Reliability and Risk Analysis. Time Series, Types of Trend Functions and Estimates of Trends

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1 Reliability and Risk Analysis

2 Stochastic process The sequence of random variables {Y t, t = 0, ±1, ±2 } is called the stochastic process The mean function of a stochastic process {Y t} is the function µ t defined by The autocovariance function is defined as µ t = E(Y t), t = 0, ±1, ±2 γ t,s = C(Y t, Y s), t, s = 0, ±1, ±2, where C(Y t, Y s) = E[(Y t µ t)(y s µ s)] = E[Y t, Y s] µ tµ s The autocorrelation function is given by C(Yt, Ys) ρ t,s = = γt,s D(Yt)D(Y s) γt,tγ s,s

3 Stacionarity A process {Y t} is said to be strictly stationary if the joint distribution Y t1, Y t2,, Y tn is the same as the joint distribution of Y t1 k, Y t2 k,, Y tn k for all choices of time lag k If a function γ s,t depends on its arguments only through their differences k = s t, we can introduce notation γ k = γ s t = γ s,t Additionally, if the mean function µ t of the process is constant for all t (µ t = µ), the process {Y t} is said to be weakly stationary

4 Stacionarity autocovariance and autocorrelation function The autocovariace function γ k of the stationary stochastic process is γ k = C(Y t, Y t k ) = E[(Y t µ)(y t k µ)], and the autocorrelation function (ACF) ρ k is given by ρ k = C(Yt, Y t k) D(Yt)D(Y t k ) = γ k γ 0

5 Partial autocorrelation function The correlation between two random variables is often caused by the correlation with another variable The partial autocorrelation provide information about the correlation values Y t a Y t k removing the effect of variables Y t 1, Y t k+1 The partial autocorrelation with a lag k is expressed by the regression coefficient φ kk in auto-regression Y t = φ k1 Y t 1 + φ k2 Y t φ kk Y t k + e t, where e t is variable uncorrelated with Y t j, j 1 φ kk is a function of the lag k, we call it the partial autocorrelation function (PACF) and denote it ρ kk

6 Partial autocorrelation function After multiplying both sides of the above equation by the variable Y t 1 and taking expectation of the equation we get γ j = φ k1 γ j 1 + φ k2 γ j φ kk γ j k, so For j = 1, 2,, k is ρ j = φ k1 ρ j 1 + φ k2 ρ j φ kk ρ j k ρ 1 = φ k1 ρ 0 + φ k2 ρ φ kk ρ k 1 ρ 2 = φ k1 ρ 1 + φ k2 ρ φ kk ρ k 2 ρ k = φ k1 ρ k 1 + φ k2 ρ k φ kk ρ 0 These equations are called Yule-Walker equations

7 Partial autocorrelation function Using Cramer s rule for k = 1, 2, we sequentially obtain ρ 11 = φ 11 = ρ 1, 1 ρ1 ρ 1 ρ 2 ρ 22 = φ 22 = 1 = ρ2 ρ2 1, ρ1 1 ρ 2 1 ρ ρ 1 ρ 2 ρ k 2 ρ 1 ρ 1 1 ρ 1 ρ k 3 ρ 2 ρ k 1 ρ k 2 ρ k 3 ρ 1 ρ k ρ kk = φ kk = 1 ρ 1 ρ 2 ρ k 2 ρ k 1 ρ 1 1 ρ 1 ρ k 3 ρ k 2 ρ k 1 ρ k 2 ρ k 3 ρ 1 1

8 Estimates Parameters µ, γ 0 and ρ k are unknown in general We use estimates µ = Y = 1 n n Y t, γ 0 = 1 n t=1 n (Y t Y ) 2 where n the number of measurements length of time series n t=k+1 ρ k = (Yt Yt)(Y t k Y t) n t=1 (Yt Y, k = 1, 2,, n 1 )2 t=1 (in R acf)

9 Estimates For the sample partial autocorrelation function we can use the recursive formula (in R pacf) ρ 11 = ρ 1 ρ kk = ˆρ k k 1 j=1 ρ k 1,j ρ k j 1 k 1 j=1 ρ, k 1,j ρ j ρ kj = ρ k 1,j ρ kk ρ k 1,k j, j = 1, 2,, k 1

