- tsfeatures
- Installation
- Usage
- List of
features
- acf_features
- arch_stat
- autocorr_features
- binarize_mean
- compengine feature set
- crossing_points
- dist_features
- entropy
- firstzero_ac
- flat_spots
- heterogeneity
- holt_parameters and hw_parameters
- hurst
- lumpiness and stability
- max_level_shift, max_var_shift and max_kl_shift
- nonlinearity
- pacf_features
- pred_features
- scal_features
- station_features
- stl_features
- unitroot_kpss and unitroot_pp
- zero_proportion

- Reproducing papers
- Resources
- License

The R package *tsfeatures* provides methods for extracting
various features from time series data.

The **stable** version on R CRAN and can be installed in
the usual way:

You can install the **development** version from Github with:

The function `tsfeatures()`

computes a tibble of time
series features from a list of time series.

```
mylist <- list(sunspot.year, WWWusage, AirPassengers, USAccDeaths)
tsfeatures(mylist)
#> # A tibble: 4 × 20
#> frequency nperiods seasonal_period trend spike linearity curvature e_acf1
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 0 1 0.125 2.10e-5 3.58 1.11 0.793
#> 2 1 0 1 0.985 3.01e-8 4.45 1.10 0.774
#> 3 12 1 12 0.991 1.46e-8 11.0 1.09 0.509
#> 4 12 1 12 0.802 9.15e-7 -2.12 2.85 0.258
#> # ℹ 12 more variables: e_acf10 <dbl>, entropy <dbl>, x_acf1 <dbl>,
#> # x_acf10 <dbl>, diff1_acf1 <dbl>, diff1_acf10 <dbl>, diff2_acf1 <dbl>,
#> # diff2_acf10 <dbl>, seasonal_strength <dbl>, peak <dbl>, trough <dbl>,
#> # seas_acf1 <dbl>
```

The default functions that `tsfeatures`

uses to compute
features are `frequency`

, `stl_features`

,
`entropy`

and `acf_features`

. Each of them can
produce one or more features. Detailed information of features included
in the *tsfeatures* package are described below. Functions from
other packages, or user-defined functions, may also be used.

We compute the autocorrelation function of the series, the
differenced series, and the twice-differenced series.
`acf_features`

produces a vector comprising the first
autocorrelation coefficient in each case, and the sum of squares of the
first 10 autocorrelation coefficients in each case.

`arch_stat`

Computes a statistic based on the Lagrange
Multiplier (LM) test of Engle (1982) for
autoregressive conditional heteroscedasticity (ARCH). The statistic
returned is the \(R^2\) value of an
autoregressive model of order specified as lags applied to \(x^2\).

The autocorrelation feature set from software package hctsa

```
autocorr_features(AirPassengers)
#> embed2_incircle_1 embed2_incircle_2 ac_9 firstmin_ac
#> 0.0000000 0.0000000 0.6709483 8.0000000
#> trev_num motiftwo_entro3 walker_propcross
#> -4902.1958042 1.1302445 0.2027972
```

`ac_9`

is the autocorrelation at lag 9.`embed2_incircle`

gives proportion of points inside a given circular boundary in a 2-d embedding space.`firstmin_ac`

returns the time of first minimum in the autocorrelation function.`trev_num`

returns the numerator of the trev function of a time series, a normalized nonlinear autocorrelation. The time lag is set to 1.`motiftwo_entro3`

finds local motifs in a binary symbolization of the time series. Coarse-graining is performed. Time-series values above its mean are given 1, and those below the mean are 0.`motiftwo_entro3`

returns the entropy of words in the binary alphabet of length 3.`walker_propcross`

simulates a hypothetical walker moving through the time domain. The hypothetical particle (or ‘walker’) moves in response to values of the time series at each point. The walker narrows the gap between its value and that of the time series by 10.`walker_propcross`

returns the fraction of time series length that walker crosses time series.

`binarize_mean`

converts an input vector into a binarized
version. Time-series values above its mean are given 1, and those below
the mean are 0.

`compengine`

calculate the features that have been used in
the CompEngine database,
using a method introduced in package `kctsa`

.

