はじめに
SVMは、オーバーフィッティングを避けて効率よく識別関数を求めることができる 手法です。
「集合知」の9章で紹介されているSVMをSageを使ってまとめてみます。
簡単な例題
いきなりSVMに進む前にクラス分けを簡単な例を使って解いてみます。
下図は、赤（-1）のグループと青（1）のグループの分布です。
# 線形クラス分類の例
# データの用意
c1 = [[1,2],[1,4],[2,4]]
c2 = [[2,1],[5,1],[4,2]]
# プロットして分布を確認
pl1 = list_plot(c1, rgbcolor='red')
pl2 = list_plot(c2, rgbcolor ='blue')
(pl1+pl2).show(xmin=0, xmax=5, ymin=0, figsize=5)
|
|
線形分類
青点の値に1、赤点の値に-1をセットし、$w_1 x_1 + w_2 x_2 + b$の線形モデルを find_fit関数を使って解いて、$w_1, w2, b$の値を求めます。
find_fitのデータは、$x_1, x_2, 値$の順にセットします。
# 各クラスに判別値をセット
v1 = [-1, -1, -1]
v2 = [1, 1, 1]
# (x, y, 判別値)のリストを作成
data = [flatten((pt, v)) for (pt, v) in zip(c1 + c2, v1 + v2)]
data
[[1, 2, -1], [1, 4, -1], [2, 4, -1], [2, 1, 1], [5, 1, 1], [4, 2, 1]] [[1, 2, -1], [1, 4, -1], [2, 4, -1], [2, 1, 1], [5, 1, 1], [4, 2, 1]] |
# 最もフィットするw1, w2, bを求める
var('x1 x2 w1 w2 b')
model(x1, x2) = w1*x1 + w2*x2 + b
fit = find_fit(data, model, solution_dict=True); print fit
# 求まった解（判別式）を返す関数を定義します
f(x1, x2) = model.subs(fit);
{b: 0.1962962962945436, w2: -0.4333333333364595, w1:
0.32592592592445524}
{b: 0.1962962962945436, w2: -0.4333333333364595, w1: 0.32592592592445524}
|
結果の表示
implicit_plot関数を使ってデータ（赤、青）と判別式が0となる線を表示します。 上手く赤と青の点を分離しているのが分かります。
# 判別式が0の線を表示
pl6 = implicit_plot(f(x1, x2) == 0, (x1, 0, 5), (x2, 0, 5))
(pl1 + pl2 +pl6).show(xmin=0, xmax=5, ymin=0, figsize=5)
|
|
求まった判別式がどのような形なのかplot3dを使って表示してみます。 データ（赤、青）の判別式の値も合わせて表示してみます。
一つの平面上として表されています。
pl3 = plot3d(f(x1, x2), (x1, 0, 5), (x2, 0, 5))
pl4 = list_plot([(x, y, f(x, y)) for (x, y) in c1], rgbcolor='red')
pl5 = list_plot([(x, y, f(x, y)) for (x, y) in c2], rgbcolor='blue')
(pl3+pl4+pl5).show(xmin=0, xmax=5)
|
Sleeping...
