3.5 Factor Analysis on Financial and Economic Time Series

Factor Analysis and Principal Component Analysis on Financial and Economic Time Series

# If you're running this on Colab, make sure to install the following packages using pip.
# On you're own computer, I recommend using conda or mamba.

# !pip install pandas-datareader
# !pip install yfinance

# !conda install pandas-datareader
# !conda install yfinance

import numpy as np
import pandas as pd
from matplotlib import pyplot as plt

import yfinance as yf
import pandas_datareader as pdr
import sklearn.decomposition
import statsmodels.multivariate.pca

import config
DATA_DIR = config.DATA_DIR

Downloading macroeconomic and financial data from FRED

fred_series_long_names = {
    'BAMLH0A0HYM2': 'ICE BofA US High Yield Index Option-Adjusted Spread',
    'NASDAQCOM': 'NASDAQ Composite Index',
    'RIFSPPFAAD90NB': '90-Day AA Financial Commercial Paper Interest Rate',
    'TB3MS': '3-Month Treasury Bill Secondary Market Rate',
    'DGS10': 'Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity',
    'VIXCLS': 'CBOE Volatility Index: VIX',
}

fred_series_short_names = {
    'BAMLH0A0HYM2': 'High Yield Index OAS',
    'NASDAQCOM': 'NASDAQ',
    'RIFSPPFAAD90NB': '90-Day AA Fin CP',
    'TB3MS': '3-Month T-Bill',
    'DGS10': '10-Year Treasury',
    'VIXCLS': 'VIX',
}

start_date = pd.to_datetime('1980-01-01') 
end_date = pd.to_datetime('today') 

df = pdr.get_data_fred(fred_series_short_names.keys(), start=start_date, end=end_date)

First, an aside about reading and writing data to disk.

df.to_csv(DATA_DIR / 'fred_panel.csv')

dff = pd.read_csv(DATA_DIR / 'fred_panel.csv')

dff.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 11882 entries, 0 to 11881
Data columns (total 7 columns):
 #   Column          Non-Null Count  Dtype  
---  ------          --------------  -----  
 0   DATE            11882 non-null  object 
 1   BAMLH0A0HYM2    7206 non-null   float64
 2   NASDAQCOM       11246 non-null  float64
 3   RIFSPPFAAD90NB  6468 non-null   float64
 4   TB3MS           535 non-null    float64
 5   DGS10           11153 non-null  float64
 6   VIXCLS          8732 non-null   float64
dtypes: float64(6), object(1)
memory usage: 649.9+ KB

dff = pd.read_csv(DATA_DIR / 'fred_panel.csv', parse_dates=['DATE'])

dff.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 11882 entries, 0 to 11881
Data columns (total 7 columns):
 #   Column          Non-Null Count  Dtype         
---  ------          --------------  -----         
 0   DATE            11882 non-null  datetime64[ns]
 1   BAMLH0A0HYM2    7206 non-null   float64       
 2   NASDAQCOM       11246 non-null  float64       
 3   RIFSPPFAAD90NB  6468 non-null   float64       
 4   TB3MS           535 non-null    float64       
 5   DGS10           11153 non-null  float64       
 6   VIXCLS          8732 non-null   float64       
dtypes: datetime64[ns](1), float64(6)
memory usage: 649.9 KB

dff = dff.set_index('DATE')

df.to_parquet(DATA_DIR / 'fred_panel.parquet')

df = pd.read_parquet(DATA_DIR / 'fred_panel.parquet')

df.info()

<class 'pandas.core.frame.DataFrame'>
DatetimeIndex: 11882 entries, 1980-01-01 to 2024-08-08
Data columns (total 6 columns):
 #   Column          Non-Null Count  Dtype  
---  ------          --------------  -----  
 0   BAMLH0A0HYM2    7206 non-null   float64
 1   NASDAQCOM       11246 non-null  float64
 2   RIFSPPFAAD90NB  6468 non-null   float64
 3   TB3MS           535 non-null    float64
 4   DGS10           11153 non-null  float64
 5   VIXCLS          8732 non-null   float64
dtypes: float64(6)
memory usage: 649.8 KB

