Teletrac modeling and estimation
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1 Teletrac modeling and estimation File 1 José Roberto Amazonas Alexandre Barbosa de Lima jra,ablima@lcs.poli.usp.br Telecommunications and Control Engineering Dept. - PTC Escola Politécnica University of São Paulo - USP São Paulo 2009 JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
2 Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
3 Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
4 Introduction Motivation Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
5 Introduction Motivation Telecommunications carriers objectives Integration of voice, data, video, image and data over a single packet switched convergent network. This network is also known as: integrated services, multi-service or next generation network. Requirements: acceptable performance for each kind of service; operational costs reduction; exibility to support current and future services; dynamic bandwidth allocation; integrated transport of all kinds of information; ecient usage of network resources by means of statistical multiplexing. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
6 Introduction Motivation Quality of service - QoS The convergent network's core consists of a single IP infrastructure supporting Quality of Service (QoS), virtual private networks and IP protocols v4 and v6; QoS = f(pdt, PDV, THRU, PLR); IETF QoS implementation proposals: MPLS constraint-based routing trac engineering IntServ DiServ JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
7 Introduction Motivation Trac control The implementation of mechanisms to monitor and to control the teletrac is a must for an ecient operation of convergent networks. Without trac control, the unbounded demand of shared resources (buers, band and processors) may seriously degrade the network performance. The trac control is necessary to protect the QoS perceived by the users and to ensure the ecient usage of network resources. Trac control functions: a) CAC - Call Admission Control b) UPC - Usage Parameter Control c) Priority control d) Congestion control Functions a), c) and d) require on-line trac prediction. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
8 Introduction Motivation Trac characteristics Measurements have shown that the data network's aggregate trac has statistical properties quite dierent from the existing trac in the PSTN. Data trac is the time series of the bytes count per unit of time. A. K. Erlang showed in 1909 that the number of generated telephone calls in a certain time interval can be modeled by the Poisson process. By contrast, the data trac traces (series) have fractal properties as long range dependence (LRD) or self-similarity, and impulsiveness (great variability), in several time aggregation scales, that are not captured by the Poisson process. The trac in some LANs and WANs is extremely impulsive because it has a heavy tail marginal distribution, i. e., non-gaussian distributions. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
9 Introduction Motivation Trac characteristics - 2 It is a known fact that the buer overow probability, with just one server working at a constant service rate, under Markovian trac is an exponential function of the buer size. an increase of the buer size produces a signicative decrease of the PLR. On the other hand, data trac long memory and impulsiveness degrade the network's performance (PLR increases) because the the buer overow probability in systems under fractal trac is a power function of the buer size, i. e., a buer queue hyperbolic decay much slower than the Markovian case. Self-similar trac produces buer overow much more frequently than the Markovian trac. Such phenomena have been observed in the beginning of the 1990 decade in ATM networks implemented with small buers (10 thru 100 cells). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
10 Introduction Objectives and contributions Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
11 Introduction Objectives and contributions Questions to be answered How can we make prediction of signals that present LRD? How can we apply the theory of time series prediction to the telecommunications network real trac? Is teletrac a linear or non-linear signal? Is it Gaussian or non-gaussian? How should we make the model's parameters estimation? How can we identify the adequate model? If the prediction can be made using a linear estimation theory, what are the available techniques? JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
