In some applications, it is important to compare the stochastic properties of two multivariate time series that have unequal dimensions. A new method is proposed to compare the spread of spectral information in two multivariate stationary processes with different dimensions. To measure discrepancies, a frequency specific spectral ratio (FS-ratio) statistic is proposed and its asymptotic properties are derived. The FS-ratio is blind to the dimension of the stationary process and captures the proportion of spectral power in various frequency bands. Here we develop a technique to automatically identify frequency bands that carry significant spectral power. We apply our method to track changes in the complexity of a 32-channel local field potential (LFP) signal from a rat following an experimentally induced stroke. At every epoch (a distinct time segment from the duration of the experiment), the nonstationary LFP signal is decomposed into stationary and nonstationary latent sources and the complexity is analyzed through these latent stationary sources and their dimensions that can change across epochs. The analysis indicates that spectral information in the Beta frequency band (12–30 Hertz) demonstrated the greatest change in structure and complexity due to the stroke.

Numerous applications require comparing two multivariate time series of unequal dimensions. Neuroscience experiments result in a stationary or nonstationary multivariate signal from different epochs (distinct non-overlapping successive time segments of the duration of the experiment). A popular approach to modeling such data decomposes the observed signal at every epoch into useful latent sources that can be stationary or nonstationary. These latent sources are lower dimensional time series obtained by linear transforms of the components of the observed multivariate series and they aim to capture important statistical properties of the observed series. At these epochs, dimension reduction techniques such as principal component analysis (PCA), factor modeling, independent component analysis (ICA), stationary subspace analysis (SSA) are often applied to extract useful lower-dimensional latent sources. Artificially setting the dimension of these latent sources to be the same across the epochs results in loss of important information since these changes could be indicative of useful brain processes such as learning (Fiecas and Ombao [

The application that motivates our methodology is the analysis of local field potentials (LFP) in an experiment that simulates ischemic stroke in humans (Data source: Stroke experiment conducted in the lab of co-author (Ron Frostig) at his Neurobiology lab;

Motivated by such applications, we propose a new method to compare spectral information in different multivariate stationary processes of varying dimensions. More specifically, the aim is to capture the amount of spectral information in various frequency bands in different stationary processes of unequal dimensions. There are already many methods and models that discuss evolution of spectral information but the key contribution of this paper is in modeling evolution of the spectrum while allowing dimension to also evolve over time. We introduce a frequency-specific spectral ratio, which we call the FS-ratio, statistic that measures the proportion of spectral power in various frequency bands. FS-ratio can be used to (i). identify frequency bands where there is significant discrepancies between pre and post stroke epochs, (ii). identify frequency bands that account for most variation within pre (and post) stroke epochs and (iii). identify the frequency bands that are consistent (vs inconsistent) across all the 600 epochs. One of the key features of this statistic is that it is blind to the dimension of the multivariate stationary process and can be used to compare successive epochs with possibly different dimensions in the stationary sources. Thus, the proposed FS-ratio is very useful in (a). discriminating between the pre and post stroke onset and (b). tracking changes over the entire course of the experiment while allowing for varying dimensions. In

In this section, we first describe our FS-ratio statistic and the method to analyze the evolution of spectral information in stationary processes with varying dimensions. Using the FS-Ratio statistic, a technique to locate the frequency bands carrying significant spectral power is discussed in

Let

The aim of this work is to compare the two spectral matrices

The data analogue of the FS-ratio parameter in (3) is then given by the FS-ratio statistic:

In this section we describe our technique that uses the FS-Ratio statistic to find the frequency bands of interest. More precisely, we aim to locate the intervals

Let

We next provide a simple illustration of the scan statistic

We consider a discretized set of frequency points

In the bottom panel of

In this section we list the required assumptions and discuss the asymptotic properties of the statistics

Please note that the

See

Please note that in finite sample situations explored using simulation examples in

In this section, we investigate the ability of the FS-ratio statistic to identify changes in the spectral properties of the local field potential (LFP) of a rat (Local field potential data on the experimental rat comes from the stroke experiment conducted at Frostig laboratory at University of California Irvine:

At 32 locations on the rat’s cortex, microelectrodes are inserted: 4 layers in the cortex, at 300

As a first step in our analysis, we applied a component-wise univariate test of second-order stationarity (Dwivedi and Subba Rao [

Next, we model the observed 32-dimensional signal as a multivariate nonstationary time series using the stationary subspace analysis (SSA) setup. We assume the observed

The next goal in the data analysis is to estimate the epoch-evolving dimension

The evolutionary dimension

We then estimated the latent stationary sources

Next, the FS-ratio statistic was evaluated on these estimated stationary sources at each of the 600 epochs at various frequency bands.

