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Nov 15th, 2017
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__Sign Up__- ## Chapter 1
- ### Time domain approach
- 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
- mulitplicative models and additvie models
- #### ARIMA / autoregressive integrated moving average
- ### Frequency Domain Approach
- assumes the primary characteristics of interest in time series analyses relate to periodic or systematic sinusoidal variations found naturally in most data.
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- ### Time Series Statistical Models
- Using discrete random variables to represent sampled time series date.
- The selection of sampling interval matters.
- #### White Noise
- ##### A white noise process is one with a mean zero and no correlation between its values at different times.
- \
- A simple kind of generated series might be a collection of uncorrelated random variables, $$w_t$$, with mean 0 and finite variance $$\sigma_w^2$$.
- 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 $$w_t\sim wn(0,\sigma_w^2)$$.
- Noticed that the variable should be indepnedent so that the calssical statiscal method suffice.
- #### Moving Averages
- We might replace the white noise series $$w_t$$ by a moving average that smooths the series.
- $$v_t=1/3(w_{t-1}+w_{t}+w_{t+1})$$
- 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.
- #### Autoregressions
- Suppose we consider the white noise series $w_t$ of Example 1.8 as input and calculate the output using the second-order equation
- $$x_t=x_{t-1}-.9x_{t-2}+w_{t}\ (1.2)$$
- 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.
- #### Random Walk with Drift
- $$x_t=\delta + x_{t-1}+w_t$$
- 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.
- Cumulative sum of white noise variates
- $$x_t = \delta t+\sum_{j=1}^t w_j$$
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- ### Measures of Dependence: Autocorrelation and Cross-Correlation
- #### The mean function
- $$\mu_{xt}=E(x_t)=\int_{-\infty}^{\infty}xf_t(x)dx$$
- #### Autocovariance function
- $$\gamma_x(s,t)=cov(x_s,x_t)=E[(x_s-\mu_s)(x_t-\mu_t)]$$
- 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 $$\gamma(s,t)$$.
- The autocovariance measures the **linear dependence** between two points on the same series observed at different times.
- 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
- $$\gamma_x(t,t)=E[(x_t-\mu_t)^2]=var(x_t)$$
- #### Autocorrelation function
- $$\rho(s,t)=\frac{\gamma(s,t)}{\sqrt{\gamma(s,s)\gamma(t,t)}}$$
- The ACF measures the **linear predictability** of the series at time t, say xt, using only the value xs.
- #### Cross-covariance function
- The cross-covariance function between **two series**, $$x_t$$ and $$y_t$$, is
- $$\gamma_{xy}(s,t)=cov(x_s,y_t)=E[(x_s-\mu_{xs})(y_t-\mu_{yt})]$$
- #### Cross-correlation function
- $$\rho_{xy}(s,t)=\frac{\gamma_{xy}(s,t)}{\sqrt{\gamma(s,s)\gamma(t,t)}}$$
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- ### Stationary Time Series
- #### Definition 1.6 A strictly stationary
- 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 {$$x_{t_1+h},x_{t_2+h},...,x_{t_k+h}$$}.
- #### Definition 1.7 A weakly stationary
- A weakly stationary time series, $$x_t$$, is a finite variance process such that
- (i) the mean value function, $$\mu_t$$, defined in the mean function is constant and does not depend on time t, and
- (ii) the autocovariance function, $$\gamma (s, t)$$, defined in (1.10) depends on s and t only through their difference $$|s-t|$$.
- ##### Mean
- $$\mu_t=\mu$$
- ##### Autocovariance function
- $$\gamma (t+h,t)=cov(x_{t+h},x_t)=cov(x_h,x_0)=\gamma (h,0)$$
- ##### Definition 1.8 Autocovariance function of a stationary time series
- $$\gamma(h)=cov(x_{t+h},x_t)=E[(x_{t+h}-\mu)(x_t-\mu)]$$
- ##### Definition 1.9 The autocorrelation function (ACF) of a stationary time series
- $$\rho(h)=^{\gamma(t+h,t)}/_{\sqrt{\gamma(t+h,t+h)\gamma(t,t)}}=^{\gamma(h)}/_{\gamma(0)}$$
- ##### Properties of stationary time series
- 1.$$\gamma(0)=E[(x_t-\mu)^2]$$
- 2.$$|\gamma(h)|<= \gamma(0)$$
- 3.$$\gamma(h)=\gamma(-h)$$
- #### Definition 1.10 jointly stationary
- 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
- $$\gamma_{xy}(h)=cov(x_{t+h},y_t)=E[(x_{t+h}-\mu_x)(y_t-\mu_y)]$$
- is a function only of lag $$h$$.
- #### Definition 1.11 cross-correlation function (CCF) of jointly stationary time series
- The cross-correlation function (CCF) of jointly stationary time series xt and yt is defined as
- $$\rho_{xy}(h)=^{\gamma_{xy}(h)}/_{\gamma_x(0)\gamma_y(0)}$$

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