df

	BAMLH0A0HYM2	NASDAQCOM	RIFSPPFAAD90NB	TB3MS	DGS10	VIXCLS
DATE
1980-01-01	NaN	NaN	NaN	12.0	NaN	NaN
1980-01-02	NaN	148.17	NaN	NaN	10.50	NaN
1980-01-03	NaN	145.97	NaN	NaN	10.60	NaN
1980-01-04	NaN	148.02	NaN	NaN	10.66	NaN
1980-01-07	NaN	148.62	NaN	NaN	10.63	NaN
...	...	...	...	...	...	...
2024-08-02	3.72	16776.16	NaN	NaN	3.80	23.39
2024-08-05	3.93	16200.08	NaN	NaN	3.78	38.57
2024-08-06	NaN	16366.85	NaN	NaN	3.90	27.71
2024-08-07	3.57	16195.81	NaN	NaN	3.96	27.85
2024-08-08	3.48	16660.02	5.16	NaN	3.99	23.79

11882 rows × 6 columns

Cleaning Data

df = dff.rename(columns=fred_series_short_names)
df

	High Yield Index OAS	NASDAQ	90-Day AA Fin CP	3-Month T-Bill	10-Year Treasury	VIX
DATE
1980-01-01	NaN	NaN	NaN	12.0	NaN	NaN
1980-01-02	NaN	148.17	NaN	NaN	10.50	NaN
1980-01-03	NaN	145.97	NaN	NaN	10.60	NaN
1980-01-04	NaN	148.02	NaN	NaN	10.66	NaN
1980-01-07	NaN	148.62	NaN	NaN	10.63	NaN
...	...	...	...	...	...	...
2024-08-02	3.72	16776.16	NaN	NaN	3.80	23.39
2024-08-05	3.93	16200.08	NaN	NaN	3.78	38.57
2024-08-06	NaN	16366.85	NaN	NaN	3.90	27.71
2024-08-07	3.57	16195.81	NaN	NaN	3.96	27.85
2024-08-08	3.48	16660.02	5.16	NaN	3.99	23.79

11882 rows × 6 columns

Balanced panel? Mixed frequencies?

df['3-Month T-Bill'].dropna()

DATE
1980-01-01    12.00
1980-02-01    12.86
1980-03-01    15.20
1980-04-01    13.20
1980-05-01     8.58
              ...  
2024-03-01     5.24
2024-04-01     5.24
2024-05-01     5.25
2024-06-01     5.24
2024-07-01     5.20
Name: 3-Month T-Bill, Length: 535, dtype: float64

Find a daily version of this series. See here: https://fred.stlouisfed.org/categories/22

We will end up using this series: https://fred.stlouisfed.org/series/DTB3

fred_series_short_names = {
    'BAMLH0A0HYM2': 'High Yield Index OAS',
    'NASDAQCOM': 'NASDAQ',
    'RIFSPPFAAD90NB': '90-Day AA Fin CP',
    'DTB3': '3-Month T-Bill',
    'DGS10': '10-Year Treasury',
    'VIXCLS': 'VIX',
}
df = pdr.get_data_fred(fred_series_short_names.keys(), start=start_date, end=end_date)
df = df.rename(columns=fred_series_short_names)

df

	High Yield Index OAS	NASDAQ	90-Day AA Fin CP	3-Month T-Bill	10-Year Treasury	VIX
DATE
1980-01-01	NaN	NaN	NaN	NaN	NaN	NaN
1980-01-02	NaN	148.17	NaN	12.17	10.50	NaN
1980-01-03	NaN	145.97	NaN	12.10	10.60	NaN
1980-01-04	NaN	148.02	NaN	12.10	10.66	NaN
1980-01-07	NaN	148.62	NaN	11.86	10.63	NaN
...	...	...	...	...	...	...
2024-08-02	3.72	16776.16	NaN	5.05	3.80	23.39
2024-08-05	3.93	16200.08	NaN	5.09	3.78	38.57
2024-08-06	NaN	16366.85	NaN	5.09	3.90	27.71
2024-08-07	3.57	16195.81	NaN	5.09	3.96	27.85
2024-08-08	3.48	16660.02	5.16	5.09	3.99	23.79

11731 rows × 6 columns

df.dropna()

