# Examples and Exercises from Think Stats, 2nd Edition

http://thinkstats2.com

``````

In [1]:

from __future__ import print_function, division

%matplotlib inline

import warnings
warnings.filterwarnings('ignore', category=FutureWarning)

import numpy as np
import pandas as pd

import random

import thinkstats2
import thinkplot

``````

## Time series analysis

Load the data from "Price of Weed".

``````

In [2]:

``````
``````

Out[2]:

.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}

.dataframe tbody tr th {
vertical-align: top;
}

text-align: right;
}

city
state
price
amount
quality
date
ppg
state.name
lat
lon

0
Annandale
VA
100
7.075
high
2010-09-02
14.13
Virginia
38.830345
-77.213870

1
Auburn
AL
60
28.300
high
2010-09-02
2.12
Alabama
32.578185
-85.472820

2
Austin
TX
60
28.300
medium
2010-09-02
2.12
Texas
30.326374
-97.771258

3
Belleville
IL
400
28.300
high
2010-09-02
14.13
Illinois
38.532311
-89.983521

4
Boone
NC
55
3.540
high
2010-09-02
15.54
North Carolina
36.217052
-81.687983

``````

The following function takes a DataFrame of transactions and compute daily averages.

``````

In [3]:

def GroupByDay(transactions, func=np.mean):
"""Groups transactions by day and compute the daily mean ppg.

transactions: DataFrame of transactions

returns: DataFrame of daily prices
"""
grouped = transactions[['date', 'ppg']].groupby('date')
daily = grouped.aggregate(func)

daily['date'] = daily.index
start = daily.date[0]
one_year = np.timedelta64(1, 'Y')
daily['years'] = (daily.date - start) / one_year

return daily

``````

The following function returns a map from quality name to a DataFrame of daily averages.

``````

In [4]:

def GroupByQualityAndDay(transactions):
"""Divides transactions by quality and computes mean daily price.

transaction: DataFrame of transactions

returns: map from quality to time series of ppg
"""
groups = transactions.groupby('quality')
dailies = {}
for name, group in groups:
dailies[name] = GroupByDay(group)

return dailies

``````

`dailies` is the map from quality name to DataFrame.

``````

In [5]:

dailies = GroupByQualityAndDay(transactions)

``````

The following plots the daily average price for each quality.

``````

In [6]:

import matplotlib.pyplot as plt

thinkplot.PrePlot(rows=3)
for i, (name, daily) in enumerate(dailies.items()):
thinkplot.SubPlot(i+1)
title = 'Price per gram (\$)' if i == 0 else ''
thinkplot.Config(ylim=[0, 20], title=title)
thinkplot.Scatter(daily.ppg, s=10, label=name)
if i == 2:
plt.xticks(rotation=30)
thinkplot.Config()
else:
thinkplot.Config(xticks=[])

``````
``````

/home/downey/anaconda3/envs/ModSimPy/lib/python3.6/site-packages/matplotlib/cbook/deprecation.py:107: MatplotlibDeprecationWarning: Adding an axes using the same arguments as a previous axes currently reuses the earlier instance.  In a future version, a new instance will always be created and returned.  Meanwhile, this warning can be suppressed, and the future behavior ensured, by passing a unique label to each axes instance.
warnings.warn(message, mplDeprecation, stacklevel=1)

``````

We can use `statsmodels` to run a linear model of price as a function of time.

``````

In [7]:

import statsmodels.formula.api as smf

def RunLinearModel(daily):
model = smf.ols('ppg ~ years', data=daily)
results = model.fit()
return model, results

``````

Here's what the results look like.

``````

In [8]:

from IPython.display import display

for name, daily in dailies.items():
model, results = RunLinearModel(daily)
print(name)
display(results.summary())

