Plotting - Statistical Tests

The main library for plotting is matplotlib, which uses the Matlab plotting capabilities.

We can also use the seaborn library on top of that to do visually nicer plots

import pandas as pd
from datetime import datetime #For handling dates
import os

import matplotlib.pyplot as plt #main plotting tool for python
import matplotlib as mpl

import seaborn as sns #A more fancy plotting library

#For presenting plots inline
%matplotlib inline
!pip install tiingo

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os.environ["TIINGO_API_KEY"] = "614c1590a592cc6696f6082f83b2666cd83882ef"

from tiingo import TiingoClient
client = TiingoClient({'api_key':'614c1590a592cc6696f6082f83b2666cd83882ef'})
start = datetime(2018,1,1)
end = datetime(2018,12,31)
stocks_data = client.get_dataframe('META',frequency='daily',startDate=start,endDate=end)
stocks_data = stocks_data[['open','close','low','high','volume']]

stocks_data.head()

	open	close	low	high	vol
date
2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891
2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495
2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357
2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388
2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216

df = stocks_data
df = df.rename(columns = {'volume':'vol'})

df['profit'] = (df.close - df.open)
for idx, row in df.iterrows():
    if row.close < row.open:
        df.loc[idx,'gain']='negative'
    elif (row.close - row.open) < 1:
        df.loc[idx,'gain']='small_gain'
    elif (row.close - row.open) < 3:
        df.loc[idx,'gain']='medium_gain'
    else:
        df.loc[idx,'gain']='large_gain'

for idx, row in df.iterrows():
    if row.vol < df.vol.mean():
        df.loc[idx,'size']='small'
    else:
        df.loc[idx,'size']='large'

df.head()

	open	close	low	high	vol	profit	gain	size
date
2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891	3.74	large_gain	small
2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495	2.79	medium_gain	small
2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357	-0.57	negative	small
2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388	1.26	medium_gain	small
2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216	1.08	medium_gain	small

gain_groups = df.groupby('gain')
gdf= df[['open','low','high','close','vol','gain']].groupby('gain').mean()
gdf = gdf.reset_index()

gdf

	gain	open	low	high	close	vol
0	large_gain	170.111111	169.610833	175.313056	174.653889	3.044808e+07
1	medium_gain	172.305769	171.410962	175.321346	174.185577	2.774962e+07
2	negative	171.605492	168.137747	172.566230	169.380246	2.731642e+07
3	small_gain	171.218049	169.827317	173.070488	171.699268	2.476326e+07

Simple plots

The documentation for the plot function for data frames can be found here: https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.plot.html

#plot a column of the dataframe against the index
df.high.plot()

<Axes: xlabel='date'>

df.high.plot()
df.low.plot(label='low values')
plt.legend(loc='best') #puts the ledgent in the best possible position

<matplotlib.legend.Legend at 0x794387df1cf0>

Histograms

#histogram for the values of a dataframe column
df.close.hist(bins=20)

<Axes: >

#histogram with the kernel density estimation (a smoothed function over the histogram)
sns.histplot(df.close,bins=20,kde=True)

<Axes: xlabel='close', ylabel='Count'>

sns.displot(df.close,bins=50,kde=True)

<seaborn.axisgrid.FacetGrid at 0x7943812a4910>

Plotting columns against each other

dff = pd.read_csv('example-functions.csv')
dfs = dff.sort_values(by='A', ascending = True) #Sorting in data frames
dfs

	A	B	C	D
0	0.631930	0.727462	1.015797	0.668071
1	1.301488	0.553052	0.416444	0.841456
2	2.495414	0.293590	0.160380	0.739603
3	3.449249	0.243811	0.078789	0.554745
4	3.655596	0.258616	0.073452	0.649052
...	...	...	...	...
95	94.612046	0.010506	0.000112	0.000078
96	94.796924	0.010510	0.000111	0.000076
97	95.209703	0.010449	0.000110	0.000073
98	95.933252	0.010419	0.000109	0.000068
99	99.169216	0.010055	0.000102	0.000049

100 rows × 4 columns

Plot columns B,C,D against A

The plt.figure() command creates a new figure for each plot

plt.figure()
dfs.plot(x = 'A', y = 'B')
plt.figure()
dfs.plot(x = 'A', y = 'C')
plt.figure()
dfs.plot(x = 'A', y = 'D')

<Axes: xlabel='A'>

<Figure size 640x480 with 0 Axes>

<Figure size 640x480 with 0 Axes>

<Figure size 640x480 with 0 Axes>

Grid of plots

Use a grid to put all the plots together using the subplots functionality

#plt.figure()
fig, ax = plt.subplots(1, 3,figsize=(20,5))
dfs.plot(x = 'A', y = 'B',ax = ax[0])
dfs.plot(x = 'A', y = 'C',ax = ax[1])
dfs.plot(x = 'A', y = 'D',ax = ax[2])

<Axes: xlabel='A'>

Plot all colums together against A.

