Introduction and Basic principles Flashcards
Define a time series
A set of observations {xt}t element of A recorded across time and is a realisation fo a sequence of random variables {Xt}t element of A. Main characteristic iod time series data has dependence and usually here A is finite as we observe over fixed time
Why does time series data need spearate analysis methods/ studies
Correlation with sampling over adjacent time points can restrict use of traditional stats methods which often assume observations are iid.
When is a time series gaussian ?
Sequence {Xt} is gaussian time series if Xt1, Xt2, … , Xtk follow a multivariate normal distribution for all time indexes
Define statistical noise
Unexplained variability with a data sample
What is independent white noise?
When White noises are all IID
What is the value of delta in a random walk
The drift. If drift = 0 Xt is simply a random walk, otherwise this drift term measures the trend of the random walk. Graph of Xt will be based loosely around line Y = Delta*x
Why is it called a random walk?
Value of time series at time t is value of the time series at t-l plus a completely random movement determined by Wt
Explain the idea of signal?
In general we employ simple additive model of some unknown signal + time series that may be white or correlated. Many realistic models for time series assume a signal with some consistent period variation which then gets contaminated by adding of random noise
Why do we rely on mean functions of time series?
A complete description of a time series (n RVs observed at K times) is provided by a multidimensional distribution function that is not easily written out. Often we compare different realisations of the time series to the mean function
What does autocovariance measure?
Linear dependence between two points on the same series observed at different times.
Explain how we can tell if a series is smooth or choppy over time through autocovariance
Smooth series autocovariance functions stay large even if t and s are far apart, choppy series will have autocovariance functions close to 0 for large separations.
If Autocovariance function is 0 what does that mean?
Xs and Xt are not linearly related
If Xs and Xt are bivariate Normal what doesn Autocovariance function of 0 mean
Xs and Xt are also independent
If X and Y are idnependent what is their covariance
0
Covariance (X,Y) = Covariance of (Y,X)?
Yes
What is the autocorrelation of a time series?
Correlation of Xs and Ys across time for s and t. The ACF measures the linear predictability of the series at time t using only the values of the xs
What value can the Autocorrelation function take?
Between -1 and 1 only
If cor(Xt , Xs)= 1 what does this imply?
We can predict xt perfectly from xs and there is a positive linear relationship xt= a+bxs
If cor(Xt , Xs)= -1 what does this imply?
We can predict xt perfectly from xs and there is a negative linear relationship xt= a-bxs