12 Introduction Objectives and contributions Contributions Development of a new fractal teletrac model in the state space. Proposal of a LRD teletrac statistical analysis methodology. Demonstration that long memory signals modeling by means of high-order AR processes may be unfeasible in practice. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
13 Time series examples Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
14 Time series examples Johnson & Johnson quarterly earnings per share Quarterly earnings per share Time JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
15 Time series examples Global warming Yearly average global temperature deviations Time JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
16 Time series examples Speech signal: aaa...hhh Speech signal Time JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
17 Time series examples New York Stock Exchange Returns of the NYSE Time JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
18 Time series examples New York Stock Exchange Earthquake EQ Time Explosion EXP Time JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
19 Operators notation Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
20 Operators notation Notation (a) 1 sample delay operator, denoted by B: Bx t = x t 1. (1) (b) m samples delay operator, denoted by B m, m Z: B m x t = x t m. (2) The impulse response h t of the m samples delay system is: h t = δ t m, (3) and its transfer function, dened as the z transform(h(z) = t= h t z t ) of the impulse response is: H(z) = z m. (4) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
21 Operators notation Notation 2 (c) dierence or backshift operator x t = x t x t 1 = (1 B)x t, (5) B j (x t ) = x t j and j (x t ) = ( j 1 (x t )), j 1, with 0 (x t ) = x t 2 x t = ( x t ) = (1 B)(1 B)x t = (1 2B + B 2 )x t = x t 2x t 1 + x t 2. The dierence operator has impulse response h t = δ t δ t 1, (6) and transfer function H(z) = (1 z 1 ). (7) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
22 Operators notation Notation 3 (d) summation operator or integrator lter, denoted by S: Sx t = x t i = x t + x t 1 + x t = i=0 = (1 + B + B )x t = (1 B) 1 x t = 1 x t. (8) The integrator lter's transfer function corresponds to the inverse of the system's function dened by (7), i. e., H(z) = (1 z 1 ) 1. (9) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
23 Stochastic processes Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
24 Stochastic processes Denitions Denition (Stochastic Process) Let T be an arbitrary set. A stochastic process is a family {x t, t T }, such that, for each t T, x t is a random variable. When the set T is the set of integer numbers Z, then {x t } is a discrete time stochastic process (or random sequence); {x t } is a continuous time stochastic process if T is taken as the set of real numbers R. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
25 Stochastic processes Denitions 2 The random variable x t is, in fact, a function of two arguments x(t, ζ), t T, ζ Ω, given that it is dened over the sample space Ω. For each ζ Ω we have a realization, trajectory or time series x t. The set of all realizations is called ensemble. Each trajectory is a function or a non-random sequence and for each xed t, x t is a number. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
26 Stochastic processes Probability distribution functions A process x t is completely specied by its nite-dimensional distributions or n-order probability distribution functions. F x (x 1, x 2,..., x n ; t 1, t 2,..., t n ) = P{x(t 1 ) x 1, x(t 2 ) x 2,..., x(t n ) x n } (10) in which t 1, t 2,..., t n are any elements of T and n 1. The rst order probability distribution function is also known as Cumulative Distribution Function - CDF. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