 |
SVMを使った分類
SVM（サポートベクターマシン）は、クラスの境界線と分離平面（超平面） の距離（マージン）を最大になるようにクラスを分類します。
先ほどの例では、下図のように境界線（半線）の中間に$y = x$の分離平面が、 あります。（線形分類の境界線とずれていることに注意して下さい）
境界線を求めるとき使った学習データのことを「サポートベクター」と呼びます。
# SVMでは、クラスの境界線と分離平面（超平面）の距離を最大になるように求めます
sv = plot(lambda x : x, (x, 0, 5))
cl1 = plot(lambda x : x + 1, (x, 0, 5), linestyle='dashed')
cl2 = plot(lambda x : x - 1, (x, 0, 5), linestyle='dashed')
(sv+cl1+cl2+pl1+pl2).show(xmin=0, xmax=5, figsize=5)
# 青の実線が分離平面で、半線がクラスの境界線です
|
|
LIBSVMのインストール
sageを使ってSVMを計算したいところですが、sageにはSVMを計算する関数がありません。 そこで、LIBSVMをインストールします。
LIBSVMの ホームページ のDownload LIBSVMから最新のtar.gzファイルをダウンロードします。
ダウンロードしたファイル(libsvm-2.9.tar.gz)を適当な場所(~/local)で解凍します。
＄ tar xzvf libsvm-2.9.tar.gz ＄ cd libsvm-2.9/python
MacOSXの場合、Makefileの一部を変更します。
#LDFLAGS = -shared # Mac OS LDFLAGS = -framework Python -bundle
また、そのままではsageで動かなかったので、svm.pyの以下の2点にfloatのキャストを 追加するように修正してください。
126c126 < svmc.svm_node_array_set(data,j,k,x[k]) --- > svmc.svm_node_array_set(data,j,k,float(x[k])) 138c138 < svmc.double_setitem(y_array,i,y[i]) --- > svmc.double_setitem(y_array,i,float(y[i]))
sage内部のpythonにLIBSVMをインストールするには、sageのpythonで seup.pyを実行します。私は、sageを~/localにインストールしているので
＄ ~/local/sage/local/bin/python setup.py installと実行しました。
正しく動くか、集合知の9.9.2の例題で確認します。 無事、同じ結果がでたので、次に進みます。
# svm.pyを2カ所修正した
# 動作を確認するために、集合知の9.9.2の例題を実行する
from svm import *
prob = svm_problem([1,-1],[[1,0,1],[-1,0,-1]])
param = svm_parameter(kernel_type = LINEAR, C=float(10))
m = svm_model(prob, param)
* optimization finished, #iter = 1 nu = 0.025000 obj = -0.250000, rho = 0.000000 nSV = 2, nBSV = 0 Total nSV = 2 * optimization finished, #iter = 1 nu = 0.025000 obj = -0.250000, rho = 0.000000 nSV = 2, nBSV = 0 Total nSV = 2 |
m.predict([1,1,1])
1.0 1.0 |
LIBSVMで簡単な例題を解く
LIBSVMへの入力は、各データどのクラスに属するかを示すclsと データのベクトル値datを引数に取ります。
# LIBSVM用にデータを加工する
cls = v1 + v2; print cls
dat = c1 + c2; print dat
[-1, -1, -1, 1, 1, 1] [[1, 2], [1, 4], [2, 4], [2, 1], [5, 1], [4, 2]] [-1, -1, -1, 1, 1, 1] [[1, 2], [1, 4], [2, 4], [2, 1], [5, 1], [4, 2]] |
# 例題にSVMを適応
prob = svm_problem(cls, dat)
m = svm_model(prob, param)
* optimization finished, #iter = 4 nu = 0.033333 obj = -1.000000, rho = 0.000000 nSV = 2, nBSV = 0 Total nSV = 2 * optimization finished, #iter = 4 nu = 0.033333 obj = -1.000000, rho = 0.000000 nSV = 2, nBSV = 0 Total nSV = 2 |
データが正しく分類されているか赤(1,4)の座標を入力すると、-1.0が返され、 正しい結果が返ってきます。
m.predict([1, 4])
-1.0 -1.0 |
結果の表示
contour_plotを使って、SVMの予想をコンタマップで表示すると、$y = x$ の線でうまく、分類されているのが、分かります。
contour_plot(lambda x, y : m.predict([x, y]), (x, 0, 5), (y, 0, 5), figsize=5)
|
|
さらに、各点（赤、青）の予想値と境界線の形状をlist_plot3で表示します。
# plot3dがうまく表示できないので、list_plot3dで代替
pl7 = list_plot3d([(x, y, m.predict([x, y])) for x in srange(0, 5, 0.1) for y in srange(0, 5, 0.1)])
pl8 = list_plot([(x, y, m.predict([x, y])) for (x, y) in c1], rgbcolor='red')
pl9 = list_plot([(x, y, m.predict([x, y])) for (x, y) in c2], rgbcolor='blue')
(pl8 + pl9 +pl7).show(xmin=0, xmax=5, ymin=0)
|
Sleeping...
