Stationary stochastic processes

Transcription

1 Chapter 1-2 Stationary stochastic processes Maria Sandsten Lecture 1 Webpage: September / 35

2 Why stochastic processes? Many signals or time series can be modelled as a stochastic process: Financial data Climate data Medical data Communication data We need the stochastic model as the deterministic model is usually too limited. This course presents basic tools for modelling time series as stationary stochastic processes. 2 / 35

3 Chapter 1-2 Introduction Stoch. var The stock price days 3 / 35

4 Temperature data 20 Lufttemperatur 15 Oktober November December grader Celsius 10 5 September dagar 4 / 35

5 Chapter 1-2 Introduction Stoch. var EEG - Electroencephalogram data 5 / 35

6 Speech data seconds 6 / 35

7 Schedule for the course Week 1: Stationary stochastic processes, mean values and covariance functions. Estimation. (4h) Week 2: Gaussian processes. Spectral densities. (4h) Week 3: Sampling. Estimation of spectral densities. (4h) Week 4: Filtering. Differentiation and integration. Cross-spectrum. (4h) Week 5: AR- and MA-processes. Optimal filters. (4h) Week 6: Summary and old exam exercises. (2h) 7 / 35

8 Webpage 8 / 35

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12 Review of basic probability theory Rules for probabilities: P(Ω) = 1 0 P(A) 1 P(A c ) = 1 P(A), where A c is the complement of A. P(A B) = P(A B)/P(B) is the conditional (betingad) probability of A given B. P(A B) = P(A) P(B) for independent events. 12 / 35

13 Review of stochastic variables A stochastic variable is a function defined at a sample space (utfallsrum), all possible outcomes, Ω. E.g., for a die (tärning), the sample space is discrete, Ω = { 1, 2, 3, 4, 5, 6 }. The probability function (sannolikhetsfunktionen) for a die is uniformly distributed, P(k) = p X (k) = 1 6, k = The distribution function (fördelningsfunktion) is found as F X (k 0 ) = P(X k 0 ) = k k 0 p X (k). Examples: 1) P(X > 4) = 1 P(X 4) = 1 F X (4). 2) P(X 4) = P(X > 3) = 1 P(X 3) = 1 F X (3). 3) P(2 X 4) = P(X 4) P(X 1) = F X (4) F X (1) = p X (2) + p X (3) + p X (4). 13 / 35

14 Review of stochastic variables The sample space can also be continuous with a density function (täthetsfunktion), f X (x), x R, and a distribution function, F X (x 0 ) = P(X x 0 ) = x0 f X (x)dx. The density function is connected to the distribution function as f X (x) = F X (x). Examples: 1) P(X > 4) = P(X 4) = 1 P(X 4) = 1 F X (4). 2) P(2 < X < 4) = P(2 X 4) = F X (4) F X (2) = 4 2 f X (x)dx. 14 / 35

15 Review of stochastic variables Expected value (väntevärde): E[X ] = k kp X (k) and E[X ] = x f X (x)dx. Variance (varians): V [X ] = k (k E[X ])2 p X (k) and V [X ] = (x E[X ])2 f X (x)dx. Standard deviation (standardavvikelse): D[X ] = V [X ]. 15 / 35

16 Review of stochastic variables Rules for expected value and variance (a and b constants): E[aX + b] = ae[x ] + b V [ax ] = a 2 V [X ] V [X + b] = V [X ] For the stochastic variables X and Y : E[X + Y ] = E[X ] + E[Y ] V [X + Y ] = C[X + Y, X + Y ] = V [X ] + C[X, Y ] + C[Y, X ] + V [Y ], where C[X, Y ] = C[Y, X ] is the covariance (kovariansen). 16 / 35

17 Example The stochastic variables X 1 and X 2 have expected values E[X 1 ] = 2 and E[X 2 ] = 1 and variances V [X 1 ] = 4 and V [X 2 ] = 6. The covariance is C[X 1, X 2 ] = 1. Calculate the expected value and variance of: Z = X 1 2X / 35