27 Stochastic processes Probability density function The probability density function - PDF is given by: f x (x 1, x 2,..., x n ; t 1, t 2,..., t n ) = n F x (x 1, x 2,..., x n ; t 1, t 2,..., t n ) x 1 x 2... x n. Applying the conditional probability density formula, (11) f x (x k x k 1,..., x 1 ) = f x(x 1,..., x k 1, x k ), (12) f x (x 1,..., x k 1 ) in which f x (x 1,..., x k 1, x k ) denotes f x (x 1,..., x k 1, x k ; t 1,..., t k 1, t k ), repeatedly over f x (x 1,..., x n 1, x n ) we get the probability chain rule f x (x 1, x 2,..., x n ) = f x (x 1 )f x (x 2 x 1 )f x (x 3 x 2, x 1 )... f x (x n x n 1,..., x 1 ). (13) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
28 Stochastic processes Independent and IID process When x t is a sequence of mutually independent random variables, (13) can be rewritten as f x (x 1, x 2,..., x n ) = f x (x 1 )f x (x 2 )... f x (x n ). (14) Denition (Purely Stochastic Process) A purely stochastic process {x t, t Z} is a sequence of mutually independent random variables. Denition (Independent and Identically Distributed Process) An Independent and Identically Distributed (IID) process {x t, t Z}, denoted by x t IID, is a purely stochastic and identically distributed process. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
29 Stochastic processes Stationarity Denition (Strict Sense Stationarity) A random process x t is stationary in the strict sense if F x (x 1, x 2,..., x n ; t 1, t 2,..., t n ) = F x (x 1, x 2,..., x n ; t 1 +c, t 2 +c,..., t n +c), for any c. (15) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
30 Stochastic processes Low order moments The mean µ x (t) of x t is the expected value of the random variable x t : µ x (t) = E[x t ] = xf x (x; t)dx, (16) in which f x (x; t) is the rst order probability density function of x t. The autocorrelation R x (t 1, t 2 ) of x t is the expected value of the product x t1 x t2 : R x (t 1, t 2 ) = E[x t1 x t2 ] = x 1 x 2 f x (x 1, x 2 ; t 1, t 2 )dx 1 dx 2, (17) in which f x (x 1, x 2 ; t 1, t 2 ) is the second order probability density function of x t. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
31 Stochastic processes Low order moments 2 The autocovariance C x (t 1, t 2 ) of x t is the covariance of the random variables x t1 and x t2 : C x (t 1, t 2 ) = R x (t 1, t 2 ) µ t1 µ t2. (18) The correlation coecient ρ x (t 1, t 2 ) of x t is the ratio: ρ x (t 1, t 2 ) = C x (t 1, t 2 ) Cx (t 1, t 1 )C x (t 2, t 2 ). (19) Many authors refer to ρ x (t 1, t 2 ) as Autocorrelation Function (ACF). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
32 Stochastic processes Wide Sense Stationarity Denition (Wide Sense Stationarity) A random process x t is wide sense stationary if its mean is constant E[x t ] = µ x, (20) and if its autocorrelation depends only on the lag τ = t 2 t 1 : R x (t 1, t 2 ) = R x (t 1, t 1 + τ) = R x (τ). (21) Observe that (21) implies an autocovariance that depends on the lag, i. e., C x (τ). In this case, the stationary process' variance is constant and given by σ 2 x = C x (0). (22) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
33 Stochastic processes Non-stationarity Real series generally present the following types of non-stationarity: (a) level-based non-stationarity: the series oscillate around and average level during a certain time and then jump to another level. The rst dierence makes these series stationary. (b) slope-based non-stationarity: the series oat around a straight line, with positive or negative slope. The second dierence makes these series stationary. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
34 Stochastic processes White noise Denition (White Independent Noise) A process w t IID is a White Independent Noise WIN when it has mean µ w and variance σ 2 w, w t WIN(µ w, σ 2 w). Denition (White Noise) A sequence {w t, t Z} of non-correlated with mean µ w and variance σ 2 w is called White Noise WN, w t WN(µ w, σ 2 w). Observation (White Gaussian Noise) A White Gaussian Noise (WGN) w t is a WIN and denoted by w t N (µ w, σ 2 w), in which N stands for the probabilities normal (Gaussian) distribution. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
35 Stochastic processes Sample mean and autocovariance Consider a stationary process x t. The sample mean of a realization x t with N points is given by x = 1 N N x t, (23) t=1 the sample autocovariance of lag τ by Ĉ τ = 1 N (x t x)(x t τ x). (24) N t=τ+1 JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
36 Stochastic processes Sample autocorrelation and variance The sample autocorrelation (SACF) of lag τ by in which Ĉ 0 (also denoted as s 2 x ) ˆρ τ = Ĉτ Ĉ 0, (25) Ĉ 0 = s 2 x = 1 N N (x t x) 2 (26) t=1 is the sample variance of x t. The sample variance can also be dened as s 2 x = 1 N 1 N t=1 (x t x) 2. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