 |
SVMのカーネルメソッドのすごさを確かめるために点の値がチェスボードのように 格子状に部分布する点を分析することにします。
まずは、チェスボードのマスの値を返すchessBox関数を作成します。 ただしく、値がセットされるか、conour_plotでみてみましょう。
# チェスボックスの例
def chessBox(x, y):
if ((int(x)+int(y))%2) == 0:
return 1
else:
return -1
#[chessBox(x, y) for x in range(0,5) for y in range(0,5)]
contour_plot(chessBox, (x, 0, 5), (y, 0, 5), figsize=5)
|
|
テストデータの作成
0から5の範囲にランダムに点を生成し、chessBox関数で赤（red）と青（blue） に振り分けます。
# ランダムな点を生成
rndPts = [[5*random(), 5*random()] for i in range(0,1000)];
|
|
Kernel関数LINEARの場合
kernel_type=LINEAR（線形を意味する）を使って分類すると、 うまく格子状のデータを分類することができません。
red = [pt for pt in rndPts if chessBox(pt[0], pt[1]) == -1];
blue = [pt for pt in rndPts if chessBox(pt[0], pt[1]) == 1];
redCls = [-1 for pt in red]
blueCls = [1 for pt in blue]
chessRedPlt = list_plot([(x, y) for (x, y) in red], rgbcolor='red')
chessBluePlt = list_plot([(x, y) for (x, y) in blue], rgbcolor='blue')
(chessRedPlt+chessBluePlt).show(xmin=0, xmax=5, figsize=5)
|
|
# カーネルメソッドをLINEARだとうまく分けられない
prob = svm_problem(redCls + blueCls, red + blue)
param = svm_parameter(kernel_type = LINEAR, C=float(1000))
m = svm_model(prob, param)
contour_plot(lambda x, y : m.predict([x, y]), (x, 0, 5), (y, 0, 5), figsize=5)
........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ .........................................................*..............\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ..........*.............................................................\ ........................................................................\ ........................................................................\ ...............................................................*........\ ........................................................................\ ..................*.....................................................\ ...................................*....................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ...................*....................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ .......*................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ .................................................. Warning: using -h 0 may be faster *.......................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ .............................*..........................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ .........*..............................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ ........................................................................\ .....................* optimization finished, #iter = 11144286 nu = 0.966000 obj = -966297.432944, rho = 0.978544 nSV = 967, nBSV = 962 Total nSV = 967 Traceback (click to the left of this block for traceback) ... ValueError: zero-size array to ufunc.reduce without identity .................................................................................................................................................................................................................................................................................................................................................................................................................................*................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................*............................................................................................................................................................................................................................................................................*..................................................................................................*........................................................................................*.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................*...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................*..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Warning: using -h 0 may be faster