The features involved can be grouped as autocorrelation, prediction,
stationarity, distribution, and scaling, which can be computed using
`autocorr_features`

, `pred_features`

,
`station_features`

, `dist_features`

, and
`scal_features`

.

x | |
---|---|

embed2_incircle_1 | 0.0000000 |

embed2_incircle_2 | 0.0000000 |

ac_9 | 0.6709483 |

firstmin_ac | 8.0000000 |

trev_num | -4902.1958042 |

motiftwo_entro3 | 1.1302445 |

walker_propcross | 0.2027972 |

localsimple_mean1 | 2.0000000 |

localsimple_lfitac | 3.0000000 |

sampen_first | Inf |

std1st_der | 33.7542815 |

spreadrandomlocal_meantaul_50 | 12.9500000 |

spreadrandomlocal_meantaul_ac2 | 38.9200000 |

histogram_mode_10 | 125.0000000 |

outlierinclude_mdrmd | 0.4166667 |

fluctanal_prop_r1 | 0.7692308 |

`crossing points`

are defined as the number of times a
time series crosses the median line.

The distribution feature set from the hctsa package.

The scaling feature set from `hctsa`

.

`histogram_mode`

measures the mode of the data vector using histograms with a given number of bins (default to 10) as suggestion.`outlierinclude_mdrmd`

measures the median as more and more outliers are included in the calculation according to a specified rule, of outliers being furthest from the mean. The threshold for including time-series data points in the analysis increases from zero to the maximum deviation, in increments of 0.01*sigma (by default), where sigma is the standard deviation of the time series. At each threshold, proportion of time series points included and median are calculated, and outputs from the algorithm measure how these statistical quantities change as more extreme points are included in the calculation.`outlierinclude_mdrmd`

essentially returns the median of the median of range indices.

The spectral `entropy`

is the Shannon entropy \[
-\int^\pi_{-\pi}\hat{f}(\lambda)\log\hat{f}(\lambda) d\lambda,
\] where \(\hat{f}(\lambda)\) is
an estimate of the spectral density of the data. This measures the
“forecastability” of a time series, where low values indicate a high
signal-to-noise ratio, and large values occur when a series is difficult
to forecast.

`firstzero_ac`

returns the first zero crossing of the
autocorrelation function.

`flat_spots`

are computed by dividing the sample space of
a time series into ten equal-sized intervals, and computing the maximum
run length within any single interval.

The `heterogeneity`

features measure the heterogeneity of
the time series. First, we pre-whiten the time series to remove the
mean, trend, and autoregressive (AR) information (Barbour & Parker
2014). Then we fit a \(GARCH(1,1)\) model to the pre-whitened time
series, \(x_t\), to measure for
autoregressive conditional heteroskedasticity (ARCH) effects. The
residuals from this model, \(z_t\), are
also measured for ARCH effects using a second \(GARCH(1,1)\) model.

`arch_acf`

is the sum of squares of the first 12 autocorrelations of \(\{x^2_t\}\).`garch_acf`

is the sum of squares of the first 12 autocorrelations of \(\{z^2_t\}\).`arch_r2`

is the \(R^2\) value of an AR model applied to \(\{x^2_t\}\).`garch_r2`

is the \(R^2\) value of an AR model applied to \(\{z^2_t\}\).

The statistics obtained from \(\{x^2_t\}\) are the ARCH effects, while those from \(\{z^2_t\}\) are the GARCH effects. Note that the two \(R^2\) values are used in the Lagrange-multiplier test of Engle (1982), and the sum of squared autocorrelations are used in the Ljung-Box test proposed by Ljung & Box (1978).

`holt_parameters`

Estimate the smoothing parameter for the
level-alpha and the smoothing parameter for the trend-beta of Holt’s
linear trend method. `hw_parameters`

considers additive
seasonal trend: ETS(A,A,A) model, returning a vector of 3 values: alpha,
beta and gamma.

We use a measure of the long-term memory of a time series
(`hurst`

), computed as 0.5 plus the maximum likelihood
estimate of the fractional differencing order \(d\) given by Haslett & Raftery (1989). We add 0.5 to make it consistent with the Hurst
coefficient. Note that the fractal dimension can be estimated as \(D = 2 - \text{hurst}\).

