= L ∪̇ R 1 Function FairDecomposition(G, L, R) 2 t = 1 /* number of sets */ 3 4 T1 = L 5 λ1, µ1 = 0, 1 6 while µ1 − λ1 ≥ 1 |L|2 do 7 Construct the graph G′ = G (λ1+µ1 2 , . . . , λt+µt 2 ; T1, . . . , [...] the form S(1), for some S ⊆ A1. Using (2) we obtain |ΓH1 (S(1))| = M · |ΓG(S)| ≥ ∑ u∈S nu ≥ |S(1)|, because nu ≥ 1 for u ∈ A1 ⊇ S. By Hall’s Theorem, there is a matching X1 in H1 covering A (1) 1 . If A2 [...] dataset λmin λ25% λ50% λ75% per1 N0 AM 0.156 1 1 1 76.18 0.860 Vui 0.0208 0.5 1 1 69.16 0.743 Cti 0.01 0.5 1 1 65.42 0.670 B 0.0128 0.5 1 1 58.63 0.630 WC 5.52e-4 0.0149 0.0417 0.1 2.17 0.0366 M3 0.0298 0.0833 …

= L ∪̇ R 1 Function FairDecomposition(G, L, R) 2 t = 1 /* number of sets */ 3 4 T1 = L 5 λ1, µ1 = 0, 1 6 while µ1 − λ1 ≥ 1 |L|2 do 7 Construct the graph G′ = G (λ1+µ1 2 , . . . , λt+µt 2 ; T1, . . . , [...] the form S(1), for some S ⊆ A1. Using (2) we obtain |ΓH1 (S(1))| = M · |ΓG(S)| ≥ ∑ u∈S nu ≥ |S(1)|, because nu ≥ 1 for u ∈ A1 ⊇ S. By Hall’s Theorem, there is a matching X1 in H1 covering A (1) 1 . If A2 [...] dataset λmin λ25% λ50% λ75% per1 N0 AM 0.156 1 1 1 76.18 0.860 Vui 0.0208 0.5 1 1 69.16 0.743 Cti 0.01 0.5 1 1 65.42 0.670 B 0.0128 0.5 1 1 58.63 0.630 WC 5.52e-4 0.0149 0.0417 0.1 2.17 0.0366 M3 0.0298 0.0833 …

Aufgabe 13.1 (5 Punkte) Es sind acht Beispiele mit jeweils zwei Attributen gegeben. Die Beispiele und die entsprechen- den Attributausprägungen sind in Fig. 1 dargestellt. 0 1 2 3 4 5 6 0 1 2 3 2 1 Figure [...] (auch Zen- troiden genannt) zufällig gewählt. Benutzen Sie jedoch bitte hier die Beispiele (1, 1) und (2, 2) als Startzentroiden! Die Nummerierung neben den Beispielen soll für die Zentroiden beibehalten [...] Betrachten Sie die Gütefunktion zur Bewertung von Clustern, die aus der Vorlesung bekannt ist: W (C) = 1 2 K∑ k=1 ∑ C(i)=k ∑ C(i′)=k D(~xi − ~xi′) Zur Auswahl welcher Cluster tendiert eine Optimierung, die auf …

Maximum Entropy Markov Models (MEMM) – Lecture Notes

Bias Problem [ LaMcPe01 ] In \(y_1\) , \(y_1\rightarrow y_2\) is most likely In \(y_2\) , \(y_2\rightarrow y_2\) is most likely \(P(\textcolor[RGB]{132,184,24}{y_1,y_1,y_1,y_1}) = .4\cdot .45 \cdot .5 = 0 [...] max}_{y^{(*)} } P(y^{(1)},\ldots, y^{(T)} \vert x^{(1)},\ldots, x^{(T)} ) & = \operatorname {arg\, max}_{y^{(*)} } \tfrac {P(y^{(1)},\ldots, y^{(T)}, x^{(1)},\ldots, x^{(T)}) }{P(x^{(1)},\ldots, x^{(T)} )} [...] r[RGB]{0,155,170}{y_1,y_2,y_1,y_2}) = .6\cdot .2 \cdot .5 = 0.06\) \(P(\textcolor[RGB]{191,2,127}{y_1,y_1,y_2,y_2}) = .4\cdot .55 \cdot .3 = 0.066\) Most likely path: always \(y_1\) Average outgoing weight …

