Time Series as Observations of a Dynamic System

The essential concept is that time series can be viewed as projections of the behavior of a dynamic system. First, the system state can be described as a point in a high-dimensional space. The axes of this space can be thought of as fundamental state variables; in an ecosystem, these variables might correspond to population abundances, resources, or environmental conditions. Second, the system state changes through time following a set of deterministic rules. In other words, the behavior of the system is not completely stochastic.

Consequently, it is possible to project the system state onto one of the coordinate axes and obtain the value of the corresponding state variable. Sequential projections over time will thus produce a time series for that variable. For example, in figure 1 the states of the canonical Lorenz Attractor (Lorenz 1963) are projected onto the x-axis, creating a time series of variable x.

Time Series Projection from the Lorenz Attractor.

Although different time series observed from a system can represent independent state variables, in general, each time series is an observation function of the system state that may convolve several different state variables.

Attractor Reconstruction / Takens’ Theorem

The goal of EDM is to reconstruct the system dynamics from time series data. As seen above, a time series can be thought of as sequential projections of the motion on an attractor; in other words, information about the behavior is encoded in the temporal ordering of the time series. Takens’ Theorem (Takens 1981) states that mathematically valid and property preserving reconstructions of the attractor can be created using lags of a single time series, then substituting those lagged time series for unknown or unobserved variables. In other words, instead of representing the system state using a complete set of state variables, we can instead use an E-dimensional lagged-coordinate embedding:

x⃗_t = ⟨x_t, x_t − τ, …, x_{t − (E − 1)τ}⟩

Attractor Reconstruction from 3 Lagged Coordinates

If sufficient lags are used, the reconstruction preserves essential mathematical properties of the original system: reconstructed states will map one-to-one to actual system states, and nearby points in the reconstruction will correspond to similar system states. Figure 2 shows a reconstruction of the Lorenz attractor where the reconstructed system state is comprised of 3 lags of variable x. Here, the visual similarity between the reconstruction and the original Lorenz Attractor is quite clear.

As a consequence of the fact that dynamical properties of the original system can be recovered from a single time series, there are multiple applications. For example, empirical models can be used for forecasting (Sugihara and May 1990), to understand nonlinear behavior (Sugihara 1994), or to uncover mechanism (Dixon, Milicich, and Sugihara 1999). Moreover, recent work describes how EDM can be used to identify causal interactions, by testing whether two time series are observed from the same system (Sugihara et al. 2012). In the next section, we demonstrate how rEDM can be used to accomplish these various tasks.

Nearest Neighbor Forecasting using Simplex Projection

As mentioned previously, the reconstruction will map one-to-one to the original attractor manifold if enough lags are used (i.e. if the reconstruction has a sufficiently large embedding dimension). If the embedding dimension is too small, then reconstructed states can overlap and appear to be the same even though they actually correspond to different states. These “singularities” will result in poor forecast performance because the system behavior cannot be uniquely determined in the reconstruction. As a consequence, we can use prediction skill as an indicator for identifying the optimal embedding dimension. In the following example we demonstrate the Simplex() projection nearest neighbor forecasting method (Sugihara and May 1990), and its’ extension EmbedDimension() that automates evaluation of an optimal embedding dimension.

library(rEDM)
str(TentMap)

## 'data.frame':    999 obs. of  2 variables:
##  $ Time   : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ TentMap: num  -0.0992 -0.6013 0.7998 -0.7944 0.798 ...

simplex_out <- Simplex(dataFrame = TentMap, lib = "1 100", pred = "201 500", columns = "TentMap",
    target = "TentMap", E = 3)
simplex_out[c(1:2, 300:301), ]

##     Time Observations Predictions Pred_Variance
## 1    201         0.94         NaN           NaN
## 2    202         0.11       0.077       0.00015
## 300  500        -1.09      -1.084       0.06366
## 301  501         0.91       0.873       0.00255

ComputeError(simplex_out$Observations, simplex_out$Predictions)

## $MAE
## [1] 0.14
## 
## $rho
## [1] 0.94
## 
## $RMSE
## [1] 0.23

Optimal embedding dimension

As noted earlier, identification of the optimal embedding dimension to best “unfold” the dynamics can be assessed with simplex prediction skill. rEDM provides the EmbedDimension() function to automate this task. EmbedDimension() parallelises function calls to Simplex(), which automatically sets values of E from 1 to maxE=10. Continuing with the previous example, we invoke EmbedDimension():

rho_E <- EmbedDimension(dataFrame = TentMap, lib = "1 100", pred = "201 500", columns = "TentMap",
    target = "TentMap")

TentMap data prediction skill vs. embedding dimension.

