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Method 2: Bivariate model mixing

In this method, we use pointwise bivariate model mixing, or precision-weighted mixing, requiring models to be evaluated at every point in the input space where we desire a prediction to be made. This method can be written succinctly as

\[ f_{\dagger} = \frac{1}{Z_P}\sum_{k=1}^{K} \frac{1}{v_k}f_k, \qquad Z_P \equiv \sum_{k=1}^{K}\frac{1}{v_k}, \]

where we can also define

\[ f_{\dagger} \sim \mathcal{N}\bigl(Z_P^{-1}\sum_k \frac{1}{v_k}f_k, Z_P^{-1}\bigr). \]

This method is precision-weighted because it uses the variances of the models at each point in the input space as the inverse weights of the corresponding model prediction, hence the model with the smallest variance at a given point will dominate the mixed model at that location.

Bivariate(loworder, highorder, error_model='informative', ci=68)

Example

Bivariate(loworder=5, highorder=10)

Parameters:

Name Type Description Default
loworder (ndarray, int, float)

The value of N_s to be used to truncate the small-g expansion.

 required
highorder (ndarray, int, float)

The value of N_l to be used to truncate the large-g expansion.

 required
error_model str

The error model to be used in the calculation. Options are 'uninformative' and 'informative'. Default is 'informative'.

'informative'
ci int

The value of the credibility interval desired (can be 68 or 95).

 68

Returns:

Type Description

None.
Source code in samba/discrepancy.py 
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def __init__(self, loworder, highorder, error_model='informative', ci=68):

    r'''
    The bivariate BMM method used to construct the mixed model of two series
    expansions. This class contains the fdagger function and the plotter.

    Example:
        Bivariate(loworder=5, highorder=10)

    Parameters:
        loworder (numpy.ndarray, int, float): The value of N_s to be 
            used to truncate the small-g expansion.

        highorder (numpy.ndarray, int, float): The value of N_l to be 
            used to truncate the large-g expansion.

        error_model (str): The error model to be used in the calculation. 
            Options are 'uninformative' and 'informative'. Default is 'informative'. 

        ci (int): The value of the credibility interval desired (can be 68 or 95).

    Returns:
        None.
    '''

    #get interval
    self.ci = ci

    #instantiate the Uncertainties class and error model
    self.u = Uncertainties(error_model)
    self.error_model = self.u.error_model

    #check type and assign class variables
    if isinstance(loworder, float) == True or isinstance(loworder, int) == True:
        loworder = np.array([loworder])

    if isinstance(highorder, float) == True or isinstance(highorder, int) == True:
        highorder = np.array([highorder])

    self.loworder = loworder 
    self.highorder = highorder

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

    return None

fdagger(g)

A do-it-all function to determine the pdf of the mixed model. Can use models indicated by inputting arrays into the loworder and highorder variables.

Example

Bivariate.fdagger(g=np.linspace(1e-6, 0.5, 100))

Parameters:

Name Type Description Default
g linspace

The linspace over which this calculation is performed.

 required

Returns:

Name Type Description
mean ndarray

The mixed model mean (either including a GP or not depending on the function arguments).
intervals ndarray

The credibility interval of the mixed model mean.
interval_low ndarray

The variance interval for the small-g expansion (calculated from the next order after the truncation).

interval_high ndarray

The variance interval for the large-g expansion (calculated from the next order after the truncation).
Source code in samba/discrepancy.py 
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def fdagger(self, g): 

    r'''
    A do-it-all function to determine the pdf of the mixed model. Can use models 
    indicated by inputting arrays into the loworder and highorder variables.

    Example:
        Bivariate.fdagger(g=np.linspace(1e-6, 0.5, 100))

    Parameters:
        g (numpy.linspace): The linspace over which this calculation is performed.

