本日は
my Bokeh シリーズが充実してきました．
- Bokeh をつかって行列演算を可視化する．
- （続き）Bokeh をつかって行列演算を可視化する(with animation)
- Bokeh対話機能を用いた画像処理結果の可視化
- Bokehを用いた2x2の行列演算の可視化
- Bokehを用いた回転行列演算可視化（手動でスライダー）
これらの続きとして
グラフ描画ツールBokehを用いて２x２行列
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
の固有ベクトルを可視化してみました．固有ベクトルの計算はNumpyのAPIを用いて求めます．
実装例
eigen.py
import numpy as np
from bokeh.plotting import figure
from bokeh.models import ColumnDataSource
from bokeh.models.widgets import Slider, Div
from bokeh.layouts import column, widgetbox, gridplot
from bokeh.io import curdoc
def matrix(a, b, c, d):
mat = np.array([[a, b],
[c, d]])
return mat
class EigenViewer(object):
def __init__(self):
xs = np.linspace(-np.pi, np.pi, 11)
ys = xs
Xs, Ys = np.meshgrid(xs, ys)
self.Xs, self.Ys = Xs.flatten(), Ys.flatten()
a, b = 1, 0
c, d = 0, 1
mat = matrix(a, b, c, d)
transXs, transYs = mat @ np.array([self.Xs, self.Ys])
TOOLS = "pan,lasso_select,save,reset"
self.source = ColumnDataSource(data=dict(Xs=self.Xs, Ys=self.Ys,
transXs=transXs, transYs=transYs))
self.evectors = ColumnDataSource(data=dict(ev0x=[0, 0], ev0y=[0, 1],
ev1x=[0, 1], ev1y=[0, 0],
transev0x=[0, 0], transev0y=[0, 1],
transev1x=[0, 1], transev1y=[0, 0]))
self.fig = figure(tools=TOOLS, title="target",
x_range=(-np.pi*np.sqrt(2)-1, np.pi*np.sqrt(2)+1),
y_range=(-np.pi*np.sqrt(2)-1, np.pi*np.sqrt(2)+1))
self.fig.scatter('Xs', 'Ys', source=self.source)
self.fig.line('ev0x', 'ev0y', source=self.evectors,
line_width=5, line_alpha=0.5, color='red')
self.fig.line('ev1x', 'ev1y', source=self.evectors,
line_width=5, line_alpha=0.5, color='blue')
self.transfig = figure(tools=TOOLS, title="transformed",
x_range=self.fig.x_range, y_range=self.fig.y_range)
self.transfig.circle('transXs', 'transYs', source=self.source)
self.transfig.line('transev0x', 'transev0y', source=self.evectors,
line_width=5, line_alpha=0.5, color='red')
self.transfig.line('transev1x', 'transev1y', source=self.evectors,
line_width=5, line_alpha=0.5, color='blue')
self.a_slider = Slider(title="a", value=a,
start=-10, end=10, step=0.1)
self.b_slider = Slider(title="b", value=b,
start=-10, end=10, step=0.1)
self.c_slider = Slider(title="c", value=c,
start=-10, end=10, step=0.1)
self.d_slider = Slider(title="d", value=d,
start=-10, end=10, step=0.1)
self.eigen0 = Div(text="eigen0")
self.eigen1 = Div(text="eigen1")
for widget in [self.a_slider, self.b_slider, self.c_slider, self.d_slider]:
widget.on_change('value', self.update_data)
box = widgetbox([self.a_slider, self.b_slider,
self.c_slider, self.d_slider])
self.plot = column(gridplot([[self.a_slider, self.b_slider, self.eigen0],
[self.c_slider, self.d_slider, self.eigen1]]),
gridplot([[self.fig, self.transfig]]))
def update_data(self, attr, old, new):
mat = matrix(self.a_slider.value, self.b_slider.value,
self.c_slider.value, self.d_slider.value)
evals, evecs = np.linalg.eig(mat)
ev0, ev1 = evals
evec0 = evecs.T[0]
evec1 = evecs.T[1]
trans0 = mat @ evec0
trans1 = mat @ evec1
self.eigen0.text = "eigen0 = " + \
str(ev0) if np.isreal(ev0) else "eigen0 = None"
self.eigen1.text = "eigen1 = " + \
str(ev1) if np.isreal(ev1) else "eigen1 = None"
transXs, transYs = mat @ np.array([self.Xs, self.Ys])
self.source.data = dict(Xs=self.Xs, Ys=self.Ys,
transXs=transXs, transYs=transYs)
data0x = [0, evec0[0]] if np.isreal(ev0) else [0, 0]
data0y = [0, evec0[1]] if np.isreal(ev0) else [0, 0]
data1x = [0, evec1[0]] if np.isreal(ev1) else [0, 0]
data1y = [0, evec1[1]] if np.isreal(ev1) else [0, 0]
trans0x = [0, trans0[0]] if np.isreal(ev0) else [0, 0]
trans0y = [0, trans0[1]] if np.isreal(ev0) else [0, 0]
trans1x = [0, trans1[0]] if np.isreal(ev1) else [0, 0]
trans1y = [0, trans1[1]] if np.isreal(ev1) else [0, 0]
self.evectors.data = dict(ev0x=data0x, ev0y=data0y,
ev1x=data1x, ev1y=data1y,
transev0x=trans0x, transev0y=trans0y,
transev1x=trans1x, transev1y=trans1y)
def main():
viewer = EigenViewer()
document = curdoc()
document.add_root(viewer.plot)
main()
実行例
$ bokeh serve --show eigen.py
赤と青のベクトルがノルムが１の固有ベクトルを表しています．右側のグラフは格子点と固有ベクトルの写り先（変換先）を表しています．
