| \form#0:\[ S (X) = \frac{Total X}{Total transactions} \] |
| \form#1:$ X,Y $ |
| \form#2:$ X $ |
| \form#3:$ Y $ |
| \form#4:$ P(Y|X) $ |
| \form#5:\[ C (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X)} \] |
| \form#6:\[ L (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X) \cdot s(Y)} \] |
| \form#7:$ \neg Y $ |
| \form#8:$ X \Rightarrow Y $ |
| \form#9:\[ Conv (X \Rightarrow Y) = \frac{1 - S(Y)}{1 - C(X \Rightarrow Y)} \] |
| \form#10:$ n $ |
| \form#11:$ n - 1 $ |
| \form#12:$ n \ge 2 $ |
| \form#13:$ A $ |
| \form#14:$ B $ |
| \form#15:$ B \Rightarrow (A - B) $ |
| \form#16:$ min(support) = .25 $ |
| \form#17:$ min(confidence) = .5 $ |
| \form#18:$ a_1, \dots, a_n $ |
| \form#19:$ c $ |
| \form#20:\[ P(C=c_i \mid A) \approx P(C=c_j \mid A) \] |
| \form#21:$ P(A_i = a \mid C=c) $ |
| \form#22:\[ P(A_i=a \mid C=c) = \frac{1}{\sqrt{2\pi\sigma^{2}_c}}exp\left(-\frac{(a-\mu_c)^{2}}{2\sigma^{2}_c}\right) \] |
| \form#23:$\mu_c$ |
| \form#24:$\sigma^{2}_c$ |
| \form#25:$c$ |
| \form#26:\[ \Pr(C = c \mid A_1 = a_1, \dots, A_n = a_n) = \frac{\Pr(C = c) \cdot \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c)} {\Pr(A_1 = a_1, \dots, A_n = a_n)} \,, \] |
| \form#27:\[ \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c) = \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \,. \] |
| \form#28:\[ \text{classify}(a_1, ..., a_n) = \arg\max_c \left\{ \Pr(C = c) \cdot \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \right\} \] |
| \form#29:$ P(A_i = a \mid C = c) $ |
| \form#30:\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a)}{\#c} \] |
| \form#31:$ \#(c,i,a) $ |
| \form#32:$ i $ |
| \form#33:$ a $ |
| \form#34:$ \#c $ |
| \form#35:\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a) + s}{\#c + s \cdot \#i} \] |
| \form#36:$ \#i $ |
| \form#37:$ s \geq 0 $ |
| \form#38:$ s = 1 $ |
| \form#39:$ s = 0 $ |
| \form#40:$ \boldsymbol Ax = \boldsymbol b $ |
| \form#41:$A$ |
| \form#42:$x$ |
| \form#43:$ \boldsymbol b $ |
| \form#44:\[ \boldsymbol Ax = \boldsymbol b \] |
| \form#45:$ \boldsymbol A $ |
| \form#46:\[ \|\boldsymbol A - \boldsymbol UV^{T} \|_2 \] |
| \form#47:$rank(\boldsymbol UV^{T}) \leq r$ |
| \form#48:$\|\cdot\|_2$ |
| \form#49:$m \times n$ |
| \form#50:$U$ |
| \form#51:$m \times r$ |
| \form#52:$V$ |
| \form#53:$n \times r$ |
| \form#54:$1 \leq r \ll \min(m, n)$ |
| \form#55:$ \Pr( \text{best label sequence} \mid \text{sequence}) $ |
| \form#56:\[ p_\lambda(\boldsymbol y | \boldsymbol x) = \frac{\exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y)}}{Z_\lambda(\boldsymbol x)} \,. \] |
| \form#57:$ F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) $ |
| \form#58:$ \boldsymbol x $ |
| \form#59:$ f_m(y_i,y_{i-1},x_i) $ |
| \form#60:$ y_i $ |
| \form#61:$ y_{i-1} $ |
| \form#62:$ x_i $ |
| \form#63:$ \lambda_m $ |
| \form#64:$ Z_\lambda(\boldsymbol x) $ |
| \form#65:\[ Z_\lambda(\boldsymbol x) = \sum_{\boldsymbol y'} \exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y')} \] |
| \form#66:$ T=\{(x_k,y_k)\}_{k=1}^N $ |
| \form#67:\[ \ell_{\lambda}=\sum_k \log p_\lambda(y_k|x_k) =\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] \] |
| \form#68:\[ \nabla \ell_{\lambda}=\sum_k[F(x_k,y_k)-E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] \] |
| \form#69:$E_{p_\lambda(Y|x)}[F(x,Y)]$ |
