ex-7: replace Fisher projection plot

It was really tricky reproduce in matplotlib but worth it.

ex-7/plots/fisher.py

@@ -0,0 +1,117 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.transforms as trans
+import io
+
+
+def line(ax, x, y, **args):
+    '''line between two points x,y'''
+    ax.plot([x[0], y[0]], [x[1], y[1]], **args)
+
+
+def angle(v):
+    '''angle of a vector wrt x axis'''
+    return np.arctan2(v[1], v[0])
+
+
+def proj_hist(a, b, w):
+    '''returns a file-like object of the w-projection histogram'''
+    fig = plt.figure(figsize=(3, 3))
+    ax = fig.add_subplot(111)
+    ax.set_aspect(0.04)
+    ax.axis('off')
+    plt.hist(a @ w, color='xkcd:dark yellow', zorder=2)
+    plt.hist(b @ w, color='xkcd:pale purple', zorder=1)
+    buf = io.BytesIO()
+    fig.savefig(buf, transparent=True)
+    plt.close(fig)
+    buf.seek(0)
+    return plt.imread(buf)
+
+
+def place_im(ax, im, transform):
+    '''place an image into an Axes by applying a transformation'''
+    im = ax.imshow(im, interpolation='none',
+                   extent=[-2, 4, -3, 2],
+                   zorder=2)
+    trans_data = transform + ax.transData
+    im.set_transform(trans_data)
+
+
+def main():
+    # rotation by π/2 matrix
+    rot = np.array([[0, -1], [1, 0]])
+
+    # covariance
+    cov = np.array([[0.33, 0.15], [0.15, 0.12]])
+
+    # means
+    m1 = np.array([-1.2,  0.5])
+    m2 = np.array([0.6,  0])
+
+    np.random.seed(3)
+    a = np.random.multivariate_normal(m1, cov, 100)
+    b = np.random.multivariate_normal(m2, cov, 100)
+
+    # w₁: naive projection onto m1-m2 plane
+    w1 = (m1 - m2) / np.linalg.norm(m1 - m2)
+
+    # w₂: Fisher projection
+    w2 = np.linalg.inv(cov) @ (m1 - m2)
+    w2 /= np.linalg.norm(w2)
+
+    #plt.rcParams['font.size'] = 8
+    fig, axes = plt.subplots(
+        1, 2, figsize=(6, 9),
+        subplot_kw=dict(aspect='equal'))
+
+    # naive projection
+    ax = axes[0]
+    ax.set_title('naïve projection', loc='right')
+    ax.set_xlim(-5.0, 3.5)
+    ax.set_ylim(-3.5, 2.0)
+
+    ax.scatter(*a.T, edgecolor='xkcd:slate grey', c='xkcd:dark yellow')
+    ax.scatter(*b.T, edgecolor='xkcd:slate grey', c='xkcd:pale purple')
+
+    dir = rot @ w1
+    line(ax, m1, m2, c='xkcd:charcoal')
+    line(ax, (m1 + m2)/2, (m1 + m2)/2 + 2.2*dir, c='xkcd:charcoal')
+    line(ax, dir*2.1 - 2.0*w1, dir*2.1 + 2.6*w1, c='xkcd:charcoal')
+    place_im(
+        ax,
+        proj_hist(a, b, w1),
+        trans.Affine2D()
+          .rotate(angle(w1))
+          .translate(*dir*3.113)
+          .translate(*w1*(-0.7)))
+
+    # fisher projection
+    ax = axes[1]
+    ax.set_title('Fisher projection', loc='right')
+    ax.set_xlim(-5.0, 3.5)
+    ax.set_ylim(-3.5, 2.0)
+
+    ax.scatter(*a.T, edgecolor='xkcd:slate grey', c='xkcd:dark yellow')
+    ax.scatter(*b.T, edgecolor='xkcd:slate grey', c='xkcd:pale purple')
+
+    dir = rot @ w2
+    line(ax, m1, m2, c='xkcd:charcoal')
+    line(ax, (m1 + m2)/2, (m1 + m2)/2 + 2.75*dir, c='xkcd:charcoal')
+    line(ax, dir*2.9 - 1.2*w2, dir*2.9 + 2.3*w2, c='xkcd:charcoal')
+    place_im(
+        ax,
+        proj_hist(a, b, w2),
+        trans.Affine2D()
+          .rotate(angle(w2))
+          .scale(0.75)
+          .translate(*dir*3.90)
+          .translate(*w2*(-0.3)))
+
+    plt.tight_layout()
+    #plt.savefig('notes/images/7-fisher.pdf', bbox_inches='tight')
+    plt.show()
+
+
+if __name__ == '__main__':
+    main()

notes/sections/7.md

@@ -104,7 +104,7 @@ histograms resulting from the  projection onto the line joining the
 class means: note the considerable overlap in the projected
 space. The right plot shows the corresponding projection based on the
 Fisher linear discriminant, showing the greatly improved classes
-separation. Figure taken from [@bishop06]](images/7-fisher.png){#fig:overlap}
+separation.](images/7-fisher.pdf){#fig:overlap}
 
 The overlap of the projections can be reduced by maximising a function that
 gives, besides a large separation, small variance within each class. The