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# Untitled

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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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