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Posted to commits@madlib.apache.org by xt...@apache.org on 2016/09/20 18:31:59 UTC
[47/51] [partial] incubator-madlib-site git commit: Update doc for
1.9.1 release
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@@ -187,361 +187,390 @@
\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:$ A = QR $
-\form#190:$ PAQ = LU $
-\form#191:$mxn$
-\form#192:$m \ge n$
-\form#193:\[ A = U \Sigma V^T, \]
-\form#194:$\Sigma$
-\form#195:$n \times n$
-\form#196:$ \sqrt{mean((X - USV^T)_{ij}^2)} $
-\form#197:$ \sqrt{mean(X_{ij}^2)} $
-\form#198:$\sigma_i, u_i, v_i$
-\form#199:$A^TA$
-\form#200:$AA^T$
-\form#201:\[ H(A) = \begin{bmatrix} 0 & A\\ A^* & 0 \end{bmatrix} \]
-\form#202:\[ A = P B Q^T, \]
-\form#203:$P$
-\form#204:$Q$
-\form#205:$B$
-\form#206:$B*B$
-\form#207:$A*A$
-\form#208:\[ B = X\Sigma Y^T, \]
-\form#209:$U = PX$
-\form#210:$V = QY$
-\form#211:\[ Ax = b \]
-\form#212:$x \in \mathbb{R}^{n}$
-\form#213:$A \in \mathbb{R}^{m \times n} $
-\form#214:$b \in \mathbb{R}^{m}$
-\form#215:$ \frac{|Ax - b|}{|b|} $
-\form#216:$ 0 \ldots n-1 $
-\form#217:$b$
-\form#218:$ N $
-\form#219:$ \boldsymbol X $
-\form#220:$ \hat{x} $
-\form#221:$ \boldsymbol{X}$
-\form#222:$ \hat{\boldsymbol X} $
-\form#223:\[ \hat{\boldsymbol X} = {\boldsymbol X} - \vec{e} \hat{x}^T \]
-\form#224:$ \vec{e} $
-\form#225:\[ \hat{\boldsymbol X} = {\boldsymbol U}{\boldsymbol \Sigma}{\boldsymbol V}^T \]
-\form#226:$ {\boldsymbol \Sigma} $
-\form#227:$ {\boldsymbol \Sigma}/(\sqrt{N-1}) $
-\form#228:$ {\boldsymbol V} $
-\form#229:$ \boldsymbol P $
-\form#230:$ {\boldsymbol X}' $
-\form#231:\begin{align*} {\boldsymbol {\hat{X}}} & = {\boldsymbol X} - \vec{e} \hat{x}^T \\ {\boldsymbol X}' & = {\boldsymbol {\hat {X}}} {\boldsymbol P}. \end{align*}
-\form#232:$\hat{x} $
-\form#233:$ \boldsymbol R $
-\form#234:\[ {\boldsymbol R} = {\boldsymbol {\hat{X}}} - {\boldsymbol X}' {\boldsymbol P}^T. \]
-\form#235:$ r $
-\form#236:\[ r = \|{\boldsymbol R}\|_F \]
-\form#237:$ \|\cdot\|_F $
-\form#238:$ r' $
-\form#239:\[ r' = \frac{ \|{\boldsymbol R}\|_F }{\|{\boldsymbol X}\|_F } \]
-\form#240:$ p \in [0,1] $
-\form#241:$ F(x) = p $
-\form#242:$ \sup \{ x \in D \mid F(x) \leq p \} $
-\form#243:$ p < 0.5 $
-\form#244:$ \inf \{ x \in D \mid F(x) \geq p \} $
-\form#245:$ p \geq 0.5 $
-\form#246:$ D $
-\form#247:$ \mathbb R $
-\form#248:$ \mathbb N_0 $
-\form#249:$ x \in \mathbb N_0 $
-\form#250:$ F(x) < p < F(x + 1) $
-\form#251:$ x + 1 $
-\form#252:$ p < F(0) $
-\form#253:$p$
-\form#254:$ 1 - p $
-\form#255:$ > x $
-\form#256:$ \leq x $
-\form#257:$ \Pr[X \leq x] $
-\form#258:$ \mathit{sp} $
-\form#259:$ \mathit{sp} \in [0,1] $
-\form#260:$ f(x) $
-\form#261:$ f $
-\form#262:$ p \leq 1 - \mathit{sp} $
-\form#263:$ \alpha > 0 $
-\form#264:$ \beta > 0 $
-\form#265:$ p = \Pr[X \leq x] $
-\form#266:$ n \in \mathbb N_0 $
-\form#267:$ p \geq \Pr[X \leq x] $
-\form#268:$ p \leq \Pr[X \leq x] $
-\form#269:$ x_0 $
-\form#270:$ \gamma > 0 $