*....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................*...........................................................................................................................................................................................................................................................................................................................................................................................................................*...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................*
optimization finished, #iter = 11144286
nu = 0.966000
obj = -966297.432944, rho = 0.978544
nSV = 967, nBSV = 962
Total nSV = 967
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_19.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("IyDjgqvjg7zjg43jg6vjg6Hjgr3jg4Pjg4njgpJMSU5FQVLjgaDjgajjgYbjgb7jgY/liIbjgZHjgonjgozjgarjgYQKcHJvYiA9IHN2bV9wcm9ibGVtKHJlZENscyArIGJsdWVDbHMsIHJlZCArIGJsdWUpCnBhcmFtID0gc3ZtX3BhcmFtZXRlcihrZXJuZWxfdHlwZSA9IExJTkVBUiwgQz1mbG9hdCgxMDAwKSkKbSA9IHN2bV9tb2RlbChwcm9iLCBwYXJhbSkKY29udG91cl9wbG90KGxhbWJkYSB4LCB5IDogbS5wcmVkaWN0KFt4LCB5XSksICh4LCAwLCA1KSwgKHksIDAsIDUpLCBmaWdzaXplPTUp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpA_PPXV/___code___.py", line 6, in <module>
exec compile(u'contour_plot(lambda x, y : m.predict([x, y]), (x, _sage_const_0 , _sage_const_5 ), (y, _sage_const_0 , _sage_const_5 ), figsize=_sage_const_5 )
File "", line 1, in <module>
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/misc/displayhook.py", line 176, in displayhook
print_obj(sys.stdout, obj)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/misc/displayhook.py", line 144, in print_obj
print >>out_stream, `obj`
File "sage_object.pyx", line 154, in sage.structure.sage_object.SageObject.__repr__ (sage/structure/sage_object.c:1496)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/plot/graphics.py", line 803, in _repr_
self.show()
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 456, in wrapper
return func(*args, **kwds)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/plot/graphics.py", line 1474, in show
self.save(**kwds)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 456, in wrapper
return func(*args, **kwds)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/plot/graphics.py", line 2199, in save
figure = self.matplotlib(**options)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/plot/graphics.py", line 1703, in matplotlib
g._render_on_subplot(subplot)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/sage/plot/contour_plot.py", line 184, in _render_on_subplot
linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/matplotlib/axes.py", line 7316, in contour
return mcontour.QuadContourSet(self, *args, **kwargs)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/matplotlib/contour.py", line 1106, in __init__
ContourSet.__init__(self, ax, *args, **kwargs)
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/matplotlib/contour.py", line 701, in __init__
self._process_levels()
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/matplotlib/contour.py", line 912, in _process_levels
self.vmin = np.amin(self.levels) # alternative would be self.layers
File "/usr/local/sage-4.7.1/local/lib/python2.7/site-packages/numpy/core/fromnumeric.py", line 1862, in amin
return amin(axis, out)
ValueError: zero-size array to ufunc.reduce without identity
|
Kernel関数RBFの場合
次に、kernel_type=RBF（ガウシアンカーネル関数）を使って分類すると、 なんとなく格子状に分類しているように見えます。
ガウシアンカーネル関数は、 $$ K(x, x') = e ^{\left( - \frac{| x - x' |^2}{\sigma^2} \right)} $$
# カーネルメソッドをRBFにするとある程度格子の形が見て取れる
prob = svm_problem(redCls + blueCls, red + blue)
param = svm_parameter(kernel_type = RBF, C=float(1000))
m = svm_model(prob, param)
contour_plot(lambda x, y : m.predict([x, y]), (x, 0, 5), (y, 0, 5), figsize=5)
...........................................*............................\ .......................*................................................\ ..* optimization finished, #iter = 143927 nu = 0.217787 obj = -170283.042235, rho = 6.634814 nSV = 244, nBSV = 194 Total nSV = 244 ...........................................*...................................................*..................................................* optimization finished, #iter = 143927 nu = 0.217787 obj = -170283.042235, rho = 6.634814 nSV = 244, nBSV = 194 Total nSV = 244 |
