Linear and Monotonic Correlation Measures

The standard measure of linear correlation in a time series is the autocorrelation given as a function of the delay parameter. The autocorrelation for each delay t is actually the Pearson correlation coefficient for the variable pair (x(i), x(i+t)). In this sense, the autocorrelation at each delay t can be estimated by other estimates of the pair-wise correlation and here the Kendall and Spearman correlation coefficients are also implemented giving the Kendall and Spearman autocorrelation, respectively. Note that the autocorrelation (Pearson, Kendall, Spearman) at different delays are considered as different measures, so that for a range of delay values the same number of measures are generated. We consider in this group also the partial autocorrelation that measures the correlation of the variable pair (x(i), x(i+t)) accounting for the intermediate variables x(i+1),...,x(i+t-1) (partial autocorrelation is the same as Pearson autocorrelation for t=1).

In addition to the autocorrelation functions, the cumulative autocorrelations (Pearson, Kendall, Spearman) are implemented as new measures that sums up the autocorrelation magnitudes for the delays up to a given delay.

There are two special values of the autocorrelation function and the respective delays are used in time series analysis: the decorrelation time, i.e. the delay for which the autocorrelation falls for the first time to the level 1/e, and the delay of zero autocorrelation, for which the autocorrelation crosses zero for the first time. These specific delays are computed for all three types of autocorrelation.

The definition of the Pearson, Kendall and Spearman correlation coefficients can be found in any standard textbook of statistics. The Pearson autocorrelation is a basic tool of linear analysis of time series and statistical description for this (estimation, inference) can be found in any standard book on time series analysis, e.g. see

Box, G.E.P., Jenkins, G.M., and Reinsel, G.C. (1994), Time Series Analysis: Forecasting and Control, Prentice-Hall, Third Edition, New Jersey.

The partial autocorrelation for each lag t is the last coefficient of the autoregressive (AR) model of order t and we implement the AR fitting to derive the partial autocorrelation measure for different given lags. The reader may refer to the same book for further details.

The implementation of Kendall and Spearman correlation coefficient on the estimation of the autocorrelation is not standard but may be useful in some cases, as when the marginal distribution of a time series deviates substantially from Gaussian. Note that the Kendall and Spearman autocorrelation  estimate correlations in the ranks of the samples in the time series and therefore cannot be considered as linear measures. We call them monotonic measures, since their values do not change under monotonic transforms of the time series.

 

Pearson Autocorrelation (PearsAutoc)

Pearson Autocorrelation is the standard autocorrelation for the given range of delays. The following parameter can be specified:

- delay (t): any valid matlab format denoting an array of positive integers or a single positive integer. The default is '1:10' meaning delays from 1 to 10.

In addition the Cumulative Pearson Autocorrelation is computed for the same range of delays. Also, if the autocorrelation for delays up to the maximum given delay crosses the 1/e or zero level the delay of decorrelation or zero-autocorrelation, respectively, is assigned a value. The cumulative Pearson autocorrelation and the delay of decorrelation and zero autocorrelation can then be simply assigned to the respective measures if these are selected with the same set of delay values.

Note that when the delay parameter is changed, the change is passed to the same parameter in the measures of Cumulative Pearson Autocorrelation, Pearson Decorrelation Time and Zero Pearson Autocorrelation Time.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '1:5 10 20', then Pearson Autocorrelation is computed for these delays and in the measure list the following measure names will appear

PearsAutoct1
PearsAutoct2
PearsAutoct3
PearsAutoct4
PearsAutoct5
PearsAutoct10
PearsAutoct20

 

Cumulative Pearson Autocorrelation (PearsCAuto)

Cumulative Pearson Autocorrelation is the cumulative function of the Pearson autocorrelation for the given range of delays. The delay parameter is determined as for the Pearson autocorrelation. The Cumulative Pearson Autocorrelation for each delay t is the sum of the absolute values of the Pearson autocorrelation up to the delay t.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Cumulative Pearson Autocorrelation is computed for these delays and in the measure list the following measure names will appear

PearsCAutot10
PearsCAutoct20

 

Pearson Decorrelation Time (PearsDecoT)

The Pearson Decorrelation Time is the delay for which the Pearson autocorrelation falls to the level 1/e. So, if the autocorrelation, computed for delays up to the maximum of the given delays (if more than one delay value is given), crosses 1/e for a delay te, then the Pearson Decorrelation Time is assigned te otherwise it has a NaN value (NaN=not a number).