Clearly they are different functions

plt.figure()
dfs.plot(x = 'A', y = ['B','C','D'])

<Axes: xlabel='A'>

<Figure size 640x480 with 0 Axes>

Plot all columns against A in log-log scale (take the logarithm for the values in both axes)

We observe straight lines for B,C while steeper drop for D. The B and C are a polynomial function of A

plt.figure(); dfs.plot(x = 'A', y = ['B','C','D'], loglog=True);

<Figure size 640x480 with 0 Axes>

Plot with log scale only on y-axis (log-linear plot).

The plot of D becomes a line, indicating that D is an exponential function of A

plt.figure()
dfs.plot(x = 'A', y = ['B','C','D'], logy=True)

<Axes: xlabel='A'>

<Figure size 640x480 with 0 Axes>

Plotting using matplotlib

Also how to put two figures in a 1x2 grid

plt.figure(figsize = (15,5)) #defines the size of figure
plt.subplot(121) #plot with 1 row, 2 columns, 1st plot
plt.plot(dfs['A'],dfs['B'],'bo-',dfs['A'],dfs['C'],'g*-',dfs['A'],dfs['D'],'rs-')
plt.subplot(122)  #plot with 1 row, 2 columns, 2nd plot
plt.loglog(dfs['A'],dfs['B'],'bo-',dfs['A'],dfs['C'],'g*-',dfs['A'],dfs['D'],'rs-')

[<matplotlib.lines.Line2D at 0x794380b9af20>,
 <matplotlib.lines.Line2D at 0x794380b9aef0>,
 <matplotlib.lines.Line2D at 0x794380b9b070>]

Using seaborn

sns.lineplot(x= 'A', y='B',data = dfs,marker='o')

<Axes: xlabel='A', ylabel='B'>

Scatter plots

Scatter plots take as imput two series X and Y and plot the points (x,y).

We will do the same plots as before as scatter plots using the dataframe functions

fig, ax = plt.subplots(1, 2, figsize=(15,5))
dff.plot(kind ='scatter', x='A', y='B', ax = ax[0])
dff.plot(kind ='scatter', x='A', y='B', loglog = True,ax = ax[1])

<Axes: xlabel='A', ylabel='B'>

plt.scatter(dff.A, dff.B)

<matplotlib.collections.PathCollection at 0x794381379b10>

fig = plt.figure()
ax = plt.gca()
ax.set_xscale('log')
ax.set_yscale('log')
plt.scatter([1,2,3],[3,2,1])

<matplotlib.collections.PathCollection at 0x79438135cc10>

Putting many scatter plots into the same plot

t = dff.plot(kind='scatter', x='A', y='B', color='DarkBlue', label='B curve', loglog=True);
dff.plot(kind='scatter', x='A', y='C',color='DarkGreen', label='C curve', ax=t, loglog = True);
dff.plot(kind='scatter', x='A', y='D',color='Red', label='D curve', ax=t, loglog = True);

Using seaborn

sns.scatterplot(x='A',y='B', data = dff)

<Axes: xlabel='A', ylabel='B'>

In log-log scale (for some reason it seems to throw away small values)

splot = sns.scatterplot(x='A',y='B', data = dff)
#splot.set(xscale="log", yscale="log")
splot.loglog()

[]

Statistical Significance

Recall the dataframe we obtained when grouping by gain

gdf

	gain	open	low	high	close	vol
0	large_gain	170.111111	169.610833	175.313056	174.653889	3.044808e+07
1	medium_gain	172.305769	171.410962	175.321346	174.185577	2.774962e+07
2	negative	171.605492	168.137747	172.566230	169.380246	2.731642e+07
3	small_gain	171.218049	169.827317	173.070488	171.699268	2.476326e+07

We see that there are differences in the volume of trading depending on the gain. But are these differences statistically significant? We can test that using the Student t-test. The Student t-test will give us a value for the differnece between the means in units of standard error, and a p-value that says how important this difference is. Usually we require the p-value to be less than 0.05 (or 0.01 if we want to be more strict). Note that for the test we will need to use all the values in the group.