``````
``````

high

OLS Regression Results

Dep. Variable:           ppg         R-squared:             0.444

Method:             Least Squares    F-statistic:           989.7

Date:             Thu, 24 Jan 2019   Prob (F-statistic): 3.69e-160

Time:                 09:23:26       Log-Likelihood:      -1510.1

No. Observations:        1241        AIC:                   3024.

Df Residuals:            1239        BIC:                   3035.

Df Model:                   1

Covariance Type:      nonrobust

coef     std err      t      P>|t|  [0.025    0.975]

Intercept    13.4496     0.045   296.080  0.000    13.361    13.539

years        -0.7082     0.023   -31.460  0.000    -0.752    -0.664

Omnibus:       56.254   Durbin-Watson:         1.847

Prob(Omnibus):  0.000   Jarque-Bera (JB):    128.992

Skew:           0.252   Prob(JB):           9.76e-29

Kurtosis:       4.497   Cond. No.               4.71

Warnings:[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

low

OLS Regression Results

Dep. Variable:           ppg         R-squared:             0.030

Method:             Least Squares    F-statistic:           35.90

Date:             Thu, 24 Jan 2019   Prob (F-statistic): 2.76e-09

Time:                 09:23:26       Log-Likelihood:      -3091.3

No. Observations:        1179        AIC:                   6187.

Df Residuals:            1177        BIC:                   6197.

Df Model:                   1

Covariance Type:      nonrobust

coef     std err      t      P>|t|  [0.025    0.975]

Intercept     5.3616     0.194    27.671  0.000     4.981     5.742

years         0.5683     0.095     5.991  0.000     0.382     0.754

Omnibus:       649.338   Durbin-Watson:         1.820

Prob(Omnibus):  0.000    Jarque-Bera (JB):   6347.614

Skew:           2.373    Prob(JB):               0.00

Kurtosis:      13.329    Cond. No.               4.85

Warnings:[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

medium

OLS Regression Results

Dep. Variable:           ppg         R-squared:             0.050

Method:             Least Squares    F-statistic:           64.92

Date:             Thu, 24 Jan 2019   Prob (F-statistic): 1.82e-15

Time:                 09:23:26       Log-Likelihood:      -2053.9

No. Observations:        1238        AIC:                   4112.

Df Residuals:            1236        BIC:                   4122.

Df Model:                   1

Covariance Type:      nonrobust

coef     std err      t      P>|t|  [0.025    0.975]

Intercept     8.8791     0.071   125.043  0.000     8.740     9.018

years         0.2832     0.035     8.057  0.000     0.214     0.352

Omnibus:       133.025   Durbin-Watson:         1.767

Prob(Omnibus):  0.000    Jarque-Bera (JB):    630.863

Skew:           0.385    Prob(JB):           1.02e-137

Kurtosis:       6.411    Cond. No.               4.73

Warnings:[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

``````

Now let's plot the fitted model with the data.

``````

In [9]:

def PlotFittedValues(model, results, label=''):
"""Plots original data and fitted values.

model: StatsModel model object
results: StatsModel results object
"""
years = model.exog[:,1]
values = model.endog
thinkplot.Scatter(years, values, s=15, label=label)
thinkplot.Plot(years, results.fittedvalues, label='model', color='#ff7f00')

``````

The following function plots the original data and the fitted curve.

``````

In [10]:

def PlotLinearModel(daily, name):
"""Plots a linear fit to a sequence of prices, and the residuals.

daily: DataFrame of daily prices
name: string
"""
model, results = RunLinearModel(daily)
PlotFittedValues(model, results, label=name)
thinkplot.Config(title='Fitted values',
xlabel='Years',
xlim=[-0.1, 3.8],
ylabel='Price per gram (\$)')

``````

Here are results for the high quality category:

``````

In [11]:

name = 'high'
daily = dailies[name]

PlotLinearModel(daily, name)

``````
``````

``````

## Moving averages

As a simple example, I'll show the rolling average of the numbers from 1 to 10.

``````

In [12]:

array = np.arange(10)

``````

With a "window" of size 3, we get the average of the previous 3 elements, or nan when there are fewer than 3.

``````

In [13]:

# It looks like this doesn't exist in recent versions of Pandas

# pd.rolling_mean(array, 3)

``````
``````

In [14]:

# But Series now provides `rolling`

series = pd.Series(array)
series.rolling(3).mean()

``````
``````

Out[14]:

0    NaN
1    NaN
2    1.0
3    2.0
4    3.0
5    4.0
6    5.0
7    6.0
8    7.0
9    8.0
dtype: float64

