T Jxtu

T¸ 0 ju(t)j < r u, the solution exists for all t ¸ 0 and furthermore satises jx(t;Ã)j · ¯(kÃk c;t) °(ku 0;t k 1) (13) Denition 26 The system (11) is inputtostate stable if there exist a classKL function¯ and a classK function° such that, for any initial stateà and.

T jxtu. Applicable, and indeed much used, in cross sectional studies In vector form the equation (1) can be written (2) Y = Xµ E ;. X t, X t 2X s N µ(t s), (t s) W t Tuesday, October 23, 12 Definition of BM • A process indexed by t for t>=0 is a Brownian motion if , and for every t and s (s. 2 Question ( marks) (a) Given that the Inverse Fourier Transform of a signal xt) is expressed as X(t) = J xwe but du 2 s Verify the differentiation property of the Fourier Transform that is stated as dx(t) jaX (0) dt (10) (b) Determine the Fourier transform of a Gaussian pulse signal a> 0 and plot clearly labelled graphs of X(t) and X(w).

µ » U0;T { If X(t) is widesense cyclostationary then Xs(t) is WSS ES150 { Harvard SEAS 15 Time averages and ergodicity † Sometimes we need to estimate the parameters of a random process through measurement † A quantity obtainable from measurements is the ensemble average For example, an estimate of the mean is m^X. T KMEANS CLUSTERINGFor all j ∈ K, initialize cluster centroids ˆr0 j randomly and set m = 1 Repeat until convergence (or until patience runs out) 1 For each t ∈ {1,,n}, set cluster identity of the point ˆcm(x t) = argmin j∈K xt − ˆrm−1 j 2 For each j ∈ K, set new representative as ˆrm j = 1 Cˆ m j xt ∈Cˆm j xt 3 m ← m 1. E(x i x j x k)=µ i µ j µ k.

Which is the same as E Z t 0 λσ (u)dB (u)− 1 2 Z t 0 (λσ 2(u)) du = 1 , The second integral is not a random variable, whence EeλI (t) = exp 1 2 λ2 Z t 0 σ2(u)du As a function of λ, the LHS is the moment generating function of I(t), while the RHS is. 2 In x3 we study a certain Bessel function J”(t) solving (11)Results of x2 guaran tee that J”(t) = t¡”J”(t) has a convergent power series P1 k=0 akt 2k, and we derive a recursion formula for the coe–cients akWe produce a solution to this recursion, and hence deflne J”(t)The solution involves the gamma function ¡(z), and wemake use of results on ¡(z) given in Appendix B. Lecture 1 11 Introduction A time series is a set of observations xt, each one being recorded at a specific time t Definition 11 A time series model for the observed data {xt} is a specifi cation of the joint distributions (or possibly only the means and covariances) of a sequence of random variables {Xt} of which {xt} is postulated to be a realization.

Tests for identiflability In another important early linear algebraic efiort, Reid 34 deflned the term sensitivity identiflabilityIf z(µ) denotes the output of a model depending on a parameter vector µ, then Reid explains sensitivity identiflability in the following way Let ¢µ denote a local perturbation about a nominal µ0, ie, ¢µ = µ ¡ µ0, which gives rise to local. 85 E(t)= (t)0 e z, (47) where !(t) is the surface charge density on a plate and ez is the unit vector parallel to the wires If Q(t) is the charge on a plate at time t, then (t)=Q(t) "a2 It "a2, (48) and E(t)= It!" 0 a 2 e z (49) b) We define the displacement current density J d with J d=!. Title (Relat rio Focus) Author dstatluciana Created Date 7/31/ PM.

φt(j)Xt−j = εt − Xq j=1 θt(j)εt−j (1) where Xt = X˜t −µt, and {εt} is a sequence of noise random variables with mean zero and scale σt such that {δt = σ−1 t εt} is independent and identically distributed The notation in (1) is consistent with that of Box and Jenkins (1976) The model parameters φt(j), θt(j), µt, and σt. Feb 01, 13 · Indeed, the fractional integrator technique tells us that 1 E C j (t ) = C j z 2 (t ) j 2 So, the global energy of the FDE is expressed as (46) x(t ) = 0 µ n ( ) z ( , t ) d is the weighted integral of the E C (t ) = E j =0 J Cj (t ) = j =0 J 1 C j z 2 (t ) j 2 (47) distributed variable z ( , t ) , so x(t ) is only a pseudo state variable. C LASSICAL S TOCHASTIC D IFFERENTIAL C ONTROL inf ↵2A E Z T 0 f(t,Xt,↵t)dt g(XT,µT) subject to dXt = b(t,Xt,↵t)dt (t,Xt,↵t)dWt;.

T µk2 cosine(w,x t µ) On average On average orthogonal parallel PCA VARIANCE MAXIMIZATION Pick directions along which data varies the most First principal component w xt = µ d j=1 ytjwj where w 1,. X 0 = x 0 I Analytic Approach (by PDEs) I HJB equation I Probabilistic Approaches (by FBSDEs) 1 Represent value function as solution of a BSDE 2 Represent the gradient of the value function as solution of a. 0 "E "t, (410) and use it in the integral equation defining Ampère’s Law (ie, equation (323.

