# Use correlation to compare two time series?

I have two time series data sets which contain hourly-intervalled, monthly, and yearly household electricity consumption in kWh. One data set is produced by a simulation, the other gathered from the real-world. My aim is to validate the simulated output by using the data gathered from the real-world.

I want to measure the similarity between these data sets, and be able to say if these are statistically similar. My first intuition is to use a correlation coefficient such a Pearson product moment correlation. But from what I read in previous posts that in general the correlation coefficient between two time-series may be a very poor metric.

I'm not very keen on statistics related to time series, but would something like a cross-correlation or maybe ARIMA do the trick? Could someone please point me in the direction for a technique which I can use in SPSS?

• Quantifying similarity is always possible. However, correlation measures linearity of relationship, not similarity of values (the correlation between $y$ and $by$ is identically 1 if $b$ is positive, but for large $b$ the two don't qualify as "similar". Concordance correlation does measure agreement. For most purposes, however, looking at the pattern of differences between observed and simulated will be as or more informative than trying to encapsulate similarity in a single number. Cross-correlation or ARIMA sound like overkill for 24 values. – Nick Cox Oct 8 '13 at 13:57
• Thank for your informative answer Nick Cox. Concordance correlation does seem like a good alternative. I understand from your explanation that measuring correlation might be useless since it measures linearity of relationship, and doesn't take into account the impact of larger values. I have no experience with Cross-correlation or ARIMA, but could you maybe explain to me why Cross-correlation and ARIMA are overkill to measure similarity? Aren't these tools more suitable to measure similarity and at the same time provide more details? – Rez Oct 8 '13 at 14:28
• Cross-correlation may have some descriptive merit if it exposes a lag structure e.g. if your simulated values lead or lag the observed values, but that should be evident too from plotting the data. On ARIMA: I am not sure that you understand what it does. It's a class of models; so do you intend to model the structure of disagreement between your simulation model and the observed values? For 24 hourly values, I wouldn't start with ARIMA, and that's a small dataset any way. – Nick Cox Oct 8 '13 at 14:36
• ARIMA is mainly used for modeling? I was advised that I could compare ARIMA coefficients. But I guess that I was pointed into the wrong direction. Do you by any chance know of any chance happen to know any useful descriptive statistics to examine such as standard deviation, median, and so on. Thank you for your help. – Rez Oct 8 '13 at 15:08
• You seem fixated on producing descriptive measures. I suggest that you focus much more on looking at your data. As you explained your problem, you have two time series, each 24 observations long. Plot the two series against time, and the difference between observed and simulated against time. You could e.g. compare the means (do they agree? how much do they differ?) and/or the differences between the means as well, but I'd advise against the idea that any single named measure gives you the answer. Note that getting the same means even would not establish that the series were identical. – Nick Cox Oct 8 '13 at 15:14