## Introduction

Calculate time-weighted summary statistics, such as averages (means) and integrals. Time weighting is used when data is unevenly sampled over time. In that case, a straight average gives misleading results, as it biases towards more frequently sampled values.

For example, a sensor might silently spend long periods of time in a steady state, and send data only when a significant change occurs. The regular mean counts the steady-state reading as only a single point, whereas a time-weighted mean accounts for the long period of time spent in the steady state. In essence, the time-weighted mean takes an integral over time, then divides by the elapsed time.

This group of functions uses the two-step aggregation pattern.

Rather than calculating the final result in one step, you first create an intermediate aggregate by using the aggregate function.

Then, use any of the accessors on the intermediate aggregate to calculate a final result. You can also roll up multiple intermediate aggregates with the rollup functions.

The two-step aggregation pattern has several advantages:

1. More efficient because multiple accessors can reuse the same aggregate
2. Easier to reason about performance, because aggregation is separate from final computation
3. Easier to understand when calculations can be rolled up into larger intervals, especially in window functions and continuous aggregates
4. Can perform retrospective analysis even when underlying data is dropped, because the intermediate aggregate stores extra information not available in the final result

## Functions in this group ###### warning

This function group includes some experimental functions. Experimental functions might change or be removed in future releases. We do not recommend using them in production. Experimental functions are marked with an Experimental tag.

### Aggregate

time_weight
Aggregate data into an intermediate time-weighted aggregate form for further calculation

### Accessor

average
Calculate the time-weighted average of values in a `TimeWeightSummary`
first_time
Get the first timestamp from a `TimeWeightSummary` aggregate
first_val
Get the first value from a `TimeWeightSummary` aggregate
integral
ExperimentalCalculate the integral from a `TimeWeightSummary`
interpolated_average
ExperimentalCalculate the time-weighted average over an interval, while interpolating the interval bounds
interpolated_integral
ExperimentalCalculate the integral over an interval, while interpolating the interval bounds
last_time
Get the last timestamp from a `TimeWeightSummary` aggregate
last_val
Get the last value from a `TimeWeightSummary` aggregate

### Rollup

rollup
Combine multiple `TimeWeightSummaries`

## Function details

`time_weight(    method TEXT,    ts TIMESTAMPTZ,    value DOUBLE PRECISION) RETURNS TimeWeightSummary`

This is the first step for performing any time-weighted calculations. Use `time_weight` to create an intermediate aggregate (`TimeWeightSummary`) from your data. This intermediate form can then be used by one or more accessors in this group to compute final results. Optionally, multiple such intermediate aggregate objects can be combined using `rollup()` before an accessor is applied.

Required arguments
NameTypeDescription
`method``TEXT`The weighting method to use. The available methods are `linear` (or its alias `trapezoidal`, for those familiar with numeric integration methods) and `LOCF`, which stands for 'last observation carried forward'. `linear` fills in missing data by interpolating linearly between the start and end points of the gap. `LOCF` fills in the gap by assuming that the value remains constant until the next value is seen. `LOCF` is most useful when a measurement is taken only when a value changes. `linear` is most useful if there are no such guarantees on the measurement. The method names are case-insensitive.
`ts``TIMESTAMPTZ`The time at each point. Null values are ignored. An aggregate evaluated on only `null` values returns `null`.
`value``DOUBLE PRECISION`The value at each point to use for the time-weighted aggregate. Null values are ignored. An aggregate evaluated on only `null` values returns `null`.
Returns
ColumnTypeDescription
`time_weight``TimeWeightSummary`A `TimeWeightSummary` object that can be passed to other functions within the time-weighting API
Examples

Aggregate data from column `val` into daily time-weighted aggregates, using the linear interpolation method:

`SELECT    time_bucket('1 day'::interval, ts) as dt,    time_weight('Linear', ts, val) AS twFROM fooGROUP BY time_bucket('1 day'::interval, ts)`
`average(    tws TimeWeightSummary) RETURNS DOUBLE PRECISION`

Calculate the time-weighted average. Equal to `integral` divided by the elapsed time.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
Returns
ColumnTypeDescription
`average``DOUBLE PRECISION`The time-weighted average.
Examples

