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/dev/null +++ b/docs/v1.11/formula.repository @@ -0,0 +1,549 @@ +\form#0:\[ S (X) = \frac{Total X}{Total transactions} \] +\form#1:\[ S (X \Rightarrow Y) = \frac{Total(X \cup Y)}{Total transactions} \] +\form#2:$ X,Y $ +\form#3:$ X $ +\form#4:$ Y $ +\form#5:$ P(Y|X) $ +\form#6:\[ C (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X)} \] +\form#7:\[ L (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X) \cdot s(Y)} \] +\form#8:$ \neg Y $ +\form#9:$ X \Rightarrow Y $ +\form#10:\[ Conv (X \Rightarrow Y) = \frac{1 - S(Y)}{1 - C(X \Rightarrow Y)} \] +\form#11:$ n $ +\form#12:$ n - 1 $ +\form#13:$ n \ge 2 $ +\form#14:$ A $ +\form#15:$ B $ +\form#16:$ B \Rightarrow (A - B) $ +\form#17:$ min(support) = .25 $ +\form#18:$ min(confidence) = .5 $ +\form#19:$ a_1, \dots, a_n $ +\form#20:$ c $ +\form#21:\[ P(C=c_i \mid A) \approx P(C=c_j \mid A) \] +\form#22:$ P(A_i = a \mid C=c) $ +\form#23:\[ P(A_i=a \mid C=c) = \frac{1}{\sqrt{2\pi\sigma^{2}_c}}exp\left(-\frac{(a-\mu_c)^{2}}{2\sigma^{2}_c}\right) \] +\form#24:$\mu_c$ +\form#25:$\sigma^{2}_c$ +\form#26:$c$ +\form#27:\[ \Pr(C = c \mid A_1 = a_1, \dots, A_n = a_n) = \frac{\Pr(C = c) \cdot \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c)} {\Pr(A_1 = a_1, \dots, A_n = a_n)} \,, \] +\form#28:\[ \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c) = \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \,. \] +\form#29:\[ \text{classify}(a_1, ..., a_n) = \arg\max_c \left\{ \Pr(C = c) \cdot \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \right\} \] +\form#30:$ P(A_i = a \mid C = c) $ +\form#31:\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a)}{\#c} \] +\form#32:$ \#(c,i,a) $ +\form#33:$ i $ +\form#34:$ a $ +\form#35:$ \#c $ +\form#36:\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a) + s}{\#c + s \cdot \#i} \] +\form#37:$ \#i $ +\form#38:$ s \geq 0 $ +\form#39:$ s = 1 $ +\form#40:$ s = 0 $ +\form#41:$ \boldsymbol Ax = \boldsymbol b $ +\form#42:$A$ +\form#43:$x$ +\form#44:$ \boldsymbol b $ +\form#45:\[ \boldsymbol Ax = \boldsymbol b \] +\form#46:$ \boldsymbol A $ +\form#47:\[ \|\boldsymbol A - \boldsymbol UV^{T} \|_2 \] +\form#48:$rank(\boldsymbol UV^{T}) \leq r$ +\form#49:$\|\cdot\|_2$ +\form#50:$m \times n$ +\form#51:$U$ +\form#52:$m \times r$ +\form#53:$V$ +\form#54:$n \times r$ +\form#55:$1 \leq r \ll \min(m, n)$ +\form#56:$ \Pr( \text{best label sequence} \mid \text{sequence}) $ +\form#57:\[ p_\lambda(\boldsymbol y | \boldsymbol x) = \frac{\exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y)}}{Z_\lambda(\boldsymbol x)} \,. \] +\form#58:$ F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) $ +\form#59:$ \boldsymbol x $ +\form#60:$ f_m(y_i,y_{i-1},x_i) $ +\form#61:$ y_i $ +\form#62:$ y_{i-1} $ +\form#63:$ x_i $ +\form#64:$ \lambda_m $ +\form#65:$ Z_\lambda(\boldsymbol x) $ +\form#66:\[ Z_\lambda(\boldsymbol