Plotting - Statistical Significance¶
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
import pandas_datareader.data as web # For accessing web data
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
os.environ["IEX_API_KEY"] = "pk_4f1eb9a770e04d2ebc44123e297618bb"#"pk_******************************"
stocks = 'FB'
data_source = 'iex'
start = datetime(2018,1,1)
end = datetime(2018,12,31)
stocks_data = web.DataReader(stocks, data_source, start, end)
#If you want to load only some of the attributes:
#stocks_data = web.DataReader(stocks, data_source, start, end)[['open','close']]
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'
df.head()
gain_groups = df.groupby('gain')
gdf= df[['open','low','high','close','vol','gain']].groupby('gain').mean()
gdf = gdf.reset_index()
Simple plots¶
df.high.plot()
df.low.plot(label='low values')
plt.legend(loc='best') #puts the ledgent in the best possible position
Histograms¶
df.close.hist(bins=20)
sns.distplot(df.close,bins=20)
Plotting columns against each other¶
dff = pd.read_excel('example-functions.xlsx')
dfs = dff.sort_values(by='A', ascending = True) #Sorting in data frames
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');
Use a grid to put all the plots together
#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]);
Plot all colums together against A.
Clearly they are different functions
plt.figure(); dfs.plot(x = 'A', y = ['B','C','D']);
Plot all columns against A in log scale
We observe straight lines for B,C while steeper drop for D
plt.figure(); dfs.plot(x = 'A', y = ['B','C','D'], loglog=True);
Plot with log scale only on y-axis.
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);
Plotting using matlab notation
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-')
Using seaborn
sns.lineplot(x= 'A', y='B',data = dfs,marker='o')
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])
plt.scatter(dff.A, dff.B)
plt.scatter([1,2,3],[3,2,1])
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)
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
AWe 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.
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))
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))
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)
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)
#Removing outliers
sns.boxplot(x='gain',y='vol', data = df, showfliers = False)
Plot the average volume over the different months
df = df.reset_index()
df.date = df.date.apply(lambda d: datetime.strptime(d, "%Y-%m-%d"))
def get_month(row):
return row.date.month
df['month'] = df.apply(get_month,axis = 1)
sns.lineplot(x='month', y = 'vol', data = df)
df['positive_profit'] = (df.profit>0)
sns.lineplot(x='month', y = 'vol', hue='positive_profit', data = df)
df.drop('date',axis=1)
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.
stocks = ['FB','GOOG','TSLA', 'MSFT','NFLX']
attr = 'close'
dfmany = web.DataReader(stocks,
data_source,
start=datetime(2018, 1, 1),
end=datetime(2018, 12, 31))[attr]
dfmany.head()
dfmany.FB.plot(label = 'facebook')
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):
rets = dfmany.pct_change(30)
rets.iloc[25:35]
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.FB.plot(label = 'facebook')
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.
data_source = 'iex'
start = datetime(2018,1,1)
end = datetime(2018,12,31)
dfb = web.DataReader('FB', data_source, start, end)
dgoog = web.DataReader('GOOG', data_source, start, end)
print(dfb.head())
print(dgoog.head())
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(dfb.profit, dgoog.profit)
fig.set_xlabel('FB profit')
fig.set_ylabel('GOOG profit')
Get all pairwise correlations in a single plot
sns.pairplot(rets.iloc[30:])
There appears to be some (fairly strong) correlation between the movement of TSLA and YELP stocks. Let's measure this.
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()
rets.corr(method='spearman')
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)
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:].FB))
print(stats.spearmanr(rets.iloc[30:].GOOG, rets.iloc[30:].FB))
print(stats.pearsonr(dfb.profit, dgoog.profit))
print(stats.spearmanr(dfb.profit, dgoog.profit))
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.