| \form#70:\[ E_{p_\lambda(Y|x)}[F(x,Y)] = \sum_y p_\lambda(y|x)F(x,y) = \sum_i\frac{\alpha_{i-1}(f_i*M_i)\beta_i^T}{Z_\lambda(x)} \] |
| \form#71:\[ Z_\lambda(x) = \alpha_n.1^T \] |
| \form#72:$\alpha_i$ |
| \form#73:$ \beta_i$ |
| \form#74:\[ \alpha_i = \begin{cases} \alpha_{i-1}M_i, & 0<i<=n\\ 1, & i=0 \end{cases}\\ \] |
| \form#75:\[ \beta_i^T = \begin{cases} M_{i+1}\beta_{i+1}^T, & 1<=i<n\\ 1, & i=n \end{cases} \] |
| \form#76:\[ \ell_{\lambda}^\prime=\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] - \frac{\lVert \lambda \rVert^2}{2\sigma ^2} \] |
| \form#77:\[ \nabla \ell_{\lambda}^\prime=\sum_k[F(x_k,y_k) - E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] - \frac{\lambda}{\sigma ^2} \] |
| \form#78:$ \boldsymbol c $ |
| \form#79:$ l(\boldsymbol c) $ |
| \form#80:$ n^2 $ |
| \form#81:\[\min_{w \in R^N} L(w) + \lambda \left(\frac{(1-\alpha)}{2} \|w\|_2^2 + \alpha \|w\|_1 \right)\] |
| \form#82:$L$ |
| \form#83:$ \alpha \in [0,1] $ |
| \form#84:$ lambda \geq 0 $ |
| \form#85:$alpha = 0$ |
| \form#86:$\alpha = 1$ |
| \form#87:\[L(\vec{w}) = \frac{1}{2}\left[\frac{1}{M} \sum_{m=1}^M (w^{t} x_m + w_{0} - y_m)^2 \right] \] |
| \form#88:\[ L(\vec{w}) = \sum_{m=1}^M\left[y_m \log\left(1 + e^{-(w_0 + \vec{w}\cdot\vec{x}_m)}\right) + (1-y_m) \log\left(1 + e^{w_0 + \vec{w}\cdot\vec{x}_m}\right)\right]\ , \] |
| \form#89:$y_m \in {0,1}$ |
| \form#90:\[ x' \leftarrow \frac{x - \bar{x}}{\sigma_x} \] |
| \form#91:\[y' \leftarrow y - \bar{y} \] |
| \form#92:$ l(\boldsymbol \beta) $ |
| \form#93:$ Y \in \{ 0,1,2 \ldots k \} $ |
| \form#94:\[ E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x) \] |
| \form#95:$ \sigma(x) = \frac{1}{1 + \exp(-x)} $ |
| \form#96:$ \boldsymbol y \in \{ 0,1 \}^{n \times k} $ |
| \form#97:$ k $ |
| \form#98:$ X \in \mathbf R^{n \times k} $ |
| \form#99:$ \boldsymbol x_i $ |
| \form#100:\[ P[Y = y_i | \boldsymbol x_i] = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \] |
| \form#101:$ \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) $ |
| \form#102:$ \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) $ |
| \form#103:\[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \] |
| \form#104:$ H = -X^T A X $ |
| \form#105:$ A = \text{diag}(a_1, \dots, a_n) $ |
| \form#106:$ a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. $ |
| \form#107:$ H $ |
| \form#108:\[ \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii} \,. \] |
| \form#109:\[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \] |
| \form#110:$ p $ |
| \form#111:$ c_i = 0 $ |
| \form#112:$ F $ |
| \form#113:\[ p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| )) \] |
| \form#114:$ Z $ |
| \form#115:$ \exp(c_i) $ |
| \form#116:$ \kappa(X^T A X) $ |
| \form#117:$ K $ |
| \form#118:$ (1, ..., K) $ |
| \form#119:$ J $ |
| \form#120:$ (0, ..., J-1) $ |
| \form#121:$ {m_{k,j}} $ |
| \form#122:$ j $ |
| \form#123:$ {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1}, \ldots m_{k_2, j_{J-1}} \ldots m_{k_K, j_{J-1}}} $ |
| \form#124:$ Y_i $ |
| \form#125:$ j = 1,.. , J$ |
| \form#126:$\pi$ |
| \form#127:$\pi_{ij}$ |
| \form#128:$i$ |
| \form#129:$j$ |
| \form#130:\[ \gamma_{ij} = \Pr(Y_i \le j)= \pi_{i1} +...+ \pi_{ij} . \] |
| \form#131:$ \mbox{logit}(\pi) = \log[\pi/(1-\pi)] $ |
| \form#132:\[ \mbox{logit}(\gamma_{ij})=\mbox{logit}(\Pr(Y_i \le j))=\log \frac{\Pr(Y_i \le j)}{1-\Pr(Y_i\le j)}, j=1,...,J−1 \] |
| \form#133:\[ \mbox{logit}(\gamma_{ij}) = \theta_j - x^T_i \beta \] |