    Returns:
        mean (numpy.ndarray): The mixed model mean (either including a GP or not 
            depending on the function arguments).

        intervals (numpy.ndarray): The credibility interval of the mixed model mean.

        interval_low (numpy.ndarray): The variance interval for the small-g expansion 
            (calculated from the next order after the truncation). 

        interval_high (numpy.ndarray): The variance interval for the large-g expansion 
            (calculated from the next order after the truncation).
    '''

    #check type
    if isinstance(self.loworder, float) == True or isinstance(self.loworder, int) == True:
        self.loworder = np.array([self.loworder])

    if isinstance(self.highorder, float) == True or isinstance(self.highorder, int) == True:
        self.highorder = np.array([self.highorder])

    #uncertainties
    v_low = np.asarray([self.u.variance_low(g, self.loworder[i]) for i in range(len(self.loworder))])
    v_high = np.asarray([self.u.variance_high(g, self.highorder[i]) for i in range(len(self.highorder))])

    #calculating models
    f_low = np.asarray([self.m.low_g(g)[i,:] for i in range(len(self.loworder))])
    f_high = np.asarray([self.m.high_g(g)[i,:] for i in range(len(self.highorder))])

    # concatenate models
    f = np.concatenate((f_low, f_high), axis=0)
    v = np.concatenate((v_low, v_high), axis=0)

    #initialise arrays
    mean_n = np.zeros([len(f), len(g)])
    mean_d = np.zeros([len(f), len(g)])
    mean = np.zeros([len(g)])
    var = np.zeros([len(f), len(g)])

    #create fdagger for each value of g
    for i in range(len(f)):
        mean_n[i] = f[i]/v[i]
        mean_d[i] = 1.0/v[i]
        var[i] = 1.0/v[i]

    #save variances for each model
    self.var_weights = var/(np.sum(var, axis=0))

    mean_n = np.sum(mean_n, axis=0)
    mean_d = np.sum(mean_d, axis=0)

    #mean, variance calculation
    mean = mean_n/mean_d
    var = 1.0/np.sum(var, axis=0)

    #which credibility interval to use
    if self.ci == 68:
        val = 1.0
    elif self.ci == 95:
        val = 1.96
    else:
        raise ValueError('Please enter either 68 or 95.')

    #initialise credibility intervals
    intervals = np.zeros([len(g), 2])
    interval_low = np.zeros([len(self.loworder), len(g), 2])
    interval_high = np.zeros([len(self.highorder), len(g), 2])

    #calculate credibility intervals
    intervals[:, 0] = (mean - val * np.sqrt(var))
    intervals[:, 1] = (mean + val * np.sqrt(var))

    for i in range(len(self.loworder)):
        interval_low[i,:,0] = (self.m.low_g(g)[i,:] - val * np.sqrt(v_low[i,:]))
        interval_low[i,:,1] = (self.m.low_g(g)[i,:] + val * np.sqrt(v_low[i,:]))

    for i in range(len(self.highorder)):
        interval_high[i,:,0] = (self.m.high_g(g)[i,:] - val * np.sqrt(v_high[i,:]))
        interval_high[i,:,1] = (self.m.high_g(g)[i,:] + val * np.sqrt(v_high[i,:]))

    return mean, intervals, interval_low, interval_high

horizontal_plot_fdagger(g1, g2, gp_dict=None, gp_dict2=None, gp_points=None, gp_points2=None, low_orders=None, high_orders=None)

Horizontal panel plotter for the paper to generate two mixed model plots.

Example

Bivariate.horizontal_plot_fdagger(g1=np.linspace(1e-6, 0.5, 100), g2=np.linspace(1e-6,1.0,100), gp_mean1=np.array([]), gp_mean2=np.array([]), gp_var1=np.array([,]), gp_var2=np.array([,]))

Parameters:

Name Type Description Default
g1 linspace

The space over which the models (and GP) were calculated for panel a.

required
g2 linspace

The space over which the models (and GP) were calculated for panel b.

required
gp_mean1 ndarray

GP mean results to be mixed with the models in panel a. Optional.

required
gp_mean2 ndarray

GP mean results to be mixed with the models in panel b. Optional.

 required
gp_var1 ndarray

GP variance results for panel a. Optional.

 required
gp_var2 ndarray

GP variance results for panel b. Optional.

required

Returns:

Type Description

None.
Source code in samba/discrepancy.py 
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def horizontal_plot_fdagger(self, g1, g2, gp_dict=None, gp_dict2=None, gp_points=None, gp_points2=None, low_orders=None, high_orders=None):

    r'''
    Horizontal panel plotter for the paper to generate two mixed model plots. 