``````

The following function plots the rolling mean.

``````

In [15]:

def PlotRollingMean(daily, name):
"""Plots rolling mean.

daily: DataFrame of daily prices
"""
dates = pd.date_range(daily.index.min(), daily.index.max())
reindexed = daily.reindex(dates)

thinkplot.Scatter(reindexed.ppg, s=15, alpha=0.2, label=name)
roll_mean = reindexed.ppg.rolling(30).mean()
thinkplot.Plot(roll_mean, label='rolling mean', color='#ff7f00')
plt.xticks(rotation=30)
thinkplot.Config(ylabel='price per gram (\$)')

``````

Here's what it looks like for the high quality category.

``````

In [16]:

PlotRollingMean(daily, name)

``````
``````

``````

The exponentially-weighted moving average gives more weight to more recent points.

``````

In [17]:

def PlotEWMA(daily, name):
"""Plots rolling mean.

daily: DataFrame of daily prices
"""
dates = pd.date_range(daily.index.min(), daily.index.max())
reindexed = daily.reindex(dates)

thinkplot.Scatter(reindexed.ppg, s=15, alpha=0.2, label=name)
roll_mean = reindexed.ppg.ewm(30).mean()
thinkplot.Plot(roll_mean, label='EWMA', color='#ff7f00')
plt.xticks(rotation=30)
thinkplot.Config(ylabel='price per gram (\$)')

``````
``````

In [18]:

PlotEWMA(daily, name)

``````
``````

``````

We can use resampling to generate missing values with the right amount of noise.

``````

In [19]:

def FillMissing(daily, span=30):
"""Fills missing values with an exponentially weighted moving average.

Resulting DataFrame has new columns 'ewma' and 'resid'.

daily: DataFrame of daily prices
span: window size (sort of) passed to ewma

returns: new DataFrame of daily prices
"""
dates = pd.date_range(daily.index.min(), daily.index.max())
reindexed = daily.reindex(dates)

ewma = reindexed.ppg.ewm(span=span).mean()

resid = (reindexed.ppg - ewma).dropna()
fake_data = ewma + thinkstats2.Resample(resid, len(reindexed))
reindexed.ppg.fillna(fake_data, inplace=True)

reindexed['ewma'] = ewma
reindexed['resid'] = reindexed.ppg - ewma
return reindexed

``````
``````

In [20]:

def PlotFilled(daily, name):
"""Plots the EWMA and filled data.

daily: DataFrame of daily prices
"""
filled = FillMissing(daily, span=30)
thinkplot.Scatter(filled.ppg, s=15, alpha=0.2, label=name)
thinkplot.Plot(filled.ewma, label='EWMA', color='#ff7f00')
plt.xticks(rotation=30)
thinkplot.Config(ylabel='Price per gram (\$)')

``````

Here's what the EWMA model looks like with missing values filled.

``````

In [21]:

PlotFilled(daily, name)

``````
``````

``````

## Serial correlation

The following function computes serial correlation with the given lag.

``````

In [22]:

def SerialCorr(series, lag=1):
xs = series[lag:]
ys = series.shift(lag)[lag:]
corr = thinkstats2.Corr(xs, ys)
return corr

``````

Before computing correlations, we'll fill missing values.

``````

In [23]:

filled_dailies = {}
for name, daily in dailies.items():
filled_dailies[name] = FillMissing(daily, span=30)

``````

Here are the serial correlations for raw price data.

``````

In [24]:

for name, filled in filled_dailies.items():
corr = thinkstats2.SerialCorr(filled.ppg, lag=1)
print(name, corr)

``````
``````

high 0.49996907065740454
low 0.10347947878427055
medium 0.16480350777650926

``````

It's not surprising that there are correlations between consecutive days, because there are obvious trends in the data.

It is more interested to see whether there are still correlations after we subtract away the trends.

``````

In [25]:

for name, filled in filled_dailies.items():
corr = thinkstats2.SerialCorr(filled.resid, lag=1)
print(name, corr)

``````
``````

high -0.01114435364441681
low 0.03606481939196412
medium -0.016501348397930337

``````

Even if the correlations between consecutive days are weak, there might be correlations across intervals of one week, one month, or one year.

``````

In [26]:

rows = []
for lag in [1, 7, 30, 365]:
print(lag, end='\t')
for name, filled in filled_dailies.items():
corr = SerialCorr(filled.resid, lag)
print('%.2g' % corr, end='\t')
print()

``````
``````

1	-0.011	0.036	-0.017
7	0.0095	-0.032	-0.045
30	-0.0055	-0.012	-0.00056
365	0.056	0.016	0.016

``````

The strongest correlation is a weekly cycle in the medium quality category.