| \form#134:$x_i$ |
| \form#135:$\beta$ |
| \form#136:$\{\theta_j\}$ |
| \form#137:$x^T_i\beta$ |
| \form#138:$ x_1, \dots, x_n \in \mathbb R^d $ |
| \form#139:$ c_1, \dots, c_k \in \mathbb R^d $ |
| \form#140:\[ (c_1, \dots, c_k) \mapsto \sum_{i=1}^n \min_{j=1}^k \operatorname{dist}(x_i, c_j) \] |
| \form#141:$ \operatorname{dist} $ |
| \form#142:$ \alpha $ |
| \form#143:$ \beta $ |
| \form#144:$ \phi_i $ |
| \form#145:$ \theta $ |
| \form#146:$\alpha$ |
| \form#147:$ z_n $ |
| \form#148:$ w_n $ |
| \form#149:$ \phi_{z_n} $ |
| \form#150:$\|\vec{a}\|_1$ |
| \form#151:$\|\vec{a}\|_2$ |
| \form#152:$\|\vec{a} - \vec{b}\|_1$ |
| \form#153:$\|\vec{a} - \vec{b}\|_2$ |
| \form#154:$\|\vec{a} - \vec{b}\|_p, p > 0$ |
| \form#155:$\|\vec{a} - \vec{b}\|_\infty$ |
| \form#156:$\|\vec{a} - \vec{b}\|_2^2$ |
| \form#157:$\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2}$ |
| \form#158:$\cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2})$ |
| \form#159:$ \vec x = (x_1, \dots, x_n) $ |
| \form#160:$ \| x \|_1 = \sum_{i=1}^n |x_i| $ |
| \form#161:$ \| x \|_2 = \sqrt{\sum_{i=1}^n x_i^2} $ |
| \form#162:$ \vec y = (y_1, \dots, y_n) $ |
| \form#163:$ \| x - y \|_\infty = \max_{i=1}^n \|x_i - y_i\| $ |
| \form#164:$ p > 0 $ |
| \form#165:$ \| x - y \|_p = (\sum_{i=1}^n \|x_i - y_i\|^p)^{\frac{1}{p}} $ |
| \form#166:$ \| x - y \|_1 = \sum_{i=1}^n |x_i - y_i| $ |
| \form#167:$ \| x - y \|_2 = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} $ |
| \form#168:$ \frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|} $ |
| \form#169:$ \| x - y \|_2^2 = \sum_{i=1}^n (x_i - y_i)^2 $ |
| \form#170:$ \arccos\left(\frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|}\right) $ |
| \form#171:$ 1 - \frac{\langle \vec x, \vec y \rangle} {\| \vec x \|^2 \cdot \| \vec y \|^2 - \langle \vec x, \vec y \rangle} $ |
| \form#172:$ \vec x = (x_1, \dots, x_m) $ |
| \form#173:$ 1 - \frac{|x \cap y|}{|x \cup y|} $ |
| \form#174:$ M $ |
| \form#175:$ \vec x $ |
| \form#176:$ M = (\vec{m_0} \dots \vec{m_{l-1}}) \in \mathbb{R}^{k \times l} $ |
| \form#177:$ \vec x \in \mathbb R^k $ |
| \form#178:$ x $ |
| \form#179:$ \arg\min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) $ |
| \form#180:$ \min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) $ |
| \form#181:$ \operatorname{dist}(\vec{m_j}, \vec x) $ |
| \form#182:$ j = $ |
| \form#183:$ x_1, \dots, x_n $ |
| \form#184:$ \frac 1n \sum_{i=1}^n x_i $ |
| \form#185:$ \widetilde{x} := \frac 1n \sum_{i=1}^n \frac{x_i}{\| x_i \|} $ |
| \form#186:$ \frac{\widetilde{x}}{\| \widetilde{x} \|} $ |
| \form#187:$ \vec x_1, \dots, \vec x_n \in \mathbb R^m $ |
| \form#188:$ ( \vec x_1 \dots \vec x_n ) \in \mathbb R^{m \times n}$ |
| \form#189:$ PA = LDL* $ |
| \form#190:$ A = QR $ |
| \form#191:$ PAQ = LU $ |
| \form#192:$mxn$ |
| \form#193:$m \ge n$ |
| \form#194:\[ A = U \Sigma V^T, \] |
| \form#195:$\Sigma$ |
| \form#196:$n \times n$ |
| \form#197:$ \sqrt{mean((X - USV^T)_{ij}^2)} $ |
| \form#198:$ \sqrt{mean(X_{ij}^2)} $ |
| \form#199:$\sigma_i, u_i, v_i$ |
| \form#200:$A^TA$ |
| \form#201:$AA^T$ |
| \form#202:\[ H(A) = \begin{bmatrix} 0 & A\\ A^* & 0 \end{bmatrix} \] |
| \form#203:\[ A = P B Q^T, \] |
| \form#204:$P$ |
| \form#205:$Q$ |
| \form#206:$B$ |
| \form#207:$B*B$ |
| \form#208:$A*A$ |
| \form#209:\[ B = X\Sigma Y^T, \] |
| \form#210:$U = PX$ |
