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# Visualizing the stock market structure

This example employs several unsupervised learning techniques to extract the stock market structure from variations in historical quotes.

The quantity that we use is the daily variation in quote price: quotes that are linked tend to fluctuate in relation to each other during a day.

    # Author: Gael Varoquaux gael.varoquaux@normalesup.org
    # License: BSD 3 clause

## Retrieve the data from Internet

The data is from 2003 - 2008. This is reasonably calm: (not too long ago so that we get high-tech firms, and before the 2008 crash). This kind of historical data can be obtained from APIs like the data.nasdaq.com and alphavantage.co.

    import sys

    import numpy as np
    import pandas as pd

    symbol_dict = {
        "TOT": "Total",
        "XOM": "Exxon",
        "CVX": "Chevron",
        "COP": "ConocoPhillips",
        "VLO": "Valero Energy",
        "MSFT": "Microsoft",
        "IBM": "IBM",
        "TWX": "Time Warner",
        "CMCSA": "Comcast",
        "CVC": "Cablevision",
        "YHOO": "Yahoo",
        "DELL": "Dell",
        "HPQ": "HP",
        "AMZN": "Amazon",
        "TM": "Toyota",
        "CAJ": "Canon",
        "SNE": "Sony",
        "F": "Ford",
        "HMC": "Honda",
        "NAV": "Navistar",
        "NOC": "Northrop Grumman",
        "BA": "Boeing",
        "KO": "Coca Cola",
        "MMM": "3M",
        "MCD": "McDonald's",
        "PEP": "Pepsi",
        "K": "Kellogg",
        "UN": "Unilever",
        "MAR": "Marriott",
        "PG": "Procter Gamble",
        "CL": "Colgate-Palmolive",
        "GE": "General Electrics",
        "WFC": "Wells Fargo",
        "JPM": "JPMorgan Chase",
        "AIG": "AIG",
        "AXP": "American express",
        "BAC": "Bank of America",
        "GS": "Goldman Sachs",
        "AAPL": "Apple",
        "SAP": "SAP",
        "CSCO": "Cisco",
        "TXN": "Texas Instruments",
        "XRX": "Xerox",
        "WMT": "Wal-Mart",
        "HD": "Home Depot",
        "GSK": "GlaxoSmithKline",
        "PFE": "Pfizer",
        "SNY": "Sanofi-Aventis",
        "NVS": "Novartis",
        "KMB": "Kimberly-Clark",
        "R": "Ryder",
        "GD": "General Dynamics",
        "RTN": "Raytheon",
        "CVS": "CVS",
        "CAT": "Caterpillar",
        "DD": "DuPont de Nemours",
    }

    symbols, names = np.array(sorted(symbol_dict.items())).T

    quotes = []

    for symbol in symbols:
        print("Fetching quote history for %r" % symbol, file=sys.stderr)
        url = (
            "https://raw.githubusercontent.com/scikit-learn/examples-data/"
            "master/financial-data/{}.csv"
        )
        quotes.append(pd.read_csv(url.format(symbol)))

    close_prices = np.vstack([q["close"] for q in quotes])
    open_prices = np.vstack([q["open"] for q in quotes])

