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Fractional Power of Rational function method (FPR) data

This class contains the data for the results of the FPR method used in Honda. It obtains a very precise mean function, but is unable to generate the uncertainty bands that Bayesian model mixing can achieve. It is used as a comparison in the paper published with this package.

FPR

Bases: Models

Source code in samba/fprdat.py 
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class FPR(Models):

    def __init__(self, g, loworder, highorder):

        r'''
        A class to calculate the FPR method curves for comparison
        to the mixed models in the three BMM methods of this package.

        Example:
            FPR(g=np.linspace(1e-6,1.0,100), loworder=np.array([5]),
                highorder=np.array([5]))

        Parameters:
            g (numpy.linspace): The input space array over which the models are 
                mixed.

            loworder (numpy.ndarray): The highest order considered in the small-g 
                expansion.

            highorder (numpy.ndarray): The highest order considered in the large-g 
                expansion.

        Returns:
            None.
        '''

        self.g = g
        self.loworder = loworder
        self.highorder = highorder 

        #instantiate Models() class here
        self.m = Models(self.loworder, self.highorder)

        return None


    def fprset(self, key):

        r'''
        Call the proper FPR function desired and obtain 
        an array of the results in the input space, g. 

        Example: 
            FPR.fprset(key='(2,4)^(1/8)')

        Parameters:
            key (str): The preferred FPR function. Enter a key in the
                convention: '(m,n)^(\alpha)', where m,n are orders
                less than or equal to N_s and N_l (loworder, highorder
                in the other classes). \alpha is the value the FPR is 
                raised to in Eq. (2.7) (Honda 2014). 

        Returns:
            fpr (numpy.ndarray): Results of the FPR function in an array. 
        '''

        #if statement for calling the proper FPR function
        fpr = np.zeros(len(self.g))

        self.keyvalue = key

        if key == '(0,0)^(1/2)':

            #FPR[1/(2*1),0,0,g_]
            fpr = np.sqrt(2 * np.pi * np.sqrt(1/(1 + (8 * self.g * np.pi)/sp.gamma(1/4)**2)))  

        elif key == '(1,1)^(1/2)':  

            #FPR[1/(2*1),1,1,g_] 
            fpr = np.sqrt(2 * np.pi * sp.gamma(1/4)) * np.sqrt((2 * np.pi * sp.gamma(-(1/4)) + 8 * \
                  self.g * np.pi * sp.gamma(1/4) + sp.gamma(1/4)**3) / (64 * self.g**2 * np.pi**2 + 2 * \
                  np.pi * sp.gamma(-(1/4)) * sp.gamma(1/4) + 8 * self.g * \
                  np.pi * sp.gamma(1/4)**2 + sp.gamma(1/4)**4))

        elif key == '(2,2)^(1/2)':
            #FPR[1/(2*1),2,2,g_] 
            fpr = 2.5066282746310002 * np.sqrt((1. + 10.1531607808241 * self.g + 37.91166947810798 * \
                  self.g**2) / (1. + 10.1531607808241 * self.g + 43.91166947810798 * \
                  self.g**2 + 72.48541321812907 * self.g**3))

        elif key == '(3,3)^(1/2)':
            #FPR[1/(2*1),3,3,g_] 
            fpr = 2.5066282746310002 * np.sqrt((1. + 16.030388229931486 * self.g + 110.26077245790317 * \
                  self.g**2 + 324.0187225963292 * self.g**3) / (1. + 16.030388229931486 * \
                  self.g + 116.26077245790317 * self.g**2 + 420.2010519759181 * \
                  self.g**3 + 619.509278307239 * self.g**4))