## Autocorrelation

The autocorrelation function is the serial correlation computed for all lags.

We can use it to replicate the results from the previous section.

``````

In [27]:

import statsmodels.tsa.stattools as smtsa

filled = filled_dailies['high']
acf = smtsa.acf(filled.resid, nlags=365, unbiased=True)
print('%0.2g, %.2g, %0.2g, %0.2g, %0.2g' %
(acf[0], acf[1], acf[7], acf[30], acf[365]))

``````
``````

1, -0.011, 0.0094, -0.0055, 0.054

``````

To get a sense of how much autocorrelation we should expect by chance, we can resample the data (which eliminates any actual autocorrelation) and compute the ACF.

``````

In [28]:

def SimulateAutocorrelation(daily, iters=1001, nlags=40):
"""Resample residuals, compute autocorrelation, and plot percentiles.

daily: DataFrame
iters: number of simulations to run
nlags: maximum lags to compute autocorrelation
"""
# run simulations
t = []
for _ in range(iters):
filled = FillMissing(daily, span=30)
resid = thinkstats2.Resample(filled.resid)
acf = smtsa.acf(resid, nlags=nlags, unbiased=True)[1:]
t.append(np.abs(acf))

high = thinkstats2.PercentileRows(t, [97.5])[0]
low = -high
lags = range(1, nlags+1)
thinkplot.FillBetween(lags, low, high, alpha=0.2, color='gray')

``````

The following function plots the actual autocorrelation for lags up to 40 days.

The flag `add_weekly` indicates whether we should add a simulated weekly cycle.

``````

In [29]:

"""Plots autocorrelation functions.

dailies: map from category name to DataFrame of daily prices
nlags: number of lags to compute
"""
thinkplot.PrePlot(3)
daily = dailies['high']
SimulateAutocorrelation(daily)

for name, daily in dailies.items():

filled = FillMissing(daily, span=30)

acf = smtsa.acf(filled.resid, nlags=nlags, unbiased=True)
lags = np.arange(len(acf))
thinkplot.Plot(lags[1:], acf[1:], label=name)

``````

To show what a strong weekly cycle would look like, we have the option of adding a price increase of 1-2 dollars on Friday and Saturdays.

``````

In [30]:

daily: DataFrame of daily prices

returns: new DataFrame of daily prices
"""
fri_or_sat = (daily.index.dayofweek==4) | (daily.index.dayofweek==5)
fake = daily.copy()
fake.loc[fri_or_sat, 'ppg'] += np.random.uniform(0, 2, fri_or_sat.sum())
return fake

``````

Here's what the real ACFs look like. The gray regions indicate the levels we expect by chance.

``````

In [31]:

axis = [0, 41, -0.2, 0.2]

thinkplot.Config(axis=axis,
loc='lower right',
ylabel='correlation',
xlabel='lag (day)')

``````
``````

``````

Here's what it would look like if there were a weekly cycle.

``````

In [32]:

thinkplot.Config(axis=axis,
loc='lower right',
xlabel='lag (days)')

``````
``````

``````

## Prediction

The simplest way to generate predictions is to use `statsmodels` to fit a model to the data, then use the `predict` method from the results.

``````

In [33]:

def GenerateSimplePrediction(results, years):
"""Generates a simple prediction.

results: results object
years: sequence of times (in years) to make predictions for

returns: sequence of predicted values
"""
n = len(years)
inter = np.ones(n)
d = dict(Intercept=inter, years=years, years2=years**2)
predict_df = pd.DataFrame(d)
predict = results.predict(predict_df)
return predict

``````
``````

In [34]:

def PlotSimplePrediction(results, years):
predict = GenerateSimplePrediction(results, years)

thinkplot.Scatter(daily.years, daily.ppg, alpha=0.2, label=name)
thinkplot.plot(years, predict, color='#ff7f00')
xlim = years[0]-0.1, years[-1]+0.1
thinkplot.Config(title='Predictions',
xlabel='Years',
xlim=xlim,
ylabel='Price per gram (\$)',
loc='upper right')