The output is a data.frame with columns E and rho detailing the embedding dimension and Pearson correlation coefficient between the simplex projected forecast at Tp = 1 timesteps ahead, and the observed data over the pred indices. Here, we observe that forecast skill peaks at E = 2, indicating that the dynamics of our data are unfolded best in 2 dimensions. Note that this optimal value does not have to correspond to the dimensionality of the original system. The forecast skill will be affected by factors such as observational noise, process error, and time series length, and so it is more useful to think of the embedding dimension as a practical measure that is dependent on properties of the data.

Prediction Decay

An important property of many natural systems is that nearby trajectories eventually diverge over time (i.e. “deterministic chaos” – the “butterfly effect”). In essence, this means that while short-term prediction is often possible, information about the predictive state of the system is diluted over time, hindering long-term forecasting. We can demonstrate this effect by examining how prediction skill changes as we increase the Tp argument, the “time to prediction”, defining the number of time steps into the future at which forecasts are made. rEDM provides the PredictInterval() function to automate this task.

Example

Using the same data with the PredictInterval() function, we supply the embedding dimension parameter with the value determined previously (E = 2):

rho_Tp <- PredictInterval(dataFrame = TentMap, lib = "1 100", pred = "201 500", target = "TentMap",
    columns = "TentMap", E = 2)

Tent map first differences simplex prediction skill as a function of forecast interval.

As above, the returned object is a data.frame with forecast skill rho and time to prediction Tp. As expected (because the parameters chosen for the tent map fall in the region for chaotic behavior), the decline in forecast skill (rho  → 0) as the forecast interval Tp increases, indicates that the system may be chaotic.

Identifying Nonlinearity

One concern is that time series may show predictability even if they are purely stochastic, they behave similarly to autocorrelated red noise. Fortunately, we can distinguish between red noise and nonlinear deterministic behavior by using S-maps as described in (Sugihara 1994).

In contrast to the nearest-neighbor interpolation of simplex projection, the S-map forecasting method (Sugihara 1994) fits local linear maps to describe the dynamics. In addition to the standard set of parameters for a lagged-coordinate reconstruction as in simplex, S-maps contain a nonlinear localisation parameter, θ, that determines the degree to which points are weighted when fitting the local linear map. For example, when θ = 0, all points are equally weighted, such that the local linear map is identical for different points in the reconstructed state-space. As such, the S-map will be identical to a global linear map (i.e. an autoregressive model). When values of θ are greater than 0, nearby points in the state space receive larger weight, and the local linear map can vary in state-space to accommodate nonlinear behavior.

Consequently, if the time series are sampled from autoregressive red noise, then the linear model (θ = 0) should produce better forecasts, because the global linear map (which will, in effect, be fitted to more data points) will reduce the effects of observation error compared to local linear maps. In contrast, if forecast skill increases for θ > 0, then the results are suggestive of nonlinear dynamics wherein better forecasts are achieved when the local linear map can change depending on the location in state-space: it is a better description of state-dependent behavior.

Example

The PredictNonlinear() function provides an evaluation of S-map forecast skill as a function of the localisation parameter theta. If unspecified, theta values will range from 0.01 to 9.

Typically, when using S-map to test for nonlinear behavior, we want to use all available points in the reconstruction of the local linear map, not just knn nearest neighbors as in simplex projection. With all points available, S-map uses the theta parameter to control the weighting assigned to individual points, thereby localising the dynamics to capture nonlinear behavior. When knn = 0, the default, SMap() will use all available points.