| \form#211:$V = QY$ |
| \form#212:\[ Ax = b \] |
| \form#213:$x \in \mathbb{R}^{n}$ |
| \form#214:$A \in \mathbb{R}^{m \times n} $ |
| \form#215:$b \in \mathbb{R}^{m}$ |
| \form#216:$ \frac{|Ax - b|}{|b|} $ |
| \form#217:$ 0 \ldots n-1 $ |
| \form#218:$b$ |
| \form#219:$ N $ |
| \form#220:$ \boldsymbol X $ |
| \form#221:$ \hat{x} $ |
| \form#222:$ \boldsymbol{X}$ |
| \form#223:$ \hat{\boldsymbol X} $ |
| \form#224:\[ \hat{\boldsymbol X} = {\boldsymbol X} - \vec{e} \hat{x}^T \] |
| \form#225:$ \vec{e} $ |
| \form#226:\[ \hat{\boldsymbol X} = {\boldsymbol U}{\boldsymbol \Sigma}{\boldsymbol V}^T \] |
| \form#227:$ {\boldsymbol \Sigma} $ |
| \form#228:$ {\boldsymbol \Sigma}/(\sqrt{(N-1)} $ |
| \form#229:$ {\boldsymbol V} $ |
| \form#230:$ \boldsymbol P $ |
| \form#231:$ {\boldsymbol X}' $ |
| \form#232:\begin{align*} {\boldsymbol {\hat{X}}} & = {\boldsymbol X} - \vec{e} \hat{x}^T \\ {\boldsymbol X}' & = {\boldsymbol {\hat {X}}} {\boldsymbol P}. \end{align*} |
| \form#233:$\hat{x} $ |
| \form#234:$ \boldsymbol R $ |
| \form#235:\[ {\boldsymbol R} = {\boldsymbol {\hat{X}}} - {\boldsymbol X}' {\boldsymbol P}^T. \] |
| \form#236:$ r $ |
| \form#237:\[ r = \|{\boldsymbol R}\|_F \] |
| \form#238:$ \|\cdot\|_F $ |
| \form#239:$ r' $ |
| \form#240:\[ r' = \frac{ \|{\boldsymbol R}\|_F }{\|{\boldsymbol X}\|_F } \] |
| \form#241:$ p \in [0,1] $ |
| \form#242:$ F(x) = p $ |
| \form#243:$ \sup \{ x \in D \mid F(x) \leq p \} $ |
| \form#244:$ p < 0.5 $ |
| \form#245:$ \inf \{ x \in D \mid F(x) \geq p \} $ |
| \form#246:$ p \geq 0.5 $ |
| \form#247:$ D $ |
| \form#248:$ \mathbb R $ |
| \form#249:$ \mathbb N_0 $ |
| \form#250:$ x \in \mathbb N_0 $ |
| \form#251:$ F(x) < p < F(x + 1) $ |
| \form#252:$ x + 1 $ |
| \form#253:$ p < F(0) $ |
| \form#254:$p$ |
| \form#255:$ 1 - p $ |
| \form#256:$ > x $ |
| \form#257:$ \leq x $ |
| \form#258:$ \Pr[X \leq x] $ |
| \form#259:$ \mathit{sp} $ |
| \form#260:$ \mathit{sp} \in [0,1] $ |
| \form#261:$ f(x) $ |
| \form#262:$ f $ |
| \form#263:$ p \leq 1 - \mathit{sp} $ |
| \form#264:$ \alpha > 0 $ |
| \form#265:$ \beta > 0 $ |
| \form#266:$ p = \Pr[X \leq x] $ |
| \form#267:$ n \in \mathbb N_0 $ |
| \form#268:$ p \geq \Pr[X \leq x] $ |
| \form#269:$ p \leq \Pr[X \leq x] $ |
| \form#270:$ x_0 $ |
| \form#271:$ \gamma > 0 $ |
| \form#272:$ \gamma $ |
| \form#273:$ \nu > 0 $ |
| \form#274:$ \nu $ |
| \form#275:$ \mu > 0 $ |
| \form#276:$ \lambda > 0 $ |
| \form#277:$ \lambda $ |
| \form#278:$ \nu_1 > 0 $ |
| \form#279:$ \nu_1 $ |
| \form#280:$ \nu_2 $ |
| \form#281:$ k > 0 $ |
| \form#282:$ \theta > 0 $ |
| \form#283:$ r \in \{ 0, 1, \dots, N \} $ |
| \form#284:$ n \in \{ 0, 1, \dots, N \} $ |
| \form#285:$ N \in \mathbb N $ |
| \form#286:$ r, n, N $ |
| \form#287:$ \mu $ |
| \form#288:$ b > 0 $ |
| \form#289:$ 2 b^2 $ |
| \form#290:$ s > 0 $ |
| \form#291:$ s $ |
| \form#292:$ m $ |
| \form#293:$ r > 0 $ |
| \form#294:$ x + r $ |
| \form#295:$ \mathit{sp} \in (0,1] $ |
| \form#296:$ r, \mathit{sp} $ |
| \form#297:$ \delta \geq 0 $ |
| \form#298:$ shape_1 $ |
| \form#299:$ shape_2 $ |
| \form#300:$ \delta $ |
| \form#301:$ \lambda \geq 0 $ |
| \form#302:$ \nu_1, \nu_2, \lambda $ |
| \form#303:$ \sigma > 0 $ |
| \form#304:$ T $ |
| \form#305:$ \sigma^2 $ |
| \form#306:$ \sigma $ |
| \form#307:$ c \geq a $ |
| \form#308:$ b \geq c $ |
| \form#309:$ b > a $ |
| \form#310:$ a, b, c $ |