The FS-ratio statistic is seen to have differences in the pre and post stroke epochs in the Theta, Alpha, and Beta bands but not in the Gamma band. It can also be seen that the biggest difference in FS-ratio between pre and post stroke is in the Beta band wherein there is a decrease in the amount of spectral information after the stroke.

The

In Fontaine et al. [

In contrast, the advantages of our method are as follows: (i). The method treats the observed LFP signal as a multivariate nonstationary time series. Using (16), we model this observed multivariate signal as a mixture of stationary and nonstationary components.

In this work, we proposed a new frequency-specific spectral ratio statistic FS-ratio that is demonstrated to be useful in comparing spectral information in two multivariate stationary processes of different dimensions. The method is motivated by applications in neuroscience wherein brain signal is recorded across several epochs and the widely used tactic is to assume the observed signal be linearly generated by latent sources of interest in lower dimensions. Applying PCA/ICA/SSA and other dimension reduction methods to the observed signal in different epochs in the experiment results in different estimates of the dimensions of latent sources. In these situations, the FS-ratio is seen to be useful because (i). It captures the proportion of spectral power in various frequency bands by means of a

Topological data analysis (TDA) methods for characterizing complexity and detecting phase transitions exist in the literature; M. Piangerelli [

The stroke experiment was conducted at the R.F.’s laboratory (Frostig laboratory at University of California Irvine:

This work is support in part by KAUST, NIH NS066001, Leducq Foundation 15CVD02 and NIH MH115697.

The authors declare no conflict of interest.

In this section, we illustrate the performance of the FS-ratio statistic in capturing spread of spectral information using simulated examples. We consider four simulation schemes and report the key summaries of the FS-ratio statistic across repetitions of each of the four schemes. In addition, 95% bootstrap confidence limits for the FS-ratio statistic are computed from

The simulation schemes presented below are designed to mimic the real data situation in

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0.5342 | 0.5391 | 0.0221 | 0.4984 | 0.6253 |

(0.08,0.16) | 0.4486 | 0.4544 | 0.0227 | 0.3566 | 0.4831 |

(0.16,0.24) | 0.0002 | 0.0002 | 0.0001 | 0.0005 | 0.0019 |

(0.24,0.32) | 0 | 0 | 0 | 0 | 0.0002 |

(0.32,0.40) | 0 | 0 | 0 | 0 | 0 |

(0.40,0.48) | 0 | 0 | 0 | 0 | 0 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0 | 0 | 0 | 0 | 0 |

(0.08,0.16) | 0 | 0 | 0 | 0 | 0 |

(0.16,0.24) | 0 | 0 | 0 | 0 | 0 |

(0.24,0.32) | 0.0003 | 0.0002 | 0.0002 | 0.0005 | 0.0017 |

(0.32,0.40) | 0.4561 | 0.4595 | 0.0181 | 0.3786 | 0.4903 |

(0.40,0.48) | 0.5205 | 0.5210 | 0.0169 | 0.4759 | 0.5826 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0.5407 | 0.5349 | 0.0286 | 0.5041 | 0.6360 |

(0.08,0.16) | 0.4423 | 0.4482 | 0.0284 | 0.3475 | 0.4778 |

(0.16,0.24) | 0.0002 | 0.0003 | 0.0002 | 0.0006 | 0.0022 |

(0.24,0.32) | 0 | 0 | 0 | 0 | 0.0002 |

(0.32,0.40) | 0 | 0 | 0 | 0 | 0 |

(0.40,0.48) | 0 | 0 | 0 | 0 | 0 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0 | 0 | 0 | 0 | 0 |