Simple Example • At date T we are asked to construct the distribution of Y, Y = 2 4 yT1 yTf 3 5, subject to a specific sequence of values of xt X = 2 4 xT1 xTf 3 5 • We expect that Y has a multivariate Normal distribution with mean a function of X and variance a function of the model. Yj = f(tj;µ) = Cx(tj;µ);. I i “tsa4_trimmed” — 17/12/8 — 1501 — page 78 — # i i i i i i 78 3 ARIMA Models where wt is white Gaussian noise with 2 w = 1 We have now assumed the current value is a particular linear function of past values The regularity that persists in.

Hansen’s two step GMM procedure⁄ Let xt be an s£1 vector of variables that are observed at date t, let µ denote the m£1 un known parameter vector, and let ut = u(xt;µ) be an r£1 covariance stationaryy vector val ued function, such that for true parameter value µ0 (1) Eut = Eu(xt;µ0) = 0In GMM function u(x;µ) deflne the moment or more generally the orthogonality condi. Nov 11, 16 · I Can always define process Y(t)=X(t)µ X(t)withµ Y (t)=0 I In such case C(t1,t2)=R(t1,t2) I Autocorrelation is a function of two variables t1 and t2 I Autocorrelation is a symmetric function R(t1,t2)=R(t2,t1) Stoch Systems Analysis Queues 15. In the usual fashion If xt and et may be correlated, one will obtain a consistent estimator by using instrumental variables (IV) estimation.

E(x i)=µ i (266) !. X (t µ) ¸ T · P 11 ¿ P 12 (µ) ¿ P T 12 (µ) ¿ P 22 (µ) ¸· x (t) x (t µ) ¸ dµ Z 0 ¡ ¿ Z 0 ¡ ¿ x T (t µ)R (µ;»)x (t »)dµ d» (4) Methodologies for constructing functionals of the abo ve type ha ve been de veloped based on solving related semidenite programmes P. ¼½ q t G / j t % j¨³ j / / / /©½ j£/ t ¸ T ¦8 5 L§ »x t \µ\¿T t¦8 m G t j ²Á² \ / t \ t m°Â j t m A ª K j¹ j q t K ªA ³ j£' L à t£x t / W v G K % t j K¨ ¨ ÅÄj¦/ t ² G x K /.

“SHORTHAND” THEORY – A SUMMARY • Cost function expressions with J 0 (x) ≡ 0 J π (x) = lim (T µ 0 T µ 1 ··· T k µ k J 0)(x), J µ (x. Reading and bibliography 1 N CesaBianchi and G Lugosi Prediction, learning, and games Cambridge University Press, 06 2 S Bubeck and N CesaBianchi. αtg(x t,µ(x t)) x 0 = x # Here, α∈(0,1) is the discount factor The expectation is taken under the assumption that actions are selected according to the policy µ In other words, at each time t, a t,µ(x t) Denote by P µ∈RX×Xthe transition probability matrix for the policy µ, whose (x,x0)th entry is P µ(x)(x,x 0) Denote by g.

(TJ)(x)= minE & g(x,u,w)αJ f(x,u,w) , ∀ x u∈U(x) w # $’ TJis the optimal cost function for the onestage problem with stage cost g and terminal cost αJ • For any stationary policy µ (T µ J)(x)=E g x,µ(x),w αJ f(x,µ(x),w) , ∀ x w & # $ # $’ 2. ∗ TJ Prob = 1 Prob = 1 J J TJ 45 Degree Line Prob = 1 Prob = J J ∗ = TJ ∗ 0 Prob = 1 1 J J J J µ1 = Tµ1Jµ1 Policy Improvement Exact Policy Evaluation Approximate Policy Evaluation Policy Improvement Exact Policy Evaluation Approximate Policy Evaluation TJ T µ1J J Policy Improvement Exact Policy Evaluation (Exact if J0 J0 J0 J0. E(x i x j)=µ i µ j " ij (267) !.

Approach1Linearization Keyidea(eg,Smithetal,1962;Ohab& Stubberud,1965) 1 Locally linearize f around mean µt 2 Compute predictive distribution (Gaussian) for. Xs(t) = X(tµ) ;. Tj This method suggests that, by continuing to iterate backward, and provided that < 1 and xt is stationary, we can represent an AR(1) model as a linear process given by1 xt = X1 j=0 jw tj (36) 1Note that lim k!1 E ⇣ xt Pk j=0 jw tj ⌘ 2 =limk!1 2k 2 k =0, so (36) exists in the mean square sense (see Appendix A for a definition).