To compute the t-test we will use the SciPy library, a Python library for scientific computing.

The Student t-test

import scipy as sp #library for scientific computations
from scipy import stats #The statistics part of the library

The t-test value is:

$$t = \frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}} $$

where $\bar x_i$ is the mean value of the $i$ dataset, $\sigma_i^2$ is the variance, and $n_i$ is the size.

#Test statistical significance of the difference in the mean volume numbers

sm = gain_groups.get_group('small_gain').vol
lg = gain_groups.get_group('large_gain').vol
med = gain_groups.get_group('medium_gain').vol
neg = gain_groups.get_group('negative').vol
print(stats.ttest_ind(sm,neg,equal_var = False))
print(stats.ttest_ind(sm,med, equal_var = False))
print(stats.ttest_ind(sm,lg, equal_var = False))
print(stats.ttest_ind(neg,med,equal_var = False))
print(stats.ttest_ind(neg,lg,equal_var = False))
print(stats.ttest_ind(med,lg, equal_var = False))

TtestResult(statistic=-0.7237664320493662, pvalue=0.4720843830832102, df=58.686780477477306)
TtestResult(statistic=-0.6532701783697626, pvalue=0.5152447681117652, df=90.15236633446038)
TtestResult(statistic=-1.2743420856982142, pvalue=0.20648370530531482, df=74.86473858994728)
TtestResult(statistic=-0.12034041075217132, pvalue=0.9045425277099464, df=73.34821305978315)
TtestResult(statistic=-0.9054964354181412, pvalue=0.36935442852925515, df=52.30908445486685)
TtestResult(statistic=-0.5972302166465407, pvalue=0.5519534365894707, df=84.35529071251472)

Kolomogorov-Smirnov Test

Test if the data for small and large gain come from the same distribution. The p-value > 0.1 inidcates that we cannot reject the null hypothesis that they do come from the same distribution.

Use scipy.stats.ks_2samp for testing two samples if they come form the same distribution.

If you want to test a single sample against a fixed distribution (e.g., normal) use the scipy.stats.kstest

import numpy as np
stats.ks_2samp(np.array(sm), np.array(lg), alternative='two-sided')

KstestResult(statistic=0.2703252032520325, pvalue=0.09473208642418271, statistic_location=26266081, statistic_sign=1)

stats.kstest(np.array(sm), 'norm')

KstestResult(statistic=1.0, pvalue=0.0, statistic_location=10464528, statistic_sign=-1)

sns.histplot(sm,bins=40,kde=True)

<Axes: xlabel='vol', ylabel='Count'>

$\chi^2$-test

We use the $\chi^2$-test to test if two random variables are independent. The larger the value of the test the farther from independence. The p-value tells us whether the value is statistically significant.

df

	open	close	low	high	vol	profit	gain	size
date
2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891	3.74	large_gain	small
2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495	2.79	medium_gain	small
2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357	-0.57	negative	small
2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388	1.26	medium_gain	small
2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216	1.08	medium_gain	small
...	...	...	...	...	...	...	...	...
2018-12-24 00:00:00+00:00	123.10	124.06	123.02	129.74	22066002	0.96	small_gain	small
2018-12-26 00:00:00+00:00	126.00	134.18	125.89	134.24	39723370	8.18	large_gain	large
2018-12-27 00:00:00+00:00	132.44	134.52	129.67	134.99	31202509	2.08	medium_gain	large
2018-12-28 00:00:00+00:00	135.34	133.20	132.20	135.92	22627569	-2.14	negative	small
2018-12-31 00:00:00+00:00	134.45	131.09	129.95	134.64	24625308	-3.36	negative	small

251 rows × 8 columns

# The crosstab methond creates the contigency table for the two attributes.
cdf = pd.crosstab(df['gain'],df['size'])
cdf

size	large	small
gain
large_gain	15	21
medium_gain	12	40
negative	39	83
small_gain	6	35

We will use the chi2_contigency function which compares the contigency table against the expected values as produced by the marginal distributions (see also the chisquare function which assumes uniform marginals).