        elif key == '(4,4)^(1/2)':
            #FPR[1/(2*1),4,4,g_] 
            fpr = 2.5066282746310002 * np.sqrt(( 1. + 22.874503929271544 * self.g + 238.23905635876318 * \
                  self.g**2 + 1303.4929582331083 * self.g**3 + 3224.5631655188554 * self.g**4)/ \
                  ( 1. + 22.874503929271544 * self.g + 244.23905635876318 * self.g**2 + 1440.7399818087376 * \
                  self.g**3 + 4575.997503671434 * self.g**4 + 6165.220279617642 * self.g**5))

        elif key == '(1,1)^(1/6)':
            #FPR[1/(2*3),1,1,g_] 
            fpr = 2.5066282746310002 * (1/( 1. + 7.086913042848253 * self.g**2 + 6.989291097242859 * \
                  self.g**3))**(1/6)

        elif key == '(2,2)^(1/6)':
            #FPR[1/(2*3),2,2,g_] 
            fpr = 2.5066282746310002 * (( 1. + 3.7875802399388747 * self.g)/( 1. + 3.7875802399388747 * \
                  self.g + 18.000000000000007 * self.g**2 + 33.83154290049999 * self.g**3 + 26.47250085109775 * \
                  self.g**4))**(1/6)

        elif key == '(3,3)^(1/6)':
            #FPR[1/(2*3),3,3,g_] 
            fpr = 2.5066282746310002 * (( 1. + 8.099704591178746 * self.g + 28.252523713142793 * \
                  self.g**2)/( 1. + 8.099704591178746 * self.g + 46.25252371314279 * \
                  self.g**2 + 145.7946826412175 * self.g**3 + 256.83437198547387 * \
                  self.g**4 + 197.46511246291166 * self.g**5))**(1/6)

        elif key == '(4,4)^(1/6)':
            #FPR[1/(2*3),4,4,g_]
            fpr = 2.5066282746310002 * (( 1. + 13.470628417091639 * self.g + 87.53888949233587 * \
                  self.g**2 + 255.1535591483978 * self.g**3)/( 1. + 13.470628417091639 * \
                  self.g + 105.53888949233587 * self.g**2 + 497.6248706560473 * self.g**3 + 1449.7000108620534 * \
                  self.g**4 + 2420.0858672492423 * self.g**5 + 1783.3424993857257 * self.g**6))**(1/6)

        elif key == '(2,2)^(1/10)':
            #FPR[1/(2*5),2,2,g_] 
            fpr = 2.5066282746310002 * (1/( 1. + 30. * self.g**2 + 32.14821200212872 * \
                  self.g**3 + 43.17787580118569 * self.g**4 + 25.54986136647706 * self.g**5))**(1/10)

        elif key == '(4,4)^(1/10)':
            #FPR[1/(2*5),4,4,g_] 
            fpr = 2.5066282746310002 * (( 1. + 5.072505666220409 * self.g + 14.43685326985559 * \
                  self.g**2)/( 1. + 5.072505666220409 * self.g + 44.43685326985559 * self.g**2 + 152.17516998661225 * \
                  self.g**3 + 403.10559809566917 * self.g**4 + 708.6889005862954 * self.g**5 + 752.9544739983669 * \
                  self.g**6 + 368.85959961298136 * self.g**7))**(1/10)

        elif key == '(3,3)^(1/14)':
            #FPR[1/(2*7),3,3,g_] 
            fpr = 2.5066282746310002 * (1/( 1. + 42. * self.g**2+155.75843284764994 * \
                  self.g**4 + 239.21559267499774 * self.g**5 + 220.9758065100799 * \
                  self.g**6 + 93.39937438057375 * self.g**7))**(1/14)

        elif key == '(4,4)^(1/18)':
            #FPR[1/(2*9),4,4,g_] 
            fpr = 2.5066282746310002 * (1/( 1. + 54. * self.g**2+594. * self.g**4+780.7879756756589 * \
                  self.g**5 + 1294.340979801729 * self.g**6 + 1475.3510504866329 * self.g**7 + 1038.5911468627608 * \
                  self.g**8 + 341.42819835916043 * self.g**9))**(1/18)

        else:
            raise KeyError('The key provided does not match any in the FPR database.')

        return fpr

    #at the moment, this is very specific to the paper plot---disassemble later for package
    def fpr_plot(self, mean, intervals, fpr_keys=None, ci=68):

        r'''
        A plotter for the overlay of the GP results and the FPR results
        from Honda (2014). 