    Example:
        Bivariate.horizontal_plot_fdagger(g1=np.linspace(1e-6, 0.5, 100), g2=np.linspace(1e-6,1.0,100),
        gp_mean1=np.array([]), gp_mean2=np.array([]), gp_var1=np.array([,]), gp_var2=np.array([,]))

    Parameters:
        g1 (numpy.linspace): The space over which the models (and GP) were calculated 
            for panel a. 

        g2 (numpy.linspace): The space over which the models (and GP) were calculated 
            for panel b. 

        gp_mean1 (numpy.ndarray): GP mean results to be mixed with the models in panel a. 
            Optional. 

        gp_mean2 (numpy.ndarray): GP mean results to be mixed with the models in panel b. 
            Optional.

        gp_var1 (numpy.ndarray): GP variance results for panel a. Optional.

        gp_var2 (numpy.ndarray): GP variance results for panel b. Optional. 

    Returns:
        None.
    '''

    #set up plot configuration        
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16,6), dpi=600, sharex=True, sharey=True, \
                     gridspec_kw={'hspace':0, 'wspace':0}, \
                     tight_layout=True)

    for ax in (ax1, ax2):
        ax.tick_params(axis='both', which='major', labelsize=20, right=True, top=True, length=6)
        ax.tick_params(axis='both', which='minor', labelsize=20, right=True, top=True, length=3)
        ax.locator_params(nbins=5)
        ax.xaxis.set_minor_locator(AutoMinorLocator())
        ax.yaxis.set_minor_locator(AutoMinorLocator())
        ax.set_xlim(0.0,1.0)
        ax.set_xticks([0.1, 0.3, 0.5, 0.7, 0.9])
        ax.set_ylim(1.0,3.0)

        #labels and true model
        ax.set_xlabel('g', fontsize=24)
        ax.set_ylabel('F(g)', fontsize=24)
        ax.plot(g1, self.m.true_model(g1), 'k', label='True model')

        #only label outer plot axes
        ax.label_outer()

    #copy class variables to force changes
    if low_orders is None and high_orders is None:
        loworder = self.loworder.copy()
        highorder = self.highorder.copy()
    else:
        loworder = low_orders
        highorder = high_orders
        self.loworder = loworder
        self.highorder = highorder

    #mixed results (panel a)
    self.loworder = np.array([loworder[0]])
    self.highorder = np.array([highorder[0]])
    self.m = Models(self.loworder, self.highorder)
    mean_1, intervals_1, interval_low_1, interval_high_1 = self.fdagger(g1)

    #mixed results (panel b)
    self.loworder = np.array([loworder[1]])
    self.highorder = np.array([highorder[1]])
    self.m = Models(self.loworder, self.highorder)
    mean_2, intervals_2, interval_low_2, interval_high_2 = self.fdagger(g2)

    #plot the small-g expansions and error bands (panel a)
    self.loworder = np.array([loworder[0]])
    self.m = Models(self.loworder, self.highorder)
    ax1.plot(g1, self.m.low_g(g1)[0,:], 'r--', \
        label=r'$f_s$ ($N_s$ = {})'.format(self.loworder[0]))
    ax1.plot(g1, interval_low_1[0, :, 0], 'r', linestyle='dotted',\
         label=r'$f_s$ {}\% CI'.format(int(self.ci)))
    ax1.plot(g1, interval_low_1[0, :, 1], 'r', linestyle='dotted')

    #plot the small-g expansions and error bands (panel b)
    self.loworder = np.array([loworder[1]])
    self.m = Models(self.loworder, self.highorder)
    ax2.plot(g2, self.m.low_g(g2)[0,:], color='r', linestyle='dashed',\
            label=r'$f_s$ ($N_s$ = {})'.format(self.loworder[0]))
    ax2.plot(g2, interval_low_2[0, :, 0], color='r', linestyle='dotted', \
        label=r'$f_s$ {}\% CI'.format(int(self.ci)))
    ax2.plot(g2, interval_low_2[0, :, 1], color='r', linestyle='dotted')

    #plot the large-g expansions and error bands (panel a)
    self.highorder = np.array([highorder[0]])
    self.m = Models(self.loworder, self.highorder)
    ax1.plot(g1, self.m.high_g(g1)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.highorder[0]))
    ax1.plot(g1, interval_high_1[0, :, 0], 'b', linestyle='dotted', label=r'$f_l$ {}\% CI'.format(int(self.ci)))
    ax1.plot(g1, interval_high_1[0, :, 1], 'b', linestyle='dotted')

