add: model approximation visualization

Approximation uses simple linear regression to determine whether
the _location information_ is significant enough (through the
calculated steepness `beta`) which can be used to determine in a
faster and more efficient way whether the calculation of the
model is necessary and helpful in the first place.

model_approximation.py

@@ -0,0 +1,105 @@
+import math
+
+import matplotlib.pyplot as plt
+import numpy as np
+from graph_tool.all import *
+
+from src import centrality
+from src import plot
+from src import fitting
+
+
+def leverage(g, weight):
+    # VertexPropertyMap
+    vp = g.new_vertex_property("double")
+    for v in g.vertices():
+        li = 0.0
+        neighbours = g.get_all_neighbours(v)
+        ki = len(neighbours)
+        # sum
+        for nv in neighbours:
+            other_neighbours = g.get_all_neighbours(nv)
+            kj = len(other_neighbours)
+            li += (ki - kj) / (ki + kj)
+        li /= ki
+        vp[v] = li
+    return vp
+
+
+def random_graph(n=5000, seed=None):
+    """
+    Uniformly random point cloud generation.
+    `n` [int] Number of points to generate. Default 5000 seems like a good starting point in point density and corresponding runtime for the subsequent calculations.
+    @return [numpy.ndarray] Array of shape(n, 2) containing the coordinates for each point of the generated point cloud.
+    """
+    if seed is None:
+        import secrets
+        seed = secrets.randbits(128)
+    rng = np.random.default_rng(seed=seed)
+    return rng.random((n, 2)), seed
+
+
+def spatial_graph(adata):
+    g, pos = graph_tool.generation.triangulation(adata, type="delaunay")
+    g.vp["pos"] = pos
+    weight = g.new_edge_property("double")
+    for e in g.edges():
+        weight[e] = math.sqrt(sum(map(abs, pos[e.source()].a - pos[e.target()].a)))**2
+    return g, weight
+
+
+points, seed = random_graph()
+g, weight = spatial_graph(points)
+g = GraphView(g)
+
+# calculate centrality values
+vp, ep = betweenness(g, weight=weight)
+vp.a = np.nan_to_num(vp.a) # correct floating point values
+ep.a = np.nan_to_num(ep.a) # correct floating point values
+
+# normalization
+min_val, max_val = vp.a.min(), vp.a.max()
+vp.a = (vp.a - min_val) / (max_val - min_val)
+
+# calculate convex hull
+convex_hull = centrality.convex_hull(g)
+
+# plot graph with convex_hull
+fig = plt.figure(figsize=(15, 5))
+ax0, ax1 = fig.subplots(1, 2)
+plot.graph_plot(fig, ax0, g, vp, convex_hull, f"Random Graph (seed: {seed})")
+
+# generate model based on convex hull and associated centrality values
+quantification = plot.quantification_data(g, vp, convex_hull)
+
+# optimize model's piece-wise linear function
+d = quantification[:, 0]
+C = quantification[:, 1]
+m_opt, c0_opt, b_opt, aic_opt = fitting.fit_piece_wise_linear(d, C)
+
+# TODO
+# should this be part of the plotting function itself, it should not be necessary for me to do this
+d_curve = np.linspace(min(d), max(d), 500)
+C_curve = np.piecewise(
+    d_curve,
+    [d_curve <= b_opt, d_curve > b_opt],
+    [lambda x: m_opt * x + c0_opt, lambda x: m_opt * b_opt + c0_opt]
+)
+# plot model containing modeled piece-wise linear function
+plot.quantification_plot(ax1, quantification, d_curve, C_curve, 'Betweenness', aic_opt)
+
+# linear regression model
+m_reg, c0_reg, aic_reg = fitting.fit_linear_regression(d, C)
+
+# TODO
+# should this be part of the plotting function itself, it should not be necessary for me to do this
+d_curve = np.linspace(min(d), max(d), 500)
+C_curve = np.piecewise(
+    d_curve,
+    [d_curve >= 0],
+    [lambda x: m_reg * x + c0_reg]
+)
+ax1.plot(d_curve, C_curve, color='k', linewidth=1, label=f"Simple Linear Regression | AIC: {aic_reg}")
+ax1.legend()
+
+fig.savefig(f"model_approximation_betweenness_5000.svg", format='svg')