Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Outline Preliminaries A brief introduction Random processes LIESSE Fourier representation of random signals Weakly stationary processes L 2 processes Weak stationarity Spectral measure Fran¸ cois Roueff Random fields with orthogonal increments Telecom ParisTech Definition Spectral representation May 17, 2018 Examples Linear filtering in the spectral domain Filtering a white noise The general case Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Examples of applications Heartbeats Time series analysis based on stochastic modeling is applied in 105 various fields : Heart frequency ⊲ Health : physiological signal analysis (image analysis). 95 ⊲ Engineering : monitoring, anomaly detection, 85 localizing/tracking. 75 ⊲ Audio data : analysis, synthesis, coding. 0 200 400 600 800 ⊲ Ecology : climatic data, hydrology. Time ⊲ Econometrics : economic/financial data. ⊲ Insurance : risk analysis. Figure: Heart rate of a resting person over a period of 900 seconds. This rate is defined as the number of heartbeats per unit of time. Here the unit is the minute and is evaluated every 0 . 5 seconds.
Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Internet traffic Speech audio data 0.15 Inter−arrival times 0.10 0.05 0.00 0 500000 1000000 1500000 Time Figure: Inter-arrival times of TCP packets, expressed in seconds, obtained from a 2 hours record of the traffic going through an Internet link. Figure: A speech audio signal with a sampling frequency equal to 8000 http://ita.ee.lbl.gov/ . Hz. Record of the unvoiced fricative phoneme sh (as in sh arp). Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Climatic data: wind speed Climatic data: temperature changes 0.8 Temperature Deviations 5 0.4 4 Wind Speed 3 0.0 2 −0.4 1 1880 1900 1920 1940 1960 1980 2000 0 1965 1970 1975 Time Time Figure: Global mean land-ocean temperature index (solid red line) and surface-air temperature index (dotted black line). Figure: Daily record of the wind speed at Kilkenny (Ireland) in knots (1 http://data.giss.nasa.gov/gistemp/graphs/ . knot = 0.5148 metres/second).
Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Gross National Product of the USA GNP quarterly rate 15000 0.06 US GNP Quart. rate 0.02 5000 −0.02 0 1950 1960 1970 1980 1990 2000 2010 1950 1960 1970 1980 1990 2000 2010 Time Time Figure: Growth national product (GNP) of the USA in Billions of $s. Figure: Quarterly rate of the US GNP. http://research.stlouisfed.org/fred2/series/GNP . Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Financial index Financial index: log returns 1500 0.10 1000 SP 500 index SP 500 log−returns 0.00 500 −0.10 −0.20 0 1950 1960 1970 1980 1990 2000 2010 1950 1960 1970 1980 1990 2000 2010 Figure: Daily open value of the Standard and Poor 500 index. This index Figure: SP500 log-returns. is computed as a weighted average of the stock prices of 500 companies traded at the New York Stock Exchange (NYSE) or NASDAQ.
Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Stochastic modelling Finite dimensional (fidi) distributions Definition : time series For all I ∈ I ( T ) (a finite subset of T ), A time series valued in ( E, E ) and indexed on T = ❩ is a collection (i) denote by Π I is the canonical projection ( x t ) t ∈ T �→ ( x t ) t ∈ I , of random variables ( X t ) t ∈ T defined on the same probability space (ii) denote by X I the random vector ( X t ) t ∈ I = Π I ◦ X , (Ω , F , P ) . (iii) denote by P X I the distribution of X I , which is defined by Definition : path �� � Let ( X t ) t ∈ T be a random process defined on (Ω , F , P ) . The path P X I = P ( X t ∈ A t , t ∈ I ) , where A t ∈ E for all t ∈ I . A t of the random experiment ω ∈ Ω is defined as ( X t ( ω )) t ∈ T viewed t ∈ I as an element of E T . Remark: P X is characterized by the collection of fidi distributions Definition : law � P X I � I ∈I ( T ) . Let X = ( X t ) t ∈ T be a random process. The law of X is defined as the image probability measure P X = P ◦ X − 1 on ( E T , E ⊗ T ) . Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain L 2 space Backshift operator, stationarity We set E = ❈ d . We denote Definition : backshift operators Let the backshift operator B : E ❩ → E ❩ be defined by � � � | X | 2 � L 2 (Ω , F , P ) = X ❈ d -valued r.v. such that E < ∞ . x = ( x t ) t ∈ ❩ ∈ E ❩ . B( x ) = ( x t − 1 ) t ∈ ❩ for all ( L 2 , � , � ) is a Hilbert space with For all τ ∈ ❩ , we define B τ by � � X T Y � X, Y � = E . x = ( x t ) t ∈ ❩ ∈ E ❩ . B τ ( x ) = ( x t − τ ) t ∈ ❩ for all A process X = ( X t ) t ∈ T is said to be stationary if X and B ◦ X Definition : L 2 Processes have the same distributions. The process X = ( X t ) t ∈ T defined on (Ω , F , P ) with values in ❈ d is an L 2 process if X t ∈ L 2 (Ω , F , P ) for all t ∈ T . Examples: constant process, i.i.d. processes, Gaussian processes, ...
Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Mean and covariance functions Scalar case E = ❈ , examples Hermitian symmetry, non-negative definiteness Let X = ( X t ) t ∈ T be an L 2 process. For all I ∈ I ( T ) , Γ I = Cov([ X ( t )] t ∈ I ) = [ γ ( s, t )] s,t ∈ I is a ⊲ Its mean function is defined by µ ( t ) = E [ X t ] , hermitian non-negative definite matrix. ⊲ Its covariance function is defined by � � − E [ X s ] E [ X t ] H . Examples Γ( s, t ) = cov( X s , X t ) = E X s X H t ⊲ L 2 independent random variables ( X t ) t ∈ ❩ have mean µ ( t ) = E ( X t ) and covariance Linear combinations → scalar case � Let X = ( X t ) t ∈ T be an L 2 process with mean function µ and var( X t ) if s = t , Γ( s, t ) = covariance function Γ . This is equivalent to say that for all 0 otherwise. u ∈ ❈ d , u H X is a scalar L 2 process with mean function u H µ and covariance function u H Γ u . ⊲ A Gaussian process is an L 2 process whose law is entirely determined by its mean and covariance functions. Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Preliminaries Weakly stationary processes Random fields with orthogonal increments Linear filtering in the spectral domain Weakly stationary processes Examples Let T = ❩ . Let X be an L 2 strictly stationary process with mean An L 2 strictly stationary process is weakly stationary. function µ and covariance function Γ . Then µ ( t ) = µ (0) and γ ( s, t ) = γ ( s − t, 0) for all s, t ∈ T . ⊲ The constant L 2 process has constant autocovariance function. Definition : Weak stationarity We say that a random process X is weakly stationary with mean µ Strong and weak white noise and autocovariance function γ : ❩ → ❈ if it is L 2 with mean ⊲ A sequence of L 2 i.i.d. random variables is called a strong function t �→ µ and covariance function ( s, t ) �→ γ ( s − t ) . white noise, denoted by X ∼ IID( µ, σ 2 ) . ⊲ An L 2 process X with constant mean µ and constant diagonal The autocorrelation function is defined (when γ (0) > 0 ) by covariance function equal to σ 2 is called a weak white noise. It ρ ( t ) = γ ( t ) is denoted by X ∼ WN( µ, σ 2 ) . (It does not have to be i.i.d.) γ (0) .
Recommend
More recommend