    #plot the large-g expansions and error bands (panel b)
    self.highorder = np.array([highorder[1]])
    self.m = Models(self.loworder, self.highorder)
    ax2.plot(g2, self.m.high_g(g2)[0,:], color='b', linestyle='dashed', \
            label=r'$f_l$ ($N_l$ = {})'.format(self.highorder[0]))
    ax2.plot(g2, interval_high_2[0, :, 0], color='b', linestyle='dotted', \
        label=r'$f_l$ {}\% CI'.format(int(self.ci)))
    ax2.plot(g2, interval_high_2[0, :, 1], color='b', linestyle='dotted')

    # error bar option
    if gp_points is not None:
        ax1.errorbar(gp_points['gs'], gp_points['datas'], yerr=gp_points['sigmas'], color="red", fmt='o', markersize=4, \
            capsize=4, label=r"Training data", zorder=10)

    if gp_points2 is not None:
        ax2.errorbar(gp_points2['gs'], gp_points2['datas'], yerr=gp_points2['sigmas'], color="red", fmt='o', markersize=4, \
            capsize=4, label=r"Training data", zorder=10)

    #uninformative model case
    if gp_dict is None:
        ax1.plot(g1, mean_1, 'g', label='Mixed model')
        ax1.plot(g1, intervals_1[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax1.plot(g1, intervals_1[:,1], 'g', linestyle='dotted')
        ax1.fill_between(g1, intervals_1[:,0], intervals_1[:,1], color='green', alpha=0.2)
    else:
        ax1.plot(gp_dict['g'], gp_dict['mean'], 'g', label='Mixed model')
        ax1.plot(gp_dict['g'], gp_dict['mean']-gp_dict['std'], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax1.plot(gp_dict['g'], gp_dict['mean']+gp_dict['std'], 'g', linestyle='dotted')
        ax1.fill_between(gp_dict['g'], gp_dict['mean']-gp_dict['std'], gp_dict['mean']+gp_dict['std'], color='green', alpha=0.2)

    #informative model case
    if gp_dict2 is None:
        ax2.plot(g2, mean_2, 'g', label='Mixed model')
        ax2.plot(g2, intervals_2[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax2.plot(g2, intervals_2[:,1], 'g', linestyle='dotted')
        ax2.fill_between(g2, intervals_2[:,0], intervals_2[:,1], color='green', alpha=0.2)
    else:
        ax2.plot(gp_dict2['g'], gp_dict2['mean'], 'g', label='Mixed model')
        ax2.plot(gp_dict2['g'], gp_dict2['mean']-gp_dict2['std'], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax2.plot(gp_dict2['g'], gp_dict2['mean']+gp_dict2['std'], 'g', linestyle='dotted')
        ax2.fill_between(gp_dict2['g'], gp_dict2['mean']-gp_dict2['std'], gp_dict2['mean']+gp_dict2['std'], color='green', alpha=0.2)

    #add panel labels
    ax1.text(0.94, 1.1, '(a)', fontsize=20)
    ax2.text(0.94, 1.1, '(b)', fontsize=20)

    ax2.legend(fontsize=16, loc='upper right')
    ax1.legend(fontsize=16, loc='upper right')
    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

plot_error_models(g)

A plotter to compare the uninformative error model results of two models to the informative error model results for the same two models. Panel a refers to the uninformative error model panel in the subplot, and panel b corresponds to the informative error model panel.

Example

Bivariate.plot_error_models(g=np.linspace(1e-6, 0.5, 100))

Parameters:

Name Type Description Default
g linspace

The space over which the models are calculated.

 required

Returns:

Type Description

None.
Source code in samba/discrepancy.py 
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def plot_error_models(self, g): 

    r'''
    A plotter to compare the uninformative error model results of two models 
    to the informative error model results for the same two models. Panel a
    refers to the uninformative error model panel in the subplot, and panel b
    corresponds to the informative error model panel. 

    Example:
        Bivariate.plot_error_models(g=np.linspace(1e-6, 0.5, 100))

    Parameters:
        g (numpy.linspace): The space over which the models are calculated.