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Pearson Decorrelation Time is computed and in the measure list the following measure name will appear (for the maximum t=20)

PearsDecoTt20

 

Zero Pearson Autocorrelation Time (PearsZeroT)

The Zero Pearson Autocorrelation Time is the delay for which the Pearson autocorrelation falls to the zero level. So, if the autocorrelation, computed for delays up to the maximum of the given delays (if more than one delay value is given), crosses zero for a delay t0, then the Zero Pearson Autocorrelation Time is assigned to t0 otherwise it has a NaN value (NaN=not a number).

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Zero Pearson Autocorrelation Time is computed and in the measure list the following measure name will appear (for the maximum t=20)

PearsZeroTt20

 

Kendall Autocorrelation (KendaAutoc)

Kendall Autocorrelation is the estimate of autocorrelation for the given range of delays making use of the Kendall correlation coefficient. For each delay t, the Kendall Autocorrelation is actually the Kendall (or so-called tau) correlation coefficient for the variable pair (x(i), x(i+t)). The following parameter can be specified:

- delay (t): any valid matlab format denoting an array of positive integers or a single positive integer. The default is '1:10' meaning delays from 1 to 10.

In addition, the Cumulative Kendall Autocorrelation is computed for the same range of delays. Also, if the Kendall autocorrelation for delays up to the maximum given delay crosses the 1/e or zero level the delay of Kendall decorrelation or zero Kendall autocorrelation, respectively, is assigned a value. The cumulative Kendall autocorrelation and the delay of Kendall decorrelation and zero Kendall autocorrelation can then be simply assigned to the respective measures, if these are selected with the same set of delay values.

Note that when the delay parameter is changed, the change is passed to the same parameter in the measures of Cumulative Kendall Autocorrelation, Kendall Decorrelation Time and Zero Kendall Autocorrelation Time.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '1:5 10 20', then Kendall Autocorrelation is computed for these delays and in the measure list the following measure names will appear

KendaAutoct1
KendaAutoct2
KendaAutoct3
KendaAutoct4
KendaAutoct5
KendaAutoct10
KendaAutoct20

 

Cumulative Kendall Autocorrelation (KendaCAuto)

Cumulative Kendall Autocorrelation is the cumulative function of the Kendall autocorrelation for the given range of delays. The delay parameter is determined as for the Kendall autocorrelation. The Cumulative Kendall Autocorrelation for each delay t is the sum of the absolute values of the Kendall autocorrelation up to the delay t.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Cumulative Kendall Autocorrelation is computed for these delays and in the measure list the following measure names will appear

KendaCAutot10
KendaCAutoct20

 

Kendall Decorrelation Time (KendaDecoT)

The Kendall Decorrelation Time is the delay for which the Kendall autocorrelation falls to the level 1/e. So, if the autocorrelation, computed for delays up to the maximum of the given delays (if more than one delay value is given), crosses 1/e for a delay te, then the Kendall Decorrelation Time is assigned te otherwise it has a NaN value (NaN = not a number).

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Kendall Decorrelation Time is computed and in the measure list the following measure name will appear (for the maximum t=20)

KendaDecoTt20

 

Zero Kendall Autocorrelation Time (KendaZeroT)

The Zero Kendall Autocorrelation Time is the delay for which the Kendall autocorrelation falls to the zero level. So, if the autocorrelation, computed for delays up to the maximum of the given delays (if more than one delay value is given), crosses zero for a delay t0, then the Zero Kendall Autocorrelation Time is assigned to t0 otherwise it has a NaN value (NaN = not a number).

