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Anomaly detection and change detection - sparse structure analysis -
1.
Anomaly Detec-on and Change Detec-on - Sparse Structure analysis - @_X37n March 3rd, 2019 in Kernel, Tokyo
2.
問題設定: 異異常箇所同定 (anomaly localiza-on) • 変数個別を⾒見見るだけでは検知できない異異常を、変数同⼠士の関係性から⾒見見たい • 例例: 市街の監視カメラ動画からの異異常検知 (個別に動く⼈人が、異異常事態で何らかの集団的⾏行行動を取るようになる。) Real-Time
Anomaly Detection and Localization in Crowded Scenes, Sabokrou et al.. 2015 CVPR
3.
問題設定: 異異常箇所同定 (anomaly localiza-on) • 正常時のデータを元にして、個々の変数関係を計算 新規データについて、変数の相関異異常度を計算すればいい。 1. 疎構造学習 2. 相関異異常度 (3. 密度⽐比推定)
4.
グラフの独⽴立性 • 最も単純な近似: 2変数が他の変数の条件付きで独⽴立であるとき、グラフでエッジがない (pairwise Markov graph; pairwise Markov random field) • Markov確率場とは graph G = (V, E) 上の確率変数族 (Xv) v in V に関して、次のMarkov性を定義 •
Global Markov であるとは、Vの互いに素な部分集合S, A, Bについて • Local Markov であるとは、任意のv in Vについて • Pairwise Markov であるとは、任意のu, v in V (uv not in E) について XA ⊥⊥ XB |XS Xv ⊥⊥ XVcl(v) |XN(v) N(v) = {u ∈ V|uv ∈ E} cl(v) = {v} ∪ N(v) Xv ⊥⊥ Xu |XV{v,u}
5.
Grow-Shrinkage Markov Network (GSNN) • Grow-Shrinkage Markov Network は、 1.) grow … 近傍 N(i) を増やす 2.) shrink … 近傍 N(i) を減らす 2段階の計算を⾏行行う。 • Local Markov 条件独⽴立として、近傍 N(i) を適切に求める。 •
Pairwise Markov 条件独⽴立の計算(|V|-2に関して指数的) よりも早い。 Efficient Markov Network Structure Discovery Using Independence Tests, Bloomberg et al., 2009
6.
Boltzmann 分布からのsparse正則化 • Boltzmann machine (2値のpairwise model) を考え、L1正則化を加えて最適化 • Boltzmann machine 確率分布関数 対数尤度関数 •
Boltzmann machine の勾配計算は⼤大変なため、辺のないグラフから⽬目的関数を ⼤大きくする辺を追加する⼿手法が提案されている。 (Su-In Lee et al. NIPS 2006: Efficient Structure Learning of Markov Networks using L1) p(x) = 1 Z exp ∑ i<j Jijxixj + ∑ i hixi xi ∈ {1, − 1} 1 N l(J, h) = ∑ i<j JijED[xixj] + ∑ i hiED[xi] − logZ(J, h) l(J, h) + λ∥J∥1
7.
Gaussian Markov 確率場からのsparse 正則化 • Gaussian Markov 確率場を考え、L1正則化を加えて最適化 (Graphical Lasso, 2008, Fredman) • Gaussian Markov 確率場 確率分布関数 •
内点法で解くと、O(|V|6) の計算時間で解ける • もうちょっと賢い⽅方法もあるらしい。 (O. Banerjee et al., Model selec-on through sparse maximum likelihood es-ma-on Mul-variate Gaussian or Binary Data, NIPS 2008) (O. Banerjee et al., Convex op-miza-on techniques for fieng sparse gaussian graphical models, NIPS 2006) p(x) = detΛ (2π)n exp ( − 1 2 (x − μ)T Λ(x − μ) ) logdetΛ−tr( ̂ΣΛ) − λ∥Λ∥1
8.
Gaussian Markov 確率場の性質 • (i, j) - pairwise Markov 独⽴立であることは、Λij = 0 と等価である。 proof. • 次で定義される偏相関係数 r が 0 でないとき、 xi と xj に直接相関があると呼ぶ •
直接相関が 0 であり、共分散⾏行行列列∑ の (i,j)成分が0であるとき、 間接相関があると呼ぶ。 Xi ⊥⊥ Xj |XV{i,j} ⇔ Λij = 0 p(xi, xj |xV{i,j}) ∝ exp ( 1 2 (Λiix2 i + Λijxixj + Λjjxjxj) + Axi + Bxj + C ) ri,j ≡ − Λi,j Λi,iΛj,j
9.