Calculate the time-weighted average of the column `val`, using the 'last observation carried forward' interpolation method:

`SELECT    id,    average(tws)FROM (    SELECT        id,        time_weight('LOCF', ts, val) AS tws    FROM foo    GROUP BY id) t`
`first_time(    tw TimeWeightSummary) RETURNS TIMESTAMPTZ`

Get the timestamp of the first point in a `TimeWeightSummary` aggregate.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
Returns
ColumnTypeDescription
`first_time``TIMESTAMPTZ`The time of the first point in the `TimeWeightSummary`
Examples

Produce a linear `TimeWeightSummary` over the column `val` and get the first timestamp:

`WITH t as (    SELECT        time_bucket('1 day'::interval, ts) as dt,        time_weight('Linear', ts, val) AS tw    FROM table    GROUP BY time_bucket('1 day'::interval, ts))SELECT    dt,    first_time(tw)FROM t;`
`first_val(    tw TimeWeightSummary) RETURNS DOUBLE PRECISION`

Get the value of the first point in a `TimeWeightSummary` aggregate.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
Returns
ColumnTypeDescription
`first_val``DOUBLE PRECISION`The value of the first point in the `TimeWeightSummary`
Examples

Produce a linear `TimeWeightSummary` over the column `val` and get the first value:

`WITH t as (    SELECT        time_bucket('1 day'::interval, ts) as dt,        time_weight('Linear', ts, val) AS tw    FROM table    GROUP BY time_bucket('1 day'::interval, ts))SELECT    dt,    first_val(tw)FROM t;`
`integral(    tws TimeWeightSummary,    unit TEXT) RETURNS DOUBLE PRECISION`

Calculate the integral, or the area under the curve formed by the data points. Equal to `average` multiplied by the elapsed time.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
Optional arguments
NameTypeDescription
`unit``TEXT`The unit of time to express the integral in. Can be `microsecond`, `millisecond`, `second`, `minute`, `hour`, or any alias for those units supported by PostgreSQL. Defaults to `second`.
Returns
ColumnTypeDescription
`integral``DOUBLE PRECISION`The time-weighted integral.
Examples