-\form#271:$ \gamma $
-\form#272:$ \nu > 0 $
-\form#273:$ \nu $
-\form#274:$ \mu > 0 $
-\form#275:$ \lambda > 0 $
-\form#276:$ \lambda $
-\form#277:$ \nu_1 > 0 $
-\form#278:$ \nu_1 $
-\form#279:$ \nu_2 $
-\form#280:$ k > 0 $
-\form#281:$ \theta > 0 $
-\form#282:$ r \in \{ 0, 1, \dots, N \} $
-\form#283:$ n \in \{ 0, 1, \dots, N \} $
-\form#284:$ N \in \mathbb N $
-\form#285:$ r, n, N $
-\form#286:$ \mu $
-\form#287:$ b > 0 $
-\form#288:$ 2 b^2 $
-\form#289:$ s > 0 $
-\form#290:$ s $
-\form#291:$ m $
-\form#292:$ r > 0 $
-\form#293:$ x + r $
-\form#294:$ \mathit{sp} \in (0,1] $
-\form#295:$ r, \mathit{sp} $
-\form#296:$ \delta \geq 0 $
-\form#297:$ shape_1 $
-\form#298:$ shape_2 $
-\form#299:$ \delta $
-\form#300:$ \lambda \geq 0 $
-\form#301:$ \nu_1, \nu_2, \lambda $
-\form#302:$ \sigma > 0 $
-\form#303:$ T $
-\form#304:$ \sigma^2 $
-\form#305:$ \sigma $
-\form#306:$ c \geq a $
-\form#307:$ b \geq c $
-\form#308:$ b > a $
-\form#309:$ a, b, c $
-\form#310:$ b $
-\form#311:$ [a, b] $
-\form#312:$ \in (0,1) $
-\form#313:$m$
-\form#314:\[ S(\vec{c}) = B(\vec{c}) M(\vec{c}) B(\vec{c}) \]
-\form#315:\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#316:$H$
-\form#317:\[ L(\vec{c}) = \sum_{i=1}^n l(y_i, \vec{x}_i, \vec{c})\ . \]
-\form#318:\[ M(\vec{c}) = \bf{A}^T\bf{A} \]
-\form#319:$\bf{A}$
-\form#320:\[ A_m = \sum_{i\in G_m}\frac{\partial l(y_i,\vec{x}_i,\vec{c})}{\partial \vec{c}} \]
-\form#321:$G_m$
-\form#322:$ {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#323:$ y $
-\form#324:$X^{*}X$
-\form#325:\[ E[Y \mid \boldsymbol x] = \boldsymbol c^T \boldsymbol x \]
-\form#326:\[ f(y \mid \boldsymbol x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot \exp\left(-\frac{1}{2 \sigma^2} \cdot (y - \boldsymbol x^T \boldsymbol c)^2 \right) \,. \]
-\form#327:$ \boldsymbol y \in \mathbf R^n $
-\form#328:$ X^T $
-\form#329:$ X^+ $
-\form#330:$ \sum_{i=1}^n \log f(y_i \mid \boldsymbol x_i) $
-\form#331:$ RSS $
-\form#332:\[ RSS = \sum_{i=1}^n ( y_i - \boldsymbol c^T \boldsymbol x_i )^2 = (\boldsymbol y - X \boldsymbol c)^T (\boldsymbol y - X \boldsymbol c) \,. \]
-\form#333:\[ \boldsymbol c = (X^T X)^+ X^T \boldsymbol y \,. \]
-\form#334:$ TSS $
-\form#335:$ ESS $
-\form#336:$ R^2 $
-\form#337:\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#338:$ R^2 = 1 - \frac{RSS}{TSS} $
-\form#339:$ TSS = RSS + ESS $
-\form#340:$ Var[Y - \boldsymbol c^T \boldsymbol x \mid \boldsymbol x] $
-\form#341:\[ \sigma^2 = \frac{RSS}{n - k} \]
-\form#342:\[ t_i = \frac{c_i}{\sqrt{\sigma^2 \cdot \left( (X^T X)^{-1} \right)_{ii} }} \,. \]
-\form#343:$ F_\nu $
-\form#344:\[ p_i = \Pr(|T| \geq |t_i|) = 2 \cdot (1 - F_{n - k}( |t_i| )) \]
-\form#345:$ \kappa(X) = \|X\|_2\cdot\|X^{-1}\|_2$
-\form#346:$X$
-\form#347:\[ \|X\|_2 = \sqrt{\lambda_{\max}\left(X^{*}X\right)}\ , \]
-\form#348:$X^{*}$
-\form#349:$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) $
-\form#350:$ \boldsymbol t $
-\form#351:$ \boldsymbol p $
-\form#352:$ X^T X $
-\form#353:$ Y \in \{ 0,1 \} $