	High Yield Index OAS	NASDAQ	90-Day AA Fin CP	3-Month T-Bill	10-Year Treasury	VIX
DATE
1997-01-02	3.06	1280.70	5.35	5.05	6.54	21.14
1997-01-03	3.09	1310.68	5.35	5.04	6.52	19.13
1997-01-06	3.10	1316.40	5.34	5.05	6.54	19.89
1997-01-07	3.10	1327.73	5.33	5.02	6.57	19.35
1997-01-08	3.07	1320.35	5.31	5.02	6.60	20.24
...	...	...	...	...	...	...
2024-07-22	3.05	18007.57	5.29	5.19	4.26	14.91
2024-07-25	3.08	17181.72	5.32	5.17	4.27	18.46
2024-07-31	3.25	17599.40	5.30	5.15	4.09	16.36
2024-08-01	3.35	17194.15	5.19	5.12	3.99	18.59
2024-08-08	3.48	16660.02	5.16	5.09	3.99	23.79

6449 rows × 6 columns

Transforming and Normalizing the data

What is transformation and normalization? Are these different things?

• Why would one transform data? What is feature engineering?

• What is normalization?

What does stationarity mean? See the the following plots. Some of these variable are stationary. Other are not? Why is this a problem?

df.plot()

<Axes: xlabel='DATE'>

df.info()

<class 'pandas.core.frame.DataFrame'>
DatetimeIndex: 11731 entries, 1980-01-01 to 2024-08-08
Data columns (total 6 columns):
 #   Column                Non-Null Count  Dtype  
---  ------                --------------  -----  
 0   High Yield Index OAS  7206 non-null   float64
 1   NASDAQ                11246 non-null  float64
 2   90-Day AA Fin CP      6468 non-null   float64
 3   3-Month T-Bill        11153 non-null  float64
 4   10-Year Treasury      11153 non-null  float64
 5   VIX                   8732 non-null   float64
dtypes: float64(6)
memory usage: 641.5 KB

df.drop(columns=['NASDAQ']).plot()

<Axes: xlabel='DATE'>

Let’s try some transformations like those used in the OFR Financial Stress Index: https://www.financialresearch.gov/financial-stress-index/files/indicators/index.html

dfn = pd.DataFrame().reindex_like(df)
dfn

	High Yield Index OAS	NASDAQ	90-Day AA Fin CP	3-Month T-Bill	10-Year Treasury	VIX
DATE
1980-01-01	NaN	NaN	NaN	NaN	NaN	NaN
1980-01-02	NaN	NaN	NaN	NaN	NaN	NaN
1980-01-03	NaN	NaN	NaN	NaN	NaN	NaN
1980-01-04	NaN	NaN	NaN	NaN	NaN	NaN
1980-01-07	NaN	NaN	NaN	NaN	NaN	NaN
...	...	...	...	...	...	...
2024-08-02	NaN	NaN	NaN	NaN	NaN	NaN
2024-08-05	NaN	NaN	NaN	NaN	NaN	NaN
2024-08-06	NaN	NaN	NaN	NaN	NaN	NaN
2024-08-07	NaN	NaN	NaN	NaN	NaN	NaN
2024-08-08	NaN	NaN	NaN	NaN	NaN	NaN

11731 rows × 6 columns

df['NASDAQ'].rolling(250).mean()

DATE
1980-01-01   NaN
1980-01-02   NaN
1980-01-03   NaN
1980-01-04   NaN
1980-01-07   NaN
              ..
2024-08-02   NaN
2024-08-05   NaN
2024-08-06   NaN
2024-08-07   NaN
2024-08-08   NaN
Name: NASDAQ, Length: 11731, dtype: float64

df = df.dropna()

df['NASDAQ'].rolling(250).mean()

DATE
1997-01-02            NaN
1997-01-03            NaN
1997-01-06            NaN
1997-01-07            NaN
1997-01-08            NaN
                 ...     
2024-07-22    13564.74956
2024-07-25    13590.21472
2024-07-31    13615.90668
2024-08-01    13640.08872
2024-08-08    13662.43556
Name: NASDAQ, Length: 6449, dtype: float64