37 Stochastic processes Ergodicity Denition (Ergodicidade) A stationary process x t is called ergodic if its main moments converge in probability a to the population's moments, i. e., if x p p µ, Ĉ τ C τ e p ˆρ τ ρ τ. a We say that the sequence {x1, x2,..., xn,...} converges in probability to x if lim P( xn x ɛ) = 0 for all ɛ > 0. n JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
38 Time Series Modeling Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
39 Time Series Modeling Introduction A time seriesx t modeling consists on estimating an invertible function h(.), called model of x t, such that in which w t IID and x t = h(..., w t 2, w t 1, w t, w t+1, w t+2,...), (27) g(..., x t 2, x t 1, x t, x t+1, x t+2,...) = w t, (28) in which g(.) = h 1 (.). The process w t is the innovation at instant t and represents the new information about the series that is obtained at instant t. In practice, the adjusted model is causal, i. e., x t = h(w t, w t 1, w t 2,...). (29) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
40 Time Series Modeling Model construction methodology The model construction methodology is based on the iterative cycle illustrated by the following steps: (a) a models general class is considered for analysis (specication); (b) there is the identication of a model, based on statistical criteria; (c) it follows the estimation phase, in which the model's parameters are obtained. In practice, it is important that the model is parsimonious 1 and (d) at last, there is the diagnostic of the adjusted model by means of a statistical analysis of the series of residues w t (is w t compatible with a WN?) 1 We say that a model is parsimonious when it uses few parameters. The use of an excessive number of parameters is undesirable because the uncertainty degree of the statistical inference procedure increases with the number of parameters. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
41 Time Series Modeling Model construction methodology 2 Postulate general class of the model Model Identification Estimation of parameters Diagnosis No Yes Figure: Box-Jenkins' iterative cycle. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
42 Time Series Modeling Linear process The process x t of (29) is linear when it corresponds to the convolution of a process w t IID and a deterministic sequence h t x t = h t w t = h k w t k k=0 = w t + h 1 w t 1 + h 2 w t (30) = (1 + h 1 B + h 2 B )w t = H(B)w t in which the symbol denotes the convolution operation and h 0 = 1. Eq. (30) is also known as the innite order moving average (MA( )) representation. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
43 Time Series Modeling ARMA model The linear lter general format of (30) is x(t) = p φ k x(t k) + w(t) k=1 q θ k w(t k). (31) k=1 The sequence h t is called impulse response of (31), also known as ARMA model of orders p and q (ARMA(p, q)). In a more compact format, we have φ(b)x t = θ(b)w t, (32) in which φ(b) is the order p auto-regressive operator φ(b) = 1 φ 1 B φ 2 B 2... φ p B p (33) and θ(b) is the q order moving average operator θ(b) = 1 θ 1 B θ 2 B 2... θ q B q. (34) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
44 Time Series Modeling ARMA model 2 If all poles and zeros of the transfer function H(z) = h k z k = θ(z) φ(z) k=0 (35) of the lter (32) are inside the unity radius circle φ(z) = 0, z < 1 (36) θ(z) = 0, z < 1 (37) then the process x t given by (30) is stationary (or non-explosive) and invertible, respectively. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
45 Time Series Modeling α-stable distribution A random variable X with α-stable distribution has a heavy tail and is dened by the following characteristic function: Z Φ X (w) = E[e jwx ] = f X (x)e jwx dx = exp{jµw σw α [1 jη sign(w)ϕ(w, α)]}, (38) in which, 8 >< tan (απ/2) se α 1 ϕ(w, α) = (39) >: 2 π ln w se α = 1, and sign(.) is the sign function, α (0 < α 2) is the characteristic exponent, µ (µ R) is the localization parameter, η ( 1 η 1) is the asymmetry parameter and σ 0 is the dispersion parameter or scale. The variance of X is innite for 0 < α < 2. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