(0.08,0.16) | 0 | 0 | 0 | 0 | 0 |

(0.16,0.24) | 0 | 0 | 0 | 0 | 0.0001 |

(0.24,0.32) | 0.0003 | 0.0002 | 0.0001 | 0.0005 | 0.0016 |

(0.32,0.40) | 0.4605 | 0.4616 | 0.0108 | 0.3862 | 0.4938 |

(0.40,0.48) | 0.5194 | 0.5186 | 0.0096 | 0.4800 | 0.5863 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0.5371 | 0.5327 | 0.0238 | 0.4977 | 0.6284 |

(0.08,0.16) | 0.4459 | 0.4504 | 0.0239 | 0.3549 | 0.4843 |

(0.16,0.24) | 0.0002 | 0.0002 | 0.0001 | 0.0005 | 0.0020 |

(0.24,0.32) | 0 | 0 | 0 | 0 | 0.0002 |

(0.32,0.40) | 0 | 0 | 0 | 0 | 0 |

(0.40,0.48) | 0 | 0 | 0 | 0 | 0 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0 | 0 | 0 | 0 | 0 |

(0.08,0.16) | 0 | 0 | 0 | 0 | 0 |

(0.16,0.24) | 0 | 0 | 0 | 0 | 0.0001 |

(0.24,0.32) | 0.0003 | 0.0003 | 0.0002 | 0.0005 | 0.0018 |

(0.32,0.40) | 0.4531 | 0.4566 | 0.0196 | 0.3758 | 0.4907 |

(0.40,0.48) | 0.5252 | 0.5225 | 0.0172 | 0.4810 | 0.5948 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0.5342 | 0.5321 | 0.0155 | 0.4871 | 0.5955 |

(0.08,0.16) | 0.4489 | 0.4510 | 0.0159 | 0.3872 | 0.4949 |

(0.16,0.24) | 0.0002 | 0.0002 | 0.0001 | 0.0003 | 0.0011 |

(0.24,0.32) | 0.0001 | 0.0001 | 0 | 0 | 0.0001 |

(0.32,0.40) | 0 | 0 | 0 | 0 | 0 |

(0.40,0.48) | 0 | 0 | 0 | 0 | 0 |

Frequency Range | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

(a,b) | CI | CI | |||

(0,0.08) | 0 | 0 | 0 | 0 | 0 |

(0.08,0.16) | 0 | 0 | 0 | 0 | 0 |

(0.16,0.24) | 0 | 0 | 0 | 0 | 0.0001 |

(0.24,0.32) | 0.0003 | 0.0003 | 0.0001 | 0.0003 | 0.0011 |

(0.32,0.40) | 0.4553 | 0.4570 | 0.0130 | 0.4021 | 0.4987 |

(0.40,0.48) | 0.5234 | 0.5219 | 0.0119 | 0.4782 | 0.5738 |

Here we present the proofs of the theoretical results in

Recall that for some

We first consider the expected value of this quantity.

First, we observe that

A sufficient condition for joint consistency of

(

Plot of estimated stationary subspace dimensions

(

(

Plot of the FS-ratio statistic

Plot of the FS-ratio statistic

Numerical summaries of FS-ratio statistic

Frequency Band | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

CI | CI | ||||

Theta (4–8 Hz) | 0.079 | 0.079 | 0.004 | 0.061 | 0.081 |

Alpha (8–12 Hz) | 0.076 | 0.077 | 0.0035 | 0.059 | 0.078 |

Beta (12–30 Hz) | 0.332 | 0.332 | 0.0129 | 0.267 | 0.341 |

Gamma (30–50 Hz) | 0.144 | 0.144 | 0.006 | 0.141 | 0.191 |

Numerical summaries of FS-ratio statistic

Frequency Band | Mean | Median | SD | Lower | Upper |
---|---|---|---|---|---|

CI | CI | ||||

Theta (4–8 Hz) | 0.062 | 0.062 | 0.004 | 0.0422 | 0.0669 |

Alpha (8–12 Hz) | 0.060 | 0.061 | 0.004 | 0.041 | 0.064 |

Beta (12–30 Hz) | 0.283 | 0.285 | 0.018 | 0.202 | 0.292 |

Gamma (30–50 Hz) | 0.146 | 0.146 | 0.006 | 0.135 | 0.187 |