Here we use an embedding dimension of E = 2 and the same parameters as in the previous examples, however, we specify the “TentMapNoise” data that adds Gaussian noise to the TentMap data as one would normally encounter with noisy observational data.

rho_theta <- PredictNonlinear(dataFrame = TentMapNoise, lib = "1 100", pred = "201 500",
    target = "TentMap", columns = "TentMap", E = 2)

Tent map first differences S-map prediction skill as a function of S-map localisation parameter.

The result is a data.frame with columns Theta and rho. Here, we see that forecast skill substantially improves as theta increases, indicating the presence of nonlinear dynamics. We also observe a degradation in forecast skill at high values of theta as the local linear map overfits to insufficient nearest neighbors.

Simplex() and SMap() functions

The functions EmbedDimension(), PredictInterval() and PredictNonlinear() are multithreaded wrapper functions for the Simplex() and SMap() algorithms. EmbedDimension() and PredictInterval() parallelise calls to Simplex() to evaluate forecast skill as a function of embedding dimension and prediction interval respectively. PredictNonlinear() parallelises calls to SMap() to assess predictive skill as a function of the nearest neighbor localisation parameter. However, one can equivalently call the underlying Simplex() and SMap() functions directly.

tentMapPredict <- Simplex(dataFrame = TentMap, lib = "1 100", pred = "201 500", target = "TentMap",
    columns = "TentMap", E = 2)

ComputeError(tentMapPredict$Observations, tentMapPredict$Predictions)$rho

## [1] 0.96

smap = SMap(dataFrame = TentMapNoise, lib = "1 100", pred = "201 500", target = "TentMap",
    columns = "TentMap", E = 2, theta = 3)

head(cbind(smap$predictions, smap$coefficients), 2)

##   Time Observations Predictions Pred_Variance Time   C0 ∂TentMap/∂TentMap(t-0)
## 1  201         0.99         NaN           NaN  201  NaN                    NaN
## 2  202         0.16       -0.14          0.28  202 -1.1                  -0.33
##   ∂TentMap/∂TentMap(t-1)
## 1                    NaN
## 2                   -1.1

tail(cbind(smap$predictions, smap$coefficients), 2)

##     Time Observations Predictions Pred_Variance Time    C0
## 300  500         -1.3       -0.63          0.36  500 -0.33
## 301  501          1.1        0.92          0.41  501  0.24
##     ∂TentMap/∂TentMap(t-0) ∂TentMap/∂TentMap(t-1)
## 300                  -0.78                  -0.16
## 301                  -0.52                  -0.03

Generalized Takens Theorem

A practical reality is that sampled observations of complex dynamics are usually composed of finite, noisy data. Additionally, the presence of stochastic, non-deterministic drivers means that “multivariate” reconstructions can often be a better description than “univariate” reconstructions. This means that in addition to creating an attractor from lags of one time series, it can be advantageous to combine different time series to create the phase-space embedding, provided they are all observed from the same system (Sauer, Yorke, and Casdagli 1991; Deyle and Sugihara 2011).

In rEDM, the Simplex() and SMap() functions allow multivariate reconstructions from any set of observation vectors. A multivariate reconstruction is defined by specifying which columns to use as coordinates in the columns argument, and which column is to be forecast in the target argument. By default, rEDM will create Takens time-delay embeddings from univariate or multivariate data, however, this can be prevented by setting the embedded parameter TRUE. In this case, the input data are assumed to already constitute a valid multidimensional embedding, and no time-delay embedding is performed.

