Nonparametric causal inference for bivariate time series

We introduce new quantities for exploratory causal inference between bivariate time series. The quantities, called penchants and leanings, are computationally straightforward to apply, follow directly from assumptions of probabilistic causality, do not depend on any assumed models for the time series generating process, and do not rely on any embedding procedures; these features may provide a clearer interpretation of the results than those from existing time series causality tools. The penchant and leaning are computed based on a structured method for computing probabilities.

Figure 1

The unitless leaning of Eq. (15) is a function of the noise, the tolerance used for terms from $\mathbf{Y}$, and the length of the signals and both ${\delta }_y$ and $B$ are unitless. See the text for an explanation of the missing data for large ${\delta }_y$. The subplots are each for different signal lengths, (a) $L=10$, (b) $L=50$, (c) $L=250$, and (d) $L=1750$.

Figure 2

Weighted mean observed leaning for Eq. (16) and a tolerance for the leaning calculation set to ${\delta }_y=B$. $\lambda$ is always positive, which implies $\mathbf{X}\stackrel{\mathrm{lean}}{\to }\mathbf{Y}$.

Figure 3

The unitless, normalized leaning, ${\lambda }^{\prime }$, of Eq. (16) can be plotted for different $l$-standard cause-effect assignments along with the cross-correlation, $\chi$ for the same lags, $l$, to show how the two values compare for this simple cyclic example.

Figure 4

Leaning, ${\lambda }_1$, of Eq. (23) computed ${\delta }_y=C$ as function of all three unitless parameters in Eq. (23), $A,B$, and $C$. The leaning agrees with intuition in this example for all the tested parameter values. The subplots are each for different values of $C$, (a) $C=0.2$, (b) $C=0.4$, (c) $C=0.6$, and (d) $C=0.8$.

Figure 5

Leaning, ${\lambda }_1$, of Eq. (24) is a function of four unitless parameters in Eq. (24), $r_x,r_y,{\beta }_{xy}$, and ${\beta }_{yx}$ (along with the initial conditions $x_0$ and $y_0$, which are fixed in this example). The tolerance domains are set as ${\delta }_x={\sigma }_{x_t-\langle x_t\rangle }$ and ${\delta }_y={\sigma }_{y_t-\langle y_t\rangle }$. The leaning is defined using the 1-standard assignment, so ${\lambda }_1>0\Rightarrow \mathbf{X}\stackrel{\mathrm{lean}}{\to }\mathbf{Y}$. The subplots are each for different parameter values, (a) $r_x=r_y=3.5$, (b) $r_x=4.0,r_y=2.0$, (c) $r_x=2.0,r_y=4.0$, and (d) $r_x=3.8,r_y=3.2$.

Figure 6

The (unitless) weighted mean observed leaning using the 1-standard cause-effect assignment, ${\lambda }_1$ only leads to causal inferences that agree with intuition for all points within the considered noise level domain, $B$, in this example for case 2. $B$ is the unitless noise parameter found in Eq. (25), and the symmetric tolerance domains for $\mathbf{Y}$ and $\mathbf{Z}$ are set to this value. All of the leanings are expected to be positive in each case. The subplots are each for different the three different cases, (a) $\mathbf{Z}=\left\{z_t\right\}$ (case 1), (b) $\mathbf{Z}=\left\{z_t^{\prime }\right\}$ (case 2), and (c) $\mathbf{Z}=\left\{z_t^{\prime \prime }\right\}$ (case 3).

Figure 7

This times series data pair is expected to have the causal relationship of $X\to Y$, where $X$ or $Y$ is marked in parentheses for each time series. Subplot (a) is the mean temperature $\left(X\right)$, and (b) is the snow fall $\left(Y\right)$.

Figure 8

The weighted mean observed leaning, ${\lambda }_l^{xy}$, using the 1-standard cause-effect assignment for $l\in [0,21]$. The time series pair ${\tau }_1=(\mathbf{X},\mathbf{Y})$ with $\mathbf{X}$ and $\mathbf{Y}$ shown in Figs. 7 and 7, respectively. The expected causal inference is $\mathbf{X}\stackrel{\mathrm{lean}}{\to }\mathbf{Y}$; i.e., the expectation is ${\lambda }_l^{xy}>0$ for every point in the plotted domain.

Figure 9

(a) A histogram of the $(B_z^{\prime },D_{st})$ set of $n={10}^4$ (unitless) weighted mean observed leanings, using the 1-standard cause-effect assignment; i.e., ${\lambda }_z^1$, show causal inferences that agree with intuition for the OMNI data set. The red dashed lines show the algebraic mean of the sets. The geomagnetic field component $B_z^{\prime }$ is calculated by taking the $B_z$ times series in the OMNI data set and then setting $B_z^{\prime }=0$ if $B_z>0$ [35, 36]. (b) The algebraic means of the aforementioned sets of weighted mean observed leanings; i.e., $\langle {\lambda }_l^z\rangle$, are positive for all leanings calculated using $l$-standard assignment given $l\in [1,20]$.

Figure 10

Eq. (31) leads to spurious leanings, i.e., weighted mean observed leanings using the 1-standard assignment, ${\lambda }_1$, that depend on both the tolerance domain ${\sigma }_y$ and the library length $L$.