The chi2_contigency returns:

1. The chi2 test statistic
2. The p-value
3. The degrees of freedom
4. The table with the expected counts <\ol>

stats.chi2_contingency(cdf)

Chi2ContingencyResult(statistic=8.364478767414871, pvalue=0.03905004813922891, dof=3, expected_freq=array([[10.32669323, 25.67330677],
       [14.91633466, 37.08366534],
       [34.99601594, 87.00398406],
       [11.76095618, 29.23904382]]))

We got p-value = 0.03905 < 0.05. Why do I get now Correlation between trading volume and gain ? Let's make a new contingency table and see what happens:

new_df = df.copy()

# Calculate the quantiles for dividing the `vol` column into five categories
quantiles = new_df['vol'].quantile([0.2, 0.4, 0.6, 0.8]).values

# Define categories based on quantile ranges
for idx, row in new_df.iterrows():
    if row.vol < quantiles[0]:
        new_df.loc[idx, 'size'] = 'very small'
    elif row.vol < quantiles[1]:
        new_df.loc[idx, 'size'] = 'small'
    elif row.vol < quantiles[2]:
        new_df.loc[idx, 'size'] = 'medium'
    elif row.vol < quantiles[3]:
        new_df.loc[idx, 'size'] = 'large'
    else:
        new_df.loc[idx, 'size'] = 'very large'

new_df.head()

	open	close	low	high	vol	profit	gain	size
date
2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891	3.74	large_gain	small
2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495	2.79	medium_gain	very small
2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357	-0.57	negative	very small
2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388	1.26	medium_gain	very small
2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216	1.08	medium_gain	very small

cdf = pd.crosstab(new_df['gain'],new_df['size'])
cdf

size	large	medium	small	very large	very small
gain
large_gain	5	3	11	12	5
medium_gain	13	13	9	7	10
negative	27	23	21	27	24
small_gain	5	11	9	5	11

stats.chi2_contingency(cdf)

Chi2ContingencyResult(statistic=17.156150315137896, pvalue=0.14381727155690543, dof=12, expected_freq=array([[ 7.17131474,  7.17131474,  7.17131474,  7.31474104,  7.17131474],
       [10.35856574, 10.35856574, 10.35856574, 10.56573705, 10.35856574],
       [24.30278884, 24.30278884, 24.30278884, 24.78884462, 24.30278884],
       [ 8.16733068,  8.16733068,  8.16733068,  8.33067729,  8.16733068]]))

The p-value now increased!

Exact fisher test

Another statistical test used to determine if there is a significant association between two categorical variables. Unlike the chi-square test, which approximates the association using large-sample assumptions, Fisher’s Exact Test calculates the exact probability, making it particularly suitable for small sample sizes or when one or more categories have low counts. Fisher’s test is typically used for 2x2 contigency tables (two categories in each variable)

df_fisher = df.copy()
df_fisher['gain'] = df['gain'].apply(lambda x: 'positive_gain' if x in ['small_gain', 'medium_gain', 'large_gain'] else 'negative')

df_fisher

	open	close	low	high	vol	profit	gain	size
date
2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891	3.74	positive_gain	small
2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495	2.79	positive_gain	small
2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357	-0.57	negative	small
2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388	1.26	positive_gain	small
2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216	1.08	positive_gain	small
...	...	...	...	...	...	...	...	...
2018-12-24 00:00:00+00:00	123.10	124.06	123.02	129.74	22066002	0.96	positive_gain	small
2018-12-26 00:00:00+00:00	126.00	134.18	125.89	134.24	39723370	8.18	positive_gain	large
2018-12-27 00:00:00+00:00	132.44	134.52	129.67	134.99	31202509	2.08	positive_gain	large
2018-12-28 00:00:00+00:00	135.34	133.20	132.20	135.92	22627569	-2.14	negative	small
2018-12-31 00:00:00+00:00	134.45	131.09	129.95	134.64	24625308	-3.36	negative	small

251 rows × 8 columns

# Create a 2x2 contingency table
contingency_table_fisher = pd.crosstab(df_fisher['gain'], df['size'])
print("Contingency Table:")
print(contingency_table_fisher)

Contingency Table:
size           large  small
gain                       
negative          39     83
positive_gain     33     96

from scipy.stats import fisher_exact

oddsratio, p_value = fisher_exact(contingency_table_fisher)
print("\nFisher's Exact Test Results:")
print(f"Odds Ratio: {oddsratio}")
print(f"P-Value: {p_value}")

Fisher's Exact Test Results:
Odds Ratio: 1.366922234392114
P-Value: 0.26848533307073885

Now p-value is greater than 0.05 and so there is no statistically significant association between gain and size. This reinforces previous findings from the Student’s t-test and Kolmogorov-Smirnov test, which also suggested that trading volume does not correlate strongly with daily gain outcomes.