Example

We begin by examining an example dataset from a coupled 3-species model system.

head(block_3sp, 3)

##   time   x_t x_t-1 x_t-2   y_t y_t-1 y_t-2  z_t z_t-1 z_t-2
## 1    3 -1.92  1.24 -0.74 -0.11  1.49 -1.27  1.5 -0.48 -1.86
## 2    4 -0.96 -1.92  1.24 -1.11 -0.11  1.49 -1.5  1.54 -0.48
## 3    5  1.33 -0.96 -1.92  2.39 -1.11 -0.11 -1.1 -1.49  1.54

Here, block_3sp is a 10-column data.frame with 9 data columns. This data has already been time-delay embedded to dimension E = 3 with a time delay of tau = -1. We use simplex forecasting based on a multivariate embedding of the three data vectors x_t x_t-1 z_t with embedded = TRUE and E = 3:

smplx_3species = Simplex(dataFrame = block_3sp, lib = "1 100", pred = "101 190",
    E = 3, columns = "x_t x_t-1 z_t", target = "x_t", embedded = TRUE)

A plot of the predictions vs. observations can be examined with:

err = ComputeError(smplx_3species$Observations, smplx_3species$Predictions)
plot(smplx_3species$Observations, smplx_3species$Predictions, pch = 19, cex = 0.5,
    xlab = "Observations", ylab = "Predictions", main = "3 Species x_t")
abline(a = 0, b = 1, lty = 2, col = "blue")
text(-1, 1, paste(capture.output(cbind(err)), collapse = "\n"))

Scatter plot of simplex forecast of x_t vs. observations.

S-map Coefficients

As described in (Deyle et al. 2016), S-map coefficients from the appropriate multivariate embedding can be interpreted as dynamic, time-varying interaction strengths. We demonstrate this with a chaotic timeseries described in (Lorenz 1996), defined for N variables k=1, … N, as

The Lorenz5D data.frame contains a N=5 dimensional system with F=8 from (Lorenz 1996). Here, we use SMap() to compute a 4-dimensional forecast at Tp=1:

smap_Lorenz <- SMap(dataFrame = Lorenz5D, lib = "1 500", pred = "601 900", E = 4,
    theta = 3, columns = "V1 V2 V3 V4", target = "V1", embedded = TRUE)

As noted earlier, SMap() returns a named list with two data frames, predictions and coefficients:

head(cbind(smap_Lorenz$predictions, smap_Lorenz$coefficients[, 2:6]), 3)

##    Time Observations Predictions Pred_Variance      C0 ∂V1/∂V1 ∂V1/∂V2
## 1 40.00        3.485         NaN           NaN     NaN     NaN     NaN
## 2 40.05        4.214       4.123         8.197 -0.4830  0.9914  0.1313
## 3 40.10        4.849       4.744         8.947 -0.5544  0.9890  0.1530
##     ∂V1/∂V3 ∂V1/∂V4
## 1       NaN     NaN
## 2 -0.011692 0.04222
## 3 -0.006055 0.02284

Here, we plot the time series for the observed (blue) and predicted (red) values of V1 in the top panel; and the inferred interactions (S-map coefficients) for the influence of V4, V3 and V2 on future values of V1 in the lower panels.

predictions = smap_Lorenz$predictions
coefficients = smap_Lorenz$coefficients
Time = predictions$Time

plot(Time, predictions$Observations, type = "l", col = "blue", ylab = "V1", xlab = "",
    lwd = 2, cex.lab = 1.3, cex.axis = 1.3)
lines(Time, predictions$Predictions, lwd = 2, col = "red")
legend("topright", legend = c("observed", "predicted"), fill = c("blue", "red"),
    bty = "n", cex = 1.3)

plot(Time, coefficients[, 6], type = "l", col = "brown", ylab = paste("∂", "V4/",
    "∂", "V1", sep = ""), xlab = "", lwd = 2, cex.lab = 1.3, cex.axis = 1.3)
plot(Time, coefficients[, 5], type = "l", col = "darkgreen", ylab = paste("∂",
    "V3/", "∂", "V1", sep = ""), xlab = "", lwd = 2, cex.lab = 1.3, cex.axis = 1.3)
plot(Time, coefficients[, 4], type = "l", col = "blue", ylab = paste("∂", "V2/",
    "∂", "V1", sep = ""), xlab = "", lwd = 2, cex.lab = 1.3, cex.axis = 1.3)

S-map prediction and coefficients of Lorenz’96 5-D system.

Multiview Embedding

The generality of Takens Theorem means that in situations with multivariate time series, there can often be many different, valid attractor reconstructions. As described in (Ye and Sugihara 2016), combining these different models can result in improved forecasts.

