Time-Series

Question 1

What is time-series ?

Answer 1

a set of time-ordered observations of a process

Question 2

how is the time organised?

Answer 2

the intervals between observations remain constant (minutes / years / anything)

Question 3

what is univariate time-series?

Answer 3

Univariate time-series = many observations originating from one source

Question 4

what is multivariate time-series?

Answer 4

Multi-variate time-series = many observations originating from multiple different sources

Question 5

What is the goal of time-series?

Answer 5

predicting and explaining the properties of time-series

Question 6

what are the key properties of time-series data?

Answer 6

Ð Variation
Ð Autocorrelation
Ð Stationarity

Question 7

what are the types of variation ?

Answer 7

(trends, seasonality, cycles, irregular variation)

Question 8

what are the types of Forecasting & Prediction?

Answer 8

Ð Predicting the evolution of a process (but also using the past)
Ð Directionality Analysis: how time-series influence / predict each other.

Question 9

what is a trend?

Answer 9

ANY systematic change in the level of a series, its long term direction / effect. (increases-decreases)

Question 10

what do we have to do with trends?

Answer 10

Ð Modelling it explicitly

Ð Detrending

Question 11

why model a trend ?

Answer 11

various characteristics of time series data can be of theoretical interest—in which case they should be modeled

Question 12

why detrend?

Answer 12

if trends are of no theoretical interest …. they should be removed so that the aspects that are of interest can be more easily analyzed.

Question 13

what is SEASONALITY ?

Answer 13

A repeating pattern of increase / decrease in the series that occurs consistently throughout its duration.

Question 14

give an example of seasonality….

Answer 14

Ð E.G – For instance, restaurant attendance may exhibit a weekly seasonal pattern such that the weekends routinely display the highest levels within the series across weeks (i.e., the time period), and the first several weekdays are consistently the lowest
Ð Or
Ð A naturally occurring time period: ‘seasonal’ factors (monthly or weekly event changes)

The underlying pattern remains fixed in seasonality, yet its magnitude may vary in effect size.

Question 15

Once a systematic component has been identified in time-series…. it is often ……

Answer 15

modelled or removed

Question 16

if seasonality is of not interest… we would thus…

Answer 16

remove it

called seasonal adjustment

Question 17

what is a cycle?

Answer 17

A cyclical component in a time series is conceptually similar to a seasonal component: It is a pattern of fluctuation (i.e., increase or decrease) that reoccurs across periods of time.

Question 18

how is a cycle unlike seasonal effects?

Answer 18

However, unlike seasonal effects whose duration is fixed across occurrences and are associated with some aspect of the calendar (e.g., days, months), the patterns represented by cyclical effects are not of fixed duration (i.e., their length often varies from cycle to cycle) and are not attributable to any naturally-occurring time periods

Question 19

WHAT IS IRREGULAR VARIATION?

Answer 19

Randomness: Any remaining variation in a time series

after removing the systematic changes in the time series (trend, seasonality, cycles).

Question 20

what is irregular variation also referred to as?

Answer 20

white noise

It constitutes any remaining variation in a time series after these three systematic components have been partitioned out. In time series parlance, when this component is completely random (i.e., not autocorrelated), it is referred to as white noise, which plays an important role in both the theory and practice of time series modeling.

Equivalent to the error term in a statistical model. Residual time series left after fitting a model to the data.

Question 21

After a model has been fit to the data, the residuals form…..

Answer 21

After a model has been fit to the data, the residuals form a time series of their own, called the residual error series. If the statistical model has been successful in accounting for all the patterns in the data (e.g., systematic components such as trend and seasonality), the residual error series should be nothing more than unrelated white noise error terms with a mean of zero and some constant variance

Question 22

what is STATIONARITY?

Answer 22

stationarity is the most important assumption when making predictions based on past observations

Our time series is stationary, if the means and variance do not change/vary over time.

A complication with time series data is that its mean, variance, or autocorrelation structure can vary over time. A time series is said to be stationary when these properties remain constant. Thus, there are many ways in which a series can be non-stationary (e.g., an increasing variance over time), but it can only be stationary in one-way (viz., when all of these features do not change).
Stationarity is a pivotal concept in time series analysis because descriptive statistics of a series (e.g., its mean and variance) are only accurate population estimates if they remain constant throughout the series. With a stationary series, it will not matter when the variable is observed: “The properties of one section of the data are much like those of any other”. As a result, a stationary series is easy to predict: Its future values will be similar to those in the past. As a result, stationarity is the most important assumption when making predictions based on past observations,

Question 23

what is the alternative assumption to stationarity?

Answer 23

weak stationarity

Question 24

what is weak stationarity?

Answer 24

Needs the mean to be stable across time (not time independent) and the auto-covariance depends only on the time diff between two time points.

Question 25

What can we do if the time-series exhibit non-stationarity?

Answer 25

transform with differencing

Question 26

what is differencing ‘basic’

Answer 26

Ð Take every single point, and subtract the next point.
Ð Take a point X and subtract a copy of X at one previous time-point, this is differencing the time-series.
Ð Then it might turn into a stationary time series (iteratively – you can stop at some point).

Question 27

any further details about differencing ?

Answer 27

Ð One or two differences might be enough. If it gets better… closer to stationarity… the variance will be reduced. If you do again and gets better again, then reduced variance. But, once variance starts to increase when differencing, you have gone too far and the last step was enough.
Ð A common mistake in time series modeling to “overdifference” the series, when more orders of differencing than are required to achieve stationarity are performed. This can complicate the process of building an adequate and parsimonious model
Ð Statistical test e.g. is the augmented Dickey–Fuller test (ADF; Said and Dickey, 1984) which tests the null hypothesis that the series is non-stationary. Thus, rejection of the null provides evidence for a stationary series.

