Interpolation

In the example above, both time series A and B had data points at every time stamp, they lined up neatly. However what happens when two series do not line up? It can be difficult, and sometimes undesired, to synchronize all sources of data to write at the exact same time. For example, if we have 10,000 servers sending 100 system metrics every 5 minutes, that would be a burst of 10M data points in a single second. We would need a pretty beefy network and cluster to accommodate that traffic. Not to mention the system would be sitting idle for the rest of 5 minutes. Instead it makes much more sense to splay the writes over time so that we have an average of 3,333 writes per second to reduce our hardware and network requirements.

How do you sum or find the avg of a number and something that doesn't exist? One option is to simply ignore the data points for all time series at the time stamp where any series is missing data. But if you have two time series and they are simply miss-aligned, your query would return an empty data set even though there is good data in storage, so that's not very useful.

Another option is to define a scalar value (e.g. 0 or the maximum value for a Long) to use whenever a data point is missing. OpenTSDB 2.0 provides a few aggregation methods that substitute a scalar value for missing data points. These are useful when working with distinct value time series such as the number of sales in at a given time.

However sometimes it doesn't make sense to define a scalar for missing data. Often you may be recording a monotonically increasing counter such as the number of bytes transmitted from a network interface. With a counter, we can use interpolation to make a guess as to what the value would be at that point in time. Interpolation takes two points and the time span between them to calculate a best guess value at the time stamp requested.

Take a look at these two time series where the data is simply offset by 10 seconds:

When OpenTSDB is calculating an aggregation it starts at the first data point found for any series, in this case it will be the data for B at ts0. We request a value for A at ts0 but there isn't any data there. We know that there is data for A at ts0+10s but since we don't have any value before that, we can't make a guess as to what it would be. Thus we simply return the value for B.

Next we run across a value for A at time ts0+10s. We request a value for ts0+10s from time series B but there isn't one. But B knows there is a value at ts0+20s and we had a value at ts0 so we can now calculate a guess for ts0+10s. The formula for linear interpolation is y = y0 + (y1 - y0) * ((x - x0) / (x1 - x0)) where, for series B, y0 = 10, y1 = 20, x = ts0+10s (or 10), x0 = ts0 (or 0) and x1 = ts0+20s (or 20). Thus we have y = 10 + (20 - 10) * ((10 - 0) / (20 - 0) which will reduce to y = 10 + 10 * (10 / 20) further reducing to y = 10 + 10 * .5 and y = 10 + 5. Therefore B will give us a guestimated value of 15 at ts0+10s.

Iteration continues over every timestamp for which a data point is found for every series returned as a part of the query. The resulting series, using the sum aggregator, will look like this:

More Examples: For the graphically inclined we have the following examples. An imaginary metric named m is recorded in OpenTSDB. The "sum of m" is the blue line at the top resulting from a query like start=1h-ago&m=sum:m. It's made of the sum of the red line for host=foo and the green line for host=bar:

It seems intuitive from the image above that if you "stack up" the red line and the green line, you'd get the blue line. At any discrete point in time, the blue line has a value that is equal to the sum of the value of the red line and the value of the green line at that time. Without interpolation, you get something rather unintuitive that is harder to make sense of, and which is also a lot less meaningful and useful:

Notice how the blue line drops down to the green data point at 18:46:48. No need to be a mathematician or to have taken advanced maths classes to see that interpolation is needed to properly aggregate multiple time series together and get meaningful results.

At the moment OpenTSDB primarily supports linear interpolation (sometimes shortened "lerp") along with some aggregators that will simply substitute zeros or the max or min value. Patches are welcome for those who would like to add other interpolation methods.

Interpolation is only performed at query time when more than one time series are found to match a query. Many metrics collection systems interpolate on write so that you original value is never recorded. OpenTSDB stores your original value and lets you retrieve it at any time.

Here is another slightly more complicated example that came from the mailing list, depicting how multiple time series are aggregated by average:

The thick blue line with triangles is the an aggregation with the avg function of multiple time series as per the query start=1h-ago&m=avg:duration_seconds. As we can see, the resulting time series has one data point at each timestamp of all the underlying time series it aggregates, and that data point is computed by taking the average of the values of all the time series at that timestamp. This is also true for the lonely data point of the squared-purple time series, that temporarily boosted the average until the next data point.

Avg

Calculates the average of all values across the time span or across multiple time series. This function will perform linear interpolation across time series. It's useful for looking at gauge metrics. Note that even though the calculation will usually result in a float, if the data points are recorded as integers, an integer will be returned losing some precision.

Count

Returns the number of data points stored in the series or range. When used to aggregate multiple series, zeros will be substituted. It's best to use this when downsampling.

Dev

Calculates the standard deviation across a span or time series. This function will perform linear interpolation across time series. It's useful for looking at gauge metrics. Note that even though the calculation will usually result in a float, if the data points are recorded as integers, an integer will be returned losing some precision.

Estimated Percentiles

Calculates various percentiles using a choice of algorithms. These are useful for series with many data points as some data may be kicked out of the calculation. When used to aggregate multiple series, the function will perform linear interpolation. See Wikipedia for details. Implementation is through the Apache Math library.

First & Last

(2.3) These aggregators will return the first or the last data point in the downsampling interval. E.g. if a downsample bucket consists of the series 2, 6, 1, 7 then the first aggregator will return 1 and last will return 7. Note that this aggregator is only useful for downsamplers. When used as a group-by aggregator, the results are indeterminate as the ordering of time series retrieved from storage and held in memory is not consistent from TSD to TSD or execution to execution.

Max

The inverse of min, it returns the largest data point from all of the time series or within a time span. This function will perform linear interpolation across time series. It's useful for looking at the upper bounds of gauge metrics.

MimMin

The "maximum if missing minimum" function returns only the smallest data point from all of the time series or within the time span. This function will not perform interpolation, instead it will return the maximum value for the type of data specified if the value is missing. This will return the Long.MaxValue for integer points or Double.MaxValue for floating point values. See Primitive Data Types for details. It's useful for looking at the lower bounds of gauge metrics.

MimMax

The "minimum if missing maximum" function returns only the largest data point from all of the time series or within the time span. This function will not perform interpolation, instead it will return the minimum value for the type of data specified if the value is missing. This will return the Long.MinValue for integer points or Double.MinValue for floating point values. See Primitive Data Types for details. It's useful for looking at the upper bounds of gauge metrics.

Min

Returns only the smallest data point from all of the time series or within the time span. This function will perform linear interpolation across time series. It's useful for looking at the lower bounds of gauge metrics.

None

(2.3) Skips group by aggregation. This aggregator is useful for fetching the raw data from storage as it will return a result set for every time series matching the filters. Note that the query will throw an exception if used with a downsampler.

Percentiles

Calculates various percentiles. When used to aggregate multiple series, the function will perform linear interpolation. Implementation is through the Apache Math library.

Sum

Calculates the sum of all data points from all of the time series or within the time span if down sampling. This is the default aggregation function for the GUI as it's often the most useful when combining multiple time series such as gauges or counters. It performs linear interpolation when data points fail to line up. If you have a distinct series of values that you want to sum and you do not need interpolation, look at zimsum

ZimSum

Calculates the sum of all data points at the specified timestamp from all of the time series or within the time span. This function does not perform interpolation, instead it substitutes a 0 for missing data points. This can be useful when working with discrete values.