Error bars

We can compute the standard error of the mean using the stats.sem method of scipy, which can also be called from the data frame

print(sm.sem())
print(neg.sem())
print(stats.sem(med))
print(stats.sem(lg))

3192415.794566366
1500801.0916460739
3272021.3263068898
3115899.1280031265

Computing confidence intervals

#confidence interval
conf = 0.95
t = stats.t.ppf((1+conf)/2.0, len(df)-1)
low = sm.mean()-sm.sem()*t
high = sm.mean()+sm.sem()*t
print(low,  ",", high)

18475802.95317895 , 31050718.510235682

We can also visualize the mean and the standard error in a bar-plot, using the barplot function of seaborn. Note that we need to apply this to the original data. The averaging is done automatically.

#sns.barplot(x='gain',y='vol', data = df, ci=95) #for older seaborn versions
sns.barplot(x='gain',y='vol', data = df, errorbar=('ci', 95))

<Axes: xlabel='gain', ylabel='vol'>

sns.pointplot(x='gain',y='vol', data = df,join = False, errorbar=('ci', 95), capsize = 0.1)

<ipython-input-32-9cf449b89e92>:1: UserWarning: 

The `join` parameter is deprecated and will be removed in v0.15.0. You can remove the line between points with `linestyle='none'`.

  sns.pointplot(x='gain',y='vol', data = df,join = False, errorbar=('ci', 95), capsize = 0.1)

<Axes: xlabel='gain', ylabel='vol'>

Visualizing distributions

We can also visualize the distribution using a box-plot. In the box plot, the box shows the quartiles of the dataset (the part between the higher 25% and lower 25%), while the whiskers extend to show the rest of the distribution, except for points that are determined to be “outliers”. The line shows the median.

sns.boxplot(x='gain',y='vol', data = df)

<Axes: xlabel='gain', ylabel='vol'>

#Removing outliers
sns.boxplot(x='gain',y='vol', data = df, showfliers = False)

<Axes: xlabel='gain', ylabel='vol'>

We can also use a violin plot to visualize the distributions

sns.violinplot(x='gain',y='vol', data = df)

<Axes: xlabel='gain', ylabel='vol'>

df['all'] = ''
sns.violinplot(x = 'all', y='profit',hue='size', split=True, data = df)

<Axes: xlabel='all', ylabel='profit'>

sns.violinplot(x='gain',y='profit', hue='size', data = df, split=True)

<Axes: xlabel='gain', ylabel='profit'>

Seaborn lineplot

Plot the average volume over the different months

df

	date	open	close	low	high	vol	profit	gain	size	all
0	2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891	3.74	large_gain	small
1	2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495	2.79	medium_gain	small
2	2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357	-0.57	negative	small
3	2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388	1.26	medium_gain	small
4	2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216	1.08	medium_gain	small
...	...	...	...	...	...	...	...	...	...	...
246	2018-12-24 00:00:00+00:00	123.10	124.06	123.02	129.74	22066002	0.96	small_gain	small
247	2018-12-26 00:00:00+00:00	126.00	134.18	125.89	134.24	39723370	8.18	large_gain	large
248	2018-12-27 00:00:00+00:00	132.44	134.52	129.67	134.99	31202509	2.08	medium_gain	large
249	2018-12-28 00:00:00+00:00	135.34	133.20	132.20	135.92	22627569	-2.14	negative	small
250	2018-12-31 00:00:00+00:00	134.45	131.09	129.95	134.64	24625308	-3.36	negative	small

251 rows × 10 columns

df = df.reset_index()
#df.date = df.date.apply(lambda d: datetime.strptime(d, "%Y-%m-%d"))
#df.date = df.date.apply(lambda d: datetime.strptime(d, "%Y-%m-%d %H:%M:%S%z").strftime("%Y-%m-%d"))

def get_month(row):
    return row.date.month

df['month'] = df.apply(get_month,axis = 1)

