Hi Anthony,

time series analysis is a wide subject and what you choose to do highly depends on your research questions.

The timeplots you posted suggest that all your series are stationary, and the most evident difference among them is in the mean of the process; in view of this, as a first step, you could simply compare the sample mean of the time series with one of the many available tests. The most common is the Student's t test, but there are other - also nonparametric- tests.

Moreover, you could check the presence of serial correlation in your time series, through an analysis of the ACF and PACF. If serial correlation is present, you could assess how it changes among the time series, specifying, for example, a different ARMA process for each one. This can be done automatically with the auto.arima() function in the R package 'forecast', but I would recommend to gain some understanding of time series analysis before doing so.

I am not sure what you mean by 'frequency distribution', but I assume you computed the periodogram. I'd say that spectral analysis is a quite refined tool in the time series framework, and you can probably take advantage of simpler techniques.

For a very good manual of basic time series analysis, see https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=2ahUKEwip9fzFt9HdAhXKyYUKHabeDlIQFjAAegQICRAC&url=https%3A%2F%2Fwww.researchgate.net%2Ffile.PostFileLoader.html%3Fid%3D5524f76cd3df3e2e7d8b4627%26assetKey%3DAS%3A273752485498895%401442279086851&usg=AOvVaw33k2rsI_PC-pIcFRZku5oV

Hi Flavio,

Thank you for that link, that is a wealth of information.

In terms of what the research question is, I essentially want to ask whether the two time-series trajectories, one in black and the other in yellow, as statistically indifferent from one another, suggesting that a particular change to the independent variable the function of interest makes no impact on the outcome.

Oh ok, so you need something to discern those two series in particular. Since they have the same mean (more or less) then you need to look at something else. I'd recommend an analysis of the correlation through an inspection of ACF and PACF, but to find differences measurable in some way I think you'll need to specify a model.

If your sample size is actually 35 I suggest you keep it really simple, if instead you have more data points use them! With small sample size you could try a simple model (for example an AR(1) or a MA(1) to start) and compare the coefficients also using their respective standard errors.

However, which model is good for a time series is not a trivial task, so have a look at the book I linked, which is one of the bibles of basic time series analysis.

Hi Flavio,

I wondered whether this could be approached differently? It's not a comparison of the quality of the time-series itself I am interested, but rather the quality of the dependent variable as a function of time. If I change the data from a time-series (as per the image) into a frequency distribution (which is highly likely to be normally distributed), could I then apply a one-way anova to determine the quality of the null hypothesis before moving to post hoc analysis?

My concern is whether you can actually apply 1-way-anova to data which is dependent on time? i.e., Just by looking at the yellow plot, each time point is probably normally distributed around a value of 750, however, sampling it is not an independent process, rather one sample is dependent on the value of that sample at time t-1.

When I look at how to implement 1-way-anova in R, this is the first thing I read:

"The observations are obtained independently and randomly from the population defined by the factor levels"

Perhaps this:

https://statistics.laerd.com/spss-tutorials/one-way-anova-repeated-measures-using-spss-statistics.php

Autocorrelation in One way Anova.

The point is that ANOVA is supposed to measure some effect on the mean of your process; I would say that, just inspecting the time plot, no significant different would be found. Sure you can use some test to decide this objectively, but it's pretty evident from the data. So, you'll need to look at differences in the second-order properties (the correlation structure), that are usually non visible or not measurable from the simple plots. ACF and PACF tell you exactly "how" the measure at time t depends on the measure at time t-1, so a difference in these indicators will tell you about " the quality of the dependent variable as a function of time".

If you want to look at some simpler stuff, you could check if the variance (although this looks pretty similar), skewness and kurtosis of the distributions change across the two samples, but it's less easy than testing a mean and I do believe that 35 data are not a sufficient sample to test third or higher order moments...

Thank you for that information Flavio. Just as a side, this was a graph put together by a collaborator. The "35" is "35 nanoseconds", however, there are in fact about 35,000 available data points in the raw data.

Oh, ok, this surely gives you a large sample size, so you could compare the distributions is some way, for example, a Kolmogorov-Smirnov test compares the distribution functions without any assumption of normality and sphericity.

On the other hand, if your plots result from 35000 data points, I guess that you will need to consider a sub-sampling, as with such a high frequency the process would result too smooth and will appear to be non-stationary even if they definitely are.