10 White noise process The white noise process {ɛ t} is an important stationary stochastic process It is a sequence of independent random variables with the same distribution with zero mean and constant variance It fulfills { 1 k = 0 ρ k = 0 k 0 { 1 k = 0 ρ kk = 0 k 0 Gaussian white noise a sequence of independent random variables with the distribution N(0, σ 2 ɛ t )

11 Deterministic trend Example: The process Y t = Y 0 + at, t = 1, n contains deterministic linear trend Y 0 denotes an initial value The graph shows given process for n = 100, Y 0 = 0, a = 1

12 Stochastic trend Example: the random walk Y t = Y t 1 + ɛ t, t = 1, n, where ɛ t WN(0, σ 2 ) Y t = Y t 1 + ɛ t = (Y t 2 + ɛ t 1) + ɛ t = = (Y t 3 + ɛ t 2) + ɛ t 1 + ɛ t = = t = Y 0 + ɛ ɛ t = Y 0 + i=1 Y 0 denotes an initial value Two possible realizations (simulation) of this process (n = 100, Y 0 = 0, ɛ t WN(0, 1)) are shown in the graphs ɛ i

13 Stochastick trend Example the random walk with the drift Y t = Y t 1 + a + ɛ t, t = 1, n, where ɛ t WN(0, σ 2 ) Y t = Y t 1 + a + ɛ t = (Y t 2 + a + ɛ t 1) + a + ɛ t = (Y t 3 + a + ɛ t 2) + 2a + ɛ t 1 + ɛ t = = t = Y 0 + at + i=1 ɛ i Y 0 denotes an initial value One possible realizations (simulation) of this process (n = 100, Y 0 = 0, ɛ t WN(0, 1)) is shown in the graph

14 Regression The basis of the classical time series analysis is its decomposition into trend T t, seasonal ingredients S t and residual component e t in the additive model in the multiplicative model then Y t = T t + S t + e t, Y t = T t S t e t Linear filters can be used to estimate the trend T t = i= λ i Y t+i

15 Regression A simple example of linear filters are moving averages with constant weights T t = 1 2a + 1 a Y t+i i= a Smoothed value of time series in time τ is obtained as the average of {y τ a,, y τ,, y τ+a} For example, for a = 2, 12 and 40 we have a = 2, λ i = { 1 5, 1 5, 1 5, 1 5, 1 5 } a = 12,λ i = { 1 25,, 1 25 } }{{} 25 krát a = 40,λ i = { 1 81,, 1 81 } }{{} 81 krát

16 Regression The graph shows the monthly production of beer in Australia from January 1956 to August 1995

17 Regression The graphs show moving averages of the length 5 (a = 2), 25 (a = 12), 81 (a = 20)

18 Regression Decoposition (in R can be computed using the function filter) are the basis of classical decomposition, in which the R performs the function decompose The function stl offers a somewhat more sophisticated method of decomposition

19 Regression Linear regression The figure shows the evolution of gross monthly wage in the Czech Republic ( , quarterly data)

20 Regression Linear regression We estimate the trend using the regression line for the time variable t = year 1999, t = 1, 13 Estimate St error t-test p-value intercept t

21 Regression Linear regression We include dummy variables q 1, q 2, q 3, q 4 into the model to describe seasonality q 1 = (1, 0, 0, 0, 1, 0, 0, 0,, 1, 0, 0, 0) q 1 = (0, 1, 0, 0, 0, 1, 0, 0,, 0, 1, 0, 0) q 1 = (0, 0, 1, 0, 0, 0, 1, 0,, 0, 0, 1, 0) q 1 = (0, 0, 0, 1, 0, 0, 0, 1,, 0, 0, 0, 1) Estimate St error t-test p-value t 484, ,8450 3,17 0,0027 t 2 111, ,3280 4,78 0,0000 t 3-5,7907 1,0430-5,55 0,0000 q , , ,30 0,0000 q , , ,35 0,0000 q , , ,63 0,0000 q , , ,09 0,0000

22 Regression Linear regression

23 Regression Linear regression The predictions for 2013 (and 95% confidence intervals) are summarized in the table prediction lower upper 2013, 1 quarter quarter quarter quarter