x) = \sum_{\boldsymbol y'} \exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y')} \] +\form#67:$ T=\{(x_k,y_k)\}_{k=1}^N $ +\form#68:\[ \ell_{\lambda}=\sum_k \log p_\lambda(y_k|x_k) =\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] \] +\form#69:\[ \nabla \ell_{\lambda}=\sum_k[F(x_k,y_k)-E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] \] +\form#70:$E_{p_\lambda(Y|x)}[F(x,Y)]$ +\form#71:\[ E_{p_\lambda(Y|x)}[F(x,Y)] = \sum_y p_\lambda(y|x)F(x,y) = \sum_i\frac{\alpha_{i-1}(f_i*M_i)\beta_i^T}{Z_\lambda(x)} \] +\form#72:\[ Z_\lambda(x) = \alpha_n.1^T \] +\form#73:$\alpha_i$ +\form#74:$ \beta_i$ +\form#75:\[ \alpha_i = \begin{cases} \alpha_{i-1}M_i, & 0<i<=n\\ 1, & i=0 \end{cases}\\ \] +\form#76:\[ \beta_i^T = \begin{cases} M_{i+1}\beta_{i+1}^T, & 1<=i<n\\ 1, & i=n \end{cases} \] +\form#77:\[ \ell_{\lambda}^\prime=\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] - \frac{\lVert \lambda \rVert^2}{2\sigma ^2} \] +\form#78:\[ \nabla \ell_{\lambda}^\prime=\sum_k[F(x_k,y_k) - E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] - \frac{\lambda}{\sigma ^2} \] +\form#79:$ \boldsymbol c $ +\form#80:$ l(\boldsymbol c) $ +\form#81:$ n^2 $ +\form#82:\[\min_{w \in R^N} L(w) + \lambda \left(\frac{(1-\alpha)}{2} \|w\|_2^2 + \alpha \|w\|_1 \right)\] +\form#83:$L$ +\form#84:$ \alpha \in [0,1] $ +\form#85:$ lambda \geq 0 $ +\form#86:$alpha = 0$ +\form#87:$\alpha = 1$ +\form#88:\[L(\vec{w}) = \frac{1}{2}\left[\frac{1}{M} \sum_{m=1}^M (w^{t} x_m + w_{0} - y_m)^2 \right] \] +\form#89:\[ L(\vec{w}) = \sum_{m=1}^M\left[y_m \log\left(1 + e^{-(w_0 + \vec{w}\cdot\vec{x}_m)}\right) + (1-y_m) \log\left(1 + e^{w_0 + \vec{w}\cdot\vec{x}_m}\right)\right]\ , \] +\form#90:$y_m \in {0,1}$ +\form#91:\[ x' \leftarrow \frac{x - \bar{x}}{\sigma_x} \] +\form#92:\[y' \leftarrow y - \bar{y} \] +\form#93:$ l(\boldsymbol \beta) $ +\form#94:$ Y \in \{ 0,1,2 \ldots k \} $ +\form#95:\[ E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x) \] +\form#96:$ \sigma(x) = \frac{1}{1 + \exp(-x)} $ +\form#97:$ \boldsymbol y \in \{ 0,1 \}^{n \times k} $ +\form#98:$ k $ +\form#99:$ X \in \mathbf R^{n \times k} $ +\form#100:$ \boldsymbol x_i $ +\form#101:\[ P[Y = y_i | \boldsymbol x_i] = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \] +\form#102:$ \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) $ +\form#103:$ \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) $ +\form#104:\[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \] +\form#105:$ H = -X^T A X $ +\form#106:$ A = \text{diag}(a_1, \dots, a_n) $ +\form#107:$ a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. $ +\form#108:$ H $ +\form#109:\[ \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii} \,. \] +\form#110:\[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \] +\form#111:$ p $ +\form#112:$ c_i = 0 $ +\form#113:$ F $ +\form#114:\[ p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| )) \] +\form#115:$ Z $ +\form#116:$ \exp(c_i) $ +\form#117:$ \kappa(X^T A X) $ +\form#118:$ K $ +\form#119:$ (1, ..., K) $ +\form#120:$ J $ +\form#121:$ (0, ..., J-1) $ +\form#122:$ {m_{k,j}} $ +\form#123:$ j $ +\form#124:$ {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#125:$ Y_i $ +\form#126:$ j = 1,.. , J$ +\form#127:$\pi$ +\form#128:$\pi_{ij}$ +\form#129:$i$ +\form#130:$j$ +\form#131:\[ \gamma_{ij} = \Pr(Y_i \le j)= \pi_{i1} +...+ \pi_{ij} . \] +\form#132:$ \mbox{logit}(\pi) = \log[\pi/(1-\pi)] $ +\form#133:\[ \mbox{logit}(\gamma_{ij})=\mbox{logit}(\Pr(Y_i \le j))=\log \frac{\Pr(Y_i \le j)}{1-\Pr(Y_i\le j)}, j=1,...,Jâ1 \] +\form#134:\[ \mbox{logit}(\gamma_{ij}) = \theta_j - x^T_i \beta \] +\form#135:$x_i$ +\form#136:$\beta$ +\form#137:$\{\theta_j\}$ +\form#138:$x^T_i\beta$ +\form#139:$ x_1, \dots, x_n \in \mathbb R^d $ +\form#140:$ c_1, \dots, c_k \in \mathbb R^d $ +\form#141:\[ (c_1, \dots, c_k) \mapsto \sum_{i=1}^n \min_{j=1}^k \operatorname{dist}(x_i, c_j) \] +\form#142:$ \operatorname{dist} $ +\form#143:$ \alpha $ +\form#144:$ \beta $ +\form#145:$ \phi_i $ +\form#146:$ \theta $ +\form#147:$\alpha$ +\form#148:$ z_n $ +\form#149:$ w_n $ +\form#150:$ \phi_{z_n} $ +\form#151:$\|\vec{a}\|_1$ +\form#152:$\|\vec{a}\|_2$ +\form#153:$\|\vec{a} - \vec{b}\|_1$ +\form#154:$\|\vec{a} - \vec{b}\|_2$ +\form#155:$\|\vec{a} - \vec{b}\|_p, p > 0$ +\form#156:$\|\vec{a} - \vec{b}\|_\infty$ +\form#157:$\|\vec{a} - \vec{b}\|_2^2$ +\form#158:$\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2}$ +\form#159:$\cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2})$ +\form#160:$ \vec x = (x_1, \dots, x_n) $ +\form#161:$ \| x \|_1 = \sum_{i=1}^n |x_i| $ +\form#162:$ \| x \|_2 = \sqrt{\sum_{i=1}^n x_i^2} $ +\form#163:$ \vec y = (y_1, \dots, y_n) $ +\form#164:$ \| x - y \|_\infty = \max_{i=1}^n \|x_i - y_i\| $ +\form#165:$ p > 0 $ +\form#166:$ \| x - y \|_p = (\sum_{i=1}^n \|x_i - y_i\|^p)^{\frac{1}{p}} $ +\form#167:$ \| x - y \|_1 = \sum_{i=1}^n |x_i - y_i| $ +\form#168:$ \| x - y \|_2 = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} $ +\form#169:$ \frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|} $ +\form#170:$ \| x - y \|_2^2 = \sum_{i=1}^n (x_i - y_i)^2 $ +\form#171:$ \arccos\left(\frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|}\right) $ +\form#172:$ 1 - \frac{\langle \vec x, \vec y \rangle} {\| \vec x \|^2 \cdot \| \vec y \|^2 - \langle \vec x, \vec y \rangle} $ +\form#173:$ \vec x = (x_1, \dots, x_m) $ +\form#174:$ 1 - \frac{|x \cap y|}{|x \cup y|} $ +\form#175:$ M $ +\form#176:$ \vec x $ +\form#177:$ M = (\vec{m_0} \dots \vec{m_{l-1}}) \in \mathbb{R}^{k \times l} $ +\form#178:$ \vec x \in \mathbb R^k $ +\form#179:$ x $ +\form#180:$ \arg\min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) $ +\form#181:$ \min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) $ +\form#182:$ \operatorname{dist}(\vec{m_j}, \vec