18 Covariance and correlation The covariance between two stochastic variables is defined by C[X, Y ] = E[(X m X )(Y m Y )] = E[XY ] m X m Y. We could note that C[X, Y ] = 0 if X and Y are independent. But, X and Y are not necessarily independent if C[X, Y ] is zero. Covariance is only a measure of linear dependence. The correlation coefficient (korrelationskoefficient) is given from ρ[x, Y ] = C[X, Y ] V [X ]V [Y ]. 18 / 35

19 Correlation-covariance example 19 / 35

20 Definition: stochastic process Definition 1.1: A stochastic process is a family of stochastic variables defined on the same sample space Ω, X (t), t T, where T is an interval of real valued numbers (continuous time) or a sequence of integers (discrete time), X t, t = 0, ±1, ±2, / 35

21 Definition: stochastic processes If X (t), t T is a stochastic process, then X (t) is a random variable and it has a certain distribution function, F X (t) (x) = P(X (t) x), Similarly (X (t 1 ), X (t 2 )) is a two-dimensional random variable, with distribution function, F X (t1 ),X (t 2 )(x 1, x 2 ) = P(X (t 1 ) x 1, X (t 2 ) x 2 ). This generalizes for the n-dimensional random variable to a n-dimensional distribution function. 21 / 35

22 Definition: stochastic processes Definition 2.1: For any stochastic process, we define a mean value function, m(t) = E[X (t)], covariance function, r(s, t) = C[X (s), X (t)], variance function, v(t) = r(t, t) = V [X (t)] = C[X (t), X (t)], correlation function, ρ(s, t) = C[X (s),x (t)]. V [X (s)]v [X (t)] 22 / 35

23 Ensemble mean The mean value of a stochastic process, m(t), can be estimated from many realizations of X (t) at the different time instants t using the ensemble mean (ensembelmedelvärde) / 35

24 Definition: stationary stochastic process Definition 2.4: A process is defined as weakly stationary (svagt stationär) when the: mean value function is constant, m(t) = m, covariance function, r(s, t) = r(t s) = r(τ), for τ = t s, variance is constant, r(t, t) = r(t t) = r(0), correlation function is the normalized covariance, ρ(τ) = r(τ)/r(0). We usually mean weakly stationary processes when we talk about stationary stochastic processes. 24 / 35

25 Temperature data 20 Lufttemperatur 15 Oktober November December grader Celsius 10 5 September dagar The four-months temperature data is not stationary. 25 / 35

26 Temperature data 10 Lufttemperatur 9 8 November grader Celsius dagar A one-month data sequence could be modelled with constant mean value. 26 / 35

27 Chapter 1-2 Introduction Stoch. var The stock price days 27 / 35

28 Chapter 1-2 Introduction Stoch. var EEG - Electroencephalogram data 28 / 35

29 Speech data seconds 29 / 35

30 Ensemble mean The mean value of a stochastic process, m(t), can be estimated from many realizations of X (t) at the different time instants t using the ensemble mean (ensembelmedelvärde) / 35

31 More ensembles When the number of ensembles approaches infinity, the mean values for all t approach m for a stationary stochastic process / 35

32 Mean value over time As we usually have just a one or a few ensemble(s), the averaged value of just one realization of data x t, t = 1... n, using the mean value over time ˆm n = 1 n X t, n t=1 gives an unbiased (väntevärdesriktigt) estimate of m, as E[ ˆm n ] = 1 n n E[X t ] = 1 (m + m m) = m. n }{{} n t=1 32 / 35

33 Mean value over time A stationary process is linearly ergodic, (linjärt ergodisk), as the ensemble mean can be estimated using the mean value over time. 33 / 35

34 Definition: stationary stochastic process Definition 2.4: A process is defined as weakly stationary (svagt stationär) when the: mean value function is constant, m(t) = m, covariance function, r(s, t) = r(t s) = r(τ), for τ = t s, variance is constant, r(t, t) = r(t t) = r(0), correlation function is the normalized covariance, ρ(τ) = r(τ)/r(0). We usually mean weakly stationary processes when we talk about stationary stochastic processes. 34 / 35

35 Example 2.5 (modified) From a stationary stochastic process U t of independent random variables with mean value, m U = m, and variance r U (0) = σ 2, we construct a new stationary stochastic process X t by X t = U t + 0.5U t 1. Calculate the mean value m X and covariance function r X (τ). 35 / 35