    Returns:
        None.
    '''

    #set up plot configuration        
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16,6), dpi=600, sharex=True, sharey=True, \
                     gridspec_kw={'hspace':0, 'wspace':0}, \
                     tight_layout=True)

    for ax in (ax1, ax2):
        ax.tick_params(axis='both', which='major', labelsize=20, right=True, top=True, length=6)
        ax.tick_params(axis='both', which='minor', labelsize=20, right=True, top=True, length=3)
        ax.locator_params(nbins=5)
        ax.xaxis.set_minor_locator(AutoMinorLocator())
        ax.yaxis.set_minor_locator(AutoMinorLocator())
        ax.set_xlim(0.0,1.0)
        ax.set_xticks([0.1, 0.3, 0.5, 0.7, 0.9])
        ax.set_ylim(1.0,3.0)

        #labels and true model
        ax.set_xlabel('g', fontsize=24)
        ax.set_ylabel('F(g)', fontsize=24)
        ax.plot(g, self.m.true_model(g), 'k', label='True model')

        #only label outer plot axes
        ax.label_outer()

    #call fdagger to calculate results (overwrite class variable)
    self.u = Uncertainties(error_model='uninformative')
    self.error_model = self.u.error_model
    mean_u, intervals_u, interval_low_u, interval_high_u = self.fdagger(g)
    self.u = Uncertainties(error_model='informative')
    self.error_model = self.u.error_model 
    mean_i, intervals_i, interval_low_i, interval_high_i = self.fdagger(g)

    #plot the small-g expansions and error bands (panel a)
    ax1.plot(g, self.m.low_g(g)[0,:], 'r--', \
        label=r'$f_s$ ($N_s$ = {})'.format(self.loworder[0]))
    ax1.plot(g, interval_low_u[0, :, 0], 'r', linestyle='dotted',\
         label=r'$f_s$ ($N_s$ = {}) {}\% CI'.format(self.loworder[0], int(self.ci)))
    ax1.plot(g, interval_low_u[0, :, 1], 'r', linestyle='dotted')

    #plot the small-g expansions and error bands (panel b)
    ax2.plot(g, self.m.low_g(g)[0,:], 'r--', label=r'$f_s$ ($N_s$ = {})'.format(self.loworder[0]))
    ax2.plot(g, interval_low_i[0, :, 0], 'r', linestyle='dotted',\
            label=r'$f_s$ {}\% CI'.format(int(self.ci)))
    ax2.plot(g, interval_low_i[0, :, 1], 'r', linestyle='dotted')

    #plot the large-g expansions and error bands (panel a)
    ax1.plot(g, self.m.high_g(g)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.highorder[0]))
    ax1.plot(g, interval_high_u[0, :, 0], 'b', linestyle='dotted', label=r'$f_l$ {}\% CI'.format(int(self.ci)))
    ax1.plot(g, interval_high_u[0, :, 1], 'b', linestyle='dotted')

    #plot the large-g expansions and error bands (panel b)
    ax2.plot(g, self.m.high_g(g)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.highorder[0]))
    ax2.plot(g, interval_high_i[0, :, 0], 'b', linestyle='dotted', \
        label=r'$f_l$ {}\% CI'.format(int(self.ci)))
    ax2.plot(g, interval_high_i[0, :, 1], 'b', linestyle='dotted')

    #PPD (panel a)
    ax1.plot(g, mean_u, 'g', label='Mixed model')
    ax1.plot(g, intervals_u[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
    ax1.plot(g, intervals_u[:,1], 'g', linestyle='dotted')
    ax1.fill_between(g, intervals_u[:,0], intervals_u[:,1], color='green', alpha=0.2)

    #PPD (panel b)
    ax2.plot(g, mean_i, 'g', label='Mixed model')
    ax2.plot(g, intervals_i[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
    ax2.plot(g, intervals_i[:,1], 'g', linestyle='dotted')
    ax2.fill_between(g, intervals_i[:,0], intervals_i[:,1], color='green', alpha=0.2)

    #ax2.legend(bbox_to_anchor=(1.0, 0.5), fontsize=16, loc='center left')
    ax2.legend(fontsize=16, loc='upper right')

    #label panels
    ax1.text(0.94, 1.1, '(a)', fontsize=20)
    ax2.text(0.94, 1.1, '(b)', fontsize=20)

    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

plot_mix(g, plot_fdagger=True, plot_true=True)

An all-in-one plotting function that will plot the results of fdagger for N numbers of models, the next orders of the expansion models, and the validation step of the model mixing in fdagger to test fdagger results.

Example

Bivariate.plot_mix(g=np.linspace(1e-6, 0.5, 100), plot_fdagger=True)

Parameters:

Name Type Description Default
g linspace

The space over which the models are calculated.

 required
plot_fdagger bool

If True, this parameter will allow for the plotting of fdagger and its credibility interval.