-\form#354:$ \boldsymbol y \in \{ 0,1 \}^n $
-\form#355:\[ P[Y = y_i | \boldsymbol x_i] = \sigma((-1)^{(1 - y_i)} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \]
-\form#356:\[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{(1 - y_i)} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \]
-\form#357:$ \boldsymbol z $
-\form#358:$ \mathit{odds}(c_1), \dots, \mathit{odds}(c_k) $
-\form#359:$ X^T A X $
-\form#360:$ \frac{1}{1 + \exp(-x)} $
-\form#361:$ \exp(x) $
-\form#362:$ 2^{-1074} $
-\form#363:$ (1 + (1 - 2^{52})) * 2^{1023}) $
-\form#364:$ 1 + \exp(x) $
-\form#365:$ 2^{-52} $
-\form#366:$X_k$
-\form#367:$y \in \{0, 1\} $
-\form#368:\[ y = G(X' \beta), \]
-\form#369:$ G $
-\form#370:\[ P = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \dots \beta_j x_j)}} = \frac{1}{1 + e^{-z}} \implies \frac{\partial P}{\partial X_k} = \beta_k \cdot \frac{1}{1 + e^{-z}} \cdot \frac{e^{-z}}{1 + e^{-z}} \\ = \beta_k \cdot P \cdot (1-P) \]
-\form#371:\[ \frac{\partial y}{\partial x_k} = \beta_k \frac{\sum_{i=1}^n P(y_i = 1)(1-P(y_i = 1))}{n}, \\ \text{where}, P(y_i=1) = g(X^{(i)}\beta) \]
-\form#372:\[ S( \boldsymbol c) = B( \boldsymbol c) M( \boldsymbol c) B( \boldsymbol c) \]
-\form#373:$ B( \boldsymbol c)$
-\form#374:$ M( \boldsymbol c)$
-\form#375:$ B( \boldsymbol c) $
-\form#376:\[ B( \boldsymbol c) = n\left(\sum_i^n -H(y_i, x_i, \boldsymbol c) \right)^{-1} \]
-\form#377:\[ M_{H} =\frac{1}{n} \sum_i^n \psi(y_i,x_i, \boldsymbol c)^T \psi(y_i,x_i, \boldsymbol c). \]
-\form#378:$ M( \boldsymbol c) $
-\form#379:$M $
-\form#380:$ (ij) $
-\form#381:$ \boldsymbol n $
-\form#382:$ \boldsymbol m $
-\form#383:$ \boldsymbol n \times m $
-\form#384:$ \boldsymbol \beta $
-\form#385:$ \boldsymbol t \in \mathbf R^{m} $
-\form#386:$ X \in \mathbf R^{m} $
-\form#387:$ R(t_i) $
-\form#388:$ t_i $
-\form#389:\[ P[T_k = t_i | \boldsymbol R(t_i)] = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}. \,. \]
-\form#390:\[ \mathcal L = \prod_{i = 1}^n \left( \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}} \right). \]
-\form#391:\[ L = \sum_{i = 1}^n \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right) \right]. \]
-\form#392:\[ \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii} \,. \]
-\form#393:$ \kappa(H) $
-\form#394:$(10^8 / m)$
-\form#395:$l(\boldsymbol \beta)$
-\form#396:$ H_0 $
-\form#397:$ H_1 $
-\form#398:$ \Gamma $
-\form#399:$ \gamma_0 \in \Gamma_0 $
-\form#400:$ \Gamma_0 \subsetneq \Gamma $
-\form#401:$ \Gamma_0 $
-\form#402:$ X_1, \dots, X_n \sim N(\mu, \sigma^2) $
-\form#403:$ H_0 : \mu \leq 0 $
-\form#404:$ H_0 : \mu = 0 $
-\form#405:$ \bar x $
-\form#406:$ s^2 $
-\form#407:\[ t = \frac{\sqrt n \cdot \bar x}{s} \]
-\form#408:$ (n - 1) $
-\form#409:$ \Pr[\bar X \geq \bar x \mid \mu = 0] $
-\form#410:$ \Pr[\bar X \geq \bar x \mid \mu \leq 0] $
-\form#411:$ \Pr[ |\bar X| \geq |\bar x| \mid \mu = 0] $
-\form#412:$ \mu_0 $
-\form#413:$ y_1, \dots, y_m $
-\form#414:$ X_1, \dots, X_n \sim N(\mu_X, \sigma^2) $
