---
title: "Multiple Series Pattern Causality Analysis: Unveiling System-Wide Interactions"
author: "Stavros Stavroglou, Athanasios Pantelous, Hui Wang"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Multiple Series Pattern Causality Analysis}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
  knitr::opts_chunk$set(
  warning = FALSE,
  collapse = TRUE,
  comment = "#>",
  fig.width = 8,
    fig.height = 6
)
```

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.

```{r message=FALSE}
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:

```{r eval=FALSE}
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.

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

```{r echo=FALSE}
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.

```{r}
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

```{r}
plot(result, "positive")
```

-   Negative causality status

```{r}
plot(result, "negative")
```

-   Dark causality status

```{r}
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.

```{r}
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

```{r}
plot(effects, status="positive")
```

-   Negative causality status

```{r}
plot(effects, status="negative")
```

-   Dark causality status

```{r}
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.

```{r}
dataset <- DJS[, -1]

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

```{r echo=FALSE}
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.

```{r eval=FALSE}
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.

```{r}
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.