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custom_title = title + '  –  sparse data' +   ' - with lowess smoothing: loose (20% bins) and looser (60% bins)'

lowess_tight = lowess(sparse_data, sparse_input, frac = .20)
lowess_loose = lowess(sparse_data, sparse_input, frac = .60)
lowess_list = [(lowess_tight[:,0], lowess_tight[:,1]), (lowess_loose[:,0], lowess_loose[:,1]) ]

graph(sparse_input, sparse_data,  custom_title, scatter=True, solid_line=False, line_viz = lowess_list)

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# ground truth linear interpolation
title_suffix = 'smooth data' + ' - linear interpolation only'
graph(in_array, curve, title_suffix, scatter=False, solid_line=True)

# two LOWESS curves
custom_title = title + '  –  noisy data' + ' with lowess smoothing: tight (12% bins) and looser (20% bins)'
from statsmodels.nonparametric.smoothers_lowess import lowess
lowess_tight = lowess(noisy_data, in_array, frac = .12)
lowess_loose = lowess(noisy_data, in_array, frac = .2)
lowess_list = [(lowess_tight[:,0], lowess_tight[:,1]), (lowess_loose[:,0], lowess_loose[:,1]) ]

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custom_title = title + '  –  noisy data' + ' - with lowess curve (Seaborn lmplot)'

import seaborn as sns
import pandas as pd

plt.figure(figsize=figure_proportions)
plot_df = pd.DataFrame({'time': in_array, 'altitude': noisy_data})
g=sns.lmplot(x='time', y='altitude', data=plot_df, \
             lowess=True, scatter_kws={'color': 'green'}, line_kws={'color': 'green'},\
             height = figure_proportions[1]*.8, aspect = figure_proportions[0]/figure_proportions[1]);

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increments = [2, 3, 5, 9]
sparse_data_series = pd.Series(sparse_data)

mvg_avg_list = [ (sparse_input, sparse_data_series.rolling(inc).mean()) for inc in increments ]
title_suffix = ' - sparse data'+ ' - moving average'
graph(sparse_input, sparse_data, title_suffix, scatter=True, solid_line=True, line_viz = mvg_avg_list)

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sampling_increment = 9
sparse_input = in_array[::sampling_increment]
sparse_data = curve[::sampling_increment]

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title_suffix = ' -  sparse data'
graph(sparse_input, sparse_data, title_suffix, scatter=True, solid_line=False, line_viz = None)

z = np.polyfit(sparse_input, sparse_data, 3)
p = np.poly1d(z)
xp = np.linspace(x_min, x_max, 100)

title_suffix = ' - sparse data'+ ' - 3rd dgr polynomial regression'
graph(sparse_input, sparse_data,  title_suffix, scatter=True, solid_line=False, line_viz = [(xp, p(xp))])

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title_suffix = ' -  sparse data'
graph(sparse_input, sparse_data, title_suffix, scatter=True, solid_line=False, line_viz = None)

m, b = np.polyfit(sparse_input, sparse_data, 1)

title_suffix =  ' - sparse data'+ ' - with linear regression'
graph(sparse_input, sparse_data,  title_suffix, scatter=True, solid_line=False, line_viz = [(in_array, m*x+b)])

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title_suffix = ' -  sparse data'
graph(sparse_input, sparse_data, title_suffix, scatter=True, solid_line=False, line_viz = None)

title_suffix = ' -  sparse data'+ '  -  with linear interpolation'
graph(sparse_input, sparse_data,  title_suffix, scatter=True, solid_line=True, line_viz = None)

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sampling_increment = 15
sparse_input = in_array[::sampling_increment]
sparse_data = curve[::sampling_increment]

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title_suffix =  ' noisy data '+   ' -  with B-Spline'

x_interp = np.linspace(x_min, x_max, 100)
BSpline = scipy.interpolate.make_interp_spline(in_array, noisy_data, 2)
y_BSpline = BSpline(x_interp)

graph(x_interp, y_BSpline, title_suffix, scatter=True, solid_line=True, line_viz = None)