df

	date	open	close	low	high	vol	profit	gain	size	all	month
0	2018-01-02 00:00:00+00:00	177.68	181.42	177.55	181.58	17694891	3.74	large_gain	small		1
1	2018-01-03 00:00:00+00:00	181.88	184.67	181.33	184.78	16595495	2.79	medium_gain	small		1
2	2018-01-04 00:00:00+00:00	184.90	184.33	184.10	186.21	13554357	-0.57	negative	small		1
3	2018-01-05 00:00:00+00:00	185.59	186.85	184.93	186.90	13042388	1.26	medium_gain	small		1
4	2018-01-08 00:00:00+00:00	187.20	188.28	186.33	188.90	14719216	1.08	medium_gain	small		1
...	...	...	...	...	...	...	...	...	...	...	...
246	2018-12-24 00:00:00+00:00	123.10	124.06	123.02	129.74	22066002	0.96	small_gain	small		12
247	2018-12-26 00:00:00+00:00	126.00	134.18	125.89	134.24	39723370	8.18	large_gain	large		12
248	2018-12-27 00:00:00+00:00	132.44	134.52	129.67	134.99	31202509	2.08	medium_gain	large		12
249	2018-12-28 00:00:00+00:00	135.34	133.20	132.20	135.92	22627569	-2.14	negative	small		12
250	2018-12-31 00:00:00+00:00	134.45	131.09	129.95	134.64	24625308	-3.36	negative	small		12

251 rows × 11 columns

#sns.lineplot(x='month', y = 'vol', data = df, ci=95)
sns.lineplot(x='month', y = 'vol', data = df, errorbar=('ci', 95))

<Axes: xlabel='month', ylabel='vol'>

df['positive_profit'] = (df.profit>0)
sns.lineplot(x='month', y = 'vol', hue='positive_profit', data = df)

<Axes: xlabel='month', ylabel='vol'>

df.drop('date',axis=1)

	open	close	low	high	vol	profit	gain	size	all	month	positive_profit
0	177.68	181.42	177.55	181.58	17694891	3.74	large_gain	small		1	True
1	181.88	184.67	181.33	184.78	16595495	2.79	medium_gain	small		1	True
2	184.90	184.33	184.10	186.21	13554357	-0.57	negative	small		1	False
3	185.59	186.85	184.93	186.90	13042388	1.26	medium_gain	small		1	True
4	187.20	188.28	186.33	188.90	14719216	1.08	medium_gain	small		1	True
...	...	...	...	...	...	...	...	...	...	...	...
246	123.10	124.06	123.02	129.74	22066002	0.96	small_gain	small		12	True
247	126.00	134.18	125.89	134.24	39723370	8.18	large_gain	large		12	True
248	132.44	134.52	129.67	134.99	31202509	2.08	medium_gain	large		12	True
249	135.34	133.20	132.20	135.92	22627569	-2.14	negative	small		12	False
250	134.45	131.09	129.95	134.64	24625308	-3.36	negative	small		12	False

251 rows × 11 columns

Comparing multiple stocks

As a last task, we will use the experience we obtained so far -- and learn some new things -- in order to compare the performance of different stocks we obtained from Yahoo finance.

from tiingo import TiingoClient
client = TiingoClient({'api_key':'614c1590a592cc6696f6082f83b2666cd83882ef'})
ticker_history = client.get_dataframe(['GOOGL', 'AAPL'],
                                      frequency='daily',
                                      metric_name='close',
                                      startDate='2017-01-01',
                                      endDate='2018-05-31')

ticker_history.head()

	GOOGL	AAPL
2017-01-03 00:00:00+00:00	808.01	116.15
2017-01-04 00:00:00+00:00	807.77	116.02
2017-01-05 00:00:00+00:00	813.02	116.61
2017-01-06 00:00:00+00:00	825.21	117.91
2017-01-09 00:00:00+00:00	827.18	118.99

stocks = ['META','GOOG','TSLA', 'MSFT','NFLX']
from tiingo import TiingoClient
client = TiingoClient({'api_key':'614c1590a592cc6696f6082f83b2666cd83882ef'})
dfmany = client.get_dataframe(stocks,
                                      frequency='daily',
                                      metric_name='close',
                                      startDate=start,
                                      endDate=end)
dfmany.head()

	META	GOOG	TSLA	MSFT	NFLX
2018-01-02 00:00:00+00:00	181.42	1065.00	320.53	85.95	201.07
2018-01-03 00:00:00+00:00	184.67	1082.48	317.25	86.35	205.05
2018-01-04 00:00:00+00:00	184.33	1086.40	314.62	87.11	205.63
2018-01-05 00:00:00+00:00	186.85	1102.23	316.58	88.19	209.99
2018-01-08 00:00:00+00:00	188.28	1106.94	336.41	88.28	212.05

dfmany.META.plot(label = 'meta')
dfmany.GOOG.plot(label = 'google')
dfmany.TSLA.plot(label = 'tesla')
dfmany.MSFT.plot(label = 'microsoft')
dfmany.NFLX.plot(label = 'netflix')
_ = plt.legend(loc='best')