Question 28

what is AUTO-CORRELATION (AC)?

Answer 28

It is a tool to find patterns in the data

Ð Specifically, the autocorrelation function tells you the correlation between points separated by various time lags

In psychological research, the current state of a variable may partially depend on prior states. That is, many psychological variables exhibit autocorrelation: when a variable is correlated with itself across different time points … Time series designs capture the effect of previous states and incorporate this potentially significant source of variance within their corresponding statistical models

Question 29

why perform autocorrelation ?

Answer 29

WHY? to determine how past and future data points are related in a time series.

Question 30

what is cross-correlation?

Answer 30

CROSS correlation – Correlations in time with another different time-series or the past of another time-series. Time lag is important.

Question 31

what are the Three types of autocorrelations?

Answer 31

1) LONG-RANGE CORRELATIONS
2) ANTI-CORRELATED (SHORT TERM AND OPPOSITE)
3) UNCORRELATED

Question 32

Name two types of pattern analysis in time-series?

Answer 32

Fourier

Wavelet

Question 33

what is fourier analysis?

Answer 33

Fourier look at oscilations through sine and cosine

Question 34

what is wavelet anaysis?

Answer 34

Ð Instead of oscillations, we look at diff types of waves.

Ð Not how much sinusoid, but how much of this pattern in data in small time windows.
Wavelet basis functions and time-frequency boxes.

Question 35

their is a trade off in wavelet?

Answer 35

Ð Good time resolution at high frequencies
Ð Good frequency resolution at low frequencies.

Ð Kind of trade-off between them

Question 36

what is Autoregressive Modelling

Answer 36

Autoregressive Modelling - An important aspect of time series analysis is forecasting: Assess the variation in a time series variable as a function of predictors and some stochastic term (error term).

an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation.

Question 37

what is the basic theory behind how Autoregressive Model (AR) works?

Answer 37

Autoregressive Model (AR) - With an autoregressive model AR the values of a time series x(t) are described by the sum of the linearly weighted values of the time series at previous time points (lagged). But usually we only need a certain number of previous time lags: model order p. FINDING P IS AN IMPORTANT THING IN TIME-SERIES

Question 38

what can Autoregressive Models be used for?

Answer 38

Ð Forecasting:
Predict the future of a process using past instances of different signals.
Ð Estimating spectral quantities:
Example: Power, Coherence. Coefficients transformed to the frequency space.
Ð Causality analysis:
Directional influence between time series.

Question 39

what is granger-causality?

Answer 39

Granger-Causality is a measure of causal or directional influence from one time series to another and is based on linear predictions of time series.

Granger-causality is not real causality, it reflects predictability.

If adding past values of signal x2 helps in predicting the evolution of x1 more than not adding those values (i.e. leads to smaller variance of the error term), then x2 Granger-causes x1.

We can assess whether having additional signals can help in predicting the evolution of a specific process we are interested in.

Question 40

what is the problem with granger causality?

Answer 40

We still have an issue with MEDIATION effects and discriminating between influential signals. In EEG, everything influences everything.

Question 41

what was developed to address this?

Answer 41

To resolve such ambiguity conditional granger causality was developed

Also, PDC is a popular index derived from Granger-causality used in multi-channel systems to disentangle direct and indirect connections

Question 42

what is a long-range temporal correlation?

Answer 42

Time series can present persistent temporal correlations that extend over many time scales: long-term memory processes.

This means, if there is an increase over some time scale, it will be very probable to be followed by an increase in the next time scale. As if some memory. The same is true of decreases, where a decrease will tend to follow a decrease. If something is growing at a time point, from here to here to here, it will also grow elsewhere in the same pattern – it is not changing randomly. Has a persistent pattern.

Question 43

example of long-range temporal correlation?

Answer 43

E.G. Persistent patterns in walking gait in parkinsons

Question 44

what is the hurst component?

Answer 44

Hurst exponent (H): is a measure of long-term memory of time series.

Question 45

what is Detrended Fluctuation Analysis (DFA) ?

Answer 45

Detrended Fluctuation Analysis (DFA)
An influential method to analyse long-range correlations in time series with non-stationarity and noise (Peng et al. 1995).

Question 46

what are the associated values with HURST component?

Answer 46

Ð 0.5 -> 1: Long-range dependency (long-range temporal correlations).
Ð 0.5: Uncorrelated series.
Ð 0 -> 0.5: Anticorrelations of values.

Question 47

again, we transform a non-stationary time series to a stationary time series via xxxxx.
In this exercise we will generate a non-stationary time series as a fractional xxxxx process with xxxxx exponent:

Answer 47

differentiation.

Brownian process

with Hurst exponent:

Question 48

The higher Hurst parameter is, the…….

Answer 48

the smoother the curve will be.

Question 49

again… what do these values mean?
 H = 0.5
 or H = 0.2
 or H =0.8

Answer 49

 H = 0.5 (true random process. Uncorrelated time serie. Brownian time series),
 or H = 0.2 (“anti-persistent behavior” or negative autocorrelation),
 or H =0.8 (long-range correlations, long memory process).

Question 50

if the auto-correlation function exhibits a positive value at time lag 1, a negative value at time lag 2, and is zero for the remaining time lags, the time series presents a…

Answer 50

anticorrelation: an increase in an interval is followed by a decrease in the subsequent interval.