| \form#311:$ b $ |
| \form#312:$ [a, b] $ |
| \form#313:$ \in (0,1) $ |
| \form#314:$m$ |
| \form#315:\[ S(\vec{c}) = B(\vec{c}) M(\vec{c}) B(\vec{c}) \] |
| \form#316:\begin{eqnarray} B(\vec{c}) & = & \left(-\sum_{i=1}^{n} H(y_i, \vec{x}_i, \vec{c})\right)^{-1}\\ & = & \left(-\sum_{i=1}^{n}\frac{\partial^2 l(y_i, \vec{x}_i, \vec{c})}{\partial c_\alpha \partial c_\beta}\right)^{-1} \end{eqnarray} |
| \form#317:$H$ |
| \form#318:\[ L(\vec{c}) = \sum_{i=1}^n l(y_i, \vec{x}_i, \vec{c})\ . \] |
| \form#319:\[ M(\vec{c}) = \bf{A}^T\bf{A} \] |
| \form#320:$\bf{A}$ |
| \form#321:\[ A_m = \sum_{i\in G_m}\frac{\partial l(y_i,\vec{x}_i,\vec{c})}{\partial \vec{c}} \] |
| \form#322:$G_m$ |
| \form#323:$ {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1} \ldots m_{k_K, j_{J-1}}} $ |
| \form#324:$ y $ |
| \form#325:$X^{*}X$ |
| \form#326:\[ E[Y \mid \boldsymbol x] = \boldsymbol c^T \boldsymbol x \] |
| \form#327:\[ f(y \mid \boldsymbol x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot \exp\left(-\frac{1}{2 \sigma^2} \cdot (y - \boldsymbol x^T \boldsymbol c)^2 \right) \,. \] |
| \form#328:$ \boldsymbol y \in \mathbf R^n $ |
| \form#329:$ X^T $ |
| \form#330:$ X^+ $ |
| \form#331:$ \sum_{i=1}^n \log f(y_i \mid \boldsymbol x_i) $ |
| \form#332:$ RSS $ |
| \form#333:\[ RSS = \sum_{i=1}^n ( y_i - \boldsymbol c^T \boldsymbol x_i )^2 = (\boldsymbol y - X \boldsymbol c)^T (\boldsymbol y - X \boldsymbol c) \,. \] |
| \form#334:\[ \boldsymbol c = (X^T X)^+ X^T \boldsymbol y \,. \] |
| \form#335:$ TSS $ |
| \form#336:$ ESS $ |
| \form#337:$ R^2 $ |
| \form#338:\begin{align*} ESS & = \boldsymbol y^T X \boldsymbol c - \frac{ \| y \|_1^2 }{n} \\ TSS & = \sum_{i=1}^n y_i^2 - \frac{ \| y \|_1^2 }{n} \\ R^2 & = \frac{ESS}{TSS} \end{align*} |
| \form#339:$ R^2 = 1 - \frac{RSS}{TSS} $ |
| \form#340:$ TSS = RSS + ESS $ |
| \form#341:$ Var[Y - \boldsymbol c^T \boldsymbol x \mid \boldsymbol x] $ |
| \form#342:\[ \sigma^2 = \frac{RSS}{n - k} \] |
| \form#343:\[ t_i = \frac{c_i}{\sqrt{\sigma^2 \cdot \left( (X^T X)^{-1} \right)_{ii} }} \,. \] |
| \form#344:$ F_\nu $ |
| \form#345:\[ p_i = \Pr(|T| \geq |t_i|) = 2 \cdot (1 - F_{n - k}( |t_i| )) \] |
| \form#346:$ \kappa(X) = \|X\|_2\cdot\|X^{-1}\|_2$ |
| \form#347:$X$ |
| \form#348:\[ \|X\|_2 = \sqrt{\lambda_{\max}\left(X^{*}X\right)}\ , \] |
| \form#349:$X^{*}$ |
| \form#350:$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) $ |
| \form#351:$ \boldsymbol t $ |
| \form#352:$ \boldsymbol p $ |
| \form#353:$ X^T X $ |
| \form#354:$ Y \in \{ 0,1 \} $ |
| \form#355:$ \boldsymbol y \in \{ 0,1 \}^n $ |
| \form#356:\[ P[Y = y_i | \boldsymbol x_i] = \sigma((-1)^{(1 - y_i)} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \] |
| \form#357:\[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{(1 - y_i)} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \] |
| \form#358:$ \boldsymbol z $ |
| \form#359:$ \mathit{odds}(c_1), \dots, \mathit{odds}(c_k) $ |
| \form#360:$ X^T A X $ |
| \form#361:$ \frac{1}{1 + \exp(-x)} $ |
| \form#362:$ \exp(x) $ |
| \form#363:$ 2^{-1074} $ |
| \form#364:$ (1 + (1 - 2^{52})) * 2^{1023}) $ |
| \form#365:$ 1 + \exp(x) $ |
| \form#366:$ 2^{-52} $ |
| \form#367:$X_k$ |
| \form#368:$y \in \{0, 1\} $ |
| \form#369:\[ y = G(X' \beta), \] |
| \form#370:$ G $ |
