class: center, middle, inverse, title-slide # IDS 702: Module 7.2 ## Stationarity and autocorrelation ### Dr. Olanrewaju Michael Akande --- ## Stationarity - The most common time series models usually assume .hlight[stationarity]. -- - .hlight[Stationarity] of a time series process means that the marginal distribution of any part of the series does not depend on time. -- - Basically, the locations in time themselves does not matter in the marginal and joint distributions; however, the differences in locations, that is, the lags, do matter! -- - Put a different way, a stationary time series is one whose properties do not depend on the particular time at which the series is observed. -- - Examples of non stationary series: + Steadily increasing trend (like the melanoma example). + Known seasonal trends, like increase in sales before Christmas. + Break in trend due to some external event. --- ## Stationarity - Denote the times series for the outcome by `\(y_t\)`. -- - Stationarity `\(\Rightarrow\)` + `\(\mathbb{Pr}(y_t)\)` is the same for all `\(t\)`, -- + `\(\mathbb{Pr}(y_t,y_{t+1})\)` is the same for all `\(t\)`, and so on... -- - .hlight[Weak stationarity] requires that only marginal moments, that is, means, variances and covariances are the same. -- - Stationarity `\(\Rightarrow\)` weak stationarity, but .hlight[the converse need not hold]. -- - For a normal distribution, the mean and variance completely characterizes the distribution, so that stationarity and weak stationarity will be equivalent. -- - Why does that matter? -- - When dealing with linear models, what distribution do we assume?? --- ## Popular stationary models - We will mainly focus on two types of stationary time series models: -- + Autoregressive models (AR models) - Value of outcome at time `\(t\)` is correlated with value at previous times. -- + Moving average models (MA models) - Value of outcome at time `\(t\)` is correlated with value of prediction errors at previous times. -- - Note: autoregressive moving average models (ARMA models) - Combination of AR and MA. -- - There are many more types of time series models (see STA 642/942). --- ## Autocorrelation - .hlight[Autocorrelation] (serial correlation) measures the strength of the linear relationship between `\(y_t\)` and its lagged values. -- - The lag `\(k\)` autocorrelation `\(\rho_k\)` measures the correlation in outcomes at time `\(t\)` and at time `\(t-k\)`, where `\(k\)` indicates how far back to go; `\(k\)` is called a lag. -- - The sample lag `\(k\)` autocorrelation `\(r_k\)` can be calculated using .block[ .small[ `$$r_k = \dfrac{\sum_{t=k+1}^T(y_t - \bar{y})(y_{t-k} - \bar{y})}{\sum_{t=1}^T(y_t - \bar{y})^2}.$$` ] ] --- ## Partial autocorrelation - Autocorrelation `\(\rho_k\)` (and the sample version `\(r_k\)`) between `\(y_t\)` and `\(y_{t-k}\)` will also include the linear relationships between `\(y_t\)` and each of `\(y_{t-1}, y_{t-2}, \ldots, y_{t-k+1}\)`. -- - As you will see, we will need to be able to assess the correlation between `\(y_t\)` and `\(y_{t-k}\)` without interference from the other lags. -- - .hlight[Partial autocorrelation] lets us do just that. -- - It is the autocorrelation between `\(y_t\)` and `\(y_{t-k}\)`, with all the linear relationships between `\(y_t\)` and each of `\(y_{t-1}, y_{t-2}, \ldots, y_{t-k+1}\)` removed. -- - In `R`, use `acf` to compute and plot autocorrelations and `pacf` to compute and plot partial autocorrelations. --- class: center, middle # What's next? ### Move on to the readings for the next module!