Author: Stavros Stavroglou, Athanasios Pantelous, Hui Wang

# Exploring Pattern Causality: An Intuitive Guide

#### Stavros Stavroglou, Athanasios Pantelous, Hui Wang

Pattern Causality is a novel approach for detecting and analyzing causal relationships within complex systems. It is particularly effective at:

1. Identifying hidden patterns in time series data: Uncovering underlying regularities in seemingly random data.
2. Quantifying different types of causality: Distinguishing between positive, negative, and “dark” causal influences, which are often difficult to observe directly.
3. Providing robust statistical analysis: Ensuring the reliability of causal inferences through rigorous statistical methods.

This algorithm is especially useful in the following domains:

- Financial market analysis: Understanding the interdependencies between different assets.
- Climate system interactions: Revealing the complex relationships between various factors in climate change.
- Medical diagnosis: Assisting in the analysis of disease progression patterns to identify potential causes.
- Complex system dynamics: Studying the interactions between components in various complex systems.

## Application in Financial Markets

First, we import stock data for Apple (AAPL) and Microsoft (MSFT). You can also import data using the yahooo API.

    library(patterncausality)
    data(DJS)
    head(DJS)
    #>         Date     X3M American.Express    Apple  Boeing Caterpillar  Chevron
    #> 1 2000-01-03 47.1875         45.88031 3.997768 40.1875    24.31250 41.81250
    #> 2 2000-01-04 45.3125         44.14794 3.660714 40.1250    24.00000 41.81250
    #> 3 2000-01-05 46.6250         42.96264 3.714286 42.6250    24.56250 42.56250
    #> 4 2000-01-06 50.3750         43.83794 3.392857 43.0625    25.81250 44.37500
    #> 5 2000-01-07 51.3750         44.47618 3.553571 44.3125    26.65625 45.15625
    #> 6 2000-01-10 51.1250         45.09618 3.491071 43.6875    25.78125 43.93750
    #>   Cisco.Systems Coca.Cola DowDuPont ExxonMobil General.Electric Goldman.Sachs
    #> 1      54.03125  28.18750  44.20833   39.15625         50.00000       88.3125
    #> 2      51.00000  28.21875  43.00000   38.40625         48.00000       82.7500
    #> 3      50.84375  28.46875  44.39583   40.50000         47.91667       78.8750
    #> 4      50.00000  28.50000  45.64583   42.59375         48.55727       82.2500
    #> 5      52.93750  30.37500  46.66667   42.46875         50.43750       82.5625
    #> 6      54.90625  29.40625  45.47917   41.87500         50.41667       84.3750
    #>        IBM    Intel Johnson...Johnson JPMorgan.Chase McDonald.s   Merck
    #> 1 116.0000 43.50000          46.09375       48.58333    39.6250 67.6250
    #> 2 112.0625 41.46875          44.40625       47.25000    38.8125 65.2500
    #> 3 116.0000 41.81250          44.87500       46.95833    39.4375 67.8125
    #> 4 114.0000 39.37500          46.28125       47.62500    38.8750 68.3750
    #> 5 113.5000 41.00000          48.25000       48.50000    39.8750 74.9375
    #> 6 118.0000 42.87500          47.03125       47.66667    40.0625 72.7500
    #>   Microsoft     Nike  Pfizer Procter...Gamble The.Home.Depot Travelers
    #> 1  58.28125 6.015625 31.8750         53.59375        65.1875   33.0000
    #> 2  56.31250 5.687500 30.6875         52.56250        61.7500   32.5625
    #> 3  56.90625 6.015625 31.1875         51.56250        63.0000   32.3125
    #> 4  55.00000 5.984375 32.3125         53.93750        60.0000   32.9375
    #> 5  55.71875 5.984375 34.5000         58.25000        63.5000   34.2500
    #> 6  56.12500 6.085938 34.4375         57.96875        63.1875   33.6250
    #>   United.Technologies UnitedHealth.Group  Verizon Walmart Walt.Disney
    #> 1            31.25000           6.718750 53.90316 66.8125    29.46252
    #> 2            29.96875           6.632813 52.16072 64.3125    31.18836
    #> 3            29.37500           6.617188 53.90316 63.0000    32.48274
    #> 4            30.78125           6.859375 53.28487 63.6875    31.18836
    #> 5            32.00000           7.664063 52.89142 68.5000    30.69527
    #> 6            32.31250           7.531250 52.61038 67.2500    35.37968

The `data(DJS)` function loads the Dow Jones stocks dataset, which includes daily stock prices for several companies, including Apple and Microsoft. The `head(DJS)` function displays the first few rows of the dataset.

Next, we visualize the stock prices to observe their trends more intuitively.

### Parameter Optimization

To achieve the best results with pattern causality analysis, it’s important to find the optimal parameters. The following code demonstrates how to search for these parameters using the `optimalParametersSearch` function (note that this code is not evaluated by default due to the time it may take).

    dataset <- DJS[,-1]
    parameter <- optimalParametersSearch(Emax = 5, tauMax = 5, metric = "euclidean", dataset = dataset)

### Calculating Causality

After determining the parameters, we can calculate the causality between the two series.

    X <- DJS$Apple
    Y <- DJS$Microsoft
    pc <- pcLightweight(X, Y, E = 3, tau = 2, metric = "euclidean", h = 1, weighted = TRUE)
    print(pc)
    #> Pattern Causality Analysis Results:
    #> Total: 0.2756
    #> Positive: 0.4095
    #> Negative: 0.1454
    #> Dark: 0.4451

The `pcLightweight` function performs a lightweight pattern causality analysis.

The `print(pc)` method displays the results of the causality analysis, including the total, positive, negative, and dark causality percentages.

Finally, we can visualize the results to better understand the causal relationships using the `plot_total` and `plot_components` functions.

    plot_total(pc)

    plot_components(pc)

The `plot_total` function visualizes the total causality, and `plot_components` visualizes the different components of causality (positive, negative, and dark).

### Detailed Analysis

For more detailed causality information, use the `pcFullDetails` function.

    X <- DJS$Apple
    Y <- DJS$Microsoft
    detail <- pcFullDetails(X, Y, E = 3, tau = 2, metric = "euclidean", h = 1, weighted = TRUE)
    # Access the causality components
    causality_real <- detail$causality_real
    causality_pred <- detail$causality_pred
    print(causality_pred)

This completes the entire process of the pattern causality algorithm.

### Parameter Description

The pattern causality functions accept several important parameters:

- `E`: Embedding dimension (integer > 1)
- `tau`: Time delay (integer > 0)
- `metric`: Distance metric (“euclidean”, “manhattan”, or “maximum”)
- `h`: Prediction horizon (integer >= 0)
- `weighted`: If TRUE, uses weighted causality strength calculation
- `relative`: If TRUE, calculates relative changes ((new-old)/old) instead of absolute changes (new-old) in signature space. Default is TRUE.

The combination of `weighted = TRUE` and `relative = TRUE` is particularly useful when: 1. You want to analyze percentage changes rather than absolute changes 2. The time series have different scales or units 3. You’re interested in the relative importance of changes

    # Example with both weighted and relative TRUE
    pc_rel_weighted <- pcLightweight(X, Y, E = 3, tau = 2, metric = "euclidean",
                                   h = 1, weighted = TRUE, relative = TRUE)
    print(pc_rel_weighted)
    #> Pattern Causality Analysis Results:
    #> Total: 0.2756
    #> Positive: 0.4095
    #> Negative: 0.1454
    #> Dark: 0.4451