46 Time Series Modeling Non-Gaussian process When the innovations in (30) are random variables with α-stable distribution, (30) denes an innite variance non-gaussian process. This fact is justied by the generalized central limit theorem, which states that, if the limit of an IID random variables sum converges, then this limit can only be a random variable with a stable distribution (the normal distribution is a particular case of the stable distribution 2 ). 2 When α = 2. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
47 Time Series Modeling Non-linearity and non-gaussianity If the innovations in (30) have nite variance, i. e., w t IWN(µ w, σ 2 w), then x t is Gaussian when h t has innite duration (central limit theorem). Any non-linearity in the function h(.) of (29) implies a non-linear process x t. In this case, x t has statistics that are necessarily non-gaussian. On the other hand, Gaussian processes are necessarily linear. The rest of this Section assumes that the innovations in (30) are of the kind w t IWN(µ w, σ 2 w). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
48 Time Series Modeling IWN's autocorrelation and SDF The IWN's w t autocorrelation in (30) is in which δ τ is the discrete time unit pulse 3. Then, its spectral density function or SDP is R w(τ) = σ 2 wδ τ + µ 2 w, (40) S w (f ) = µ 2 wδ(f ) + σ 2 w, 1/2 f 1/2, (41) in which f is the normalized frequency and δ(f ) is the Impulse generalized function (or Dirac's Delta). The w t 's SDF is dened as the Discrete Time Fourier Transform (DFT) of its autocorrelation R w(τ), i. e., S w (f ) = 3 δ τ = 1 for τ = 0, δ τ = 0 para τ 0, τ Z. m= R w(m) e j2πfm JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
49 Time Series Modeling The linear process x t 's SDF and auto-covariance The linear process x t 's SDF is S x (f ) = H(f ) 2 S w (f ) = µ 2 w H(0) 2 δ(f ) + σ 2 w H(f ) 2, 1/2 f 1/2, (42) in which H(f ) = H(z) z=e j2πf is the lter's frequency response. The x t 's auto-covariance is given by C x(τ) = σ 2 w h t h t+τ, (43) t=0 and its variance by σ 2 x = σ 2 w h 2 t. (44) t=0 JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
50 Time Series Modeling AR( ) model As, in practice, the estimated models are invertible (i. e., (37) is valid), we can dene the inverse operator G(B) = H 1 (B) and rewrite (30) in the innite order auto-regressive format (AR( )) x t = g 1 x t 1 + g 2 x t w t = g k x t k + w t. k=1 (45) So, x t may be interpreted as a weighted sum of its past values x t 1, x t 2,... plus an innovation w t. The equivalent model AR( ) suggests that we can compute the probability of a future value x t+k being between two specied bounds, i. e., (45) states that it is possible to make inferences or predictions of the series' future values. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
51 Auto-regressive models Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
52 Auto-regressive models AR(p) model An order p auto-regressive model satises the equation in which φ(b) is an order p polynomial. φ(b)x t = w t. (46) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
53 Auto-regressive models Auto-correlation function Multiplying both sides of (46) and x t k and taking the expectation we get E[x t x t k ] = φ 1 E[x t 1 x t k ] + φ 2 E[x t 2 x t k ] φ pe[x t p x t k ] + E[w t x t k ], as x t k does not depend on w t, but only on the noise up to instant t k, that are not correlated with w t, then E[w t x t k] = 0, k > 0, and C x (k) = φ 1 C x (k 1) + φ 2 C x (k 2) φ p C x (k p), k > 0. (47) Dividing (47) by C x (0) = σ 2, we get x ρ x (k) = φ 1 ρ x (k 1) + φ 2 ρ x (k 2) φ p ρ x (k p), k > 0, (48) or φ(b)ρ x (k) = 0. (49) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
54 Auto-regressive models Auto-correlation function 2 Let G 1, i = 1,..., p, be the roots of φ(b) = 0 (model's characteristic i equation). Then, we can write p φ(b) = (1 G i B), i=1 and it can be shown that the general solution of (49) is ρ x (k) = A 1 G k 1 + A 2 G k A p G k, (50) p in which the constants A i, i = 1, 2,..., p, are determined by initial conditions over ρ x (0), ρ x (1),..., ρ x (p 1). As the roots of φ(b) = 0 must be out of unit radius circle, we must have G k < 1, k = 1,..., p. So, the ACF plot of an AR(p) process will normally show a mixture of damping sine and cosine patterns and exponential decays. As B = z 1, then Eqs. 36 and 37 may be written as φ(b) = 0 for B > 1 and θ(b) = 0 for B > 1. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