Graphical Gaussian Model (GGM)によるグラフ定義 • 6 変数の場合の例例
10.
伝統的⼿手法 • 素朴な⽅方法: 共分散の逆⾏行行列列計算して、閾値以下を0としてしまう • 確率モデルでなくなる。 •
精度⾏行行列列は正定値でなくなる • 直接相関と間接相関を分ける性質がなくなる • 閾値の設定が実⽤用上簡単でない • Λが疎な解となるような事前分布を付し、MAP推定する >> Graphical Lasso 2008 Fredman • 伝統的⽅方法: 共分散構造選択 (Dempster 1972) • 以下の繰り返し • (1) ⼩小さい⾏行行列列要素をひとつ 0 にする • (2) その拘束上で確率モデルを推定
11.
ブロック座標降下法によるGraphical Lasso計算 • ⽬目的関数 これは次と等価 (既出) • f の勾配 •
座標降下法: Λ のほかを既知として、⼀一つの列列・⾏行行について解く。 —> 疎な精度⾏行行列列を、逆⾏行行列列計算なしに求められる。 とし、L, Wを既知、 の制約条件のもとで、 fの勾配 = 0 を計算する。 Λ* = argmaxΛ{ln p(Λ)ΠN n=1N(x(n) |0,Λ)} Λ* = argmaxΛ f(Λ; Σ, ρ) p(Λ) = ρ 2 exp(−ρ∥Λ∥1) f(Λ; Σ, ρ) ≡ ln|Λ|−tr(ΣΛ) − ρ∥Λ∥1 ∂f ∂Λ = Λ−1 − Σ − ρsign(Λ) ˜⇤˜⇤ 1 = I ˜⇤ = ✓ L l lT ◆ ˜⇤ 1 = ✓ W w wT ◆ ˜⌃ = ✓ R s sT si,i ◆
12.
• (fの勾配) = 0 • 拘束条件 •
を導⼊入する。 により β が求まり、λ, l, w, σ も以下から計算できる。 • (a) のβ計算は、次の最適化問題と等しい。 実務的には、Lasso の solver でこちらを解く。 ブロック座標降下法によるGraphical Lasso計算 Wl + λw = 0 wl + σλ = 1 W − R − ρsign(L) = 0 w − s − ρsign(l) = 0 σ − si,i − ρsign(λ) = 0 ˜⇤ 1 ˜⇤ = ✓ WL + wlT Wl + w lT W + wT wT l + ◆ β ≡ W−1 w l = − λW−1 w = − λβWβ − s − ρsign(β) = 0 (a) λ = 1 σ − βTWβ l = β σ − βTWβ w = Wβ σ = si,i + ρ min 1 2 ∥W 1 2 β − b∥2 + ρ∥β∥1
13.
外れ値解析 • 異異常値を次のように定義 Graphical Lasso で求めた精度⾏行行列列Λから計算できる。 • 意味: 各他の変数から期待される予想値に対して、どれだけ期待から外れてい るかを求めることになる。 αi
≡ − ln(x′i |x′−i, D) α(x′) = 1 2 ln 2π Λi,i + 1 2Λi,i ∑ Λi,jx′j
14.
異異常解析 • 異異常値を次のように定義 Graphical Lasso で求めた精度⾏行行列列Λから計算できる。 • 意味: 正常データ D に対して、新たに得たデータD’ がどのように異異常度を持っ ているかを計測する。 αi
≡ ∫ dx−i p(x−i |D) ∫ dxi p(xi |x−i, D)ln p(x′i |x′−i, D) p(x′i |x′−i, D′) α(x′) = 1 2 ln Λi,i Λ′i,i − 1 2 ( [ΛSΛ]i,i Λi,i − [Λ′SΛ′]i,i Λ′i,i )
15.
参考資料料 • 潜在的グラフ構造からの異異常検知 (井⼿手剛) hop://latent-dynamics.net/01/2010_LD_Ide.pdf • グラフィカルモデル (機械学習プロフェッショナルシリーズ) (渡辺 有祐) •
異異常検知と変化検知 (機械学習プロフェッショナルシリーズ) (井⼿手剛, 杉⼭山将)
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