`Stability`

and `lumpiness`

are two time series
features based on tiled (non-overlapping) windows. Means or variances
are produced for all tiled windows. Then `stability`

is the
variance of the means, while `lumpiness`

is the variance of
the variances.

These three features compute features of a time series based on
sliding (overlapping) windows. `max_level_shift`

finds the
largest mean shift between two consecutive windows.
`max_var_shift`

finds the largest variance shift between two
consecutive windows. `max_kl_shift`

finds the largest shift
in Kulback-Leibler divergence between two consecutive windows. Each
feature returns a vector of 2 values: the size of the shift, and the
time index of the shift.

The `nonlinearity`

coefficient is computed using a
modification of the statistic used in Teräsvirta’s nonlinearity test.
Teräsvirta’s test uses a statistic \(X^2=T\log(\text{SSE}1/\text{SSE}0)\) where
SSE1 and SSE0 are the sum of squared residuals from a nonlinear and
linear autoregression respectively. This is non-ergodic, so instead, we
define it as \(10X^2/T\) which will
converge to a value indicating the extent of nonlinearity as \(T\rightarrow\infty\). This takes large
values when the series is nonlinear, and values around 0 when the series
is linear.

We compute the partial autocorrelation function of the series, the
differenced series, and the second-order differenced series. Then
`pacf_features`

produces a vector comprising the sum of
squares of the first 5 partial autocorrelation coefficients in each
case.

The prediction feature set from the `hctsa`

package. The
first two elements are obtained from `localsimple_taurus`

with different forecast methods (the mean, and an LS fit). The third is
from `sampen_first`

.

- Simple predictors using the past trainLength values of the time
series to predict its next value.
`localsimple_taures`

returns the first zero crossing of the autocorrelation function of the residuals from this Simple local time-series forecasting. `sampen_first`

returns the first Sample Entropy of a time series where the embedding dimension is set to 5 and the threshold is set to 0.3.`sampenc`

is the underlying function to calculate the first sample entropy with optional dimension and threshold settings.

The scaling feature set from `hctsa`

.

`fluctanal_prop_r1`

implements fluctuation analysis. It fits a polynomial of order 1 and then returns the range. The order of fluctuations is 2, corresponding to root mean square fluctuations.

The stationary feature set from `hctsa`

.

```
station_features(AirPassengers)
#> std1st_der spreadrandomlocal_meantaul_50
#> 33.75428 12.45000
#> spreadrandomlocal_meantaul_ac2
#> 38.88000
```

`std1st_der`

returns the standard deviation of the first derivative of the time series.- 100 time-series segments of length l are selected at random from the
time series and the mean of the first zero-crossings of the
autocorrelation function in each segment is calculated using
`spreadrandomlocal_meantaul`

.

`stl_features`

Computes various measures of trend and
seasonality of a time series based on an STL decomposition. The
`mstl`

function is used to do the decomposition.

`nperiods`

is the number of seasonal periods in the data
(determined by the frequency of observation, not the observations
themselves) and set to 1 for non-seasonal data.
`seasonal_period`

is a vector of seasonal periods and set to
1 for non-seasonal data.

The size and location of the peaks and troughs in the seasonal
component are used to compute strength of peaks (`peak`

) and
strength of trough (`trough`

).

The rest of the features are modifications of features used in Kang, Hyndman & Smith-Miles (2017). We extend the STL decomposition approach (Cleveland et al.1990) to handle multiple seasonalities. Thus, the decomposition contains a trend, up to \(M\) seasonal components and a remainder component: \[ x_t=f_t+s_{1,t}+\cdots+s_{M.t}+e_t, \] where \(f_t\) is the smoothed trend component, \(s_{i,t}\) is the \(i\)th seasonal component and \(e_t\) is a remainder component. The components are estimated iteratively. Let \(s^{(k)}_{i,t}\) be the estimate of \(s_i,t\) at the \(k\)th iteration, with initial values given as \(s^{(0)}_{i,t}=0\). The we apply an STL decomposition to \(x_t-\sum^{j=1}_{j\neq1}{}^{^{M}}s^{k-1}_{j,t}\) to obtained updated estimates \(s^{(k)}_{i,t}\) for \(k=1,2,\ldots\). In practice, this converges quickly and only two iterations are required. To allow the procedure to be applied automatically, we set the seasonal window span for STL to be 21 in all cases. For a non-seasonal time series, we simply estimate \(x_t=f_t+e_t\) where \(f_t\) is computed using Friedman’s “super smoother” (Friedman 1984).

