According to wiki, time series is a sequence of data points that
- consists of successive measurements made over a time interval
- the time interval is continuous
- the distance in this time interval between any two consecutive data point is the same
- each time unit in the time interval has at most one data point
In probability theory, a stochastic process, or often random process, is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system).
In mathematics and statistics, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean, variance and autocorrelation, if they are present, also do not change over time and do not follow any trends.
Following are three ways to achieve a stationary series:
- differentiate the data, to remove the trend
- make a simple fit, then model on the residuals from the fit, so as to remove the long term trend
- for non-constant variance series, taking the logarithm or square root of the series may stabilize the variance
In statistics, a unit root test tests whether a time series variable is non-stationary using an autoregressive model. A well-known test that is valid in large samples is the augmented Dickey–Fuller test. The optimal finite sample tests for a unit root in autoregressive models were developed by Denis Sargan and Alok Bhargava. Another test is the Phillips–Perron test. These tests use the existence of a unit root as the null hypothesis.
In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. The reasoning is that any collection of random samples from a process must represent the average statistical properties of the entire process. In other words, regardless of what the individual samples are, a birds-eye view of the collection of samples must represent the whole process. Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent rate.
Univariate time series
Let \( \{y_t\} = \{ \dots, y_{t-1}, y_t, y_{t+1}, \dots \} \) denotes a time series. And suppose \( \{y_t\} \) is covariance stationary, then
- \( \operatorname{E}[y_t] = \mu \quad \forall t \)
- \( \operatorname{cov}(y_t, y_{t-j}) = \operatorname{E}[(y_t - \mu)(y_{t-j} - \mu)] = \gamma_j \quad \forall t, j \)
Stationary time series have time invariant first and second moments. And \( \gamma_j \) is called the \( j^{th} \) order or lag \( j \) autocovariance of \( \{ y_t \} \), and a plot of \( \gamma_j \) against \( j \) is called the autocovariance function.
The autocorrelations of \( \{ y_t \} \) are defined by:
And a plot of \( \rho_j \) against \( j \) is called the autocorrelation function (ACF).
Any function of a stationary time series is also stationary, that is, if \( \{ y_t \} \) is stationary, then \( \{ z_t \} = \{ g(y_t) \} \) is stationary for any function \( g(\dot) \).
The lag \( j \) sample autocovariance and lag \( j \) sample autocorrelation are defined as
where \( \bar{y} = \frac{1}{T}\sum_{t=1}^Ty_t \) is the sample mean. To see why \( \hat{\gamma}_j \) is achieved by deviding \( T \) instead of \( T - j \), click here.
A stationary time series \( \{ y_t \} \) is ergodic if sample moments converge in probability to population moments, i.e. \( \bar{y} \overset{P}{\to} \mu \), \( \hat{\gamma}_j \overset{P}{\to} \gamma_j \), and \( \hat{\rho}_j \overset{P}{\to} \rho_j \).
Fitting time series in R
Model fitting
Once a model has been fitted to the data, the deviations from the model are the residuals. If the model is appropriate, then the residual mimic the true errors.
A complete data analysis involves:
-
step1: Finding a good model to fit the signal based on the data
-
step2: Finding a good model to fit the noise, based on the residuals from the model
-
step3: Adjusting variances, test statistics, confidence intervals, and predictions, based on the model for noise.
What to look for in a time series:
-
The signal: trend, intervention, periodic, …
-
The structure of the random elements
Stationary
A stationary time series is one that has had trend elements removed and that has a time invariant pattern in the random noise.
1
Box.test(series, lag=Int, type='Ljung-Box')
It examines whether there is significant evidence for non-zero correlations at lags 1-lag, small p-value (i.e. less than 0.05) suggests that the series is stationary,
1
adf.test(series, alternative='stationary')
It is a t-statistic test, small p-values suggest the data is stationary
1
kpss.test(series)
accepting the null hypothesis means that the series is stationary, and small p-values suggest that the series is not stationary.
And external link check time series stationary, significance of the correlation coefficient
Periodicity
The spectrum measures the intensity of periodic patterns of difference frequencies.
Models
A random walk model is very widely used for non-stationary data, particularly finance and economic data. Random walks typically have:
-
long periods of apparent trends up or down
-
sudden and unpredictable changes in direction
Some Exponential Smoothing based methods: HoltWinters, bats, tbats, msts, dshw, stlf …
The AutoRegressive Integrated Moving Average model
The main disadvantage of the ARIMA approach is that the seasonality is forced to be periodic whereas tbats model allows for dynamic seasonality.
GARCH
GARCH
Goodness of fit
JarqueBera.test
Components of model
1
2
3
4
5
anova(fit) # analysis of variance
names(fit) # gives all names associated with fit
summary(fit)
Some snippets
1
2
3
4
5
6
7
data <- read.csv(file)
data$GMT2 <- strptime(data$GMT, "%Y-%m-%d %H:%M:%S")
ts <- timeSeries(data$value, data$GMT2)
result <- interpNA(ts, method='linear')
ts3 <- ts(ts$`TS.1`, freq=1440)
stl(ts3, s.window='periodic')
result$`time.series`