        Example:
            FPR.fpr_plot(mean=np.array(), intervals=np.array([,]), 
                fpr_keys=['(3,3)^(1/6)'], ci=95)

        Parameters:
            mean (numpy.ndarray): A PPD mean to be compared to the FPR 
                results.

            intervals (numpy.ndarray): A 2D array to plot a UQ band around 
                the PPD. 

            fpr_keys (list): A list of strings of fpr keys to be read in 
                by the function and calculated using the fprset()
                function above.

            ci (int): The uncertainty calculated on the expansions. Can
                be either 68 or 95. 

        Returns:
            None.
        '''

        #set up plot configuration
        fig, ax1 = plt.subplots(figsize=(8,6), dpi=600)
        ax1.tick_params(axis='x', labelsize=18)
        ax1.tick_params(axis='y', labelsize=18)
        ax1.locator_params(nbins=8)
        ax1.xaxis.set_minor_locator(AutoMinorLocator())
        ax1.xaxis.set_label_coords(.5, -.05)
        ax1.yaxis.set_minor_locator(AutoMinorLocator())
        ax1.yaxis.set_label_coords(-.05, .5)

        #set up x and y limits
        ax1.set_xlim(0,1)
        ax1.set_xticks([0, 0.2, 0.4, 0.6, 0.8, 1])
        ax1.set_ylim(1.2,3.2)
        ax1.set_yticks([1.2, 1.6, 2.0, 2.4, 2.8, 3.2])

        #labels and true model
        ax1.set_xlabel('g', fontsize=22)
        ax1.set_ylabel('F(g)', fontsize=22)
        ax1.plot(self.g, self.m.true_model(self.g), 'k', label='True model')

        #unpack ci
        self.ci = ci 

        #plot the small-g expansions and error bands
        ax1.plot(self.g, self.m.low_g(self.g)[0,:], 'r--', label=r'$f_s$ ($N_s$ = {})'.format(self.m.loworder[0]))

        #plot the large-g expansions and error bands
        ax1.plot(self.g, self.m.high_g(self.g)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.m.highorder[0]))

        #plot the GP results (mixed model)
        ax1.plot(self.g, mean, 'g', label='Mean')
        ax1.plot(self.g, intervals[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax1.plot(self.g, intervals[:,1], 'g', linestyle='dotted')
        ax1.fill_between(self.g, intervals[:,0], intervals[:,1], color='green', alpha=0.2)

        #FPR results
        if fpr_keys is not None:
            ax1.set_prop_cycle(cycler('color', ['darkviolet', 'deepskyblue', 'darkorange', 'gold']))
            for k in fpr_keys:
                mn = k[0:5]
                alpha = k[6:]
                fpr = self.fprset(k)
                ax1.plot(self.g, fpr, linestyle='dashed', label=r'$F_{{{}}}^{{{}}} (g)$'.format(mn, alpha))

        ax1.legend(fontsize=16, loc='upper right')