Here, we demonstrate this idea using the Multiview() function with the 3-species data used above. Multiview() operates by constructing all possible embeddings of dimension E with lag up to E-1. These embeddings are ranked by forecast skill (rho) over the lib portion of the data. The individual forecasts for the top multiview embeddings are then averaged together. If multiview is not specified it is set to sqrt(C) where C is the number of E-dimensional combinations created from all data vectors.

Mview = Multiview(dataFrame = block_3sp, lib = "1 100", pred = "101 190", E = 3,
    columns = "x_t y_t z_t", target = "x_t")

Multiview() returns a named list with two data.frames: View, and Predictions. View lists the various combinations of data embedding vectors used for the forecasts along with their prediction statistics. Predictions returns the final averaged multiview projections.

Mview$View[which(Mview$View$rho > 0.91), ]

##   Col_1 Col_2 Col_3    rho    MAE   RMSE   name_1   name_2   name_3
## 1     1     2     7 0.9320 0.2391 0.2996 x_t(t-0) x_t(t-1) z_t(t-0)
## 3     1     2     3 0.9395 0.2221 0.2815 x_t(t-0) x_t(t-1) x_t(t-2)
## 7     1     2     8 0.9215 0.2484 0.3202 x_t(t-0) x_t(t-1) z_t(t-1)

Causality Inference and Cross Mapping

One of the corollaries to the Generalized Takens Theorem is that it should be possible to cross predict or cross map between variables that are observed from the same system. Consider two variables, x and y that interact in a dynamic system. Then the univariate reconstructions based on x or y alone should uniquely identify the system state and and thus the corresponding value of the other variable.

Cross Mapping Between Reconstructions of the Lorenz Attractor

In the case of unidirectional causality, e.g. x causes y, the causal variable (x) leaves a signature on the affected variable (y). Consequently, the reconstructed states based on y can be used to cross predict the values of x (because the reconstruction based on y must be complete, it must include information about the value of x). Note that this cross prediction is in the opposite direction of the causal effect. At the same time, cross prediction from x to y will fail, because the time series of x behaves independently of y, so a univariate reconstruction using only lags of x is necessarily incomplete.

Although x has incomplete information for predicting y, it does affect the values of y, and therefore will likely to have nonzero predictive skill. However, this cross mapping will be limited to the statistical association between x and y and will generally not improve as longer time series are used for reconstruction. In contrast, the cross prediction of x from y will generally improve. This convergence is therefore a crucial property for inferring causality. For practical reasons, the sensitivity of detecting causality this way is improved if, instead of predicting the future value of another variable, we estimate the concurrent value of another variable. We refer to this modified method as cross mapping, because we are not “predicting” the future.

For a more detailed description of using cross mapping to infer causation, see (Sugihara et al. 2012).

Convergent Cross Mapping (CCM)

In rEDM, convergent cross mapping is implemented as the CCM() function, which provides a wrapper to compute cross map skill for different subsamples of the data. In the following example, we reproduce the analysis from (Sugihara et al. 2012) to identify causality between anchovy landings in California and Newport Pier sea-surface temperature. For this example, a previously identified value of 3 for the embedding dimension will be used.

To quantify convergence, we compute the cross map skill over many random subsamples of the time series. The libSizes argument specifies the size of the library set, and sample specifies the number of subsamples generated at each library size. random and replacement specify how the subsamples will be generated. The default is random sampling without replacement.

cmap <- CCM(dataFrame = sardine_anchovy_sst, E = 3, Tp = 0, columns = "anchovy",
    target = "np_sst", libSizes = "10 70 5", sample = 100, showPlot = TRUE)

Convergent cross mapping of Newport sea surface temperature with anchovy landings.

The output is a data.frame with statistics for each model run (in this case, 100 models at each library size) as a function of library size. Recalling that cross mapping indicates causal influence in the reverse direction (target to source), we see that the cross mapping anchovy:np_sst converges at a positive value of ρ, indicating that Newport sea surface temperature influences anchovy landings. Because average cross map skill less than 0 means there is no prediction skill, (predictions should not be anticorrelated with observations), we infer from the np_sst:anchovy cross mapping that anchovy landings do not effect sea surface temperatures.