# 'High Yield Index OAS': Leave as is
dfn['High Yield Index OAS'] = df['High Yield Index OAS']
dfn['CP - Treasury Spread, 3m'] = df['90-Day AA Fin CP'] - df['3-Month T-Bill']
# 'NASDAQ':  # We're using something different, but still apply rolling mean transformation
dfn['NASDAQ'] = np.log(df['NASDAQ']) - np.log(df['NASDAQ'].rolling(250).mean())
dfn['10-Year Treasury'] = df['10-Year Treasury'] - df['10-Year Treasury'].rolling(250).mean()
# 'VIX': Leave as is
dfn['VIX'] = df['VIX']

dfn = dfn.drop(columns=['90-Day AA Fin CP', '3-Month T-Bill'])
dfn = dfn.dropna()

dfn.info()

<class 'pandas.core.frame.DataFrame'>
DatetimeIndex: 6200 entries, 1998-01-05 to 2024-08-08
Data columns (total 5 columns):
 #   Column                    Non-Null Count  Dtype  
---  ------                    --------------  -----  
 0   High Yield Index OAS      6200 non-null   float64
 1   NASDAQ                    6200 non-null   float64
 2   10-Year Treasury          6200 non-null   float64
 3   VIX                       6200 non-null   float64
 4   CP - Treasury Spread, 3m  6200 non-null   float64
dtypes: float64(5)
memory usage: 290.6 KB

We finished with our transformations. Now, let’s normalize. First, why is it important?

dfn.plot()

<Axes: xlabel='DATE'>

Now, normalize each column, $\( z = \frac{x - \bar x}{\text{std}(x)} \)$

dfn = (dfn - dfn.mean()) / dfn.std()

dfn.plot()

<Axes: xlabel='DATE'>

def pca(dfn, module='scikitlearn'):
    if module == 'statsmodels':
        _pc1, _loadings, projection, rsquare, _, _, _ = statsmodels.multivariate.pca.pca(dfn,
            ncomp=1, standardize=True, demean=True, normalize=True, gls=False,
            weights=None, method='svd')
        _loadings = _loadings['comp_0']
        loadings = np.std(_pc1) * _loadings
        pc1 = _pc1 / np.std(_pc1)
        pc1 = pc1.rename(columns={'comp_0':'PC1'})['PC1']

    elif module == 'scikitlearn':
        pca = sklearn.decomposition.PCA(n_components=1)
        _pc1 = pd.Series(pca.fit_transform(dfn)[:,0], index=dfn.index, name='PC1')
        _loadings = pca.components_.T * np.sqrt(pca.explained_variance_)
        _loadings = pd.Series(_loadings[:,0], index=dfn.columns)

        loadings = np.std(_pc1) * _loadings
        pc1 = _pc1 / np.std(_pc1)
        pc1.name = 'PC1'
    else:
        raise ValueError



    loadings.name = "loadings"

    return pc1, loadings

def stacked_plot(df, filename=None):
    """
    df=category_contributions
    # category_contributions.sum(axis=1).plot()
    """

    df_pos = df[df >= 0]
    df_neg = df[df < 0]

    alpha = .3
    linewidth = .5

    ax = df_pos.plot.area(alpha=alpha, linewidth=linewidth, legend=False)
    pc1 = df.sum(axis=1)
    pc1.name = 'pc1'
    pc1.plot(color="Black", label='pc1', linewidth=1)


    plt.legend()
    ax.set_prop_cycle(None)
    df_neg.plot.area(alpha=alpha, ax=ax, linewidth=linewidth, legend=False, ylim=(-3,3))
    # recompute the ax.dataLim
    ax.relim()
    # update ax.viewLim using the new dataLim
    ax.autoscale()
    # ax.set_ylabel('Standard Deviations')
    # ax.set_ylim(-3,4)
    # ax.set_ylim(-30,30)

    if not (filename is None):
        filename = Path(filename)
        figure = plt.gcf() # get current figure
        figure.set_size_inches(8, 6)
        plt.savefig(filename, dpi=300)

pc1, loadings = pca(dfn, module='scikitlearn')

pc1.plot()

<Axes: xlabel='DATE'>

stacked_plot(dfn)

Let’s compare solutions from two different packages

def root_mean_squared_error(sa, sb):
    return np.sqrt(np.mean((sa - sb)**2))

pc1_sk, loadings_sk = pca(dfn, module='scikitlearn')
pc1_sm, loadings_sm = pca(dfn, module='statsmodels')
root_mean_squared_error(pc1_sm, pc1_sk)