Next, we will calculate returns over a period of length $T$, defined as:

$$r(t) = \frac{f(t)-f(t-T)}{f(t)} $$

The returns can be computed with a simple DataFrame method pct_change(). Note that for the first $T$ timesteps, this value is not defined (of course): Here we calculate for $T$ = 30

rets = dfmany.pct_change(30)
rets.iloc[25:35]

	META	GOOG	TSLA	MSFT	NFLX
2018-02-07 00:00:00+00:00	NaN	NaN	NaN	NaN	NaN
2018-02-08 00:00:00+00:00	NaN	NaN	NaN	NaN	NaN
2018-02-09 00:00:00+00:00	NaN	NaN	NaN	NaN	NaN
2018-02-12 00:00:00+00:00	NaN	NaN	NaN	NaN	NaN
2018-02-13 00:00:00+00:00	NaN	NaN	NaN	NaN	NaN
2018-02-14 00:00:00+00:00	-0.010473	0.004413	0.005553	0.056545	0.322922
2018-02-15 00:00:00+00:00	-0.025505	0.006504	0.053018	0.073075	0.366837
2018-02-16 00:00:00+00:00	-0.037813	0.007732	0.066334	0.056136	0.354472
2018-02-20 00:00:00+00:00	-0.058014	0.000209	0.057458	0.051366	0.326492
2018-02-21 00:00:00+00:00	-0.055078	0.003975	-0.009245	0.036362	0.325348

Now we'll plot the timeseries of the returns of the different stocks.

Notice that the NaN values are gracefully dropped by the plotting function.

rets.META.plot(label = 'meta')
rets.GOOG.plot(label = 'google')
rets.TSLA.plot(label = 'tesla')
rets.MSFT.plot(label = 'microsoft')
rets.NFLX.plot(label = 'netflix')
_ = plt.legend(loc='best')

plt.scatter(rets.TSLA, rets.GOOG)
plt.xlabel('TESLA 30-day returns')
_ = plt.ylabel('GOOGLE 30-day returns')

We can also use the seaborn library for doing the scatterplot. Note that this method returns an object which we can use to set different parameters of the plot. In the example below we use it to set the x and y labels of the plot. Read online for more options.

dfb = client.get_dataframe('META',frequency='daily', startDate=start, endDate=end)[['open','high','low','close','volume']]
dgoog = client.get_dataframe('GOOG',frequency='daily', startDate=start, endDate=end)[['open','high','low','close','volume']]

start = datetime(2018,1,1)
end = datetime(2018,12,31)

dfb = client.get_dataframe('META',frequency='daily',startDate=start,endDate=end)
dfb = dfb[['open','close','low','high','volume']]

dgoog = client.get_dataframe('GOOG',frequency='daily',startDate=start,endDate=end)
dgoog = dgoog[['open','close','low','high','volume']]


print(dfb.head())
print(dgoog.head())

                             open   close     low    high    volume
date                                                               
2018-01-02 00:00:00+00:00  177.68  181.42  177.55  181.58  17694891
2018-01-03 00:00:00+00:00  181.88  184.67  181.33  184.78  16595495
2018-01-04 00:00:00+00:00  184.90  184.33  184.10  186.21  13554357
2018-01-05 00:00:00+00:00  185.59  186.85  184.93  186.90  13042388
2018-01-08 00:00:00+00:00  187.20  188.28  186.33  188.90  14719216
                              open    close       low     high   volume
date                                                                   
2018-01-02 00:00:00+00:00  1048.34  1065.00  1045.230  1066.94  1223114
2018-01-03 00:00:00+00:00  1064.31  1082.48  1063.210  1086.29  1416093
2018-01-04 00:00:00+00:00  1088.00  1086.40  1084.002  1093.57   990510
2018-01-05 00:00:00+00:00  1094.00  1102.23  1092.000  1104.25  1210974
2018-01-08 00:00:00+00:00  1102.23  1106.94  1101.620  1111.27  1003098

def gainrow(row):
    if row.close < row.open:
        return 'negative'
    elif (row.close - row.open) < 1:
        return 'small_gain'
    elif (row.close - row.open) < 3:
        return 'medium_gain'
    else:
        return 'large_gain'

dfb['gain'] = dfb.apply(gainrow, axis = 1)
dgoog['gain'] = dgoog.apply(gainrow, axis = 1)
dfb['profit'] = dfb.close-dfb.open
dgoog['profit'] = dgoog.close-dgoog.open