Create a table to track irregularly sampled storage usage in bytes, and get the total storage used in byte-hours. Use the 'last observation carried forward' interpolation method:

```-- Create a table to track irregularly sampled storage usageCREATE TABLE user_storage_usage(ts TIMESTAMP, storage_bytes BIGINT);INSERT INTO user_storage_usage(ts, storage_bytes) VALUES    ('01-01-2022 00:00', 0),    ('01-01-2022 00:30', 100),    ('01-01-2022 03:00', 300),    ('01-01-2022 03:10', 1000),    ('01-01-2022 03:25', 817);
-- Get the total byte-hours usedSELECT    toolkit_experimental.integral(time_weight('LOCF', ts, storage_bytes), 'hours')FROM    user_storage_usage;```
`interpolated_average(    tws TimeWeightSummary,    start TIMESTAMPTZ,    interval INTERVAL    [, prev TimeWeightSummary]    [, next TimeWeightSummary]) RETURNS DOUBLE PRECISION`

Calculate the time-weighted average over an interval, while interpolating the interval bounds.

Similar to `average`, but allows an accurate calculation across interval bounds when data has been bucketed into separate time intervals, and there is no data point precisely at the interval bound. For example, this is useful in a window function.

Values from the previous and next buckets are used to interpolate the values at the bounds, using the same interpolation method used within the `TimeWeightSummary` itself.

Equal to `interpolated_integral` divided by the elapsed time.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
`start``TIMESTAMPTZ`The start of the interval which the time-weighted average should cover (if there is a preceeding point).
`interval``INTERVAL`The length of the interval which the time-weighted average should cover.
Optional arguments
NameTypeDescription
`prev``TimeWeightSummary`The `TimeWeightSummary` from the prior interval, used to interpolate the value at `start`. If NULL, the first timestamp in `tws` is used for the starting value. The prior interval can be determined from the PostgreSQL `lag()` function.
`next``TimeWeightSummary`The `TimeWeightSummary` from the next interval, used to interpolate the value at `start` + `interval`. If NULL, the first timestamp in `tws` is used for the starting value. The next interval can be determined from the PostgreSQL `lead()` function.
Returns
ColumnTypeDescription
`average``DOUBLE PRECISION`The time-weighted average for the interval (`start`, `start` + `interval`), computed from the `TimeWeightSummary` plus end points interpolated from `prev` and `next`
Examples

Calculate the time-weighted daily average of the column `val`, interpolating over bucket bounds using the 'last observation carried forward' method:

`SELECT    id,    time,    interpolated_average(        tws,        time,        '1 day',        LAG(tws) OVER (PARTITION BY id ORDER by time),        LEAD(tws) OVER (PARTITION BY id ORDER by time)    )FROM (    SELECT        id,        time_bucket('1 day', ts) AS time,        time_weight('LOCF', ts, val) AS tws    FROM foo    GROUP BY id, time) t`
`interpolated_integral(    tws TimeWeightSummary,    start TIMESTAMPTZ,    interval INTERVAL    [, prev TimeWeightSummary]    [, next TimeWeightSummary]) RETURNS DOUBLE PRECISION`

Calculate the integral over an interval, while interpolating the interval bounds.

Similar to `integral`, but allows an accurate calculation across interval bounds when data has been bucketed into separate time intervals, and there is no data point precisely at the interval bound. For example, this is useful in a window function.

Values from the previous and next buckets are used to interpolate the values at the bounds, using the same interpolation method used within the `TimeWeightSummary` itself.

Equal to `interpolated_average` multiplied by the elapsed time.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
`start``TIMESTAMPTZ`The start of the interval which the time-weighted average should cover (if there is a preceding point).
`interval``INTERVAL`The length of the interval which the time-weighted average should cover.
Optional arguments
NameTypeDescription
`prev``TimeWeightSummary`The `TimeWeightSummary` from the prior interval, used to interpolate the value at `start`. If NULL, the first timestamp in `tws` is used for the starting value. The prior interval can be determined from the PostgreSQL `lag()` function.
`next``TimeWeightSummary`The `TimeWeightSummary` from the next interval, used to interpolate the value at `start` + `interval`. If NULL, the first timestamp in `tws` is used for the starting value. The next interval can be determined from the PostgreSQL `lead()` function.
Returns
ColumnTypeDescription
`integral``DOUBLE PRECISION`The time-weighted integral for the interval (`start`, `start` + `interval`), computed from the `TimeWeightSummary` plus end points interpolated from `prev` and `next`
Examples

Create a table to track irregularly sampled storage usage in bytes, and get the total storage used in byte-hours between January 1 and January 6. Use the 'last observation carried forward' interpolation method:

```-- Create a table to track irregularly sampled storage usageCREATE TABLE user_storage_usage(ts TIMESTAMP, storage_bytes BIGINT);INSERT INTO user_storage_usage(ts, storage_bytes) VALUES    ('01-01-2022 20:55', 27),    ('01-02-2022 18:33', 100),    ('01-03-2022 03:05', 300),    ('01-04-2022 12:13', 1000),    ('01-05-2022 07:26', 817);

-- Get the total byte-hours used between Jan. 1 and Jan. 6SELECT    toolkit_experimental.interpolated_integral(        time_weight('LOCF', ts, storage_bytes),        '01-01-2022',        '5 days',        NULL,        NULL,        'hours'    )FROM    user_storage_usage;```
`last_time(    tw TimeWeightSummary) RETURNS TIMESTAMPTZ`

Get the timestamp of the last point in a `TimeWeightSummary` aggregate.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
Returns
ColumnTypeDescription
`last_time``TIMESTAMPTZ`The time of the last point in the `TimeWeightSummary`
Examples

Produce a linear `TimeWeightSummary` over the column `val` and get the last timestamp:

`WITH t as (    SELECT        time_bucket('1 day'::interval, ts) as dt,        time_weight('Linear', ts, val) AS tw    FROM table    GROUP BY time_bucket('1 day'::interval, ts))SELECT    dt,    last_time(tw)FROM t;`
`last_val(    tw TimeWeightSummary) RETURNS DOUBLE PRECISION`

Get the value of the last point in a `TimeWeightSummary` aggregate.

