mra {fractalRegression}

Multiscale Regression Analysis (MRA)

Description

Fast function for computing multiscale regression analysis (MRA) on long time series. Combining DFA with ordinary least square regression, MRA is a form of fractal regression that can be used to estimate asymmetric and multiscale regression coefficients between two variables.

Usage

mra(x, y, order, scales)

Arguments

x

A real valued vector (i.e., time series data) to be analyzed. A key difference between DCCA and MRA is that MRA produces asymmetric estiamtes. That is, x is assumed to be an independent variable and y is assumed to be a dependent variable. MRA should be used when one of the time series in question is usefully cast as the independent variable. That is, x is assumed to effect change in y. If no such causal relationship is anticipated, use DCCA instead.

y

A real valued vector (i.e., time series data) to be analyzed.

order

is an integer indicating the polynomial order used for detrending the local windows (e.g, 1 = linear, 2 = quadratic, etc.). There is not a pre-determined limit on the order of the polynomial order but the user should avoid using a large polynomial on small windows. This can result in overfitting and non-meaningful estimates.

scales

An integer vector of scales over which to compute correlation. Unlike univariate DFA, MRA does not require that scales be in log units. Scale intervals can be sequential, for example, when the analysis is exploratory and no a priori hypotheses have been made about the scale of correlation. A small subset of targeted scales may also be investigated where scale-specific research questions exist. We have found that windows smaller than say 8 observations create stability problems due to overfitting. This is espcially when the order of the fitting polynomial is large.

Details

Mathematical treatment of the MRA algorithm and its performance can be found in Kristoufek (2015) and Likens et al. (2019).

Note that under conditions with linear and quadratic trends, Likens et al. (2019) found that there was a systematic positive bias in the \beta estimates for larger scales. Using a polynomial detrending order of 2 or greater was shown to attenuate this bias.

Value

The object returned from the mra() function is a list including the following:

• scales indicates the values of the scales used for estimates

• betas are the scale specific \beta estimates of the influence of x on y

• r2 is the scale specific r-squared value of the model fit (i.e., variance in y accounted for by x at that scale)

• t_observed is the estimated t-statistic for a given \beta at a given scale.

References

Kristoufek, L. (2015). Detrended fluctuation analysis as a regression framework: Estimating dependence at different scales. Physical Review E, 91(2), 022802.

Likens, A. D., Amazeen, P. G., West, S. G., & Gibbons, C. T. (2019). Statistical properties of Multiscale Regression Analysis: Simulation and application to human postural control. Physica A: Statistical Mechanics and its Applications, 532, 121580.

Examples

# Here is a simple example for running MRA using a white noise and pink noise time series.
# For more detailed examples, see the vignette. 

noise <- rnorm(5000)

pink.noise <- fgn_sim(n = 5000, H = 0.9)

scales <- logscale(scale_min = 10, scale_max = 1250, scale_ratio = 1.1)

mra.out <- mra(x = noise, y = pink.noise, order = 1, scales = scales)