    # The daily variations of the quotes are what carry the most information
    variation = close_prices - open_prices

    Fetching quote history for 'AAPL'
    Fetching quote history for 'AIG'
    Fetching quote history for 'AMZN'
    Fetching quote history for 'AXP'
    Fetching quote history for 'BA'
    Fetching quote history for 'BAC'
    Fetching quote history for 'CAJ'
    Fetching quote history for 'CAT'
    Fetching quote history for 'CL'
    Fetching quote history for 'CMCSA'
    Fetching quote history for 'COP'
    Fetching quote history for 'CSCO'
    Fetching quote history for 'CVC'
    Fetching quote history for 'CVS'
    Fetching quote history for 'CVX'
    Fetching quote history for 'DD'
    Fetching quote history for 'DELL'
    Fetching quote history for 'F'
    Fetching quote history for 'GD'
    Fetching quote history for 'GE'
    Fetching quote history for 'GS'
    Fetching quote history for 'GSK'
    Fetching quote history for 'HD'
    Fetching quote history for 'HMC'
    Fetching quote history for 'HPQ'
    Fetching quote history for 'IBM'
    Fetching quote history for 'JPM'
    Fetching quote history for 'K'
    Fetching quote history for 'KMB'
    Fetching quote history for 'KO'
    Fetching quote history for 'MAR'
    Fetching quote history for 'MCD'
    Fetching quote history for 'MMM'
    Fetching quote history for 'MSFT'
    Fetching quote history for 'NAV'
    Fetching quote history for 'NOC'
    Fetching quote history for 'NVS'
    Fetching quote history for 'PEP'
    Fetching quote history for 'PFE'
    Fetching quote history for 'PG'
    Fetching quote history for 'R'
    Fetching quote history for 'RTN'
    Fetching quote history for 'SAP'
    Fetching quote history for 'SNE'
    Fetching quote history for 'SNY'
    Fetching quote history for 'TM'
    Fetching quote history for 'TOT'
    Fetching quote history for 'TWX'
    Fetching quote history for 'TXN'
    Fetching quote history for 'UN'
    Fetching quote history for 'VLO'
    Fetching quote history for 'WFC'
    Fetching quote history for 'WMT'
    Fetching quote history for 'XOM'
    Fetching quote history for 'XRX'
    Fetching quote history for 'YHOO'

## Learning a graph structure

We use sparse inverse covariance estimation to find which quotes are correlated conditionally on the others. Specifically, sparse inverse covariance gives us a graph, that is a list of connections. For each symbol, the symbols that it is connected to are those useful to explain its fluctuations.

    from sklearn import covariance

    alphas = np.logspace(-1.5, 1, num=10)
    edge_model = covariance.GraphicalLassoCV(alphas=alphas)

    # standardize the time series: using correlations rather than covariance
    # former is more efficient for structure recovery
    X = variation.copy().T
    X /= X.std(axis=0)
    edge_model.fit(X)

    GraphicalLassoCV(alphas=array([ 0.03162278,  0.05994843,  0.11364637,  0.21544347,  0.40842387,
            0.77426368,  1.46779927,  2.7825594 ,  5.27499706, 10.        ]))

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
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GraphicalLassoCV

    GraphicalLassoCV(alphas=array([ 0.03162278,  0.05994843,  0.11364637,  0.21544347,  0.40842387,
            0.77426368,  1.46779927,  2.7825594 ,  5.27499706, 10.        ]))

## Clustering using affinity propagation

We use clustering to group together quotes that behave similarly. Here, amongst the various clustering techniques available in the scikit-learn, we use Affinity Propagation as it does not enforce equal-size clusters, and it can choose automatically the number of clusters from the data.

Note that this gives us a different indication than the graph, as the graph reflects conditional relations between variables, while the clustering reflects marginal properties: variables clustered together can be considered as having a similar impact at the level of the full stock market.

    from sklearn import cluster

    _, labels = cluster.affinity_propagation(edge_model.covariance_, random_state=0)
    n_labels = labels.max()

    for i in range(n_labels + 1):
        print(f"Cluster {i + 1}: {', '.join(names[labels == i])}")

    Cluster 1: Apple, Amazon, Yahoo
    Cluster 2: Comcast, Cablevision, Time Warner
    Cluster 3: ConocoPhillips, Chevron, Total, Valero Energy, Exxon
    Cluster 4: Cisco, Dell, HP, IBM, Microsoft, SAP, Texas Instruments
    Cluster 5: Boeing, General Dynamics, Northrop Grumman, Raytheon
    Cluster 6: AIG, American express, Bank of America, Caterpillar, CVS, DuPont de Nemours, Ford, General Electrics, Goldman Sachs, Home Depot, JPMorgan Chase, Marriott, McDonald's, 3M, Ryder, Wells Fargo, Wal-Mart
    Cluster 7: GlaxoSmithKline, Novartis, Pfizer, Sanofi-Aventis, Unilever
    Cluster 8: Kellogg, Coca Cola, Pepsi
    Cluster 9: Colgate-Palmolive, Kimberly-Clark, Procter Gamble
    Cluster 10: Canon, Honda, Navistar, Sony, Toyota, Xerox

## Embedding in 2D space

For visualization purposes, we need to lay out the different symbols on a 2D canvas. For this we use Manifold learning techniques to retrieve 2D embedding. We use a dense eigen_solver to achieve reproducibility (arpack is initiated with the random vectors that we don’t control). In addition, we use a large number of neighbors to capture the large-scale structure.