        #inset plot parameters
        x1 = 0.26
        x2 = 0.31
        y1 = 2.15
        y2 = 2.25
        axins = zoomed_inset_axes(ax1, 6, loc=9) 
        axins.plot(self.g, self.m.true_model(self.g), 'k', label='True model')
        axins.plot(self.g, self.m.low_g(self.g)[0,:], 'r--', label=r'$f_s$ ($N_s$ = {})'.format(self.m.loworder[0]))
        axins.plot(self.g, self.m.high_g(self.g)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.m.highorder[0]))
        axins.plot(self.g, mean, 'g', label='Mean')
        axins.plot(self.g, intervals[:,0], 'g', linestyle='dotted', label=r'{}$\%$ interval'.format(int(self.ci)))
        axins.plot(self.g, intervals[:,1], 'g', linestyle='dotted')
        axins.fill_between(self.g, intervals[:,0], intervals[:,1], color='green', alpha=0.2)
        if fpr_keys is not None:
            axins.set_prop_cycle(cycler('color', ['darkviolet', 'deepskyblue', 'darkorange', 'gold']))
        for k in fpr_keys:
                mn = k[0:5]
                alpha = k[6:]
                fpr = self.fprset(k)
                axins.plot(self.g, fpr, linestyle='dashed', label=r'$F_{{{}}}^{{{}}} (g)$'.format(mn, alpha))

        axins.set_xlim(x1, x2)
        axins.set_ylim(y1, y2)
        mark_inset(ax1, axins, loc1=2, loc2=4, fc="none", ec="0.5")
        plt.draw()
        plt.show()

        #save figure option
        # response = input('Would you like to save this figure? (yes/no)')

        # if response == 'yes':
        #     name = input('Enter a file name (include .jpg, .png, etc.)')
        #     fig.savefig(name, bbox_inches='tight')

        return None

__init__(g, loworder, highorder)

A class to calculate the FPR method curves for comparison to the mixed models in the three BMM methods of this package.

Example

FPR(g=np.linspace(1e-6,1.0,100), loworder=np.array([5]), highorder=np.array([5]))

Parameters:

Name Type Description Default
g linspace

The input space array over which the models are mixed.

 required
loworder ndarray

The highest order considered in the small-g expansion.

 required
highorder ndarray

The highest order considered in the large-g expansion.

 required

Returns:

Type Description

None.
Source code in samba/fprdat.py 
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def __init__(self, g, loworder, highorder):

    r'''
    A class to calculate the FPR method curves for comparison
    to the mixed models in the three BMM methods of this package.

    Example:
        FPR(g=np.linspace(1e-6,1.0,100), loworder=np.array([5]),
            highorder=np.array([5]))

    Parameters:
        g (numpy.linspace): The input space array over which the models are 
            mixed.

        loworder (numpy.ndarray): The highest order considered in the small-g 
            expansion.

        highorder (numpy.ndarray): The highest order considered in the large-g 
            expansion.

    Returns:
        None.
    '''

    self.g = g
    self.loworder = loworder
    self.highorder = highorder 

    #instantiate Models() class here
    self.m = Models(self.loworder, self.highorder)

    return None

fpr_plot(mean, intervals, fpr_keys=None, ci=68)

A plotter for the overlay of the GP results and the FPR results from Honda (2014).

Example

FPR.fpr_plot(mean=np.array(), intervals=np.array([,]), fpr_keys=['(3,3)^(1/6)'], ci=95)

Parameters:

Name Type Description Default
mean ndarray

A PPD mean to be compared to the FPR results.

 required
intervals ndarray

A 2D array to plot a UQ band around the PPD.

required
fpr_keys list

A list of strings of fpr keys to be read in by the function and calculated using the fprset() function above.

 None
ci int

The uncertainty calculated on the expansions. Can be either 68 or 95.

68

Returns:

Type Description

None.
Source code in samba/fprdat.py 
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def fpr_plot(self, mean, intervals, fpr_keys=None, ci=68):

    r'''
    A plotter for the overlay of the GP results and the FPR results
    from Honda (2014). 

    Example:
        FPR.fpr_plot(mean=np.array(), intervals=np.array([,]), 
            fpr_keys=['(3,3)^(1/6)'], ci=95)

    Parameters:
        mean (numpy.ndarray): A PPD mean to be compared to the FPR 
            results.

        intervals (numpy.ndarray): A 2D array to plot a UQ band around 
            the PPD. 

        fpr_keys (list): A list of strings of fpr keys to be read in 
            by the function and calculated using the fprset()
            function above.

        ci (int): The uncertainty calculated on the expansions. Can
            be either 68 or 95. 