55 Auto-regressive models Auto-correlation function 3 For example, consider an AR(2) model x t = φ 1 x t 1 + φ 2 x t 2 + w t and its ACF, which satises the second order dierence equation ρ k = φ 1 ρ k 1 + φ 2 ρ k 2, k > 0 with initial values ρ 0 = 1 e ρ 1 = φ 1 1 φ 2. From (50), the general solution of this equation is ρ k = G 1(1 G 2 2 )G k 1 G 2(1 G 2 1 )G k 2 (G 1 G 2 )(1 + G 1 G 2 ). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
56 Auto-regressive models Identication In practice, the order p of an AR series is unknown and must be empirically specied. There are two approaches: i) use of the Partial Auto-correlation Function (FACP); ii) use of some model's selection (identication) criterium. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
57 Auto-regressive models PACF of AR(p) models Let φ mi be the i-th coecient of an AR(m) process, so the last coecient is φ mm. Making k = 1,..., m in (48) (in the following, we adopt the simplied notation ρ x (k) = ρ k ) and considering that ρ k = ρ k (ACF's even symmetry), we get the Yule-Walker equations ρ 1 = φ m1 + φ m2 ρ φ mm ρ m 1, ρ 2 = φ m1 ρ 1 + φ m φ mm ρ m 2,. ρ m = φ m1 ρ m 1 + φ m2 ρ m φ mm, (51) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
58 Auto-regressive models PACF of AR(p) models 2 that may be written in matrix format 1 ρ 1... ρ m 1 ρ ρ m ρ m 1 ρ m φ m1 φ m2. φ mm ρ 1 ρ 2 ρ m =. (52) or in its compact format R m φ m = ρ m, (53) in which R m is the order m autocorrelations matrix, φ m the model's parameters vector and ρ m is the autocorrelations vector. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
59 Auto-regressive models PACF of AR(p) models 3 Solving (53) for m = 1, 2,..., we get φ 11 = ρ 1 1 ρ 1 ρ 1 ρ 2 φ 22 = 1 ρ 1 ρ ρ 1 ρ 1 ρ 1 1 ρ 2 ρ 2 ρ 1 ρ 3 φ 33 = 1 ρ 1 ρ 2 ρ 1 1 ρ 1 ρ 2 ρ 1 1 (54) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
60 Auto-regressive models PACF of AR(p) models 4 And, in general, φ mm = R m R m, (55) in which R m is the m order autocorrelations matrix' determinant and R m is matrix R m with the last column replaced by the autocorrelations vector. The sequence {φ mm, m = 1, 2,...} is the PACF. It can be shown that an AR(p) model has φ mm 0 for m p and φ mm = 0 for m > p. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
61 Auto-regressive models PACF of AR(p) models 5 The PACF may be estimated by adjusting the AR(m), m = 1, 2,... models sequence x t = φ 11 x t 1 + w 1t x t = φ 21 x t 1 + φ 22 x t 2 + w 2t. x t = φ m1 x t 1 + φ m2 x t φ mm x t m + w mt, (56). by the least square method. The sequence { ˆφ mm, m = 1, 2,...} is the Sample Partial Autocorrelation Function (SPACF). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
62 Auto-regressive models Model identication The basic idea of an ARMA model selection criterium (or information criterion) is to choose the orders k and l that minimize the quantity P(k, l) = ln ˆσ 2 k,l + (k + l)c(n) N, (57) in which ˆσ 2 k,l is a residual variance estimate obtained by adjusting an ARMA(k, l) model to the N series observations. C(N) is a functions of the series size. The quantity (k + l) C(N) is called penalty term and it increases when N the number of parameters increases, while ˆσ 2 decreases. k,l JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
63 Auto-regressive models Akaike's criterium Akaike proposed the information criterium AIC(k, l) = ln ˆσ 2 k,l 2(k + l) +, (58) N known as AIC, in which ˆσ 2 k,l is the maximum likelihood estimator of σ 2 w for an ARMA(k, l) model. Upper bounds K and L for k and l must be specied. (58) has to evaluated for all possible (k, l) combinations with 0 k K 0 l L. In general, K and L are functions of N, for example, K = L = ln N. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
64 Auto-regressive models AIC and BIC For the case of AR(p) models, (58) reduces to AIC(k) = ln ˆσ 2 + 2k k, k K. (59) N Another criterium that is much used is the (Schwarz) Bayesian Information Criteria (BIC) BIC(k, l) = ln ˆσ 2 + ln N k,l (k + l). (60) N For the case of AR(p) models, (60) reduces to BIC(k) = ln ˆσ 2 k + k ln N N. (61) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