``````

Here's what the prediction looks like for the high quality category, using the linear model.

``````

In [35]:

name = 'high'
daily = dailies[name]

_, results = RunLinearModel(daily)
years = np.linspace(0, 5, 101)
PlotSimplePrediction(results, years)

``````
``````

``````

When we generate predictions, we want to quatify the uncertainty in the prediction. We can do that by resampling. The following function fits a model to the data, computes residuals, then resamples from the residuals to general fake datasets. It fits the same model to each fake dataset and returns a list of results.

``````

In [36]:

def SimulateResults(daily, iters=101, func=RunLinearModel):
"""Run simulations based on resampling residuals.

daily: DataFrame of daily prices
iters: number of simulations
func: function that fits a model to the data

returns: list of result objects
"""
_, results = func(daily)
fake = daily.copy()

result_seq = []
for _ in range(iters):
fake.ppg = results.fittedvalues + thinkstats2.Resample(results.resid)
_, fake_results = func(fake)
result_seq.append(fake_results)

return result_seq

``````

To generate predictions, we take the list of results fitted to resampled data. For each model, we use the `predict` method to generate predictions, and return a sequence of predictions.

If `add_resid` is true, we add resampled residuals to the predicted values, which generates predictions that include predictive uncertainty (due to random noise) as well as modeling uncertainty (due to random sampling).

``````

In [37]:

"""Generates an array of predicted values from a list of model results.

When add_resid is False, predictions represent sampling error only.

When add_resid is True, they also include residual error (which is
more relevant to prediction).

result_seq: list of model results
years: sequence of times (in years) to make predictions for

returns: sequence of predictions
"""
n = len(years)
d = dict(Intercept=np.ones(n), years=years, years2=years**2)
predict_df = pd.DataFrame(d)

predict_seq = []
for fake_results in result_seq:
predict = fake_results.predict(predict_df)
predict += thinkstats2.Resample(fake_results.resid, n)
predict_seq.append(predict)

return predict_seq

``````

To visualize predictions, I show a darker region that quantifies modeling uncertainty and a lighter region that quantifies predictive uncertainty.

``````

In [38]:

def PlotPredictions(daily, years, iters=101, percent=90, func=RunLinearModel):
"""Plots predictions.

daily: DataFrame of daily prices
years: sequence of times (in years) to make predictions for
iters: number of simulations
percent: what percentile range to show
func: function that fits a model to the data
"""
result_seq = SimulateResults(daily, iters=iters, func=func)
p = (100 - percent) / 2
percents = p, 100-p

low, high = thinkstats2.PercentileRows(predict_seq, percents)
thinkplot.FillBetween(years, low, high, alpha=0.3, color='gray')

low, high = thinkstats2.PercentileRows(predict_seq, percents)
thinkplot.FillBetween(years, low, high, alpha=0.5, color='gray')

``````

Here are the results for the high quality category.

``````

In [39]:

years = np.linspace(0, 5, 101)
thinkplot.Scatter(daily.years, daily.ppg, alpha=0.1, label=name)
PlotPredictions(daily, years)
xlim = years[0]-0.1, years[-1]+0.1
thinkplot.Config(title='Predictions',
xlabel='Years',
xlim=xlim,
ylabel='Price per gram (\$)')

``````
``````

``````

But there is one more source of uncertainty: how much past data should we use to build the model?

The following function generates a sequence of models based on different amounts of past data.

``````

In [40]:

def SimulateIntervals(daily, iters=101, func=RunLinearModel):
"""Run simulations based on different subsets of the data.

daily: DataFrame of daily prices
iters: number of simulations
func: function that fits a model to the data

returns: list of result objects
"""
result_seq = []
starts = np.linspace(0, len(daily), iters).astype(int)

for start in starts[:-2]:
subset = daily[start:]
_, results = func(subset)
fake = subset.copy()

for _ in range(iters):
fake.ppg = (results.fittedvalues +
thinkstats2.Resample(results.resid))
_, fake_results = func(fake)
result_seq.append(fake_results)

return result_seq