True
plot_true bool

Determines whether or not to plot the true model curve. Default is True.

 True

Returns:

Name Type Description
mean ndarray

The mean of the mixed model at each point in g.
intervals ndarray

The values of the credibility intervals at each point in g.
Source code in samba/discrepancy.py 
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def plot_mix(self, g, plot_fdagger=True, plot_true=True):

    r'''
    An all-in-one plotting function that will plot the results of fdagger for N numbers
    of models, the next orders of the expansion models, and the validation step of the 
    model mixing in fdagger to test fdagger results.

    Example:
        Bivariate.plot_mix(g=np.linspace(1e-6, 0.5, 100), plot_fdagger=True)

    Parameters:
        g (numpy.linspace): The space over which the models are calculated.

        plot_fdagger (bool): If True, this parameter will allow for the 
            plotting of fdagger and its credibility interval. 

        plot_true (bool): Determines whether or not to plot the true model curve. 
            Default is True.

    Returns:
        mean (numpy.ndarray): The mean of the mixed model at each point in g.

        intervals (numpy.ndarray): The values of the credibility intervals at each
            point in g. 
    '''

    #set up plot configuration
    fig = plt.figure(figsize=(8,6), dpi=600)
    ax = plt.axes()
    ax.tick_params(axis='x', labelsize=18)
    ax.tick_params(axis='y', labelsize=18)
    ax.locator_params(nbins=8)
    ax.xaxis.set_minor_locator(AutoMinorLocator())
    ax.yaxis.set_minor_locator(AutoMinorLocator())

    #set up x and y limits
    ax.set_xlim(0.0,1.0)
   # ax.set_xlim(0.0,0.5)
    ax.set_ylim(1.2,3.2)
   # ax.set_ylim(1.0,3.0)
    ax.set_yticks([1.2, 1.6, 2.0, 2.4, 2.8, 3.2])
  #  ax.set_ylim(2.0,2.8)
   # ax.set_yticks([2.0, 2.2, 2.4, 2.6, 2.8])

    #labels and true model
    ax.set_xlabel('g', fontsize=22)
    ax.set_ylabel('F(g)', fontsize=22)

    if plot_true is True:
        ax.plot(g, self.m.true_model(g), 'k', label='True model')

    # call fdagger to calculate results
    mean, intervals, interval_low, interval_high = self.fdagger(g)

    # plot the small-g expansions and error bands
    for i,j in zip(range(len(self.loworder)), self.loworder):
        ax.plot(g, self.m.low_g(g)[i,:], 'r--', label=r'$f_s$ ($N_s$ = {})'.format(j))

    for i in range(len(self.loworder)):
        ax.plot(g, interval_low[i, :, 0], 'r', linestyle='dotted', \
            label=r'$f_s$ ($N_s$ = {}) {}\% CI'.format(self.loworder[i], int(self.ci)))
        ax.plot(g, interval_low[i, :, 1], 'r', linestyle='dotted')

    # for each large-g order, calculate and plot
    for i,j in zip(range(len(self.highorder)), self.highorder):
        ax.plot(g, self.high_g(g)[i,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(j))

    for i in range(len(self.highorder)):
        ax.plot(g, interval_high[i, :, 0], 'b', linestyle='dotted', \
            label=r'$f_l$ ($N_l$ = {}) {}\% CI'.format(self.highorder[i], int(self.ci)))
        ax.plot(g, interval_high[i, :, 1], 'b', linestyle='dotted')

    if plot_fdagger == True:
        ax.plot(g, mean, 'g', label='Mean')
        ax.plot(g, intervals[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax.plot(g, intervals[:,1], 'g', linestyle='dotted')
        ax.fill_between(g, intervals[:,0], intervals[:,1], color='green', alpha=0.2)

    ax.legend(fontsize=16, loc='upper right')
    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 mean, intervals

vertical_plot_fdagger(g1, g2, gp_dict=None, gp_dict2=None, gp_points=None, gp_points2=None, low_orders=None, high_orders=None)

Vertical panel plotter for the paper to generate two mixed model plots.