    # Finding a low-dimension embedding for visualization: find the best position of
    # the nodes (the stocks) on a 2D plane

    from sklearn import manifold

    node_position_model = manifold.LocallyLinearEmbedding(
        n_components=2, eigen_solver="dense", n_neighbors=6
    )

    embedding = node_position_model.fit_transform(X.T).T

## Visualization

The output of the 3 models are combined in a 2D graph where nodes represents the stocks and edges the:

- cluster labels are used to define the color of the nodes

- the sparse covariance model is used to display the strength of the edges

- the 2D embedding is used to position the nodes in the plan

This example has a fair amount of visualization-related code, as visualization is crucial here to display the graph. One of the challenge is to position the labels minimizing overlap. For this we use an heuristic based on the direction of the nearest neighbor along each axis.

    import matplotlib.pyplot as plt
    from matplotlib.collections import LineCollection

    plt.figure(1, facecolor="w", figsize=(10, 8))
    plt.clf()
    ax = plt.axes([0.0, 0.0, 1.0, 1.0])
    plt.axis("off")

    # Plot the graph of partial correlations
    partial_correlations = edge_model.precision_.copy()
    d = 1 / np.sqrt(np.diag(partial_correlations))
    partial_correlations *= d
    partial_correlations *= d[:, np.newaxis]
    non_zero = np.abs(np.triu(partial_correlations, k=1)) > 0.02

    # Plot the nodes using the coordinates of our embedding
    plt.scatter(
        embedding[0], embedding[1], s=100 * d**2, c=labels, cmap=plt.cm.nipy_spectral
    )

    # Plot the edges
    start_idx, end_idx = np.where(non_zero)
    # a sequence of (*line0*, *line1*, *line2*), where::
    #            linen = (x0, y0), (x1, y1), ... (xm, ym)
    segments = [
        [embedding[:, start], embedding[:, stop]] for start, stop in zip(start_idx, end_idx)
    ]
    values = np.abs(partial_correlations[non_zero])
    lc = LineCollection(
        segments, zorder=0, cmap=plt.cm.hot_r, norm=plt.Normalize(0, 0.7 * values.max())
    )
    lc.set_array(values)
    lc.set_linewidths(15 * values)
    ax.add_collection(lc)

    # Add a label to each node. The challenge here is that we want to
    # position the labels to avoid overlap with other labels
    for index, (name, label, (x, y)) in enumerate(zip(names, labels, embedding.T)):
        dx = x - embedding[0]
        dx[index] = 1
        dy = y - embedding[1]
        dy[index] = 1
        this_dx = dx[np.argmin(np.abs(dy))]
        this_dy = dy[np.argmin(np.abs(dx))]
        if this_dx > 0:
            horizontalalignment = "left"
            x = x + 0.002
        else:
            horizontalalignment = "right"
            x = x - 0.002
        if this_dy > 0:
            verticalalignment = "bottom"
            y = y + 0.002
        else:
            verticalalignment = "top"
            y = y - 0.002
        plt.text(
            x,
            y,
            name,
            size=10,
            horizontalalignment=horizontalalignment,
            verticalalignment=verticalalignment,
            bbox=dict(
                facecolor="w",
                edgecolor=plt.cm.nipy_spectral(label / float(n_labels)),
                alpha=0.6,
            ),
        )

    plt.xlim(
        embedding[0].min() - 0.15 * embedding[0].ptp(),
        embedding[0].max() + 0.10 * embedding[0].ptp(),
    )
    plt.ylim(
        embedding[1].min() - 0.03 * embedding[1].ptp(),
        embedding[1].max() + 0.03 * embedding[1].ptp(),
    )

    plt.show()

Total running time of the script: (0 minutes 35.668 seconds)

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