This section of the user guide provides an overview of time series **aggregation**, **smoothing** and **differencing**.

## Time Series Aggregation

The `timeseries`

function performs fast, distributed time series aggregation leveraging Solr’s builtin faceting and date math capabilities.

The example below performs a monthly time series aggregation:

```
timeseries(collection1,
q=*:*,
field="recdate_dt",
start="2012-01-20T17:33:18Z",
end="2012-12-20T17:33:18Z",
gap="+1MONTH",
format="YYYY-MM",
count(*))
```

When this expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"recdate_dt": "2012-01",
"count(*)": 8703
},
{
"recdate_dt": "2012-02",
"count(*)": 8648
},
{
"recdate_dt": "2012-03",
"count(*)": 8621
},
{
"recdate_dt": "2012-04",
"count(*)": 8533
},
{
"recdate_dt": "2012-05",
"count(*)": 8792
},
{
"recdate_dt": "2012-06",
"count(*)": 8598
},
{
"recdate_dt": "2012-07",
"count(*)": 8679
},
{
"recdate_dt": "2012-08",
"count(*)": 8469
},
{
"recdate_dt": "2012-09",
"count(*)": 8637
},
{
"recdate_dt": "2012-10",
"count(*)": 8536
},
{
"recdate_dt": "2012-11",
"count(*)": 8785
},
{
"EOF": true,
"RESPONSE_TIME": 16
}
]
}
}
```

## Vectorizing the Time Series

Before a time series result can be operated on by math expressions the data will need to be vectorized. Specifically in the example above, the aggregation field count(*) will need to by moved into an array. As described in the Streams and Vectorization section of the user guide, the `col`

function can be used to copy a numeric column from a list of tuples into an array.

The expression below demonstrates the vectorization of the count(*) field.

```
let(a=timeseries(collection1,
q=*:*,
field="test_dt",
start="2012-01-20T17:33:18Z",
end="2012-12-20T17:33:18Z",
gap="+1MONTH",
format="YYYY-MM",
count(*)),
b=col(a, count(*)))
```

When this expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"b": [
8703,
8648,
8621,
8533,
8792,
8598,
8679,
8469,
8637,
8536,
8785
]
},
{
"EOF": true,
"RESPONSE_TIME": 5
}
]
}
}
```

## Smoothing

Time series smoothing is often used to remove the noise from a time series and help spot the underlying trends. The math expressions library has three **sliding window** approaches for time series smoothing. The **sliding window** approaches use a summary value from a sliding window of the data to calculate a new set of smoothed data points.

The three **sliding window** functions are lagging indicators, which means they don’t start to move in the direction of the trend until the trend effects the summary value of the sliding window. Because of this lagging quality these smoothing functions are often used to confirm the direction of the trend.

### Moving Average

The `movingAvg`

function computes a simple moving average over a sliding window of data. The example below generates a time series, vectorizes the count(*) field and computes the moving average with a window size of 3.

The moving average function returns an array that is of shorter length then the original data set. This is because results are generated only when a full window of data is available for computing the average. With a window size of three the moving average will begin generating results at the 3rd value. The prior values are not included in the result.

This is true for all the sliding window functions.

```
let(a=timeseries(collection1,
q=*:*,
field="test_dt",
start="2012-01-20T17:33:18Z",
end="2012-12-20T17:33:18Z",
gap="+1MONTH",
format="YYYY-MM",
count(*)),
b=col(a, count(*)),
c=movingAvg(b, 3))
```

When this expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"c": [
8657.333333333334,
8600.666666666666,
8648.666666666666,
8641,
8689.666666666666,
8582,
8595,
8547.333333333334,
8652.666666666666
]
},
{
"EOF": true,
"RESPONSE_TIME": 7
}
]
}
}
```

### Exponential Moving Average

The `expMovingAvg`

function uses a different formula for computing the moving average that responds faster to changes in the underlying data. This means that it is less of a lagging indicator then the simple moving average.

Below is an example that computes an exponential moving average:

```
let(a=timeseries(collection1, q=*:*,
field="test_dt",
start="2012-01-20T17:33:18Z",
end="2012-12-20T17:33:18Z",
gap="+1MONTH",
format="YYYY-MM",
count(*)),
b=col(a, count(*)),
c=expMovingAvg(b, 3))
```

When this expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"c": [
8657.333333333334,
8595.166666666668,
8693.583333333334,
8645.791666666668,
8662.395833333334,
8565.697916666668,
8601.348958333334,
8568.674479166668,
8676.837239583334
]
},
{
"EOF": true,
"RESPONSE_TIME": 5
}
]
}
}
```

### Moving Median

The `movingMedian`

function uses the median of the sliding window rather than the average. In many cases the moving median will be more **robust** to outliers then moving averages.

Below is an example computing the moving median:

```
let(a=timeseries(collection1,
q=*:*,
field="test_dt",
start="2012-01-20T17:33:18Z",
end="2012-12-20T17:33:18Z",
gap="+1MONTH",
format="YYYY-MM",
count(*)),
b=col(a, count(*)),
c=movingMedian(b, 3))
```

When this expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"c": [
8648,
8621,
8621,
8598,
8679,
8598,
8637,
8536,
8637
]
},
{
"EOF": true,
"RESPONSE_TIME": 7
}
]
}
}
```

## Differencing

Differencing is often used to remove the trend or seasonality from a time series. This is known as making a time series **stationary**.

### First Difference

The actual technique of differencing is to use the difference between values rather than the original values. The **first difference** takes the difference between a value and the value that came directly before it. The first difference is often used to remove the trend from a time series.

In the example below, the `diff`

function computes the first difference of a time series. The result array length is one value smaller then the original array. This is because the `diff`

function only returns a result for values where the prior value has been subtracted.

```
let(a=timeseries(collection1,
q=*:*,
field="test_dt",
start="2012-01-20T17:33:18Z",
end="2012-12-20T17:33:18Z",
gap="+1MONTH",
format="YYYY-MM",
count(*)),
b=col(a, count(*)),
c=diff(b))
```

When this expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"c": [
-55,
-27,
-88,
259,
-194,
81,
-210,
168,
-101,
249
]
},
{
"EOF": true,
"RESPONSE_TIME": 11
}
]
}
}
```

### Lagged Differences

The `diff`

function has an optional second parameter to specify a lag in the difference. If a lag is specified the difference is taken between a value and the value at a specified lag in the past. Lagged differences are often used to remove seasonality from a time series.

The simple example below demonstrates how lagged differencing works. Notice that the array in the example follows a simple repeated pattern. This type of pattern is often displayed with seasonality. In this example we can remove this pattern using the `diff`

function with a lag of 4. This will subtract the value lagging four indexes behind the current index. Notice that result set size is the original array size minus the lag. This is because the `diff`

function only returns results for values where the lag of 4 is possible to compute.

```
let(a=array(1,2,5,2,1,2,5,2,1,2,5),
b=diff(a, 4))
```

Expression is sent to the `/stream`

handler it responds with:

```
{
"result-set": {
"docs": [
{
"b": [
0,
0,
0,
0,
0,
0,
0
]
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}
```