Apple-Blossom Thrips

In this example, we use EDM to re-examine the classic apple-blossom thrips (Thrips imaginis) time series from the Wait Institute in Australia (Davidson and Andrewartha 1948a, 1948b). Seasonal outbreaks of Thrips imaginis were observed to vary greatly in magnitude from year to year, but large outbreaks tended to coincide across large spatial domains. This lead to the hypothesis that regional-scale climatic factors were responsible for controlling the size of the seasonal outbreaks (what might now be called the Moran effect (Moran 1953)).

head(Thrips, 2)

##   Year Month Thrips_imaginis maxT_degC Rain_mm Season
## 1 1932     4             4.5      19.2   140.1 -0.500
## 2 1932     5            23.4      19.1    53.7 -0.866

The data column Thrips_imaginis contains counts of Thrips imaginis obtained from the Global Population Dynamics Database (GPDD) (NERC Centre for Population Biology 2010). maxT_degC is the mean maximum daily temperature (^∘C) taken over each month and Rain_mm is the monthly rainfall (mm), both from the Waite Institute. The final column Season is a simple annual sinusoid that peaks in December (the Austral summer) to emulate an indicator of season.

First, we plot the data. Note that all the time-series variables, particularly the mean maximum daily temperature, show marked seasonality.

Thrips abundance and environmental variables.

Univariate Analysis

We first examine the dependence of simplex predictability on the embedding dimension.

rho_E <- EmbedDimension(dataFrame = Thrips, columns = "Thrips_imaginis", target = "Thrips_imaginis",
    lib = "1 72", pred = "1 72", showPlot = TRUE)

Simplex embedding dimension for Thrips abundance.

While there is an initial peak in the simplex prediction at E = 3, the global maximum is at E = 8. This suggests that both E = 3 and E = 8 are practical embedding dimensions, although E = 8 is preferrable with a higher predictive skill.

To test for nonlinearity we use the S-map PredictNonlinear() function.

E = 8
rho_theta_e3 = PredictNonlinear(dataFrame = Thrips, columns = "Thrips_imaginis",
    target = "Thrips_imaginis", lib = "1 73", pred = "1 73", E = E)

SMap localisation parameter for Thrips abundance.

The S-map results demonstrate clear nonlinearity in the Thrips time series, as nonlinear models theta > 0 give substantially better predictions than the linear model theta = 0. This suggests that Thrips, despite the strong seasonal dynamics, do not simply track the environment passively, but have some intrinsic dynamics. To look more closely at the issue of seasonal drivers, however, we turn to convergent cross-mapping (CCM).

Seasonal Drivers

Recall that there is a two-part criterion for CCM to be a rigorous test of causality:

1. The cross map prediction skill is statistically significant when using the full time series as the library.
2. Cross map prediction demonstrates convergence, i.e. prediction skill increases as more of the time series is used for the library and the reconstructed attractor becomes more dense.

For an initial summary, we first compute the cross map skill (measured with Pearsons ρ) for each variable pair. Note that CCM() computes the cross map in both “directions”.

vars = colnames(Thrips[3:6])
var_pairs = combn(vars, 2)  # Combinations of vars, 2 at a time
libSize = paste(NROW(Thrips) - E, NROW(Thrips) - E, 10, collapse = " ")
ccm_matrix = array(NA, dim = c(length(vars), length(vars)), dimnames = list(vars,
    vars))

for (i in 1:ncol(var_pairs)) {
    ccm_out = CCM(dataFrame = Thrips, columns = var_pairs[1, i], target = var_pairs[2,
        i], libSizes = libSize, Tp = 0, E = E, sample = 100)

    outVars = names(ccm_out)

    var_out = unlist(strsplit(outVars[2], ":"))
    ccm_matrix[var_out[2], var_out[1]] = ccm_out[1, 2]

    var_out = unlist(strsplit(outVars[3], ":"))
    ccm_matrix[var_out[2], var_out[1]] = ccm_out[1, 3]
}

We note that ccm_matrix rows are the second of the CCM() returned variables, while columns are the first variable. As outlined earlier, influences are quantified from the target to the columns variable so that here, rows are considered the ’target, influencing variable, and columns thecolumns` influenced variable.