E(X(t)) = µ, independent of t RX(t1,t2) is a function only of the time difference t2 −t1 aiajR(ti −tj) ≥ 0 To see why this is necessary, recall that the correlation matrix for a random vector must be nonnegative definite, so if we take a set of n samples from the. April 10, 04 2216 WSPC/191IJHR International Journal of Humanoid Robotics Vol 1, No 1 (04) 29–43 c World Scientific Publishing Company WHOLE. ⇥(a)/⇥t = J a Da/Dt = J⇤ a ˙⇥ a J a au J a diusion flux Lagrangian form of conservation equations convection flux ( a)/⇤t = J a ˙⇥ a D Dt t material (convective) derivative u mass is a conserved scalar /t = J 1 D Dt = u continuity equation (classical mechanics)!.

JNi(t)j denote the degree (number of neighbors) of node i at time t In this paper, the sequence of communication graphs fG(t)g1 t=0 can be either deterministic or stochastic Let Gi = (Ei;V), i = 1;;r, denote a nite collection of graphs with common vertex set V Their union is a graph G with the same vertex set and a edge set that is the. Filtering Random Processes Let X(t,e) be a random processFor the moment we show the outcome e of the underlying random experiment Let Y(t,e)=LX(t,e) be the output of a linear system when X(t,e) is the input Clearly, Y(t,e) is an ensemble of functions selected by e, and is a random process What can we say about Y when we have a statistical description of X and a description. (definition of the diusion flux in the equation for.

T } J t } v } J ï X & Ì } } X } u } M K o J v } r µ o X t > } µ u u } v M t o } µ J o W } µ P o J t E } } W } } o M t E } U > } X u u } > } X } X. Ex i kx j lx k (mL)="k "t i k "l "t j l "m "t k m LM N t=0 That leads to the moments of the elements of the vector x (265) !. Cw(t i,t j)=Ew(t)w∗(t) = 0 for t = t The autocovariance must be of the form Cw(t i,t j)=q(t)δ(t − t)whereq(t)=Ew(t)2 ≥ 0 is the meansquared value at time ti Unless specifically stated to be otherwise, it is assumed that the mean value of white noise is zero In that case, Rw (ti,tj)=Cw i j) Examples A coin tossing sequence.

(spherically symmetric), so it is reasonable to believe that standard Brownian motion is also rotationally invariant if O is a 2 2 orthogonal matrix and Z is a BM in R2, then so is OZ We can also consider scaling if Bt is a Brownian motion, then σBt is equal in distribution to Bσ2tThis is true. Sup1•t•n E(ju tj qjF ¡1) < 1 for some q > 2 This assumption is satisfled by a variety of data generating processes Under condition (a), (xt) is predetermined The condition can simply be met by choosing the natural flltration (ut;xt1) for (Ft) The martingale difierence assumption for the regression errors. T(J re) (x) = (TJ() x) αr, ∀ x, T µ (J re) (x) = (T µ J)(x) αr, ∀ x, where e is the unit function e(x) ≡ 1 (holds for most DP models) • A third important property that holds for some (but not all) DP models is that T and T.

1 Introduction This paper analyzes nonlinear least squares estimation of the nonlinear cointegrating regression yt = f(xt;µ0)ut (1) where ut is assumed to be stationary and the scalar xt = Pt j=1 vt is assumed to be an inte grated process For the case of a linear response function f(x;µ) = x0µ, the analysis of such cointegrating regressions is wellestablished. 3 General features of ecological/environmental time series Examples 1 Mauna Loa (CO 2,, Oct `58Sept `90) CO2 1960 1970 1980 1990 3 330 340 350. Machine Learning Srihari Topics 1 Multivariate Gaussian Basic Parameterization 2 Covariance and Information Form 3 Operations on Gaussians.

For statement (d) notice that 1 T2 XT t=1 y 2 t¡1et = 1 T2 XT t=1 y2 t¡1Ee 2 t 1 T2 XT t=1 y 2 t¡1 et ¡Ee 2 t ¢ The flrst term converges to!2(Ee2 t) R1 0 J2 c (s)ds, while the second term is negligible Indeed, according to direct generalization of Theorems 42 and 44 in Hansen (1992). (2) where the solutions x(t;µ) in general depend on unknowns q and the (possibly unknown) initial conditions x0 so that µ = (q;x0) In addition to computing estimates µ^ for the unknown parameters using observations fyjg, it is widely accepted that quantifying. Dt = g (xt) µ t J÷(xt 1,rt) J÷(xt,rt) (4) The parameter vector is then updated according to rt 1 = rt %tdt$ (xt), (5) where the components of r0 are initialized to arbitrary values and %t is a sequence of scalar step sizes The average rew ard estimate is updated according to µ t 1 = (1 %t)µ t %tg (xt), where µ 0 is an.

(TJ)(x)= min u∈U(x) E w g(x,u,w)αJ f(x,u,w), ∀x TJ is the optimal cost function for the onestage problem with stage cost g and terminal cost αJ • For any stationary policy µ (TµJ)(x)=E w g x,µ(x),w αJ f(x,µ(x),w), ∀x.