-\form#415:$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma^2) $
-\form#416:$ \mu_X, \mu_Y, $
-\form#417:$ H_0 : \mu_X \leq \mu_Y $
-\form#418:$ H_0 : \mu_X = \mu_Y $
-\form#419:$ \bar x, \bar y $
-\form#420:$ s_X^2, s_Y^2 $
-\form#421:\[ t = \frac{\bar x - \bar y}{s_p \sqrt{1/n + 1/m}} \]
-\form#422:\[ s_p^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2 + \sum_{i=1}^m (y_i - \bar y)^2} {n + m - 2} \]
-\form#423:$ (n + m - 2) $
-\form#424:$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] $
-\form#425:$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] $
-\form#426:$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] $
-\form#427:$ X_1, \dots, X_n \sim N(\mu_X, \sigma_X^2) $
-\form#428:$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma_Y^2) $
-\form#429:$ \mu_X, \mu_Y, \sigma_X^2, $
-\form#430:$ \sigma_Y^2 $
-\form#431:\[ t = \frac{\bar x - \bar y}{\sqrt{s_X^2/n + s_Y^2/m}} \]
-\form#432:\[ \frac{(s_X^2 / n + s_Y^2 / m)^2}{(s_X^2 / n)^2/(n-1) + (s_Y^2 / m)^2/(m-1)} \]
-\form#433:$ x_1, \dots, x_m $
-\form#434:$ y_1, \dots, y_n $
-\form#435:$ X_1, \dots, X_m \sim N(\mu_X, \sigma^2) $
-\form#436:$ Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) $
-\form#437:$ H_0 : \sigma_X < \sigma_Y $
-\form#438:$ H_0 : \sigma_X = \sigma_Y $
-\form#439:\[ f = \frac{s_Y^2}{s_X^2} \]
-\form#440:$ (m - 1) $
-\form#441:$ \Pr[F \geq f \mid \sigma_X = \sigma_Y] $
-\form#442:$ \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] $
-\form#443:$ 2 \cdot \min \{ p, 1 - p \} $
-\form#444:$ p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] $
-\form#445:$ n_1, \dots, n_k $
-\form#446:$ N = (N_1, \dots, N_k) $
-\form#447:$ p = (p_1, \dots, p_k) $
-\form#448:$ H_0 : p = p^0 $
-\form#449:$ n_i $
-\form#450:$ p^0_i $
-\form#451:$ p^0 = (\frac 1k, \dots, \frac 1k) $
-\form#452:$ (k - 1) $
-\form#453:$ n = \sum_{i=1}^n n_i $
-\form#454:\[ \chi^2 = \sum_{i=1}^k \frac{(n_i - np_i)^2}{np_i} \]
-\form#455:$ \Pr[X^2 \geq \chi^2 \mid p = p^0] $
-\form#456:$ \phi = \sqrt{\frac{\chi^2}{n}} $
-\form#457:$ \sqrt{\frac{\chi^2}{n + \chi^2}} $
-\form#458:$ X_1, \dots, X_m $
-\form#459:$ Y_1, \dots, Y_n $
-\form#460:$ F_X, F_Y $
-\form#461:$ H_0 : F_X = F_Y $
-\form#462:\[ d = \max_{t \in \mathbb R} |F_x(t) - F_y(t)| \]
-\form#463:$ F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| $
-\form#464:$ F_y $
-\form#465:$ k = (r + 0.12 + \frac{0.11}{r}) \cdot d $
-\form#466:$ r = \sqrt{\frac{m n}{m+n}}. $
-\form#467:$ d $
-\form#468:$ \Pr[D \geq d \mid F_X = F_Y] $
-\form#469:$ H_0 : \forall i,j: \Pr[X_i > Y_j] + \frac{\Pr[X_i = Y_j]}{2} = \frac 12 $
-\form#470:\[ z = \frac{u - \bar x}{\sqrt{\frac{mn(m+n+1)}{12}}} \]
-\form#471:$ u $
-\form#472:$ u = \min \{ u_x, u_y \} $
-\form#473:\[ u_x = mn + \binom{m+1}{2} - \sum_{i=1}^m r_{x,i} \]
-\form#474:\[ r_{x,i} = \{ j \mid x_j < x_i \} + \{ j \mid y_j < x_i \} + \frac{\{ j \mid x_j = x_i \} + \{ j \mid y_j = x_i \} + 1}{2} \]
-\form#475:$ m+n $
-\form#476:$ \Pr[Z \geq z \mid H_0] $