For comparison we also compute the lagged cross-correlation, allowing lags of up to ±6 months.

corr_matrix <- array(NA, dim = c(length(vars), length(vars)), dimnames = list(vars,
    vars))

for (ccm_from in vars) {
    for (ccm_to in vars[vars != ccm_from]) {
        ccf_out <- ccf(Thrips[, ccm_from], Thrips[, ccm_to], type = "correlation",
            lag.max = 6, plot = FALSE)$acf
        corr_matrix[ccm_from, ccm_to] <- max(abs(ccf_out))
    }
}

We compare the two matrices.

ccm_matrix

##                 Thrips_imaginis maxT_degC Rain_mm Season
## Thrips_imaginis              NA    0.6056  0.4265 0.5611
## maxT_degC                0.9233        NA  0.8213 0.9625
## Rain_mm                  0.5110    0.4626      NA 0.3930
## Season                   0.9546    0.9918  0.7773     NA

corr_matrix

##                 Thrips_imaginis maxT_degC Rain_mm Season
## Thrips_imaginis              NA    0.4490  0.2668 0.4488
## maxT_degC                0.4490        NA  0.5949 0.9453
## Rain_mm                  0.2668    0.5949      NA 0.5333
## Season                   0.4488    0.9453  0.5333     NA

We can see that the cross map strengths are not symmetric. In general CCM(X1 : X2) != CCM(X2 : X1). We also notice that the cross map and correlation between temperature and the seasonal indicator are high, with the cross map results suggesting that the seasonal variable can almost perfectly recover the temperature, ρ = 0.9918. This makes interpretation more complicated, because we have to consider the possibility that cross mapping is simply identifying the shared seasonality between two time series. In other words, cross mapping between temperature and any variable with a seasonal cycle, might suggest an interaction even if there is no actual causal mechanism.

Convergent Cross-Mapping

With this in mind, we examine convergence in cross-map predictability, i.e. we compute rho as a function of library size L. The magnitude of the cross-correlation between Thrips and the cross mapped variable is shown as a black dashed line for comparison.

thrips_xmap_maxT <- CCM(dataFrame = Thrips, E = E, Tp = 0, columns = "Thrips_imaginis",
    target = "maxT_degC", libSizes = "13 73 3", sample = 300, showPlot = FALSE)
CCMPlot(thrips_xmap_maxT, E)

## [1] "Thrips_imaginis:maxT_degC  :  maxT_degC:Thrips_imaginis \nE= 8"
##  [1] 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

abline(h = corr_matrix["Thrips_imaginis", "maxT_degC"], col = "black", lty = 2)

thrips_xmap_Rain <- CCM(dataFrame = Thrips, E = E, Tp = 0, columns = "Thrips_imaginis",
    target = "Rain_mm", libSizes = "13 73 3", sample = 300, showPlot = FALSE)
CCMPlot(thrips_xmap_Rain, E)

## [1] "Thrips_imaginis:Rain_mm  :  Rain_mm:Thrips_imaginis \nE= 8"
##  [1] 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

abline(h = corr_matrix["Thrips_imaginis", "Rain_mm"], col = "black", lty = 2)

thrips_xmap_Season <- CCM(dataFrame = Thrips, E = E, Tp = 0, columns = "Thrips_imaginis",
    target = "Season", libSizes = "13 73 3", sample = 300, showPlot = FALSE)
CCMPlot(thrips_xmap_Season, E)

## [1] "Thrips_imaginis:Season  :  Season:Thrips_imaginis \nE= 8"
##  [1] 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73

abline(h = corr_matrix["Thrips_imaginis", "Season"], col = "black", lty = 2)

Thrips cross mapped to climatic variables. Vertical axis is cross map prediction skill (rho).

The results show evidence of convergence for Thrips cross mapping to temperature and season variables, with the ρ at maximum library size L significantly exceeding linear correlation. The rain variable does not indicate a substantially different cross map interaction, appearing confounded as to causal influence.