| \form#371:\[ P = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \dots \beta_j x_j)}} = \frac{1}{1 + e^{-z}} \implies \frac{\partial P}{\partial X_k} = \beta_k \cdot \frac{1}{1 + e^{-z}} \cdot \frac{e^{-z}}{1 + e^{-z}} \\ = \beta_k \cdot P \cdot (1-P) \] |
| \form#372:\[ \frac{\partial y}{\partial x_k} = \beta_k \frac{\sum_{i=1}^n P(y_i = 1)(1-P(y_i = 1))}{n}, \\ \text{where}, P(y_i=1) = g(X^{(i)}\beta) \] |
| \form#373:\[ S( \boldsymbol c) = B( \boldsymbol c) M( \boldsymbol c) B( \boldsymbol c) \] |
| \form#374:$ B( \boldsymbol c)$ |
| \form#375:$ M( \boldsymbol c)$ |
| \form#376:$ B( \boldsymbol c) $ |
| \form#377:\[ B( \boldsymbol c) = n\left(\sum_i^n -H(y_i, x_i, \boldsymbol c) \right)^{-1} \] |
| \form#378:\[ M_{H} =\frac{1}{n} \sum_i^n \psi(y_i,x_i, \boldsymbol c)^T \psi(y_i,x_i, \boldsymbol c). \] |
| \form#379:$ M( \boldsymbol c) $ |
| \form#380:$M $ |
| \form#381:$ (ij) $ |
| \form#382:$ \boldsymbol n $ |
| \form#383:$ \boldsymbol m $ |
| \form#384:$ \boldsymbol n \times m $ |
| \form#385:$ \boldsymbol \beta $ |
| \form#386:$ \boldsymbol t \in \mathbf R^{m} $ |
| \form#387:$ X \in \mathbf R^{m} $ |
| \form#388:$ R(t_i) $ |
| \form#389:$ t_i $ |
| \form#390:\[ P[T_k = t_i | \boldsymbol R(t_i)] = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}. \,. \] |
| \form#391:\[ \mathcal L = \prod_{i = 1}^n \left( \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}} \right). \] |
| \form#392:\[ L = \sum_{i = 1}^n \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right) \right]. \] |
| \form#393:\[ \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii} \,. \] |
| \form#394:$ \kappa(H) $ |
| \form#395:$(10^8 / m)$ |
| \form#396:$l(\boldsymbol \beta)$ |
| \form#397:$ H_0 $ |
| \form#398:$ H_1 $ |
| \form#399:$ \Gamma $ |
| \form#400:$ \gamma_0 \in \Gamma_0 $ |
| \form#401:$ \Gamma_0 \subsetneq \Gamma $ |
| \form#402:$ \Gamma_0 $ |
| \form#403:$ X_1, \dots, X_n \sim N(\mu, \sigma^2) $ |
| \form#404:$ H_0 : \mu \leq 0 $ |
| \form#405:$ H_0 : \mu = 0 $ |
| \form#406:$ \bar x $ |
| \form#407:$ s^2 $ |
| \form#408:\[ t = \frac{\sqrt n \cdot \bar x}{s} \] |
| \form#409:$ (n - 1) $ |
| \form#410:$ \Pr[\bar X \geq \bar x \mid \mu = 0] $ |
| \form#411:$ \Pr[\bar X \geq \bar x \mid \mu \leq 0] $ |
| \form#412:$ \Pr[ |\bar X| \geq |\bar x| \mid \mu = 0] $ |
| \form#413:$ \mu_0 $ |
| \form#414:$ y_1, \dots, y_m $ |
| \form#415:$ X_1, \dots, X_n \sim N(\mu_X, \sigma^2) $ |
| \form#416:$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma^2) $ |
| \form#417:$ \mu_X, \mu_Y, $ |
| \form#418:$ H_0 : \mu_X \leq \mu_Y $ |
| \form#419:$ H_0 : \mu_X = \mu_Y $ |
| \form#420:$ \bar x, \bar y $ |
| \form#421:$ s_X^2, s_Y^2 $ |
| \form#422:\[ t = \frac{\bar x - \bar y}{s_p \sqrt{1/n + 1/m}} \] |
| \form#423:\[ s_p^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2 + \sum_{i=1}^m (y_i - \bar y)^2} {n + m - 2} \] |
| \form#424:$ (n + m - 2) $ |
| \form#425:$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] $ |
| \form#426:$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] $ |
| \form#427:$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] $ |
| \form#428:$ X_1, \dots, X_n \sim N(\mu_X, \sigma_X^2) $ |
| \form#429:$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma_Y^2) $ |
| \form#430:$ \mu_X, \mu_Y, \sigma_X^2, $ |
| \form#431:$ \sigma_Y^2 $ |
| \form#432:\[ t = \frac{\bar x - \bar y}{\sqrt{s_X^2/n + s_Y^2/m}} \] |
| \form#433:\[ \frac{(s_X^2 / n + s_Y^2 / m)^2}{(s_X^2 / n)^2/(n-1) + (s_Y^2 / m)^2/(m-1)} \] |