Required arguments
NameTypeDescription
`tws``TimeWeightSummary`The input `TimeWeightSummary` from a `time_weight()` call.
Returns
ColumnTypeDescription
`last_val``DOUBLE PRECISION`The value of the last point in the `TimeWeightSummary`
Examples

Produce a linear `TimeWeightSummary` over the column `val` and get the last value:

`WITH t as (    SELECT        time_bucket('1 day'::interval, ts) as dt,        time_weight('Linear', ts, val) AS tw    FROM table    GROUP BY time_bucket('1 day'::interval, ts))SELECT    dt,    last_val(tw)FROM t;`
`rollup(    tws TimeWeightSummary) RETURNS TimeWeightSummary`

Combine multiple intermediate time-weighted aggregate (`TimeWeightSummary`) objects produced by `time_weight()` into a single intermediate `TimeWeightSummary` object. For example, you can use `rollup` to combine time-weighted aggregates from 15-minute buckets into daily buckets.

Required arguments
NameTypeDescription
`time_weight``TimeWeightSummary`The `TimeWeightSummary` aggregate produced by a `time_weight` call
Returns
ColumnTypeDescription
`rollup``StatsSummary1D`A new `TimeWeightSummary` aggregate produced by combining the input `TimeWeightSummary` aggregates

## Extended examples

### Aggregate data into a `TimeWeightSummary` and calculate the average

Given a table `foo` with data in a column `val`, aggregate data into a daily `TimeWeightSummary`. Use that to calculate the average for column `val`:

`WITH t as (    SELECT        time_bucket('1 day'::interval, ts) as dt,        time_weight('Linear', ts, val) AS tw    FROM foo    WHERE measure_id = 10    GROUP BY time_bucket('1 day'::interval, ts))SELECT    dt,    average(tw)FROM t;`

### Parallelism and ordering

Time-weighted average calculations are not strictly parallelizable, as defined by PostgreSQL. These calculations require inputs to be strictly ordered, but in general, PostgreSQL parallelizes by assigning rows randomly to workers.

However, the algorithm can be parallelized if it is guaranteed that all rows within some time range go to the same worker. This is the case for both continuous aggregates and distributed hypertables. (Note that the partitioning keys of the distributed hypertable must be within the `GROUP BY` clause, but this is usually the case.)

### Combining aggregates across measurement series

If you try to combine overlapping `TimeWeightSummaries`, an error is thrown. For example, you might create a `TimeWeightSummary` for `device_1` and a separate `TimeWeightSummary` for `device_2`, both covering the same period of time. You can't combine these because the interpolation techniques only make sense when restricted to a single measurement series.

If you want to calculate a single summary statistic across all devices, use a simple average, like this:

`WITH t as (SELECT measure_id,        average(            time_weight('LOCF', ts, val)        ) as time_weighted_average    FROM foo    GROUP BY measure_id)SELECT avg(time_weighted_average) -- use the normal avg function to average the time-weighted averagesFROM t;`

### Parallelism in multi-node

The time-weighted average functions are not strictly parallelizable in the PostgreSQL sense. PostgreSQL requires that parallelizable functions accept potentially overlapping input. As explained above, the time-weighted functions do not. However, they do support partial aggregation and partition-wise aggregation in multi-node setups.

### Reducing memory usage

Because the time-weighted aggregates require ordered sets, they build up a buffer of input data, sort it, and then perform the aggregation steps. When memory is too small to build up a buffer of points, you might see Out of Memory failures or other issues. In these cases, try using a multi-level aggregate. For example:

`WITH t as (SELECT measure_id,    time_bucket('1 day'::interval, ts),    time_weight('LOCF', ts, val)    FROM foo    GROUP BY measure_id, time_bucket('1 day'::interval, ts)    )SELECT measure_id,    average(        rollup(time_weight)    )FROM tGROUP BY measure_id;`