    Returns:
        None.
    '''

    #set up plot configuration
    fig, ax1 = plt.subplots(figsize=(8,6), dpi=600)
    ax1.tick_params(axis='x', labelsize=18)
    ax1.tick_params(axis='y', labelsize=18)
    ax1.locator_params(nbins=8)
    ax1.xaxis.set_minor_locator(AutoMinorLocator())
    ax1.xaxis.set_label_coords(.5, -.05)
    ax1.yaxis.set_minor_locator(AutoMinorLocator())
    ax1.yaxis.set_label_coords(-.05, .5)

    #set up x and y limits
    ax1.set_xlim(0,1)
    ax1.set_xticks([0, 0.2, 0.4, 0.6, 0.8, 1])
    ax1.set_ylim(1.2,3.2)
    ax1.set_yticks([1.2, 1.6, 2.0, 2.4, 2.8, 3.2])

    #labels and true model
    ax1.set_xlabel('g', fontsize=22)
    ax1.set_ylabel('F(g)', fontsize=22)
    ax1.plot(self.g, self.m.true_model(self.g), 'k', label='True model')

    #unpack ci
    self.ci = ci 

    #plot the small-g expansions and error bands
    ax1.plot(self.g, self.m.low_g(self.g)[0,:], 'r--', label=r'$f_s$ ($N_s$ = {})'.format(self.m.loworder[0]))

    #plot the large-g expansions and error bands
    ax1.plot(self.g, self.m.high_g(self.g)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.m.highorder[0]))

    #plot the GP results (mixed model)
    ax1.plot(self.g, mean, 'g', label='Mean')
    ax1.plot(self.g, intervals[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
    ax1.plot(self.g, intervals[:,1], 'g', linestyle='dotted')
    ax1.fill_between(self.g, intervals[:,0], intervals[:,1], color='green', alpha=0.2)

    #FPR results
    if fpr_keys is not None:
        ax1.set_prop_cycle(cycler('color', ['darkviolet', 'deepskyblue', 'darkorange', 'gold']))
        for k in fpr_keys:
            mn = k[0:5]
            alpha = k[6:]
            fpr = self.fprset(k)
            ax1.plot(self.g, fpr, linestyle='dashed', label=r'$F_{{{}}}^{{{}}} (g)$'.format(mn, alpha))

    ax1.legend(fontsize=16, loc='upper right')

    #inset plot parameters
    x1 = 0.26
    x2 = 0.31
    y1 = 2.15
    y2 = 2.25
    axins = zoomed_inset_axes(ax1, 6, loc=9) 
    axins.plot(self.g, self.m.true_model(self.g), 'k', label='True model')
    axins.plot(self.g, self.m.low_g(self.g)[0,:], 'r--', label=r'$f_s$ ($N_s$ = {})'.format(self.m.loworder[0]))
    axins.plot(self.g, self.m.high_g(self.g)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.m.highorder[0]))
    axins.plot(self.g, mean, 'g', label='Mean')
    axins.plot(self.g, intervals[:,0], 'g', linestyle='dotted', label=r'{}$\%$ interval'.format(int(self.ci)))
    axins.plot(self.g, intervals[:,1], 'g', linestyle='dotted')
    axins.fill_between(self.g, intervals[:,0], intervals[:,1], color='green', alpha=0.2)
    if fpr_keys is not None:
        axins.set_prop_cycle(cycler('color', ['darkviolet', 'deepskyblue', 'darkorange', 'gold']))
    for k in fpr_keys:
            mn = k[0:5]
            alpha = k[6:]
            fpr = self.fprset(k)
            axins.plot(self.g, fpr, linestyle='dashed', label=r'$F_{{{}}}^{{{}}} (g)$'.format(mn, alpha))

    axins.set_xlim(x1, x2)
    axins.set_ylim(y1, y2)
    mark_inset(ax1, axins, loc1=2, loc2=4, fc="none", ec="0.5")
    plt.draw()
    plt.show()

    #save figure option
    # response = input('Would you like to save this figure? (yes/no)')

    # if response == 'yes':
    #     name = input('Enter a file name (include .jpg, .png, etc.)')
    #     fig.savefig(name, bbox_inches='tight')

    return None

fprset(key)

Call the proper FPR function desired and obtain an array of the results in the input space, g.