/opt/homebrew/Caskroom/mambaforge/base/envs/finm/lib/python3.12/site-packages/numpy/core/fromnumeric.py:3643: FutureWarning: The behavior of DataFrame.std with axis=None is deprecated, in a future version this will reduce over both axes and return a scalar. To retain the old behavior, pass axis=0 (or do not pass axis)
  return std(axis=axis, dtype=dtype, out=out, ddof=ddof, **kwargs)

6.819735805196269e-16

Factor Analysis of a Panel of Stock Returns?

sample = yf.download("SPY AAPL MSFT", start="2017-01-01", end="2017-04-30")

[                       0%%                      ]

[**********************67%%******                ]  2 of 3 completed

[*********************100%%**********************]  3 of 3 completed

sample

Price	Adj Close			Close			High			Low			Open			Volume
Ticker	AAPL	MSFT	SPY	AAPL	MSFT	SPY	AAPL	MSFT	SPY	AAPL	MSFT	SPY	AAPL	MSFT	SPY	AAPL	MSFT	SPY
Date
2017-01-03	26.952709	56.930573	198.560013	29.037500	62.580002	225.240005	29.082500	62.840000	225.830002	28.690001	62.130001	223.880005	28.950001	62.790001	225.039993	115127600	20694100	91366500
2017-01-04	26.922539	56.675858	199.741302	29.004999	62.299999	226.580002	29.127501	62.750000	226.750000	28.937500	62.119999	225.610001	28.962500	62.480000	225.619995	84472400	21340000	78744400
2017-01-05	27.059452	56.675858	199.582642	29.152500	62.299999	226.399994	29.215000	62.660000	226.580002	28.952499	62.029999	225.479996	28.980000	62.189999	226.270004	88774400	24876000	78379000
2017-01-06	27.361116	57.167107	200.296692	29.477501	62.840000	227.210007	29.540001	63.150002	227.750000	29.117500	62.040001	225.899994	29.195000	62.299999	226.529999	127007600	19922900	71559900
2017-01-09	27.611732	56.985157	199.635513	29.747499	62.639999	226.460007	29.857500	63.080002	227.070007	29.485001	62.540001	226.419998	29.487499	62.759998	226.910004	134247600	20382700	46939700
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2017-04-24	33.476292	61.806141	209.986481	35.910000	67.529999	237.169998	35.987499	67.660004	237.410004	35.794998	67.099998	234.559998	35.875000	67.480003	237.179993	68537200	29770000	119209900
2017-04-25	33.683716	62.163116	211.208298	36.132500	67.919998	238.550003	36.224998	68.040001	238.949997	35.967499	67.599998	237.809998	35.977501	67.900002	237.910004	75486000	30242700	76698300
2017-04-26	33.485619	62.080715	211.075500	35.919998	67.830002	238.399994	36.150002	68.309998	239.529999	35.845001	67.620003	238.350006	36.117500	68.080002	238.509995	80164800	26190800	84702500
2017-04-27	33.511253	62.483406	211.252609	35.947498	68.269997	238.600006	36.040001	68.379997	238.949997	35.827499	67.580002	237.979996	35.980000	68.150002	238.770004	56985200	34971000	57410300
2017-04-28	33.478622	62.657337	210.792191	35.912498	68.459999	238.080002	36.075001	69.139999	238.929993	35.817501	67.690002	237.929993	36.022499	68.910004	238.899994	83441600	39548800	63532800

81 rows × 18 columns

sample['Adj Close']

Ticker	AAPL	MSFT	SPY
Date
2017-01-03	26.952709	56.930573	198.560013
2017-01-04	26.922539	56.675858	199.741302
2017-01-05	27.059452	56.675858	199.582642
2017-01-06	27.361116	57.167107	200.296692
2017-01-09	27.611732	56.985157	199.635513
...	...	...	...
2017-04-24	33.476292	61.806141	209.986481
2017-04-25	33.683716	62.163116	211.208298
2017-04-26	33.485619	62.080715	211.075500
2017-04-27	33.511253	62.483406	211.252609
2017-04-28	33.478622	62.657337	210.792191