-\form#477:$ \Pr[|Z| \geq |z| \mid H_0] $
-\form#478:$ X_1, \dots, X_n $
-\form#479:$ \epsilon_i $
-\form#480:$ v_i $
-\form#481:$ v_{i-1} $
-\form#482:$ v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j $
-\form#483:$ w^+ = \sum_{i \mid x_i > 0} r_i $
-\form#484:$ w^- = \sum_{i \mid x_i < 0} r_i $
-\form#485:\[ r_i = \{ j \mid |x_j| < |x_i| \} + \frac{\{ j \mid |x_j| = |x_i| \} + 1}{2}. \]
-\form#486:$ w = \min \{ w^+, w^- \} $
-\form#487:$ w^+ $
-\form#488:$ w^- $
-\form#489:\[ z = \frac{w^+ - \frac{n(n+1)}{4}} {\sqrt{\frac{n(n+1)(2n+1)}{24} - \sum_{i=1}^n \frac{t_i^2 - 1}{48}}} \]
-\form#490:$ |x_i| $
-\form#491:$ \Pr[Z \geq z \mid \mu \leq 0] $
-\form#492:$ \Pr[ |Z| \geq |z| \mid \mu = 0] $
-\form#493:$ x_{1,1}, \dots, x_{1, n_1}, x_{2,1}, \dots, x_{2,n_2}, \dots, x_{k,n_k} $
-\form#494:$ X_{i,j} \sim N(\mu_i, \sigma^2) $
-\form#495:$ \mu_1, \dots, \mu_k $
-\form#496:$ H_0 : \mu_1 = \dots = \mu_k $
-\form#497:$ x_{i,j} $
-\form#498:$ n := \sum_{i=1}^k n_i $
-\form#499:$ \overline{x_i} $
-\form#500:$ s_i^2 $
-\form#501:$ \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2. $
-\form#502:$ \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2. $
-\form#503:$ (k-1) $
-\form#504:$ (n-k) $
-\form#505:$ s_b^2 := \frac{\mathit{SS}_b}{k-1} $
-\form#506:$ s_w^2 := \frac{\mathit{SS}_w}{n-k} $
-\form#507:\[ f = \frac{s_b^2}{s_w^2}. \]
-\form#508:$ \Pr[ F \geq f \mid H_0] $
-\form#509:\[ ||\boldsymbol A - \boldsymbol UV ||_2 \]
-\form#510:$ rank(\boldsymbol UV) \leq k $
-\form#511:$ ||\cdot||_2 $
-\form#512:$ k \leq rank(\boldsymbol A)$
-\form#513:$ m \times n $
-\form#514:$ m \times k $
-\form#515:$ k \times n $
-\form#516:$\frac{1}{num\_features}$
-\form#517:$q$
-\form#518:$d$
-\form#519:$ (\langle x,y\rangle + q)^d $
-\form#520:$\epsilon$
-\form#521:\[ \underset{w,b}{\text{Minimize }} \lambda||w||^2 + \frac{1}{n}\sum_{i=1}^n \ell(y_i,f_{w,b}(x_i)) \]
-\form#522:$(x_1,y_1),\ldots,(x_n,y_n)$
-\form#523:$\ell(y,f(x))$
-\form#524:$\ell(y,f(x)) = \max(0,1-yf(x))$
-\form#525:$\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)$
-\form#526:$ f_{w,b}(x) = \langle w, x\rangle + b$
-\form#527:\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \]
-\form#528:$ t $
-\form#529:$ 1 $
-\form#530:$ X_t $
-\form#531:$ q $
-\form#532:$ \phi(B) $
-\form#533:$ \theta(B) $
-\form#534:$ Y_{t} $
-\form#535:$ Y_{t} = (1-B)^{d}(X_{t} - \mu) $
-\form#536:$ d>0 $
-\form#537:$ Z_t $
-\form#538:\[ \phi(B) Y_t= \phi_1 Y_{t-1} + \dots + \phi_{p} Y_{t-p} \]
-\form#539:\[ \theta(B) Z_t = \theta_{1} Z_{t-1} + \dots + \theta_{q} Z_{t-q}. \]
-\form#540:$\tau, \epsilon_1, \epsilon_2, \epsilon_3,$
-\form#541:$\exp(-\gamma||x-y||^2)$
-\form#542:$\gamma$
-\form#543:$ (\langle x,y\rangle + q)^r $
-\form#544:$r$
-\form#545:$ PA = LDL* $
-\form#546:$ {\boldsymbol \Sigma}/(\sqrt{(N-1)} $
+\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:$\textit{acc}=(\textit{tp}+\textit{tn})/(\textit{tp}+\textit{tn}+\textit{fp} +\textit{fn})$