x) $ +\form#183:$ j = $ +\form#184:$ x_1, \dots, x_n $ +\form#185:$ \frac 1n \sum_{i=1}^n x_i $ +\form#186:$ \widetilde{x} := \frac 1n \sum_{i=1}^n \frac{x_i}{\| x_i \|} $ +\form#187:$ \frac{\widetilde{x}}{\| \widetilde{x} \|} $ +\form#188:$ \vec x_1, \dots, \vec x_n \in \mathbb R^m $ +\form#189:$ ( \vec x_1 \dots \vec x_n ) \in \mathbb R^{m \times n}$ +\form#190:$ PA = LDL* $ +\form#191:$ A = QR $ +\form#192:$ PAQ = LU $ +\form#193:$mxn$ +\form#194:$m \ge n$ +\form#195:\[ A = U \Sigma V^T, \] +\form#196:$\Sigma$ +\form#197:$n \times n$ +\form#198:$ \sqrt{mean((X - USV^T)_{ij}^2)} $ +\form#199:$ \sqrt{mean(X_{ij}^2)} $ +\form#200:$\sigma_i, u_i, v_i$ +\form#201:$A^TA$ +\form#202:$AA^T$ +\form#203:\[ H(A) = \begin{bmatrix} 0 & A\\ A^* & 0 \end{bmatrix} \] +\form#204:\[ A = P B Q^T, \] +\form#205:$P$ +\form#206:$Q$ +\form#207:$B$ +\form#208:$B*B$ +\form#209:$A*A$ +\form#210:\[ B = X\Sigma Y^T, \] +\form#211:$U = PX$ +\form#212:$V = QY$ +\form#213:\[ Ax = b \] +\form#214:$x \in \mathbb{R}^{n}$ +\form#215:$A \in \mathbb{R}^{m \times n} $ +\form#216:$b \in \mathbb{R}^{m}$ +\form#217:$ \frac{|Ax - b|}{|b|} $ +\form#218:$ 0 \ldots n-1 $ +\form#219:$b$ +\form#220:$ N $ +\form#221:$ \boldsymbol X $ +\form#222:$ \hat{x} $ +\form#223:$ \boldsymbol{X}$ +\form#224:$ \hat{\boldsymbol X} $ +\form#225:\[ \hat{\boldsymbol X} = {\boldsymbol X} - \vec{e} \hat{x}^T \] +\form#226:$ \vec{e} $ +\form#227:\[ \hat{\boldsymbol X} = {\boldsymbol U}{\boldsymbol \Sigma}{\boldsymbol V}^T \] +\form#228:$ {\boldsymbol \Sigma} $ +\form#229:$ {\boldsymbol \Sigma}/(\sqrt{(N-1)} $ +\form#230:$ {\boldsymbol V} $ +\form#231:$ \boldsymbol P $ +\form#232:$ {\boldsymbol X}' $ +\form#233:\begin{align*} {\boldsymbol {\hat{X}}} & = {\boldsymbol X} - \vec{e} \hat{x}^T \\ {\boldsymbol X}' & = {\boldsymbol {\hat {X}}} {\boldsymbol P}. \end{align*} +\form#234:$\hat{x} $ +\form#235:$ \boldsymbol R $ +\form#236:\[ {\boldsymbol R} = {\boldsymbol {\hat{X}}} - {\boldsymbol X}' {\boldsymbol P}^T. \] +\form#237:$ r $ +\form#238:\[ r = \|{\boldsymbol R}\|_F \] +\form#239:$ \|\cdot\|_F $ +\form#240:$ r' $ +\form#241:\[ r' = \frac{ \|{\boldsymbol R}\|_F }{\|{\boldsymbol X}\|_F } \] +\form#242:$ p \in [0,1] $ +\form#243:$ F(x) = p $ +\form#244:$ \sup \{ x \in D \mid F(x) \leq p \} $ +\form#245:$ p < 0.5 $ +\form#246:$ \inf \{ x \in D \mid F(x) \geq p \} $ +\form#247:$ p \geq 0.5 $ +\form#248:$ D $ +\form#249:$ \mathbb R $ +\form#250:$ \mathbb N_0 $ +\form#251:$ x \in \mathbb N_0 $ +\form#252:$ F(x) < p < F(x + 1) $ +\form#253:$ x + 1 $ +\form#254:$ p < F(0) $ +\form#255:$p$ +\form#256:$ 1 - p $ +\form#257:$ > x $ +\form#258:$ \leq x $ +\form#259:$ \Pr[X \leq x] $ +\form#260:$ \mathit{sp} $ +\form#261:$ \mathit{sp} \in [0,1] $ +\form#262:$ f(x) $ +\form#263:$ f $ +\form#264:$ p \leq 1 - \mathit{sp} $ +\form#265:$ \alpha > 0 $ +\form#266:$ \beta > 0 $ +\form#267:$ p = \Pr[X \leq x] $ +\form#268:$ n \in \mathbb N_0 $ +\form#269:$ p \geq \Pr[X \leq x] $ +\form#270:$ p \leq \Pr[X \leq x] $ +\form#271:$ x_0 $ +\form#272:$ \gamma > 0 $ +\form#273:$ \gamma $ +\form#274:$ \nu > 0 $ +\form#275:$ \nu $ +\form#276:$ \mu > 0 $ +\form#277:$ \lambda > 0 $ +\form#278:$ \lambda $ +\form#279:$ \nu_1 > 0 $ +\form#280:$ \nu_1 $ +\form#281:$ \nu_2 $ +\form#282:$ k > 0 $ +\form#283:$ \theta > 0 $ +\form#284:$ r \in \{ 0, 1, \dots, N \} $ +\form#285:$ n \in \{ 0, 1, \dots, N \} $ +\form#286:$ N \in \mathbb N $ +\form#287:$ r, n, N $ +\form#288:$ \mu $ +\form#289:$ b > 0 $ +\form#290:$ 2 b^2 $ +\form#291:$ s > 0 $ +\form#292:$ s $ +\form#293:$ m $ +\form#294:$ r > 0 $ +\form#295:$ x + r $ +\form#296:$ \mathit{sp} \in (0,1] $ +\form#297:$ r, \mathit{sp} $ +\form#298:$ \delta \geq 0 $ +\form#299:$ shape_1 $ +\form#300:$ shape_2 $ +\form#301:$ \delta $ +\form#302:$ \lambda \geq 0 $ +\form#303:$ \nu_1, \nu_2, \lambda $ +\form#304:$ \sigma > 0 $ +\form#305:$ T $ +\form#306:$ \sigma^2 $ +\form#307:$ \sigma $ +\form#308:$ c \geq a $ +\form#309:$ b \geq c $ +\form#310:$ b > a $ +\form#311:$ a, b, c $ +\form#312:$ b $ +\form#313:$ [a, b] $ +\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:$\textit{tp}$ +\form#511:$\textit{tn}$ +\form#512:$\textit{fp}$ +\form#513:$\textit{fn}$ +\form#514:$\textit{tpr}=\textit{tp}/(\textit{tp}+\textit{fn})$ +\form#515:$\textit{tnr}=\textit{tn}/(\textit{fp}+\textit{tn})$ +\form#516:$\textit{ppv}=\textit{tp}/(\textit{tp}+\textit{fp})$ +\form#517:$\textit{npv}=\textit{tn}/(\textit{tn}+\textit{fn})$ +\form#518:$\textit{fpr}=\textit{fp}/(\textit{fp}+\textit{tn})$ +\form#519:$\textit{fdr}=1-\textit{ppv}$ +\form#520:$\textit{fnr}=\textit{fn}/(\textit{fn}+\textit{tp})$ +\form#521:$\textit{acc}=(\textit{tp}+\textit{tn})/(\textit{tp}+\textit{tn}+\textit{fp} +\textit{fn})$ +\form#522:$\textit{f1}=2*\textit{tp}/(2*\textit{tp}+\textit{fp}+\textit{fn})$ +\form#523:$\gamma$ +\form#524:$\exp(-\gamma||x-y||^2)$ +\form#525:$q$ +\form#526:$ (\langle x,y\rangle + q)^r $ +\form#527:$r$ +\form#528:$\epsilon$ +\form#529:\[ \underset{w,b}{\text{Minimize }} \lambda||w||^2 + \frac{1}{n}\sum_{i=1}^n \ell(y_i,f_{w,b}(x_i)) \] +\form#530:$(x_1,y_1),\ldots,(x_n,y_n)$ +\form#531:$\ell(y,f(x))$ +\form#532:$\ell(y,f(x)) = \max(0,1-yf(x))$ +\form#533:$\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)$ +\form#534:$ f_{w,b}(x) = \langle w, x\rangle + b$ +\form#535:\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \] +\form#536:$ t $ +\form#537:$ 1 $ +\form#538:$ X_t $ +\form#539:$ q $ +\form#540:$ \phi(B) $ +\form#541:$ \theta(B) $ +\form#542:$ Y_{t} $ +\form#543:$ Y_{t} = (1-B)^{d}(X_{t} - \mu) $ +\form#544:$ d>0 $ +\form#545:$ Z_t $ +\form#546:\[ \phi(B) Y_t= \phi_1 Y_{t-1} + \dots + \phi_{p} Y_{t-p} \] +\form#547:\[ \theta(B) Z_t = \theta_{1} Z_{t-1} + \dots + \theta_{q} Z_{t-q}. \] +\form#548:$\tau, \epsilon_1, \epsilon_2, \epsilon_3,$
