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FXデータ分析の練習（２）〜１変量データの統計量（基礎）〜

FXデータ分析の練習（３）〜多変量データの統計量（基礎）〜

python3 データ分析 pandas python scipy numpy

投稿日：2019年12月1日

このエントリーをはてなブックマークに追加
pandasはデータ分析に非常に便利なライブラリです。この記事のシリーズではpandasを使ってFXのデータを分析する練習をしてみました。

はじめに

この記事について

この記事ではFXのサンプルデータを使って多変量データの分析の基礎についてまとめます。
pandasやscipyには分析をする便利な関数が用意されていますが、計算できるものは一応自分で計算してからそれらの関数を使用しています。

この記事の前提

言語はPython3.6.3でJupyterNotebookを使って実行しています。ライブラリは以下のものを使用しています。

  • numpy==1.17.3
  • pandas==0.25.3
  • scipy==1.3.2

データの準備

サンプルデータの取得

ここでは多変量解析ということで、２つの異なるカテゴリのデータを使いたいと思います。Part1で準備したEURUSDのデータ

これに加えてGDPUSDのデータをダウンロードしてきます。リンク先の『Download Historical Data Here』からダウンロードします。

ディレクトリの構成

ダウンロードできたら作業ディレクトリ直下に解凍して以下の様なディレクトリ構成にします。

.
├── HISTDATA_COM_MT_EURUSD_M12017
│   ├── DAT_MT_EURUSD_M1_2017.csv
│   └── DAT_MT_EURUSD_M1_2017.txt
├── HISTDATA_COM_MT_EURUSD_M12018
│   ├── DAT_MT_EURUSD_M1_2018.csv
│   └── DAT_MT_EURUSD_M1_2018.txt
└── HISTDATA_COM_MT_GBPUSD_M12017
    ├── DAT_MT_GBPUSD_M1_2017.csv
    └── DAT_MT_GBPUSD_M1_2017.txt

多変量データの基礎

データの読み込み

まずはデータを読み込みます。
今回はEURUSDの2017年のデータとGBPUSDの2017年のデータを使ってみます。

%matplotlib inline
import pandas as pd
import matplotlib.pyplot as plt

data1 = pd.read_csv(
    "./HISTDATA_COM_MT_EURUSD_M12017/DAT_MT_EURUSD_M1_2017.csv",
    names=["date","time","open","high","low","close","spread"])
data2 = pd.read_csv(
    "./HISTDATA_COM_MT_NZDCAD_M12017/DAT_MT_NZDCAD_M1_2017.csv",
    names=["date","time","open","high","low","close","spread"])

共分散（covariance）

共分散（covariance）は２つの変数の関係の強さを表す指標です。共分散は以下の式で表されます。

s_{xy} = \frac {\displaystyle \sum_{i = 1}^n {(x_i - \overline{x})(y_{i} - \overline{y})}}{n-1}{ \qquad { \text{（共分散）}}} \\ \quad \\ \qquad x:変数１ \\ \qquad y:変数２ \\ \qquad n:xのデータ数(yのデータ数)

例：ある数人の身長と体重を計測したとしましょう。

無題のプレゼンテーション.png
  • Aさん【身長：165cm；体重61.3kg】
  • Bさん【身長：168cm；体重62.0kg】
  • Cさん【身長：172cm；体重64.5kg】

身長が大きい方が体重も大きい値をとりがちですね？その様なデータの場合、共分散は正の値を取ります。
例えばこの３人の場合身長の平均が168.3cm、体重の平均が62.6kgです。共分散を計算すると、

{ ( 165 - 168.3 ) × ( 61.3 - 62.6 ) + ( 168 - 168.3 ) × ( 62.0 - 62.6 ) + ( 172 - 168.3 ) × ( 64.5 - 62.6 ) } ÷ ( 3 -1 ) = 0.25

ではpythonで計算してみます。

# それぞれの『open』のデータを取得
data1_open = data1["open"]
data2_open = data2["open"]

data_len = len(data1_open)

# それぞれのデータの平均値を計算
data1_mean = data1_open.mean()
data2_mean = data2_open.mean()

# 全てのデータに対して平均との差を計算
# Noneは0に置換
data1_diff = (data1_open - data1_mean).fillna(0)
data2_diff = (data2_open - data2_mean).fillna(0)