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/bed9253d/docs/latest/glm_8sql__in.html
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diff --git a/docs/latest/glm_8sql__in.html b/docs/latest/glm_8sql__in.html
index 7a8dbf3..90a198e 100644
--- a/docs/latest/glm_8sql__in.html
+++ b/docs/latest/glm_8sql__in.html
@@ -47,7 +47,7 @@
<td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.9</span>
+ <span id="projectnumber">1.9.1</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -1809,7 +1809,7 @@ Functions</h2></td></tr>
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a class="el" href="dir_68267d1309a1af8e8297ef4c3efbcdba.html">src</a></li><li class="navelem"><a class="el" href="dir_efbcf68973d247bbf15f9eecae7f24e3.html">ports</a></li><li class="navelem"><a class="el" href="dir_a4a48839224ef8488facbffa8a397967.html">postgres</a></li><li class="navelem"><a class="el" href="dir_dc596537ad427a4d866006d1a3e1fe29.html">modules</a></li><li class="navelem"><a class="el" href="dir_e502304d8bc7f1ef2456a474c6d92acd.html">glm</a></li><li class="navelem"><a class="el" href="glm_8sql__in.html">glm.sql_in</a></li>
- <li class="footer">Generated on Thu Apr 7 2016 14:24:10 for MADlib by
+ <li class="footer">Generated on Tue Sep 20 2016 11:27:01 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
</ul>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/bed9253d/docs/latest/graph_legend.html
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diff --git a/docs/latest/graph_legend.html b/docs/latest/graph_legend.html
index 3152b79..dcd5260 100644
--- a/docs/latest/graph_legend.html
+++ b/docs/latest/graph_legend.html
@@ -47,7 +47,7 @@
<td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.9</span>
+ <span id="projectnumber">1.9.1</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -174,7 +174,7 @@ A yellow dashed arrow denotes a relation between a template instance and the tem
<!-- start footer part -->
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
- <li class="footer">Generated on Thu Apr 7 2016 14:24:11 for MADlib by
+ <li class="footer">Generated on Tue Sep 20 2016 11:27:01 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
</ul>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/bed9253d/docs/latest/group__grp__arima.html
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diff --git a/docs/latest/group__grp__arima.html b/docs/latest/group__grp__arima.html
index 3ef557a..341d112 100644
--- a/docs/latest/group__grp__arima.html
+++ b/docs/latest/group__grp__arima.html
@@ -47,7 +47,7 @@
<td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.9</span>
+ <span id="projectnumber">1.9.1</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -372,29 +372,29 @@ SELECT * FROM arima_beer_forecast_output;
</ol>
</dd></dl>
<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd>An ARIMA model is an <em>a</em>uto-<em>r</em>egressive <em>i</em>ntegrated <em>m</em>oving <em>a</em>verage model. An ARIMA model is typically expressed in the form <p class="formulaDsp">