Strength of trend (`trend`

) and strength of seasonality
(`seasonal.strength`

) are defined as \[
\text{trend} = 1-\frac{\text{Var}(e_t)}{\text{Var}(f_t+e_t)}\quad
\text{and}\quad
\text{seasonal.strength}=1-\frac{\text{Var}(e_t)}{\text{Var}(s_{i,t}+e_t)}.
\] If their values are less than 0, they are set to 0, while
values greater than 1 are set to 1. For non-seasonal time series
`seasonal.strength`

is 0. For seasonal time series,
`seasonal.strength`

is an M-vector, where M is the number of
periods. This is analogous to the way the strength of trend and
seasonality were defined in Wang, Smith & Hyndman (2006), Hyndman, Wang & Laptev (2015) and Kang, Hyndman & Smith-Miles (2017).

`spike`

measures the “spikiness” of a time series, and is
computed as the variance of the leave-one-out variances of the remainder
component \(e_t\).

`linearity`

and `curvature`

measures the
linearity and curvature of a time series calculated based on the
coefficients of an orthogonal quadratic regression.

We compute the autocorrelation function of \(e_t\), and `e_acf1`

and
`e_acf10`

contain the first autocorrelation coefficient and
the sum of the first ten squared autocorrelation coefficients.

`unitroot_kpss`

is a vector comprising the statistic for
the KPSS unit root test with linear trend and lag one, and
`unitroot_pp`

is the statistic for the “Z-alpha” version of
PP unit root test with constant trend and lag one.

Here we replicate the analysis in Hyndman, Wang & Laptev (ICDM
2015). However, note that crossing_points, peak and trough are defined
differently in the *tsfeatures* package than in the Hyndman et al
(2015) paper. Other features are the same.

```
hwl <- bind_cols(
tsfeatures(yahoo,
c("acf_features","entropy","lumpiness",
"flat_spots","crossing_points")),
tsfeatures(yahoo,"stl_features", s.window='periodic', robust=TRUE),
tsfeatures(yahoo, "max_kl_shift", width=48),
tsfeatures(yahoo,
c("mean","var"), scale=FALSE, na.rm=TRUE),
tsfeatures(yahoo,
c("max_level_shift","max_var_shift"), trim=TRUE)) %>%
select(mean, var, x_acf1, trend, linearity, curvature,
seasonal_strength, peak, trough,
entropy, lumpiness, spike, max_level_shift, max_var_shift, flat_spots,
crossing_points, max_kl_shift, time_kl_shift)
```

Compute the features used in Kang, Hyndman & Smith-Miles (IJF
2017). Note that the trend and ACF1 are computed differently for
non-seasonal data in the *tsfeatures* package than in the Kang et
al (2017). `tsfeatures`

uses `mstl`

which uses
`supsmu`

for the trend calculation with non-seasonal data,
whereas Kang et al used a penalized regression spline computed using
`mgcv`

instead. Other features are the same.

```
library(tsfeatures)
library(dplyr)
library(tidyr)
library(forecast)
M3data <- purrr::map(Mcomp::M3,
function(x) {
tspx <- tsp(x$x)
ts(c(x$x,x$xx), start=tspx[1], frequency=tspx[3])
})
khs_stl <- function(x,...) {
lambda <- BoxCox.lambda(x, lower=0, upper=1, method='loglik')
y <- BoxCox(x, lambda)
c(stl_features(y, s.window='periodic', robust=TRUE, ...), lambda=lambda)
}
```

```
khs <- bind_cols(
tsfeatures(M3data, c("frequency", "entropy")),
tsfeatures(M3data, "khs_stl", scale=FALSE)) %>%
select(frequency, entropy, trend, seasonal_strength, e_acf1, lambda) %>%
replace_na(list(seasonal_strength=0)) %>%
rename(
Frequency = frequency,
Entropy = entropy,
Trend = trend,
Season = seasonal_strength,
ACF1 = e_acf1,
Lambda = lambda) %>%
mutate(Period = as.factor(Frequency))
```

This package is free and open source software, licensed under GPL-3.