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  1. ## Chapter 1
  2. ### Time domain approach
  3. This is generally motivated by the presumption that correlation between adjacent points in time is best explained in terms of a dependence of the current value on past values
  4.  
  5. mulitplicative models and additvie models
  6.  
  7. #### ARIMA / autoregressive integrated moving average
  8. ### Frequency Domain Approach
  9. assumes the primary characteristics of interest in time series analyses relate to periodic or systematic sinusoidal variations found naturally in most data.
  10.  
  11. -------
  12. ### Time Series Statistical Models
  13. Using discrete random variables to represent sampled time series date.
  14.  
  15. The selection of sampling interval matters.
  16.  
  17. #### White Noise
  18.  
  19. ##### A white noise process is one with a mean zero and no correlation between its values at different times.
  20.   \
  21.  
  22. A simple kind of generated series might be a collection of uncorrelated random variables, $$w_t$$, with mean 0 and finite variance $$\sigma_w^2$$.
  23.  
  24.  
  25.  
  26. The time series generated from uncorrelated variables is used as a model for noise in en- gineering applications, where it is called white noise; we shall sometimes denote this process as $$w_t\sim wn(0,\sigma_w^2)$$.
  27.  
  28. Noticed that the variable should be indepnedent so that the calssical statiscal method suffice.
  29.  
  30. #### Moving Averages
  31. We might replace the white noise series $$w_t$$ by a moving average that smooths the series.
  32.  
  33. $$v_t=1/3(w_{t-1}+w_{t}+w_{t+1})$$
  34. Inspecting the series shows a smoother version of the first series, reflecting the fact that the slower oscillations are more apparent and some of the faster oscillations are taken out.
  35.  
  36. #### Autoregressions
  37. Suppose we consider the white noise series $w_t$ of Example 1.8 as input and calculate the output using the second-order equation
  38. $$x_t=x_{t-1}-.9x_{t-2}+w_{t}\ (1.2)$$
  39. Equation (1.2) represents a regression or prediction of the current value $$x_t$$ of a time series as a function of the past two values of the series, and, hence, the term autoregression is suggested for this model.
  40.  
  41. #### Random Walk with Drift
  42.  
  43. $$x_t=\delta + x_{t-1}+w_t$$
  44. for t = 1, 2, . . ., with initial condition x0 = 0, and where wt is white noise. The constant $$\delta$$ is called the drift and when $$\delta=0$$, this is called simply a random walk.
  45. Cumulative sum of white noise variates
  46. $$x_t = \delta t+\sum_{j=1}^t w_j$$
  47.  
  48.  
  49. ----
  50. ### Measures of Dependence: Autocorrelation and Cross-Correlation
  51. #### The mean function
  52. $$\mu_{xt}=E(x_t)=\int_{-\infty}^{\infty}xf_t(x)dx$$
  53.  
  54. #### Autocovariance function
  55. $$\gamma_x(s,t)=cov(x_s,x_t)=E[(x_s-\mu_s)(x_t-\mu_t)]$$
  56.  
  57. for all s and t. When no possible confusion exists about which time series we are referring to, we will drop the subscript and write as $$\gamma(s,t)$$.
  58.  
  59. The autocovariance measures the **linear dependence** between two points on the same series observed at different times.
  60.  
  61. Recall from classical statistics that if  x(s,t) = 0, xs and xt are not linearly related, but there still may be some dependence structure between them. If, however, xs and xt are bivariate normal,  x(s,t) = 0 ensures their independence. It is clear that, for s = t, the autocovariance reduces to the (assumed finite) variance, because
  62.  
  63. $$\gamma_x(t,t)=E[(x_t-\mu_t)^2]=var(x_t)$$
  64. #### Autocorrelation function
  65. $$\rho(s,t)=\frac{\gamma(s,t)}{\sqrt{\gamma(s,s)\gamma(t,t)}}$$
  66. The ACF measures the **linear predictability** of the series at time t, say xt, using only the value xs.
  67.  
  68. #### Cross-covariance function
  69. The cross-covariance function between **two series**, $$x_t$$ and $$y_t$$, is
  70. $$\gamma_{xy}(s,t)=cov(x_s,y_t)=E[(x_s-\mu_{xs})(y_t-\mu_{yt})]$$
  71. #### Cross-correlation function
  72. $$\rho_{xy}(s,t)=\frac{\gamma_{xy}(s,t)}{\sqrt{\gamma(s,s)\gamma(t,t)}}$$
  73. -------
  74. ### Stationary Time Series
  75. #### Definition 1.6 A strictly stationary
  76. A strictly stationary time series is one for which the probabilistic behavior of every collection of values {$$x_{t_1},x_{t_2},...,x_{t_k}$$} is identical to that of the time shifted set {$$x_{t_1+h},x_{t_2+h},...,x_{t_k+h}$$}.
  77.  
  78. #### Definition 1.7 A weakly stationary
  79. A weakly stationary time series, $$x_t$$, is a finite variance process such that
  80. (i) the mean value function, $$\mu_t$$, defined in the mean function is constant and does not depend on time t, and
  81. (ii) the autocovariance function,  $$\gamma (s, t)$$, defined in (1.10) depends on s and t only through their difference $$|s-t|$$.
  82.  
  83. ##### Mean
  84. $$\mu_t=\mu$$
  85. ##### Autocovariance function
  86. $$\gamma (t+h,t)=cov(x_{t+h},x_t)=cov(x_h,x_0)=\gamma (h,0)$$
  87.  
  88. ##### Definition 1.8 Autocovariance function of a stationary time series
  89. $$\gamma(h)=cov(x_{t+h},x_t)=E[(x_{t+h}-\mu)(x_t-\mu)]$$
  90.  
  91. ##### Definition 1.9 The autocorrelation function (ACF) of a stationary time series
  92. $$\rho(h)=^{\gamma(t+h,t)}/_{\sqrt{\gamma(t+h,t+h)\gamma(t,t)}}=^{\gamma(h)}/_{\gamma(0)}$$
  93. ##### Properties of stationary time series
  94. 1.$$\gamma(0)=E[(x_t-\mu)^2]$$
  95. 2.$$|\gamma(h)|<= \gamma(0)$$
  96. 3.$$\gamma(h)=\gamma(-h)$$
  97.  
  98. #### Definition 1.10 jointly stationary
  99. Two time series, say, $$x_t$$ and $$y_t$$ , are said to be jointly stationary if they are each stationary, and the cross-covariance function
  100. $$\gamma_{xy}(h)=cov(x_{t+h},y_t)=E[(x_{t+h}-\mu_x)(y_t-\mu_y)]$$
  101. is a function only of lag $$h$$.
  102. #### Definition 1.11 cross-correlation function (CCF) of jointly stationary time series
  103. The cross-correlation function (CCF) of jointly stationary time series xt and yt is defined as
  104. $$\rho_{xy}(h)=^{\gamma_{xy}(h)}/_{\gamma_x(0)\gamma_y(0)}$$
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