Example

FPR.fprset(key='(2,4)^(1/8)')

Parameters:

Name Type Description Default
key str

The preferred FPR function. Enter a key in the convention: '(m,n)^(\alpha)', where m,n are orders less than or equal to N_s and N_l (loworder, highorder in the other classes). \alpha is the value the FPR is raised to in Eq. (2.7) (Honda 2014).

required

Returns:

Name Type Description
fpr ndarray

Results of the FPR function in an array.
Source code in samba/fprdat.py 
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def fprset(self, key):

    r'''
    Call the proper FPR function desired and obtain 
    an array of the results in the input space, g. 

    Example: 
        FPR.fprset(key='(2,4)^(1/8)')

    Parameters:
        key (str): The preferred FPR function. Enter a key in the
            convention: '(m,n)^(\alpha)', where m,n are orders
            less than or equal to N_s and N_l (loworder, highorder
            in the other classes). \alpha is the value the FPR is 
            raised to in Eq. (2.7) (Honda 2014). 

    Returns:
        fpr (numpy.ndarray): Results of the FPR function in an array. 
    '''

    #if statement for calling the proper FPR function
    fpr = np.zeros(len(self.g))

    self.keyvalue = key

    if key == '(0,0)^(1/2)':

        #FPR[1/(2*1),0,0,g_]
        fpr = np.sqrt(2 * np.pi * np.sqrt(1/(1 + (8 * self.g * np.pi)/sp.gamma(1/4)**2)))  

    elif key == '(1,1)^(1/2)':  

        #FPR[1/(2*1),1,1,g_] 
        fpr = np.sqrt(2 * np.pi * sp.gamma(1/4)) * np.sqrt((2 * np.pi * sp.gamma(-(1/4)) + 8 * \
              self.g * np.pi * sp.gamma(1/4) + sp.gamma(1/4)**3) / (64 * self.g**2 * np.pi**2 + 2 * \
              np.pi * sp.gamma(-(1/4)) * sp.gamma(1/4) + 8 * self.g * \
              np.pi * sp.gamma(1/4)**2 + sp.gamma(1/4)**4))

    elif key == '(2,2)^(1/2)':
        #FPR[1/(2*1),2,2,g_] 
        fpr = 2.5066282746310002 * np.sqrt((1. + 10.1531607808241 * self.g + 37.91166947810798 * \
              self.g**2) / (1. + 10.1531607808241 * self.g + 43.91166947810798 * \
              self.g**2 + 72.48541321812907 * self.g**3))

    elif key == '(3,3)^(1/2)':
        #FPR[1/(2*1),3,3,g_] 
        fpr = 2.5066282746310002 * np.sqrt((1. + 16.030388229931486 * self.g + 110.26077245790317 * \
              self.g**2 + 324.0187225963292 * self.g**3) / (1. + 16.030388229931486 * \
              self.g + 116.26077245790317 * self.g**2 + 420.2010519759181 * \
              self.g**3 + 619.509278307239 * self.g**4))

    elif key == '(4,4)^(1/2)':
        #FPR[1/(2*1),4,4,g_] 
        fpr = 2.5066282746310002 * np.sqrt(( 1. + 22.874503929271544 * self.g + 238.23905635876318 * \
              self.g**2 + 1303.4929582331083 * self.g**3 + 3224.5631655188554 * self.g**4)/ \
              ( 1. + 22.874503929271544 * self.g + 244.23905635876318 * self.g**2 + 1440.7399818087376 * \
              self.g**3 + 4575.997503671434 * self.g**4 + 6165.220279617642 * self.g**5))