-<img class="formulaDsp" alt="\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \]" src="form_527.png"/>
+<img class="formulaDsp" alt="\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \]" src="form_529.png"/>
</p>
</dd></dl>
-<p>where <img class="formulaInl" alt="$B$" src="form_205.png"/> is the backshift operator. The time <img class="formulaInl" alt="$ t $" src="form_528.png"/> is from <img class="formulaInl" alt="$ 1 $" src="form_529.png"/> to <img class="formulaInl" alt="$ N $" src="form_218.png"/>.</p>
+<p>where <img class="formulaInl" alt="$B$" src="form_206.png"/> is the backshift operator. The time <img class="formulaInl" alt="$ t $" src="form_530.png"/> is from <img class="formulaInl" alt="$ 1 $" src="form_531.png"/> to <img class="formulaInl" alt="$ N $" src="form_219.png"/>.</p>
<p>ARIMA models involve the following variables:</p><ul>
-<li>The values of the time series: <img class="formulaInl" alt="$ X_t $" src="form_530.png"/>.</li>
-<li>Parameters of the model: <img class="formulaInl" alt="$ p $" src="form_110.png"/>, <img class="formulaInl" alt="$ q $" src="form_531.png"/>, and <img class="formulaInl" alt="$ d $" src="form_467.png"/>; <img class="formulaInl" alt="$ d $" src="form_467.png"/> is the differencing order, <img class="formulaInl" alt="$ p $" src="form_110.png"/> is the order of the AR operator, and <img class="formulaInl" alt="$ q $" src="form_531.png"/> is the order of the MA operator.</li>
-<li>The AR operator: <img class="formulaInl" alt="$ \phi(B) $" src="form_532.png"/>.</li>
-<li>The MA operator: <img class="formulaInl" alt="$ \theta(B) $" src="form_533.png"/>.</li>
-<li>The lag difference: <img class="formulaInl" alt="$ Y_{t} $" src="form_534.png"/>, where <img class="formulaInl" alt="$ Y_{t} = (1-B)^{d}(X_{t} - \mu) $" src="form_535.png"/>.</li>
-<li>The mean value: <img class="formulaInl" alt="$ \mu $" src="form_286.png"/>, which is set to be zero for <img class="formulaInl" alt="$ d>0 $" src="form_536.png"/> and estimated from the data when d=0.</li>
-<li>The error terms: <img class="formulaInl" alt="$ Z_t $" src="form_537.png"/>.</li>
+<li>The values of the time series: <img class="formulaInl" alt="$ X_t $" src="form_532.png"/>.</li>
+<li>Parameters of the model: <img class="formulaInl" alt="$ p $" src="form_110.png"/>, <img class="formulaInl" alt="$ q $" src="form_533.png"/>, and <img class="formulaInl" alt="$ d $" src="form_468.png"/>; <img class="formulaInl" alt="$ d $" src="form_468.png"/> is the differencing order, <img class="formulaInl" alt="$ p $" src="form_110.png"/> is the order of the AR operator, and <img class="formulaInl" alt="$ q $" src="form_533.png"/> is the order of the MA operator.</li>
+<li>The AR operator: <img class="formulaInl" alt="$ \phi(B) $" src="form_534.png"/>.</li>
+<li>The MA operator: <img class="formulaInl" alt="$ \theta(B) $" src="form_535.png"/>.</li>
+<li>The lag difference: <img class="formulaInl" alt="$ Y_{t} $" src="form_536.png"/>, where <img class="formulaInl" alt="$ Y_{t} = (1-B)^{d}(X_{t} - \mu) $" src="form_537.png"/>.</li>