Maximum Entropy Markov Models (MEMM) – Lecture Notes

Bias Problem [ LaMcPe01 ] In \(y_1\) , \(y_1\rightarrow y_2\) is most likely In \(y_2\) , \(y_2\rightarrow y_2\) is most likely \(P(\textcolor[RGB]{132,184,24}{y_1,y_1,y_1,y_1}) = .4\cdot .45 \cdot .5 = 0 [...] max}_{y^{(*)} } P(y^{(1)},\ldots, y^{(T)} \vert x^{(1)},\ldots, x^{(T)} ) & = \operatorname {arg\, max}_{y^{(*)} } \tfrac {P(y^{(1)},\ldots, y^{(T)}, x^{(1)},\ldots, x^{(T)}) }{P(x^{(1)},\ldots, x^{(T)} )} [...] r[RGB]{0,155,170}{y_1,y_2,y_1,y_2}) = .6\cdot .2 \cdot .5 = 0.06\) \(P(\textcolor[RGB]{191,2,127}{y_1,y_1,y_2,y_2}) = .4\cdot .55 \cdot .3 = 0.066\) Most likely path: always \(y_1\) Average outgoing weight …

Decision Tree Splitting – Lecture Notes

desirable) Simple examples: \(C_1\) 0 \(C_2\) 6 \(\text{Gini}=0.0\) \(C_1\) 1 \(C_2\) 5 \(\text{Gini}\approx 0.278\) \(C_1\) 2 \(C_2\) 4 \(\text{Gini}\approx 0.444\) \(C_1\) 3 \(C_2\) 3 \(\text{Gini}=0.5\) [...] \[\text{information-gain}(T_1,\ldots ,T_m)=\text{entropy}(T) - \sum \nolimits _{i=1}^m\tfrac {|T_i|}{|T|} \cdot \text{entropy}(T_i)\] \(C_1\) 0 \(C_2\) 6 \(\text{entropy}=0\) \(C_1\) 1 \(C_2\) 5 \(\text{ent [...] \text{gain-ratio}(T_1,\ldots ,T_m) =& \frac{\text{information-gain}(T_1,\ldots ,T_m)}{\text{split-info}(T_1,\ldots ,T_m)} \\ \text{split-info}(T_1,\ldots ,T_m)=& -\sum \nolimits _{i=1}^m\frac{|T_i|}{|T|} …

Decision Tree Splitting – Lecture Notes

desirable) Simple examples: \(C_1\) 0 \(C_2\) 6 \(\text{Gini}=0.0\) \(C_1\) 1 \(C_2\) 5 \(\text{Gini}\approx 0.278\) \(C_1\) 2 \(C_2\) 4 \(\text{Gini}\approx 0.444\) \(C_1\) 3 \(C_2\) 3 \(\text{Gini}=0.5\) [...] \[\text{information-gain}(T_1,\ldots ,T_m)=\text{entropy}(T) - \sum \nolimits _{i=1}^m\tfrac {|T_i|}{|T|} \cdot \text{entropy}(T_i)\] \(C_1\) 0 \(C_2\) 6 \(\text{entropy}=0\) \(C_1\) 1 \(C_2\) 5 \(\text{ent [...] \text{gain-ratio}(T_1,\ldots ,T_m) =& \frac{\text{information-gain}(T_1,\ldots ,T_m)}{\text{split-info}(T_1,\ldots ,T_m)} \\ \text{split-info}(T_1,\ldots ,T_m)=& -\sum \nolimits _{i=1}^m\frac{|T_i|}{|T|} …