : int
        Determines whether intermediate computations and/or plots
        are displayed. The valid codes are defined in
        numina.array.display.pause_debugplot.

    Return
    ------
    poly : instance of Polynomial (numpy)
        Result from the polynomial fit using numpy Polynomial. Only
        points not flagged as rejected are employed in the fit.
    yres : 1d numpy array, float
        Residuals from polynomial fit. Note that the residuals are
        computed for all the points, including the rejected ones. In
        this way the dimension of this array is the same as the
        dimensions of the input 'x' and 'y' arrays.
    reject : 1d numpy array, bool
        Boolean array indicating rejected points.