# Ergodic Subspace Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Definition of Ergodicity

#### Conditional Equivalence

## 3. Ergodic Subspace Analysis

## 4. Performance in Artificial Situations

#### 4.1. Computation Examples

#### 4.1.1. Manual Computational Example

#### 4.1.2. Larger-Scale Computational Example

#### 4.2. Simulations

#### 4.2.1. Simulation 1

#### 4.2.2. Simulation 2

#### 4.2.3. Simulation 3

## 5. Application to COGITO Data

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. ESA in R Code

`ergodicSubspaceAnalysis <-`

`function(betweenCovariance, withinCovariance, cutoff = 0.1) {`

`# BEGIN ESA`

`# Step 1: Combining the matrices`

`combinedCovariance <- 0.5 * (betweenCovariance + withinCovariance)`

`# Step 2`

`eigendecompositionResult1 <- eigen(combinedCovariance);`

`QWhite <- t(eigendecompositionResult1$vectors);`

`for (i in 1:nrow(QWhite))`

`QWhite[i,] <- QWhite[i,] / sqrt(eigendecompositionResult1$values[i])`

`# Step 3`

`transformedBetweenCovariance <- QWhite %*% betweenCovariance %*% t(QWhite)`

`eigendecompositionResult2 <- eigen(0.5*transformedBetweenCovariance)`

`Q2 <- t(eigendecompositionResult2$vectors)`

`ergodicity <-`

`eigendecompositionResult2$values - rev(eigendecompositionResult2$values)`

`QESA <- Q2 %*% QWhite`

`# Step 4`

`cutoff <- 0.1`

`VBetween <- QESA[ergodicity > cutoff,]`

`VErgodic <- QESA[(ergodicity < cutoff) & (ergodicity > -cutoff),]`

`VWithin <- QESA[ergodicity < -cutoff,]`

`# Prepare return element`

`result <- list("QESA" = QESA, "ergodicity" = ergodicity,`

`"VBetween" = VBetween, "VErgodic" = VErgodic,`

`"VWithin" = VWithin)`

`return(result)`

`}`

`# Manual example from the article`

`betweenCovariance <- matrix(c(1,0.5,0.5,1), nrow = 2, byrow = T)`

`withinCovariance <- matrix(c(1,-0.5,-0.5,1), nrow = 2, byrow = T)`

`ergodicSubspaceAnalysis(betweenCovariance, withinCovariance)`

`# Sports example from the article, with~a fixed x and using`

`# expected covariance`

`factors <- matrix(c(1,1,1,1,`

`3,-1,-1,-1,`

`0,1,-1,0,`

`0,1,1,-2), nrow = 4, byrow = T)`

`for (i in 1:4) factors[i,] <- factors[i,] / norm(as.matrix(factors[i,]),"2")`

`x <- 0 betweenLoadings <- diag(c(10,x,5,1))`

`betweenCovariance <- t(factors) %*% betweenLoadings %*% factors`

`withinLoadings <- diag(c(x,10,5,1))`

`withinCovariance <- t(factors) %*% withinLoadings %*% factors`

`ergodicSubspaceAnalysis(betweenCovariance, withinCovariance)`

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1 | Strictly speaking, ergodicity would even require the variances to be equal and then, consequently, the covariances instead of the correlations; however, it is sometimes also useful to concentrate on the correlations and allow differences in variances. |

2 | Equivalence in this article is used very similarly to the term ergodicity; we will use this term from here on to avoid confusion. |

**Figure 1.**Expectation of the first estimated ergodicity value for different x values in the simulation. The ergodicity values follow ${(1-x)}^{2}$.

**Figure 2.**The expectation of the first estimated ergodicity value for different sample sizes N and time point T.

**Figure 3.**Scree plot for three situations of ergodicity: fully ergodic (independently sampled covariance matrices), fully non-ergodic (identical covariance matrices), and an intermediate case with a 50% mixture.

**Figure 4.**Scree plot for three artificial situations of ergodicity, fully ergodic (independently sampled covariance matrices), fully non-ergodic (identical covariance matrices), and an intermediate case with a 50% mixture, together with the ergodicity values of the nine cognitive tasks from the COGITO study (raw and de-trended).

**Table 1.**Reconstruction precision of the factor with dominant variance between participants and its corresponding ergodicity value. The value x gives the degree to which the data is ergodic, with $x=0$ indicating strong differences between and within participants and $x=10$ indicating a perfectly ergodic situation. The angle $\alpha $ is the angle between the true first factor and the reconstructed factor; a cosine of one means perfect reconstruction, a cosine of 0 means orthogonal vectors.

x | $cos\left(\mathit{\alpha}\right)$ | ${\mathit{erg}}_{1}$ | stdv(${\mathit{erg}}_{1}$) |
---|---|---|---|

0 | 1 | 1 | 0.144 |

1 | 0.999 | 0.823 | 0.129 |

2 | 0.996 | 0.670 | 0.119 |

3 | 0.988 | 0.542 | 0.111 |

4 | 0.972 | 0.432 | 0.103 |

5 | 0.938 | 0.337 | 0.095 |

6 | 0.875 | 0.253 | 0.090 |

7 | 0.764 | 0.182 | 0.084 |

8 | 0.594 | 0.115 | 0.080 |

9 | 0.421 | 0.056 | 0.075 |

10 | 0.277 | 0.005 | 0.072 |

**Table 2.**Standard error of the ergodicity value of the factor with dominant variance between participants for different sample sizes and time series lengths. High values indicate uncertain measurements, while lower values indicate a more precise measurement of the degree of ergodicity in the data.

N | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{T}$ | 25 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 | 250 |

25 | 0.197 | 0.139 | 0.114 | 0.098 | 0.087 | 0.078 | 0.075 | 0.068 | 0.065 | 0.061 |

50 | 0.197 | 0.139 | 0.113 | 0.094 | 0.085 | 0.079 | 0.071 | 0.069 | 0.066 | 0.060 |

75 | 0.194 | 0.136 | 0.111 | 0.097 | 0.085 | 0.079 | 0.071 | 0.066 | 0.064 | 0.061 |

100 | 0.194 | 0.137 | 0.109 | 0.097 | 0.085 | 0.077 | 0.072 | 0.067 | 0.063 | 0.060 |

125 | 0.194 | 0.133 | 0.111 | 0.094 | 0.086 | 0.078 | 0.072 | 0.068 | 0.064 | 0.060 |

150 | 0.193 | 0.131 | 0.109 | 0.097 | 0.084 | 0.078 | 0.073 | 0.067 | 0.062 | 0.060 |

175 | 0.194 | 0.135 | 0.110 | 0.095 | 0.085 | 0.078 | 0.070 | 0.067 | 0.063 | 0.060 |

200 | 0.195 | 0.135 | 0.111 | 0.096 | 0.084 | 0.078 | 0.070 | 0.066 | 0.063 | 0.060 |

225 | 0.193 | 0.133 | 0.111 | 0.095 | 0.086 | 0.076 | 0.072 | 0.068 | 0.064 | 0.060 |

250 | 0.194 | 0.136 | 0.109 | 0.094 | 0.085 | 0.078 | 0.073 | 0.066 | 0.063 | 0.060 |

**Table 3.**The ergodicity values and the corresponding nine factors of the ergodic subspace analysis (ESA) on the de-trended cognitive data from the COGITO study.

$\mathit{erg}$ | Processing Speed | Episodic Memory | Working Memory | ||||||
---|---|---|---|---|---|---|---|---|---|

Numerical | Verbal | Figural | Numerical | Verbal | Figural | Numerical | Verbal | Figural | |

0.314 | 0.200 | 0.155 | 0.130 | 0.144 | 0.191 | 0.141 | 0.113 | 0.166 | 0.214 |

0.181 | 0.125 | 0.201 | 0.165 | 0.040 | 0.009 | 0.194 | −0.370 | −0.334 | −0.037 |

0.004 | 0.246 | 0.076 | 0.206 | −0.331 | −0.237 | −0.379 | 0.140 | 0.092 | 0.066 |

−0.038 | −0.041 | −0.201 | 0.120 | −0.343 | −0.234 | 0.405 | −0.065 | −0.076 | 0.398 |

−0.081 | 0.032 | 0.170 | −0.229 | 0.393 | −0.551 | −0.019 | −0.004 | −0.009 | 0.163 |

−0.163 | 0.177 | 0.400 | −0.619 | −0.334 | 0.095 | 0.096 | −0.118 | 0.120 | −0.004 |

−0.264 | 0.119 | 0.016 | 0.062 | −0.088 | −0.245 | 0.421 | 0.282 | 0.116 | −0.58 |

−0.460 | 0.115 | 0.043 | −0.169 | −0.017 | 0.083 | −0.035 | 0.591 | −0.671 | 0.078 |

−0.542 | 0.785 | −0.652 | −0.251 | 0.108 | 0.039 | 0.009 | −0.119 | −0.020 | −0.013 |

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**MDPI and ACS Style**

von Oertzen, T.; Schmiedek, F.; Voelkle, M.C. Ergodic Subspace Analysis. *J. Intell.* **2020**, *8*, 3.
https://doi.org/10.3390/jintelligence8010003

**AMA Style**

von Oertzen T, Schmiedek F, Voelkle MC. Ergodic Subspace Analysis. *Journal of Intelligence*. 2020; 8(1):3.
https://doi.org/10.3390/jintelligence8010003

**Chicago/Turabian Style**

von Oertzen, Timo, Florian Schmiedek, and Manuel C. Voelkle. 2020. "Ergodic Subspace Analysis" *Journal of Intelligence* 8, no. 1: 3.
https://doi.org/10.3390/jintelligence8010003