| \form#434:$ x_1, \dots, x_m $ |
| \form#435:$ y_1, \dots, y_n $ |
| \form#436:$ X_1, \dots, X_m \sim N(\mu_X, \sigma^2) $ |
| \form#437:$ Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) $ |
| \form#438:$ H_0 : \sigma_X < \sigma_Y $ |
| \form#439:$ H_0 : \sigma_X = \sigma_Y $ |
| \form#440:\[ f = \frac{s_Y^2}{s_X^2} \] |
| \form#441:$ (m - 1) $ |
| \form#442:$ \Pr[F \geq f \mid \sigma_X = \sigma_Y] $ |
| \form#443:$ \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] $ |
| \form#444:$ 2 \cdot \min \{ p, 1 - p \} $ |
| \form#445:$ p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] $ |
| \form#446:$ n_1, \dots, n_k $ |
| \form#447:$ N = (N_1, \dots, N_k) $ |
| \form#448:$ p = (p_1, \dots, p_k) $ |
| \form#449:$ H_0 : p = p^0 $ |
| \form#450:$ n_i $ |
| \form#451:$ p^0_i $ |
| \form#452:$ p^0 = (\frac 1k, \dots, \frac 1k) $ |
| \form#453:$ (k - 1) $ |
| \form#454:$ n = \sum_{i=1}^n n_i $ |
| \form#455:\[ \chi^2 = \sum_{i=1}^k \frac{(n_i - np_i)^2}{np_i} \] |
| \form#456:$ \Pr[X^2 \geq \chi^2 \mid p = p^0] $ |
| \form#457:$ \phi = \sqrt{\frac{\chi^2}{n}} $ |
| \form#458:$ \sqrt{\frac{\chi^2}{n + \chi^2}} $ |
| \form#459:$ X_1, \dots, X_m $ |
| \form#460:$ Y_1, \dots, Y_n $ |
| \form#461:$ F_X, F_Y $ |
| \form#462:$ H_0 : F_X = F_Y $ |
| \form#463:\[ d = \max_{t \in \mathbb R} |F_x(t) - F_y(t)| \] |
| \form#464:$ F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| $ |
| \form#465:$ F_y $ |
| \form#466:$ k = (r + 0.12 + \frac{0.11}{r}) \cdot d $ |
| \form#467:$ r = \sqrt{\frac{m n}{m+n}}. $ |
| \form#468:$ d $ |
| \form#469:$ \Pr[D \geq d \mid F_X = F_Y] $ |
| \form#470:$ H_0 : \forall i,j: \Pr[X_i > Y_j] + \frac{\Pr[X_i = Y_j]}{2} = \frac 12 $ |
| \form#471:\[ z = \frac{u - \bar x}{\sqrt{\frac{mn(m+n+1)}{12}}} \] |
| \form#472:$ u $ |
| \form#473:$ u = \min \{ u_x, u_y \} $ |
| \form#474:\[ u_x = mn + \binom{m+1}{2} - \sum_{i=1}^m r_{x,i} \] |
| \form#475:\[ r_{x,i} = \{ j \mid x_j < x_i \} + \{ j \mid y_j < x_i \} + \frac{\{ j \mid x_j = x_i \} + \{ j \mid y_j = x_i \} + 1}{2} \] |
| \form#476:$ m+n $ |
| \form#477:$ \Pr[Z \geq z \mid H_0] $ |
| \form#478:$ \Pr[|Z| \geq |z| \mid H_0] $ |
| \form#479:$ X_1, \dots, X_n $ |
| \form#480:$ \epsilon_i $ |
| \form#481:$ v_i $ |
| \form#482:$ v_{i-1} $ |
| \form#483:$ v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j $ |
| \form#484:$ w^+ = \sum_{i \mid x_i > 0} r_i $ |
| \form#485:$ w^- = \sum_{i \mid x_i < 0} r_i $ |
| \form#486:\[ r_i = \{ j \mid |x_j| < |x_i| \} + \frac{\{ j \mid |x_j| = |x_i| \} + 1}{2}. \] |
| \form#487:$ w = \min \{ w^+, w^- \} $ |
| \form#488:$ w^+ $ |
| \form#489:$ w^- $ |
| \form#490:\[ z = \frac{w^+ - \frac{n(n+1)}{4}} {\sqrt{\frac{n(n+1)(2n+1)}{24} - \sum_{i=1}^n \frac{t_i^2 - 1}{48}}} \] |
| \form#491:$ |x_i| $ |
| \form#492:$ \Pr[Z \geq z \mid \mu \leq 0] $ |
| \form#493:$ \Pr[ |Z| \geq |z| \mid \mu = 0] $ |
| \form#494:$ x_{1,1}, \dots, x_{1, n_1}, x_{2,1}, \dots, x_{2,n_2}, \dots, x_{k,n_k} $ |
| \form#495:$ X_{i,j} \sim N(\mu_i, \sigma^2) $ |
| \form#496:$ \mu_1, \dots, \mu_k $ |
| \form#497:$ H_0 : \mu_1 = \dots = \mu_k $ |
| \form#498:$ x_{i,j} $ |
| \form#499:$ n := \sum_{i=1}^k n_i $ |
| \form#500:$ \overline{x_i} $ |
| \form#501:$ s_i^2 $ |
| \form#502:$ \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2. $ |
| \form#503:$ \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2. $ |
| \form#504:$ (k-1) $ |