+<li>The mean value: <img class="formulaInl" alt="$ \mu $" src="form_287.png"/>, which is set to be zero for <img class="formulaInl" alt="$ d>0 $" src="form_538.png"/> and estimated from the data when d=0.</li>
+<li>The error terms: <img class="formulaInl" alt="$ Z_t $" src="form_539.png"/>.</li>
</ul>
<p>The auto regression operator models the prediction for the next observation as some linear combination of the previous observations. More formally, an AR operator of order <img class="formulaInl" alt="$ p $" src="form_110.png"/> is defined as</p>
<p class="formulaDsp">
-<img class="formulaDsp" alt="\[ \phi(B) Y_t= \phi_1 Y_{t-1} + \dots + \phi_{p} Y_{t-p} \]" src="form_538.png"/>
+<img class="formulaDsp" alt="\[ \phi(B) Y_t= \phi_1 Y_{t-1} + \dots + \phi_{p} Y_{t-p} \]" src="form_540.png"/>
</p>
-<p>The moving average operator is similar, and it models the prediction for the next observation as a linear combination of the errors in the previous prediction errors. More formally, the MA operator of order <img class="formulaInl" alt="$ q $" src="form_531.png"/> is defined as</p>
+<p>The moving average operator is similar, and it models the prediction for the next observation as a linear combination of the errors in the previous prediction errors. More formally, the MA operator of order <img class="formulaInl" alt="$ q $" src="form_533.png"/> is defined as</p>
<p class="formulaDsp">
-<img class="formulaDsp" alt="\[ \theta(B) Z_t = \theta_{1} Z_{t-1} + \dots + \theta_{q} Z_{t-q}. \]" src="form_539.png"/>
+<img class="formulaDsp" alt="\[ \theta(B) Z_t = \theta_{1} Z_{t-1} + \dots + \theta_{q} Z_{t-q}. \]" src="form_541.png"/>
</p>
<p>We estimate the parameters using the Levenberg-Marquardt Algorithm. In mathematics and computing, the Levenberg-Marquardt algorithm (LMA), also known as the damped least-squares (DLS) method, provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function.</p>
-<p>Like other numeric minimization algorithms, LMA is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector, $p$, as well as some tuning parameters <img class="formulaInl" alt="$\tau, \epsilon_1, \epsilon_2, \epsilon_3,$" src="form_540.png"/>.</p>
+<p>Like other numeric minimization algorithms, LMA is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector, $p$, as well as some tuning parameters <img class="formulaInl" alt="$\tau, \epsilon_1, \epsilon_2, \epsilon_3,$" src="form_542.png"/>.</p>
<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
<p>[1] Rob J Hyndman and George Athanasopoulos: Forecasting: principles and practice, <a href="http://otexts.com/fpp/">http://otexts.com/fpp/</a></p>
<p>[2] Robert H. Shumway, David S. Stoffer: Time Series Analysis and Its Applications With R Examples, Third edition Springer Texts in Statistics, 2010</p>
@@ -406,7 +406,7 @@ SELECT * FROM arima_beer_forecast_output;
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