Multiple Series Pattern Causality Analysis: Unveiling System-Wide Interactions

In patterncausality: Pattern Causality Algorithm

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This vignette provides a comprehensive demonstration of multivariate pattern causality analysis, a powerful technique for investigating complex interactions within large-scale systems. This approach is particularly useful when dealing with multiple interconnected time series, allowing us to move beyond pairwise analysis and understand system-wide dynamics. The key aspects of this analysis include:

1. Matrix-based causality assessment: Quantifying causal relationships between all pairs of time series in the system, represented in a matrix format.
2. System-wide pattern identification: Identifying recurring patterns of behavior across the entire system, revealing underlying dynamics.
3. Visualization of complex causal relationships: Using graphical representations to make intricate causal networks more understandable.
4. Analysis of effects across the entire system: Measuring the overall impact of causality within the system, providing insights into the collective behavior.

The matrix-based approach is particularly well-suited for the study of:

• Financial market networks: Analyzing the interconnectedness of stock prices and other financial instruments.
• Economic systems: Understanding the relationships between various economic indicators and their impact on the overall economy.
• Social networks: Investigating the spread of information and influence within social structures.
• Complex ecological systems: Studying the interactions between different species and environmental factors.

Pattern Causality Matrix

In this vignette, we will utilize the DJS dataset, which comprises 29 stock price series. This dataset provides a sufficiently large and complex system to demonstrate the capabilities of our multivariate pattern causality analysis.

library(patterncausality)
data(DJS)
#head(DJS)

Before estimating the causality matrix, it is crucial to determine the optimal parameters for our analysis. These parameters, including the embedding dimension (E) and time delay (tau), significantly influence the accuracy and reliability of the results. We use the optimalParametersSearch function to identify these optimal values:

dataset <- DJS[,-1] # remove the date column
params <- optimalParametersSearch(
  Emax = 3, 
  tauMax = 3, 
  metric = "euclidean", 
  dataset = dataset,
  verbose = FALSE
)
print(params)

With the optimal parameters identified, we can now estimate the pattern causality matrix using the pcMatrix function. This function calculates the causality between all pairs of time series in the dataset, resulting in a matrix representation of the system's causal structure.

result <- pcMatrix(
  dataset = dataset, 
  E = 3,           # Embedding dimension
  tau = 1,         # Time delay
  metric = "euclidean",
  h = 1,           # Prediction horizon
  weighted = FALSE  # Unweighted analysis
)

result <- readRDS("DJSm.rds")
result$is_square <- TRUE

The resulting analysis yields three matrices, each representing a different aspect of causality: positive, negative, and dark causality. These matrices can be accessed through the pc_matrix object.

print(result)

The plot function for object pc_matrix provides a powerful tool for visualizing these complex matrices. By plotting each causality type separately, we can gain a deeper understanding of the system's dynamics.

• Positive causality status

plot(result, "positive")

• Negative causality status

plot(result, "negative")

• Dark causality status

plot(result, "dark")

The visualization reveals a clear positive connection within the system, indicating a tendency for stocks to influence each other positively.

Pattern Causality Effect

Following the matrix calculation, we can quantify the total effect within the system using the pcEffect function. This function aggregates the causality measures to provide a system-wide perspective on the overall impact of pattern causality.

effects <- pcEffect(result)
print(effects)

The total effect of pattern causality can be observed, providing a measure of the overall strength of causal interactions within the system.

• Positive causality status

plot(effects, status="positive")

• Negative causality status

plot(effects, status="negative")

• Dark causality status

plot(effects, status="dark")

Cross Matrix analysis

Sometimes, we also need to face the problem that X has multiple series, and Y also has multiple series, we want to know the causality between each series in X and each series in Y, to save the computation time, we can use the pcCrossMatrix function to get the causality matrix from each series in X to Y.

This time we construct the datasets for X and Y in stock dataset.

dataset <- DJS[, -1]

X <- dataset[, 1:10]
Y <- dataset[, 11:29]

result_cross <- readRDS("djscross.rds")

Then we can estimate the causality matrix from each series in X to Y and give the new matrix from X to Y.

result_cross <- pcCrossMatrix(
  X = X,
  Y = Y,
  E = 3,
  tau = 1,
  metric = "euclidean",
  h = 1,
  weighted = FALSE,
  verbose = FALSE
)

The new matrix will be saved in the result_cross object, we can also plot the matrix by the plot function.

plot(result_cross, "positive")

This will show the causality matrix from each series in X to Y and the color of the matrix represents the causality strength in the whole period.

We provide the multiple types of multi-series pattern causality analysis here and it would be useful to face many different situatiuons for the matrix analysis and network analysis.

patterncausality documentation built on April 3, 2025, 6:57 p.m.