Solved – How to visualize changes in time series variance

data visualizationtime seriesvariance

I am exploring a time series with the aim of detecting obvious changes in variance over time (I have little experience in time series analysis). The hypothesis is that the series has become more stable over time (stable=decreased dispersion).

I am wondering whether there are any strategies to visually assess how a measure of dispersion (does not have to be variance) changes as the series progresses. I understand there are tests for the constant variance assumption. My outcome is a daily measure captured over 10 years, with changes between months being the relevant measure.

As a first step, I have plotted the monthly MAD (median absolute deviations) over time. However, I'm not sure that's a sensible approach.

Best Answer

I think this is an good question with a number of possible approaches. Sadly the change in variation over time has received far less study than the change in mean/median value in time series data.

I think there is a good R package for variance changes in time series called 'changepoint'

This has statistical tools for analysing if you have change-points where the variance alters during a series. Its visualisations are, however, much better for means than for variance. It also is dealing more with the times when the variance changes rather than trends over time.

You also seem interested in trends, I think your approach of plotting a variance measure (variance, deviation, MAD etc) is a good place to go. You then of course have another time series! you can analyse this with regression, ARIMA etc and generate trends and forecasts.

I hope this helps

Related Solutions

Solved – Removing a ‘trend in variance’ from a time series

No one will be able to help you without an example of the data or more information, however a few things can be said. I will not elaborate on the algebra of things though, you best research that for yourself.

The books of Greene and/or Hayashi concerning Econometrics are a good source about the topics of heteroscedasticity, autocorrelation and (F)GLS. An easier example is Stock/Watson.

First, the variance trend can probably be modeled by a matrix of covariances, ie. $V[\epsilon|X] = \Omega$
In that case you can use the GLS estimator to eliminate the heteroscedasticity. Especially useful to you should be the FGLS techniques. In that case you estimate the matrix from residuals of a regular OLS and then iteratively refine your estimates. Wikipedia FGLS

As far as your dependency on the magnitude of your data is concerned, this is also eliminated by FGLS. You can see that the GLS estimator (in terms of General Method Of Moments) comes down to the following problem:
${E}[\,x_t(y_t - x_t'\beta)/\sigma^2(x_t)\,]=0$
So you should transform your model accordingly (this is equivalent to using the GLS estimator, using the matrix $\Omega$
In addition to the wikipedia article I wrote about GLS / heteroscedasticity here

If you are using R, another strategy can be to use Newey-West variance/covariance robust error term estimators. This can be achieved by adding vcov=vcovHAC as an argument to your R-functions. If your dataset is large enough, this will give you consistent estimates of your error terms.

Solved – Trend Analysis: How to tell random fluctuations from actual changes in trends

Expanding on @Aleksendr Blekh's answer. You can use the below code for decomposing trend, using the decompose function in R:

births <- scan("http://robjhyndman.com/tsdldata/data/nybirths.dat")
birthstimeseries <- ts(births, frequency=12, start=c(1946,1))
if(!require("TTR")) { install.packages("TTR");  require("TTR") }
birthstimeseriescomponents <- decompose(birthstimeseries)
birthstimeseriescomponents$seasonal # the seasonal component
birthstimeseriescomponents$trend # the trend component
birthstimeseriescomponents$random # random (irregular / remainder) component
# plot the seasonality, trend and random (epsilon) components
plot(birthstimeseriescomponents)

For changes in trends, relevant reading: Hidden Markov Models, Change Point Analysis.

This post may also be helpful to find fluctuations from the underlying trend. The approach here is fitting a polynomial instead of time series decomposition and reporting points that fall outside the confidence intervals.

Best Answer

Related Solutions

Solved – Removing a ‘trend in variance’ from a time series

Solved – Trend Analysis: How to tell random fluctuations from actual changes in trends

Related Question