| \form#505:$ (n-k) $ |
| \form#506:$ s_b^2 := \frac{\mathit{SS}_b}{k-1} $ |
| \form#507:$ s_w^2 := \frac{\mathit{SS}_w}{n-k} $ |
| \form#508:\[ f = \frac{s_b^2}{s_w^2}. \] |
| \form#509:$ \Pr[ F \geq f \mid H_0] $ |
| \form#510:\[ ||\boldsymbol A - \boldsymbol UV ||_2 \] |
| \form#511:$ rank(\boldsymbol UV) \leq k $ |
| \form#512:$ ||\cdot||_2 $ |
| \form#513:$ k \leq rank(\boldsymbol A)$ |
| \form#514:$ m \times n $ |
| \form#515:$ m \times k $ |
| \form#516:$ k \times n $ |
| \form#517:$\gamma$ |
| \form#518:$\exp(-\gamma||x-y||^2)$ |
| \form#519:$q$ |
| \form#520:$ (\langle x,y\rangle + q)^r $ |
| \form#521:$r$ |
| \form#522:$\epsilon$ |
| \form#523:\[ \underset{w,b}{\text{Minimize }} \lambda||w||^2 + \frac{1}{n}\sum_{i=1}^n \ell(y_i,f_{w,b}(x_i)) \] |
| \form#524:$(x_1,y_1),\ldots,(x_n,y_n)$ |
| \form#525:$\ell(y,f(x))$ |
| \form#526:$\ell(y,f(x)) = \max(0,1-yf(x))$ |
| \form#527:$\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)$ |
| \form#528:$ f_{w,b}(x) = \langle w, x\rangle + b$ |
| \form#529:\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \] |
| \form#530:$ t $ |
| \form#531:$ 1 $ |
| \form#532:$ X_t $ |
| \form#533:$ q $ |
| \form#534:$ \phi(B) $ |
| \form#535:$ \theta(B) $ |
| \form#536:$ Y_{t} $ |
| \form#537:$ Y_{t} = (1-B)^{d}(X_{t} - \mu) $ |
| \form#538:$ d>0 $ |
| \form#539:$ Z_t $ |
| \form#540:\[ \phi(B) Y_t= \phi_1 Y_{t-1} + \dots + \phi_{p} Y_{t-p} \] |
| \form#541:\[ \theta(B) Z_t = \theta_{1} Z_{t-1} + \dots + \theta_{q} Z_{t-q}. \] |
| \form#542:$\tau, \epsilon_1, \epsilon_2, \epsilon_3,$ |
| \form#543:$ {\boldsymbol \Sigma}/(\sqrt{N-1}) $ |
| \form#544:$f(x)$ |
| \form#545:\[ f(x) = \sum_i \alpha_i k(x_i,x), \] |
| \form#546:$ \alpha_i $ |
| \form#547:$ k(\cdot, \cdot) $ |
| \form#548:$ f(\boldsymbol x) $ |
| \form#549:$ f(\boldsymbol x) \geq 0 $ |
| \form#550:\[ f'(\boldsymbol x) = \langle \boldsymbol w, \boldsymbol x \rangle, \] |
| \form#551:$ \boldsymbol w $ |
| \form#552:$ k(\boldsymbol x_i, \boldsymbol x_j) $ |
| \form#553:\[ k(\boldsymbol x_i, \boldsymbol x_j) = \langle \phi(\boldsymbol x_i), \phi(\boldsymbol x_j) \rangle, \] |
| \form#554:$ \phi(\boldsymbol x) $ |
| \form#555:$l(z) = \max(0, 1-z)$ |
| \form#556:$ \boldsymbol y $ |
| \form#557:$ K(\boldsymbol x,\boldsymbol y)=(\boldsymbol x \cdot \boldsymbol y)^d $ |
| \form#558:$ K(\boldsymbol x,\boldsymbol y)=exp(-\gamma || \boldsymbol x \cdot \boldsymbol y ||^2 ) $ |
| \form#559:$\frac{1}{num\_features}$ |
| \form#560:$d$ |
| \form#561:$ (\langle x,y\rangle + q)^d $ |
| \form#562:$\textit{tp}$ |
| \form#563:$\textit{tn}$ |
| \form#564:$\textit{fp}$ |
| \form#565:$\textit{fn}$ |
| \form#566:$\textit{tpr}=\textit{tp}/(\textit{tp}+\textit{fn})$ |
| \form#567:$\textit{tnr}=\textit{tn}/(\textit{fp}+\textit{tn})$ |
| \form#568:$\textit{ppv}=\textit{tp}/(\textit{tp}+\textit{fp})$ |
| \form#569:$\textit{npv}=\textit{tn}/(\textit{tn}+\textit{fn})$ |
| \form#570:$\textit{fpr}=\textit{fp}/(\textit{fp}+\textit{tn})$ |
| \form#571:$\textit{fdr}=1-\textit{ppv}$ |
| \form#572:$\textit{fnr}=\textit{fn}/(\textit{fn}+\textit{tp})$ |
| \form#573:$\textit{acc}=(\textit{tp}+\textit{tn})/(\textit{tp}+\textit{tn}+\textit{fp}+\textit{fn})$ |
| \form#574:$\textit{f1}=2*\textit{tp}/(2*\textit{tp}+\textit{fp}+\textit{fn})$ |
| \form#575:$\textit{acc}=(\textit{tp}+\textit{tn})/(\textit{tp}+\textit{tn}+\textit{fp} +\textit{fn})$ |