#Also using seaborn
fig = sns.scatterplot(x = dfb.profit, y = dgoog.profit)
fig.set_xlabel('FB profit')
fig.set_ylabel('GOOG profit')

Text(0, 0.5, 'GOOG profit')

Get all pairwise correlations in a single plot

sns.pairplot(rets.iloc[30:])

<seaborn.axisgrid.PairGrid at 0x7f2445c8ba00>

There appears to be some (fairly strong) correlation between the movement of Google and Microsoft stocks. Let's measure this.

Correlation Coefficients

The correlation coefficient between variables $X$ and $Y$ is defined as follows:

$$\text{Corr}(X,Y) = \frac{E\left[(X-\mu_X)(Y-\mu_Y)\right]}{\sigma_X\sigma_Y}$$

Pandas provides a DataFrame method to compute the correlation coefficient of all pairs of columns: corr().

rets.corr()

	META	GOOG	TSLA	MSFT	NFLX
META	1.000000	0.598776	0.226680	0.470696	0.546996
GOOG	0.598776	1.000000	0.210441	0.790085	0.348008
TSLA	0.226680	0.210441	1.000000	-0.041910	-0.120763
MSFT	0.470696	0.790085	-0.041910	1.000000	0.489569
NFLX	0.546996	0.348008	-0.120763	0.489569	1.000000

rets.corr(method='spearman')

	META	GOOG	TSLA	MSFT	NFLX
META	1.000000	0.540949	0.271608	0.457852	0.641344
GOOG	0.540949	1.000000	0.288135	0.803731	0.382466
TSLA	0.271608	0.288135	1.000000	0.042190	-0.065939
MSFT	0.457852	0.803731	0.042190	1.000000	0.456912
NFLX	0.641344	0.382466	-0.065939	0.456912	1.000000

It takes a bit of time to examine that table and draw conclusions.

To speed that process up it helps to visualize the table using a heatmap.

_ = sns.heatmap(rets.corr(), annot=True)

Computing p-values

Use the scipy.stats library to obtain the p-values for the pearson and spearman rank correlations

print(stats.pearsonr(rets.iloc[30:].NFLX, rets.iloc[30:].TSLA))
print(stats.spearmanr(rets.iloc[30:].NFLX, rets.iloc[30:].TSLA))
print(stats.pearsonr(rets.iloc[30:].GOOG, rets.iloc[30:].META))
print(stats.spearmanr(rets.iloc[30:].GOOG, rets.iloc[30:].META))

PearsonRResult(statistic=-0.120762891951422, pvalue=0.07318882534649845)
SignificanceResult(statistic=-0.065938830644713, pvalue=0.32918605296193537)
PearsonRResult(statistic=0.5987760976044885, pvalue=6.856639483413373e-23)
SignificanceResult(statistic=0.5409485585956174, pvalue=3.3888933351952313e-18)

print(stats.pearsonr(dfb.profit, dgoog.profit))
print(stats.spearmanr(dfb.profit, dgoog.profit))

PearsonRResult(statistic=0.7546239297103, pvalue=1.817184238439732e-47)
SignificanceResult(statistic=0.724266383357056, pvalue=4.366810528478797e-42)

Matplotlib

Finally, it is important to know that the plotting performed by Pandas is just a layer on top of matplotlib (i.e., the plt package).

So Panda's plots can (and should) be replaced or improved by using additional functions from matplotlib.

For example, suppose we want to know both the returns as well as the standard deviation of the returns of a stock (i.e., its risk).

Here is visualization of the result of such an analysis, and we construct the plot using only functions from matplotlib.

_ = plt.scatter(rets.mean(), rets.std())
plt.xlabel('Expected returns')
plt.ylabel('Standard Deviation (Risk)')
for label, x, y in zip(rets.columns, rets.mean(), rets.std()):
    plt.annotate(
        label,
        xy = (x, y), xytext = (20, -20),
        textcoords = 'offset points', ha = 'right', va = 'bottom',
        bbox = dict(boxstyle = 'round,pad=0.5', fc = 'yellow', alpha = 0.5),
        arrowprops = dict(arrowstyle = '->', connectionstyle = 'arc3,rad=0'))

To understand what these functions are doing, (especially the annotate function), you will need to consult the online documentation for matplotlib. Just use Google to find it.