簡単な画像認識
最後に簡単な画像認識をLIBSVMを使って確かめてみます。
5x5のマス目に数字の0から4までを書いた画像5個（本当に少ないです！） をテストデータとして使用します。
# 文字認識
m0 = [[0,1,1,1,0], [1,0,0,0,1], [1,0,0,0,1], [1,0,0,0,1], [0,1,1,1,0]]
m1 = [[0,0,1,0,0], [0,0,1,0,0], [0,0,1,0,0], [0,0,1,0,0], [0,0,1,0,0]]
m2 = [[0,1,1,1,1], [1,0,0,1,0], [0,0,1,0,0], [0,1,0,0,0], [1,1,1,1,1]]
m3 = [[0,1,1,1,0], [1,0,0,0,1], [0,0,1,1,0], [1,0,0,0,1], [0,1,1,1,0]]
m4 = [[0,0,1,0,0], [0,1,0,0,0], [1,0,0,1,0], [1,1,1,1,1], [0,0,0,1,0]]
p0 = flatten(m0); p1 = flatten(m1); p2 = flatten(m2); p3 = flatten(m3)
p4 = flatten(m4)
v = [0,1,2,3,4]
|
|
作成したデータをcontour_plotを使って表示して、確認します。 なんとなくそれらしく見えるでしょう。（笑）
# 画像作成用の関数
def mesh(x, y, tbl):
idX = int(x)
idY = int(4.9-y)
return tbl[idY][idX]
# 学習用画像を表示
trn0 = contour_plot(lambda x, y : mesh(x, y, m0), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
trn1 = contour_plot(lambda x, y : mesh(x, y, m1), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
trn2 = contour_plot(lambda x, y : mesh(x, y, m2), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
trn3 = contour_plot(lambda x, y : mesh(x, y, m3), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
trn4 = contour_plot(lambda x, y : mesh(x, y, m4), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
html.table([[trn0, trn1, trn2], [trn3, trn4]])
|
|
画像認識
いよいよ画像認識の準備が整いました。 えいや〜で、モデルを作成します。
# 画像認識
param = svm_parameter(kernel_type = RBF, C=float(10))
prob = svm_problem(v,[p0,p1,p2,p3,p4])
m = svm_model(prob, param)
* optimization finished, #iter = 1 nu = 0.246622 obj = -2.466216, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.280928 obj = -2.809276, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.676333 obj = -6.763327, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.262318 obj = -2.623181, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.262318 obj = -2.623181, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.280928 obj = -2.809276, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.280928 obj = -2.809276, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.330771 obj = -3.307713, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.202682 obj = -2.026823, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.262318 obj = -2.623181, rho = 0.000000 nSV = 2, nBSV = 0 Total nSV = 5 * optimization finished, #iter = 1 nu = 0.246622 obj = -2.466216, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.280928 obj = -2.809276, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.676333 obj = -6.763327, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.262318 obj = -2.623181, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.262318 obj = -2.623181, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.280928 obj = -2.809276, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.280928 obj = -2.809276, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.330771 obj = -3.307713, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.202682 obj = -2.026823, rho = 0.000000 nSV = 2, nBSV = 0 * optimization finished, #iter = 1 nu = 0.262318 obj = -2.623181, rho = 0.000000 nSV = 2, nBSV = 0 Total nSV = 5 |
テスト用の画像
テストに使う画像は、以下の3個です。 判別結果は、ただしく0, 1, 1となっています。（めでたし、めでたし）
# テスト用の画像を作成する
m0_1 = [[0,1,1,1,0], [0,1,0,0,1], [0,1,0,0,1], [0,1,0,0,1], [0,1,1,1,0]]
m1_1 = [[0,0,1,0,0], [0,1,1,0,0], [0,0,1,0,0], [0,0,1,0,0], [0,1,1,1,0]]
m1_2 = [[0,0,0,1,0], [0,0,0,1,0], [0,0,1,0,0], [0,0,1,0,0], [0,0,1,0,0]]
t0_1 = flatten(m0_1)
t1_1 = flatten(m1_1)
t1_2 = flatten(m1_2)
|
|
# テスト用画像の表示
cnt1 = contour_plot(lambda x, y : mesh(x, y, m0_1), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
cnt2 = contour_plot(lambda x, y : mesh(x, y, m1_1), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
cnt3 = contour_plot(lambda x, y : mesh(x, y, m1_2), (x, 0, 4.9), (y, 0, 4.9), aspect_ratio=1, figsize=(2, 1.23))
html.table([[cnt1, cnt2, cnt3]])
|
|
# テスト用画像の認識
print m.predict(t0_1)
print m.predict(t1_1)
print m.predict(t1_1)
0.0 1.0 1.0 0.0 1.0 1.0 |
念のため、学習に使ったデータが正しく識別されるかもみてみました。
こんなに少ないデータでよく認識できるものだと感心しました。これがSVMのすごさなのでしょうか？
# 学習用画像の認識
print m.predict(p0)
print m.predict(p1)
print m.predict(p2)
print m.predict(p3)
print m.predict(p4)
0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 |
|
|