# ２変数の積の和を計算
# Noneは0に置換
diff_sum = sum((data1_diff * data2_diff).fillna(0))

# 共分散を計算
conv_sample = diff_sum / data_len

print("共分散：{}".format(conv_sample))
出力
共分散：0.001699501944684368

共分散行列（covariance matrix）

共分散行列（covariance matrix）または分散共分散行列（variance-covariance matrix）は複数の変数において、分散と共分散の一覧を行列の形式でまとめたものです。

\textbf{Cov}= \left[ \begin{array}{rr} \sigma_{x}^{2} & Cov(x,y) \\ Cov(x,y) & \sigma_{y}^{2} \end{array} \right] \\ \quad \\ \qquad \sigma_{x}^2:xの分散 \\ \qquad \sigma_{y}^2:yの分散 \\ \qquad Cov(x,y):xとyの共分散

分散共分散行列では対角成分がそれぞれの変数の分散になり、その他の成分が共分散になるわけですね。
pandasのDataFrameやscipyではcov()メソッドでこの分散共分散行列を取得できます。

data_concated = pd.concat([data1_open,data2_open], axis=1)
data_concated.cov()
▲出力された分散共分散行列

相関（correlation）

相関（correlation）とは二つの確率変数の分布法則の関係の事を言います。例えばXとYという２つの変量（ともに測定データ）があった時を考えます。この時、Xの大きな値とYの大きな値、Xの小さな値とYの小さな値が対応している場合、この２つの変量には正の相関があるといえます。
この相関の程度は相関係数（correlation coefficient）で表すことができます。普通、単に相関係数と言った時ときにはピアソンの積率相関係数（Pearson product-moment correlation coefficient）のことを指し、以下の式であらわされます。

r_{xy} = { \frac{{\displaystyle \sum_{i = 1}^n (x_i - \overline{x}) (y_i - \overline{y})}}{\sqrt{{\displaystyle \sum_{i = 1}^n (x_i - \overline{x})^2}} \sqrt{{\displaystyle \sum_{i = 1}^n (y_i - \overline{y})^2}}} = \frac{s_{xy}}{s_xs_y}}{ \qquad { \text{（ピアソンの積率相関係数）}}} \\\quad \\ \qquad \overline{x}:x=\{x_{i}\}の算術平均 \\ \qquad \overline{y}:y=\{y_{i}\}の算術平均 \\ \qquad {s_{xy}}:共分散 \\ \qquad {s_{x}}:xの標本標準偏差 \\ \qquad {x_{y}}:yの標本標準偏差

相関係数は共分散を−１〜＋１の範囲で標準化したものだとみなすこともできます。相関係数などを使いそれぞれの変数の相関を調べることを相関分析（correlation analysis）といいます。
実際にpythonで一つずつ計算してみます。

# それぞれの『open』のデータを取得
data1_open = data1["open"]
data2_open = data2["open"]

data_len = len(data1_open)

# それぞれのデータの平均値を計算
data1_mean = data1_open.mean()
data2_mean = data2_open.mean()

# 全てのデータに対して平均との差を計算
# Noneは0に置換
data1_diff = (data1_open - data1_mean).fillna(0)
data2_diff = (data2_open - data2_mean).fillna(0)

# ２変数の積の和を計算
# Noneは0に置換
diff_sum = (data1_diff * data2_diff).fillna(0).sum()

# 差の２乗を計算
data1_diff_sq = (data1_diff * data1_diff).fillna(0)
data2_diff_sq = (data2_diff * data2_diff).fillna(0)

# 差の２乗の和の積を計算
data_diff_sum_sq = (data1_diff_sq.sum() * data2_diff_sq.sum())

# 相関係数を計算
correlation = diff_sum/np.sqrt(data_diff_sum_sq)
print("相関係数：{}".format(correlation))
出力
相関係数：0.8925594922892961

非常に強い正の相関がでました。実際にグラフを見てみます。

data1_open.plot()
data2_open.plot()
▲EURUSDのグラフ
▲GBPUSDのグラフ

片方のグラフの増加に対してもう片方のグラフも増加していることがわかります。
実際のデータでは相関係数のみで関係性が強い、無関係の判断するのは危険ですが一つの指標になりそうですね。

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FXデータ分析の練習（２）〜１変量データの統計量（基礎）〜
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