Example

Bivariate.vertical_plot_fdagger(g1=np.linspace(1e-6, 0.5, 100), g2=np.linspace(1e-6,1.0,100), gp_mean1=np.array([]), gp_mean2=np.array([]), gp_var1=np.array([,]), gp_var2=np.array([,]))

Parameters:

Name Type Description Default
g1 linspace

The space over which the models (and GP) were calculated for panel a.

required
g2 linspace

The space over which the models (and GP) were calculated for panel b.

required
gp_mean1 ndarray

GP mean results to be mixed with the models in panel a. Optional.

required
gp_mean2 ndarray

GP mean results to be mixed with the models in panel b. Optional.

 required
gp_var1 ndarray

GP variance results for panel a. Optional.

 required
gp_var2 ndarray

GP variance results for panel b. Optional.

required

Returns:

Type Description

None.
Source code in samba/discrepancy.py 
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def vertical_plot_fdagger(self, g1, g2, gp_dict=None, gp_dict2=None, gp_points=None, gp_points2=None, low_orders=None, high_orders=None):

    r'''
    Vertical panel plotter for the paper to generate two mixed model plots. 

    Example:
        Bivariate.vertical_plot_fdagger(g1=np.linspace(1e-6, 0.5, 100), g2=np.linspace(1e-6,1.0,100),
        gp_mean1=np.array([]), gp_mean2=np.array([]), gp_var1=np.array([,]), gp_var2=np.array([,]))

    Parameters:
        g1 (numpy.linspace): The space over which the models (and GP) were calculated 
            for panel a. 

        g2 (numpy.linspace): The space over which the models (and GP) were calculated 
            for panel b. 

        gp_mean1 (numpy.ndarray): GP mean results to be mixed with the models in panel a. 
            Optional. 

        gp_mean2 (numpy.ndarray): GP mean results to be mixed with the models in panel b. 
            Optional.

        gp_var1 (numpy.ndarray): GP variance results for panel a. Optional.

        gp_var2 (numpy.ndarray): GP variance results for panel b. Optional. 

    Returns:
        None.
    '''

    #set up plot configuration
    fig = plt.figure(figsize=(8,12), dpi=600)
    gs = fig.add_gridspec(2,1, hspace=0, wspace=0)
    (ax1, ax2) = gs.subplots(sharex=True, sharey=True)

    for ax in (ax1, ax2):

        ax.tick_params(axis='x', right=True, which='both', labelsize=18)
        ax.tick_params(axis='y', top=True, which='both', labelsize=18)
        ax.locator_params(nbins=5)
        ax.xaxis.set_minor_locator(AutoMinorLocator())
        ax.yaxis.set_minor_locator(AutoMinorLocator())
        ax.set_xlim(0.0,1.0)
        ax.set_ylim(1.2,3.2)
        ax.set_yticks([1.4, 1.8, 2.2, 2.6, 3.0])

        #labels and true model
        ax.set_xlabel('g', fontsize=22)
        ax.set_ylabel('F(g)', fontsize=22)
        ax.plot(g1, self.m.true_model(g1), 'k', label='True model')

        #only label outer plot axes
        ax.label_outer()

    #copy class variables to force changes
    if low_orders is None and high_orders is None:
        loworder = self.loworder.copy()
        highorder = self.highorder.copy()
    else:
        loworder = low_orders
        highorder = high_orders
        self.loworder = loworder
        self.highorder = highorder

   # else:
    #mixed results (panel a)
    self.loworder = np.array([loworder[0]])
    self.highorder = np.array([highorder[0]])
    self.m = Models(self.loworder, self.highorder)
    mean_1, intervals_1, interval_low_1, interval_high_1 = self.fdagger(g1)

    #mixed results (panel b)
    self.loworder = np.array([loworder[1]])
    self.highorder = np.array([highorder[1]])
    self.m = Models(self.loworder, self.highorder)
    mean_2, intervals_2, interval_low_2, interval_high_2 = self.fdagger(g2)

    #plot the small-g expansions and error bands (panel a)
    self.loworder = np.array([loworder[0]])
    self.m = Models(self.loworder, self.highorder)
    ax1.plot(g1, self.m.low_g(g1)[0,:], 'r--', \
        label=r'$f_s$ ($N_s$ = {})'.format(self.loworder[0]))
    ax1.plot(g1, interval_low_1[0, :, 0], 'r', linestyle='dotted',\
         label=r'$f_s$ {}\% CI'.format(int(self.ci)))
    ax1.plot(g1, interval_low_1[0, :, 1], 'r', linestyle='dotted')

