*Carl Haines – carl.haines@pgr.reading.ac.uk*

Space weather is a natural threat that requires good quality forecasting with as much lead time as possible. In this post I outline the simple and understandable analogue ensemble (AnEn) or “similar day” approach to forecasting. I focus mainly on exploring the method itself and, although this work forecasts space weather through a timeseries of ground level observations, AnEn can be applied to many prediction tasks, particularly time series with strong auto-correlation. AnEn has previously been used to predict wind speed [1], temperature [1] and solar wind [2]. The code for AnEn is available at https://github.com/Carl-Haines/AnalogueEnsemble should you wish to try out the method for you own application.

The idea behind AnEn is to take a set of recent observations, look back in a historic dataset for analogous periods, then take what happened following those analogous periods as the forecast. If multiple analogous periods are used, then an ensemble of forecasts can be created giving a distribution of possible outcomes with probabilistic information.

*Figure 1 – An example of AnEn applied to a space weather event with forecast time t_{0}. The black line shows the observations, the grey line shows the ensemble members, the red line shows the median of the ensemble and the yellow and green lines are reference forecasts. *

Figure 1 is an example of a forecast made using the AnEn method where the forecast is made at t_{0}. The 24-hours of observations (black) prior to t_{0 }are matched to similar periods in the historic dataset (grey). Here I have chosen to give the most recent observations the most weighting as they hold the most relevant information. The grey analogue lines then flow on after t_{0} forming the forecast. Combined, these form an ensemble and the median of these is shown in red. The forecast can be chosen to be the median (or any percentile) of the ensemble or a probability of an event occurring can be given by counting how many of the ensemble member do/don’t experience the event.

Figure 1 also shows two reference forecasts, namely 27-day recurrence and climatology, as benchmarks to beat. 27-day recurrence uses the observation from 27-days ago as the forecast for today. This is reasonable because the Sun rotates every 27-days as seen from earth so broadly speaking the same part of the Sun is emitting the relevant solar wind on timescales larger than 27-days.

To quantify how well AnEn works as a forecast I ran the forecast on the entire dataset by repeatedly changing the forecast time t_{0} and applied two metrics, namely mean absolute error (MAE) and skill, to the median of the ensemble members. MAE is the size of the mean difference between the forecast made by AnEn and what was actually observed. The mean of the absolute errors over all the forecasts (taken as median of the ensemble) is taken and we end up with a value for each lead time. Figure 2 shows the MAE for AnEn median and the reference forecasts. We see that AnEn has the smallest (best) MAE at short lead times and outperforms the reference forecasts for all lead times up to a week.

*Figure 2 – The mean absolute error of the AnEn median and reference forecasts.*

An error metric such as MAE cannot take into account that certain conditions are inherently more difficult to forecast such as storm times. For this we can use a skill metric defined by

where in this case we use climatology as the reference forecast. Skill can take any value between and where a perfect forecast would receive a value of and an unskilful forecast would receive a value of . A negative value of skill signifies that the forecast is worse than the reference forecast.

Figure 3 shows the skill of AnEn and 27-day recurrence with respect to climatology. We see that AnEn is most skilful for short lead times and outperforms 27-day recurrence for all lead times considered.

*Figure 3 – The skill of the AnEn median and 27-day recurrence with respect to climatology.*

In summary, the analogue ensemble forecast method matches current conditions with historical events and lifts the previously seen timeseries as the prediction. AnEn seems to perform well for this application and outperforms the reference forecasts of climatology and 27-day recurrence. The code for AnEn is available at https://github.com/Carl-Haines/AnalogueEnsemble.

The work presented here makes up a part of a paper that is under review in the journal of Space Weather.

Here, AnEn has been applied to a dataset from the space weather domain. If you would like to find out more about space weather then take a look at these previous blog posts from Shannon Jones (https://socialmetwork.blog/2018/04/13/the-solar-stormwatch-citizen-science-project/) and I (https://socialmetwork.blog/2019/11/15/the-variation-of-geomagnetic-storm-duration-with-intensity/).

[1] Delle Monache, L., Eckel, F. A., Rife, D. L., Nagarajan, B., & Searight, K.(2013) Probabilistic Weather Prediction with an Analog Ensemble. doi: 10.1175/mwr-d-12-00281.1

[2] Owens, M. J., Riley, P., & Horbury, T. S. (2017a). Probabilistic Solar Wind and Ge-704omagnetic Forecasting Using an Analogue Ensemble or “Similar Day” Approach. doi: 10.1007/s11207-017-1090-7