In addition, we are still left with the conundrum that temperature and to a lesser extent, rainfall, are easily predicted from the seasonal cycle, and so we cannot immediately ignore the possibility that the cross map results are an artifact of shared seasonal forcing.

To reframe, we wish to reject the null hypothesis that the level of cross mapping we obtain for maxT_degC and Rain_mm can be solely explained by shared seasonality. This hypothesis can be tested using randomization tests based on surrogate data. The idea here is to generate surrogate time series with the same level of shared seasonality. Cross mapping between the real time series and these surrogates thus generates a null distribution for ρ, against which the actual cross map ρ value can be compared.

Seasonal Surrogate Test

# Create matrix with temperature and rain surrogates (1000 time series vectors)
surr_maxT = SurrogateData(Thrips$maxT_degC, method = "seasonal", T_period = 12, num_surr = 1000,
    alpha = 3)
surr_rain = SurrogateData(Thrips$Rain_mm, method = "seasonal", T_period = 12, num_surr = 1000,
    alpha = 3)

# Rain cannot be negative
surr_rain = apply(surr_rain, 2, function(x) {
    i = which(x < 0)
    x[i] = 0
    x
})

# data.frame to hold CCM rho values between Thrips abundance and variable
rho_surr <- data.frame(maxT = numeric(1000), Rain = numeric(1000))

# data.frames with time, Thrips, and 1000 surrogate climate variables for CCM()
maxT_data = as.data.frame(cbind(seq(1:nrow(Thrips)), Thrips$Thrips_imaginis, surr_maxT))
names(maxT_data) = c("time", "Thrips_imaginis", paste("T", as.character(seq(1, 1000)),
    sep = ""))

rain_data = as.data.frame(cbind(seq(1:nrow(Thrips)), Thrips$Thrips_imaginis, surr_rain))
names(rain_data) = c("time", "Thrips_imaginis", paste("R", as.character(seq(1, 1000)),
    sep = ""))

# Cross mapping
for (i in 1:1000) {
    targetCol = paste("T", i, sep = "")  # as in maxT_data

    ccm_out = CCM(dataFrame = maxT_data, E = E, Tp = 0, columns = "Thrips_imaginis",
        target = targetCol, libSizes = "73 73 5", sample = 1)

    col = paste("Thrips_imaginis", ":", targetCol, sep = "")

    rho_surr$maxT[i] = ccm_out[1, col]
}

for (i in 1:1000) {
    targetCol = paste("R", i, sep = "")  # as in rain_data

    ccm_out = CCM(dataFrame = rain_data, E = E, Tp = 0, columns = "Thrips_imaginis",
        target = targetCol, libSizes = "73 73 5", sample = 1)

    col = paste("Thrips_imaginis", ":", targetCol, sep = "")

    rho_surr$Rain[i] = ccm_out[1, col]
}

We now have a null distribution, and can estimate a p-value for rejecting the null hypothesis of mutual seasonality.

1 - ecdf(rho_surr$maxT)(ccm_matrix["maxT_degC", "Thrips_imaginis"])

## [1] 0

1 - ecdf(rho_surr$Rain)(ccm_matrix["Rain_mm", "Thrips_imaginis"])

## [1] 0.047

In the case of temperature, the CCM influence we estimated (0.9233) is higher than the linear correlation (0.449), and is highly significant in relation to a surrogate null distribution. Regarding rainfall, the CCM influence (0.511) is higher than the linear correlate (0.2668), but not significant at the 95th percentile of the surrogate null distribution. We note that the original Thrips data collections were at a much higher frequency than those available through the GPDD, and that monthly accumulated rainfall may be inadequate to resolve lifecycle influences on a species with a lifecycle of approximately one month. With more highly resolved data, it may well be possible to establish significance.

Data Input

Data can be input as an R data.frame, or read from a .csv file. In either case, the first column must define a time vector, and all columns are expected to be named. The time vector can be a string encoding a Date, or Datetime format. All subsequent columns are expected to be numeric.