81 rows × 3 columns

tickers = [
    'AAPL','ABBV','ABT','ACN','ADP','ADSK','AES','AET','AFL','AMAT','AMGN','AMZN','APA',
    'APHA','APD','APTV','ARE','ASML','ATVI','AXP','BA','BAC','BAX','BDX','BIIB','BK',
    'BKNG','BMY','BRKB','BRK.A','COG','COST','CPB','CRM','CSCO','CVS','DAL','DD','DHR',
    'DIS','DOW','DUK','EMR','EPD','EQT','ESRT','EXPD','FFIV','FLS','FLT','FRT','GE',
    'GILD','GOOGL','GOOG','GS','HAL','HD','HON','IBM','INTC','IP','JNJ','JPM','KEY',
    'KHC','KIM','KO','LLY','LMT','LOW','MCD','MCHP','MDT','MMM','MO','MRK','MSFT',
    'MTD','NEE','NFLX','NKE','NOV','ORCL','OXY','PEP','PFE','PG','RTN','RTX','SBUX',
    'SHW','SLB','SO','SPG','STT','T','TGT','TXN','UNH','UPS','USB','UTX','V','VZ',
    'WMT','XOM',
]

" ".join(tickers)

'AAPL ABBV ABT ACN ADP ADSK AES AET AFL AMAT AMGN AMZN APA APHA APD APTV ARE ASML ATVI AXP BA BAC BAX BDX BIIB BK BKNG BMY BRKB BRK.A COG COST CPB CRM CSCO CVS DAL DD DHR DIS DOW DUK EMR EPD EQT ESRT EXPD FFIV FLS FLT FRT GE GILD GOOGL GOOG GS HAL HD HON IBM INTC IP JNJ JPM KEY KHC KIM KO LLY LMT LOW MCD MCHP MDT MMM MO MRK MSFT MTD NEE NFLX NKE NOV ORCL OXY PEP PFE PG RTN RTX SBUX SHW SLB SO SPG STT T TGT TXN UNH UPS USB UTX V VZ WMT XOM'

data = yf.download(" ".join(tickers), start="1980-01-01", end="2024-08-01", progress=False)

8 Failed downloads:

['APHA', 'COG', 'ATVI', 'FLT', 'RTN', 'BRKB', 'UTX', 'BRK.A']: YFTzMissingError('$%ticker%: possibly delisted; No timezone found')

data['Adj Close']['AAPL'].plot()

<Axes: xlabel='Date'>

cols_with_many_nas = [
    "BRK.A",
    "APHA",
    "UTX",
    "RTN",
    "COG",
    "BRKB",
    "ATVI",
    "FLT",
    "DOW",
    "KHC",
    "V",
    "APTV",
    "ABBV",
    "ESRT",
]
df = data['Adj Close']
df = df.drop(columns=cols_with_many_nas).dropna().pct_change().dropna()

df.columns

Index(['AAPL', 'ABT', 'ACN', 'ADP', 'ADSK', 'AES', 'AET', 'AFL', 'AMAT',
       'AMGN', 'AMZN', 'APA', 'APD', 'ARE', 'ASML', 'AXP', 'BA', 'BAC', 'BAX',
       'BDX', 'BIIB', 'BK', 'BKNG', 'BMY', 'COST', 'CPB', 'CRM', 'CSCO', 'CVS',
       'DAL', 'DD', 'DHR', 'DIS', 'DUK', 'EMR', 'EPD', 'EQT', 'EXPD', 'FFIV',
       'FLS', 'FRT', 'GE', 'GILD', 'GOOG', 'GOOGL', 'GS', 'HAL', 'HD', 'HON',
       'IBM', 'INTC', 'IP', 'JNJ', 'JPM', 'KEY', 'KIM', 'KO', 'LLY', 'LMT',
       'LOW', 'MCD', 'MCHP', 'MDT', 'MMM', 'MO', 'MRK', 'MSFT', 'MTD', 'NEE',
       'NFLX', 'NKE', 'NOV', 'ORCL', 'OXY', 'PEP', 'PFE', 'PG', 'RTX', 'SBUX',
       'SHW', 'SLB', 'SO', 'SPG', 'STT', 'T', 'TGT', 'TXN', 'UNH', 'UPS',
       'USB', 'VZ', 'WMT', 'XOM'],
      dtype='object', name='Ticker')

df['AAPL'].plot()

<Axes: xlabel='Date'>

pc1, loadings = pca(df, module='scikitlearn')

pc1.plot()

<Axes: xlabel='Date'>

pc1.cumsum().plot()

<Axes: xlabel='Date'>