65 Auto-regressive models AR models estimation Having identied the AR model's order p, we can go to the parameters estimation phase. The methods of moments, Least Squares and Maximum Likelihood may be used. As, in general, the moments estimators are not good, statistical packages as S-PLUS, E-VIEWS, etc., use some Least Squares or Maximum Likelihood estimator. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
66 Non-stationary series modeling Outline 1 Introduction Motivation Objectives and contributions 2 Time series basic concepts Time series examples Operators notation Stochastic processes Time Series Modeling Auto-regressive models Non-stationary series modeling JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
67 Non-stationary series modeling Introduction A non-stationary process has time dependent moments. Common types of non-stationarity: time dependent mean time dependent variance JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
68 Non-stationary series modeling Stationary series with respect to trends A process y t is called stationary with respect to trends if it is of the type y t = TD t + x t, (62) in which TD t denotes the term of deterministic trends (constant, trend, season) that depends on t and x t is a stationary process. For example, the process y t given by y t = µ + δt + x t, x t = φx t 1 + w t y t µ δt = φ(y t 1 µ δ(t 1)) + w t y t = c + βt + φy t 1 + w t (63) in which φ < 1, c = µ(1 φ) + δ, β = δ(1 φ)t and w t is a WGN with null mean and power σ 2, is a stationary AR(1) process with respect to trends. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
69 Non-stationary series modeling ARIMA model If a process corresponds to the dierence of order d = 1, 2,... of x t y t = (1 B) d x t = d x t (64) is stationary, then y t can be represented by an ARMA(p, q) model φ(b)y t = θ(b)w t. (65) In this case, φ(b) d x t = θ(b)w t (66) is an ARIMA(p, d, q) model and we say that x t is an integral of y t because x t = S d y t. (67) JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
70 Non-stationary series modeling ARIMA model 2 As the ARIMA(p, d, q) model H(z) = θ(z) φ(z)(1 z 1 ) d (68) is marginally stable, as it has d roots on the unit circle, x t of (66) is a homogeneous non-stationary process (meaning non-explosive) or having unit roots JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
71 Non-stationary series modeling ARIMA model 3 Observe that: (a) d = 1 corresponds to homogeneous non-stationary series with respect to the level (they oscillate around a mean level during a certain time and then jump to another temporary level); (b) d = 2 corresponds to homogeneous non-stationary series with respect to the trend (they oscillate along a direction for a certain time and then change to another temporary direction). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
72 Non-stationary series modeling ARIMA model 3 The ARIMA model (66) may be represented in three ways: (a) ARMA(p + d, q) (similar to Eq. (31)) p+d q x(t) = ϕ k x(t k) + w(t) θ k w(t k); (69) k=1 k=1 (b) AR( ) (inverted format), given by (45) or (c) MA( ), according to (30). JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
73 Non-stationary series modeling Random walk Consider the model y t I (1) y t = y t 1 + x t, (70) in which x t is a stationary process. If we assume the initial condition y 0, (70) can be rewritten as an integrated sum y t = y 0 + t x j. (71) The integrated sum t j=1 x j is called stochastic trend and it is denoted by TS t. Observe that TS t = TS t 1 + x t, (72) in which TS 0 = 0. j=1 JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
74 Non-stationary series modeling Random walk 2 If x t N (0, σ 2 x) in (70), then y t is known as random walk. Including a constant in the right side of (70), we have a random walk with drift, y t = θ 0 + y t 1 + x t. (73) Given the initial condition y 0, we can write y t = y 0 + θ 0 t + t x j (74) j=1 = TD t + TS t JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
75 Non-stationary series modeling Random walk 3 The mean, variance, autocovariance and ACF of y t are given by µ t = y 0 + tθ 0 (75) σ 2 (t) = tσ 2 x (76) C k (t) = (t k)σ 2 x (77) ρ k (t) = t k. (78) t Observe that ρ k (t) 1 when t >> k and the literature states that the random walk has strong memory. The random walk's SACF decays linearly for large lags. JRAA-ABL (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo / 74
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