Time Series Flashcards
Discrete Time Series
If the Index set “T” is a countable set.
A Time Series is Discrete or Continuous by the adjective of the Index set “T”, not the random variable “X”.
Continuous Time Series
If the Index set “T” is an interval.
A Time Series is Discrete or Continuous by the adjective of the Index set “T”, not the random variable “X”.
Time Series
Time series is a measurement of same Random Variables over a Time Index set T which might be a finite, countably infinite or uncountable set.
Stationary Time Series
- Constant mean
- Constant variance
- No Auto correlation.
Weakly Stationary Time Series
- Constant mean
- Constant variance
- The covariance (and also correlation) between Xt and Xt-h is the same for all t at each lag h = 1, 2, 3, etc.
Intuition behind Fourier transform on Time Series
What is stationarity?
Real life example of a stationary time series
Exogenous variables
Removing outliers
Time series Clustering
Seasonality
Repetition of pattern at certain interval.
E.g. - Weeks, Months, Quarters, Seasons
Cyclicality
Repetition of pattern without any fixed period.
E.g. - Some stocks are cyclical in nature.
Noise
Random variation or irregularity that remains after removing the trend and seasonal components.
Some kind of irregularity that happened in the past, but not expected to repeat in the future.
White Noise
Uncorrelated values.
ACF of white noise has a spike at zero, and then nothing else.
Why do we decompose Time Series?
To breakdown a time series data set into individual components - Trend, Seasonality and Noise.
To get a understanding of the underlying patterns, trends, seasonality, and cyclical variations
Additive Decomposition
Components of the time series are additive, can be expressed as the sum of Trend, Cycle, Seasonality, and Noise.
Y(t) =T(t) + S(t) + ε(t)
Used when the seasonal variation is relatively constant over time.
Multiplicative Decomposition
Components are considered to be multiplicative, can be expressed as the product of Trend, Cycle, Seasonality, and Noise.
Y(t) =T(t) * S(t) * ε(t)
Used when the seasonal variation increases over time.
How is linear regression deflect from time series?
Data are not necessarily independent and not necessarily identically distributed.
Time series is a list of observations where the ordering matters. Ordering is very important because there is dependency and changing the order could change the meaning of the data.
Trend
On average, the measurements tend to increase (or decrease) over time
AR(p) model
A linear model to predict the value at the present time using the value at the previous time.
Order (p) of the model indicates how many previous times we use to predict the present time.
Autocorrelation Function (ACF)
Autocorrelation function (ACF) for a series gives correlations between the series and lagged values of the series for lags of 1, 2, 3, and so on.
MA(q) model
A linear model to predict the value at the present time using the value of past errors.
Order (q) of the model indicates how many past error we use to predict the present time.
ACF & PACF for AR(p)
ACF: Taper to zero
PACF: First q lags = Non-zero; lags > q = 0