``````

And this function plots the results.

``````

In [41]:

def PlotIntervals(daily, years, iters=101, percent=90, func=RunLinearModel):
"""Plots predictions based on different intervals.

daily: DataFrame of daily prices
years: sequence of times (in years) to make predictions for
iters: number of simulations
percent: what percentile range to show
func: function that fits a model to the data
"""
result_seq = SimulateIntervals(daily, iters=iters, func=func)
p = (100 - percent) / 2
percents = p, 100-p

low, high = thinkstats2.PercentileRows(predict_seq, percents)
thinkplot.FillBetween(years, low, high, alpha=0.2, color='gray')

``````

Here's what the high quality category looks like if we take into account uncertainty about how much past data to use.

``````

In [42]:

name = 'high'
daily = dailies[name]

thinkplot.Scatter(daily.years, daily.ppg, alpha=0.1, label=name)
PlotIntervals(daily, years)
PlotPredictions(daily, years)
xlim = years[0]-0.1, years[-1]+0.1
thinkplot.Config(title='Predictions',
xlabel='Years',
xlim=xlim,
ylabel='Price per gram (\$)')

``````
``````

``````

## Exercises

Exercise: The linear model I used in this chapter has the obvious drawback that it is linear, and there is no reason to expect prices to change linearly over time. We can add flexibility to the model by adding a quadratic term, as we did in Section 11.3.

Use a quadratic model to fit the time series of daily prices, and use the model to generate predictions. You will have to write a version of `RunLinearModel` that runs that quadratic model, but after that you should be able to reuse code from the chapter to generate predictions.

``````

In [43]:

# Solution goes here

``````
``````

In [44]:

# Solution goes here

``````
``````

In [45]:

# Solution goes here

``````
``````

In [46]:

# Solution goes here

``````

Exercise: Write a definition for a class named `SerialCorrelationTest` that extends `HypothesisTest` from Section 9.2. It should take a series and a lag as data, compute the serial correlation of the series with the given lag, and then compute the p-value of the observed correlation.

Use this class to test whether the serial correlation in raw price data is statistically significant. Also test the residuals of the linear model and (if you did the previous exercise), the quadratic model.

``````

In [47]:

# Solution goes here

``````
``````

In [48]:

# Solution goes here

``````
``````

In [49]:

# Solution goes here

``````
``````

In [50]:

# Solution goes here

``````

Worked example: There are several ways to extend the EWMA model to generate predictions. One of the simplest is something like this:

1. Compute the EWMA of the time series and use the last point as an intercept, `inter`.

2. Compute the EWMA of differences between successive elements in the time series and use the last point as a slope, `slope`.

3. To predict values at future times, compute `inter + slope * dt`, where `dt` is the difference between the time of the prediction and the time of the last observation.

``````

In [51]:

name = 'high'
daily = dailies[name]

filled = FillMissing(daily)
diffs = filled.ppg.diff()

thinkplot.plot(diffs)
plt.xticks(rotation=30)
thinkplot.Config(ylabel='Daily change in price per gram (\$)')

``````
``````

``````
``````

In [52]:

filled['slope'] = diffs.ewm(span=365).mean()
thinkplot.plot(filled.slope[-365:])
plt.xticks(rotation=30)
thinkplot.Config(ylabel='EWMA of diff (\$)')

``````
``````

``````
``````

In [53]:

# extract the last inter and the mean of the last 30 slopes
start = filled.index[-1]
inter = filled.ewma[-1]
slope = filled.slope[-30:].mean()

start, inter, slope

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``````

Out[53]:

(Timestamp('2014-05-13 00:00:00', freq='D'),
10.92951876545549,
-0.00262422491642447)

``````
``````

In [54]:

# reindex the DataFrame, adding a year to the end
dates = pd.date_range(filled.index.min(),
filled.index.max() + np.timedelta64(365, 'D'))
predicted = filled.reindex(dates)

``````
``````

In [55]:

# generate predicted values and add them to the end
predicted['date'] = predicted.index
one_day = np.timedelta64(1, 'D')
predicted['days'] = (predicted.date - start) / one_day
predict = inter + slope * predicted.days
predicted.ewma.fillna(predict, inplace=True)

``````
``````

In [56]:

# plot the actual values and predictions
thinkplot.Scatter(daily.ppg, alpha=0.1, label=name)
thinkplot.Plot(predicted.ewma, color='#ff7f00')

``````
``````

``````

As an exercise, run this analysis again for the other quality categories.

``````

In [ ]:

``````