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Zero Kendall Autocorrelation Time is computed and in the measure list the following measure name will appear (for the maximum t=20)

KendaZeroTt20

 

Spearman Autocorrelation (SpearAutoc)

Spearman Autocorrelation is the estimate of autocorrelation for the given range of delays making use of the Spearman correlation coefficient. For each delay t, the Spearman Autocorrelation is actually the Spearman correlation coefficient for the variable pair (x(i), x(i+t)). The following parameter can be specified:

- delay (t): any valid matlab format denoting an array of positive integers or a single positive integer. The default is '1:10' meaning delays from 1 to 10.

In addition the Cumulative Spearman Autocorrelation is computed for the same range of delays. Also, if the Spearman autocorrelation, computed for delays up to the maximum given delay, crosses the 1/e or zero level the delay of Spearman decorrelation or zero Spearman autocorrelation, respectively, is assigned a value. The cumulative Spearman autocorrelation and the delay of Spearman decorrelation and zero Spearman autocorrelation can then be simply assigned to the respective measures, if these are selected with the same set of delay values.

Note that when the delay parameter is changed, the change is passed to the same parameter in the measures of Cumulative Spearman Autocorrelation, Spearman Decorrelation Time and Zero Spearman Autocorrelation Time.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '1:5 10 20', then Spearman Autocorrelation is computed for these delays and in the measure list the following measure names will appear

SpearAutoct1
SpearAutoct2
SpearAutoct3
SpearAutoct4
SpearAutoct5
SpearAutoct10
SpearAutoct20

 

Cumulative Spearman Autocorrelation (SpearCAuto)

Cumulative Spearman Autocorrelation is the cumulative function of the Spearman autocorrelation for the given range of delays. The delay parameter is determined as for the Spearman autocorrelation. The Cumulative Spearman Autocorrelation for each delay t is the sum of the absolute values of the Spearman autocorrelation up to the delay t.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Cumulative Spearman Autocorrelation is computed for these delays and in the measure list the following measure names will appear

SpearCAutot10
SpearCAutoct20

 

Spearman Decorrelation Time (SpearDecoT)

The Spearman Decorrelation Time is the delay for which the Spearman autocorrelation falls to the level 1/e. So, if the autocorrelation, computed for delays up to the maximum of the given delays (if more than one delay value is given), crosses 1/e for a delay te, then the Spearman Decorrelation Time is assigned te otherwise it has a NaN value (NaN = not a number).

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Spearman Decorrelation Time is computed and in the measure list the following measure name will appear (for the maximum t=20)

SpearDecoTt20

 

Zero Spearman Autocorrelation Time (SpearZeroT)

The Zero Spearman Autocorrelation Time is the delay for which the Spearman autocorrelation falls to the zero level. So, if the autocorrelation, computed for delays up to the maximum of the given delays (if more than one delay value is given), crosses zero for a delay t0, then the Zero Spearman Autocorrelation Time is assigned to t0 otherwise it has a NaN value (NaN = not a number).

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '10 20', then Zero Spearman Autocorrelation Time is computed and in the measure list the following measure name will appear (for the maximum t=20)

SpearZeroTt20

 

Partial Autocorrelation (PartialAut)

Partial Autocorrelation for a lag t is the correlation of the variable pair (x(i), x(i+t)) accounting for the intermediate variables x(i+1),...,x(i+t-1). Here it can be computed for the given range of delays. The following parameter can be specified:

- delay (t): any valid matlab format denoting an array of positive integers or a single positive integer. The default is '1:10' meaning delays from 1 to 10.

Example: If the user selects this measure by activating the check box in the beginning of the measure line and sets for delay (t) '1:5 10 20', then Partial Autocorrelation is computed for these delays and in the measure list the following measure names will appear

PartialAutt1
PartialAutt2
PartialAutt3
PartialAutt4
PartialAutt5
PartialAutt10
PartialAutt20

 

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