    #plot the small-g expansions and error bands (panel b)
    self.loworder = np.array([loworder[1]])
    self.m = Models(self.loworder, self.highorder)
    ax2.plot(g2, self.m.low_g(g2)[0,:], color='r', linestyle='dashed',\
            label=r'$f_s$ ($N_s$ = {})'.format(self.loworder[0]))
    ax2.plot(g2, interval_low_2[0, :, 0], color='r', linestyle='dotted', \
        label=r'$f_s$ {}\% CI'.format(int(self.ci)))
    ax2.plot(g2, interval_low_2[0, :, 1], color='r', linestyle='dotted')

    #plot the large-g expansions and error bands (panel a)
    self.highorder = np.array([highorder[0]])
    self.m = Models(self.loworder, self.highorder)
    ax1.plot(g1, self.m.high_g(g1)[0,:], 'b--', label=r'$f_l$ ($N_l$ = {})'.format(self.highorder[0]))
    ax1.plot(g1, interval_high_1[0, :, 0], 'b', linestyle='dotted', label=r'$f_l$ {}\% CI'.format(int(self.ci)))
    ax1.plot(g1, interval_high_1[0, :, 1], 'b', linestyle='dotted')

    #plot the large-g expansions and error bands (panel b)
    self.highorder = np.array([highorder[1]])
    self.m = Models(self.loworder, self.highorder)
    ax2.plot(g2, self.m.high_g(g2)[0,:], color='b', linestyle='dashed', \
            label=r'$f_l$ ($N_l$ = {})'.format(self.highorder[0]))
    ax2.plot(g2, interval_high_2[0, :, 0], color='b', linestyle='dotted', \
        label=r'$f_l$ {}\% CI'.format(int(self.ci)))
    ax2.plot(g2, interval_high_2[0, :, 1], color='b', linestyle='dotted')

    # error bar option
    if gp_points is not None:
        ax1.errorbar(gp_points['gs'], gp_points['datas'], yerr=gp_points['sigmas'], color="red", fmt='o', markersize=4, \
            capsize=4, label=r"Training data", zorder=10)

    if gp_points2 is not None:
        ax2.errorbar(gp_points2['gs'], gp_points2['datas'], yerr=gp_points2['sigmas'], color="red", fmt='o', markersize=4, \
            capsize=4, label=r"Training data", zorder=10)

    #uninformative model case
    if gp_dict is None:
        ax1.plot(g1, mean_1, 'g', label='Mixed model')
        ax1.plot(g1, intervals_1[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax1.plot(g1, intervals_1[:,1], 'g', linestyle='dotted')
        ax1.fill_between(g1, intervals_1[:,0], intervals_1[:,1], color='green', alpha=0.2)
    else:
        ax1.plot(gp_dict['g'], gp_dict['mean'], 'g', label='Mixed model')
        ax1.plot(gp_dict['g'], gp_dict['mean']-gp_dict['std'], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax1.plot(gp_dict['g'], gp_dict['mean']+gp_dict['std'], 'g', linestyle='dotted')
        ax1.fill_between(gp_dict['g'], gp_dict['mean']-gp_dict['std'], gp_dict['mean']+gp_dict['std'], color='green', alpha=0.2)

    #informative model case
    if gp_dict2 is None:
        ax2.plot(g2, mean_2, 'g', label='Mixed model')
        ax2.plot(g2, intervals_2[:,0], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax2.plot(g2, intervals_2[:,1], 'g', linestyle='dotted')
        ax2.fill_between(g2, intervals_2[:,0], intervals_2[:,1], color='green', alpha=0.2)
    else:
        ax2.plot(gp_dict2['g'], gp_dict2['mean'], 'g', label='Mixed model')
        ax2.plot(gp_dict2['g'], gp_dict2['mean']-gp_dict2['std'], 'g', linestyle='dotted', label=r'{}$\%$ CI'.format(int(self.ci)))
        ax2.plot(gp_dict2['g'], gp_dict2['mean']+gp_dict2['std'], 'g', linestyle='dotted')
        ax2.fill_between(gp_dict2['g'], gp_dict2['mean']-gp_dict2['std'], gp_dict2['mean']+gp_dict2['std'], color='green', alpha=0.2)

    #add panel labels
    ax1.text(0.94, 1.3, '(a)', fontsize=18)
    ax2.text(0.94, 1.3, '(b)', fontsize=18)

    ax2.legend(fontsize=18, loc='upper right')
    ax1.legend(fontsize=18, loc='upper right')
    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