    elif key == '(1,1)^(1/6)':
        #FPR[1/(2*3),1,1,g_] 
        fpr = 2.5066282746310002 * (1/( 1. + 7.086913042848253 * self.g**2 + 6.989291097242859 * \
              self.g**3))**(1/6)

    elif key == '(2,2)^(1/6)':
        #FPR[1/(2*3),2,2,g_] 
        fpr = 2.5066282746310002 * (( 1. + 3.7875802399388747 * self.g)/( 1. + 3.7875802399388747 * \
              self.g + 18.000000000000007 * self.g**2 + 33.83154290049999 * self.g**3 + 26.47250085109775 * \
              self.g**4))**(1/6)

    elif key == '(3,3)^(1/6)':
        #FPR[1/(2*3),3,3,g_] 
        fpr = 2.5066282746310002 * (( 1. + 8.099704591178746 * self.g + 28.252523713142793 * \
              self.g**2)/( 1. + 8.099704591178746 * self.g + 46.25252371314279 * \
              self.g**2 + 145.7946826412175 * self.g**3 + 256.83437198547387 * \
              self.g**4 + 197.46511246291166 * self.g**5))**(1/6)

    elif key == '(4,4)^(1/6)':
        #FPR[1/(2*3),4,4,g_]
        fpr = 2.5066282746310002 * (( 1. + 13.470628417091639 * self.g + 87.53888949233587 * \
              self.g**2 + 255.1535591483978 * self.g**3)/( 1. + 13.470628417091639 * \
              self.g + 105.53888949233587 * self.g**2 + 497.6248706560473 * self.g**3 + 1449.7000108620534 * \
              self.g**4 + 2420.0858672492423 * self.g**5 + 1783.3424993857257 * self.g**6))**(1/6)

    elif key == '(2,2)^(1/10)':
        #FPR[1/(2*5),2,2,g_] 
        fpr = 2.5066282746310002 * (1/( 1. + 30. * self.g**2 + 32.14821200212872 * \
              self.g**3 + 43.17787580118569 * self.g**4 + 25.54986136647706 * self.g**5))**(1/10)

    elif key == '(4,4)^(1/10)':
        #FPR[1/(2*5),4,4,g_] 
        fpr = 2.5066282746310002 * (( 1. + 5.072505666220409 * self.g + 14.43685326985559 * \
              self.g**2)/( 1. + 5.072505666220409 * self.g + 44.43685326985559 * self.g**2 + 152.17516998661225 * \
              self.g**3 + 403.10559809566917 * self.g**4 + 708.6889005862954 * self.g**5 + 752.9544739983669 * \
              self.g**6 + 368.85959961298136 * self.g**7))**(1/10)

    elif key == '(3,3)^(1/14)':
        #FPR[1/(2*7),3,3,g_] 
        fpr = 2.5066282746310002 * (1/( 1. + 42. * self.g**2+155.75843284764994 * \
              self.g**4 + 239.21559267499774 * self.g**5 + 220.9758065100799 * \
              self.g**6 + 93.39937438057375 * self.g**7))**(1/14)

    elif key == '(4,4)^(1/18)':
        #FPR[1/(2*9),4,4,g_] 
        fpr = 2.5066282746310002 * (1/( 1. + 54. * self.g**2+594. * self.g**4+780.7879756756589 * \
              self.g**5 + 1294.340979801729 * self.g**6 + 1475.3510504866329 * self.g**7 + 1038.5911468627608 * \
              self.g**8 + 341.42819835916043 * self.g**9))**(1/18)

    else:
        raise KeyError('The key provided does not match any in the FPR database.')

    return fpr