# CS-109A Introduction to Data Science

## Lecture 10 Additional Code: Multiple and Polynomial Regression (September 26, 2019 version)¶

Harvard University
Fall 2020
Instructors: Pavlos Protopapas, Kevin Rader, and Chris Tanner
Lab Instructor: Chris Tanner and Eleni Kaxiras
Authors: Rahul Dave, David Sondak, Will Claybaugh, Pavlos Protopapas, Chris Tanner

In [7]:
## RUN THIS CELL TO GET THE RIGHT FORMATTING
import requests
from IPython.core.display import HTML
styles = requests.get("https://raw.githubusercontent.com/Harvard-IACS/2018-CS109A/master/content/styles/cs109.css").text
HTML(styles)

Out[7]:

1. Learning Goals / Tip of the Week / Terminology
2. Training/Validation/Testing Splits (slides + interactive warm-up)
3. Polynomial Regression, and Revisiting the Cab Data
4. Multiple regression and exploring the Football data
5. A nice trick for forward-backwards

## Learning Goals¶

After this lab, you should be able to

• Explain the difference between train/validation/test data and WHY we have each.
• Implement cross-validation on a dataset
• Implement arbitrary multiple regression models in both SK-learn and Statsmodels.
• Interpret the coefficent estimates produced by each model, including transformed and dummy variables
In [8]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

import statsmodels.api as sm
from statsmodels.api import OLS

from sklearn import preprocessing
from sklearn.preprocessing import PolynomialFeatures
from sklearn.metrics import r2_score
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler

from pandas.plotting import scatter_matrix

import seaborn as sns

%matplotlib inline


## Extra Tip of the Week¶

Within your terminal (aka console aka command prompt), most shell environments support useful shortcuts:

• press the [up arrow] to navigate through your most recent commands
• press [CTRL + A] to go to the beginning of the line
• press [CTRL + E] to go to the end of the line
• press [CTRL + K] to clear the line
• type history to see the last commands you've run

## Terminology¶

Say we have input features $X$, which via some function $f()$, approximates outputs $Y$. That is, $Y = f(X) + \epsilon$ (where $\epsilon$ represents our unmeasurable variation (i.e., irreducible error).

• Inference: estimates the function $f$, but the goal isn't to make predictions for $Y$; rather, it is more concerned with understanding the relationship between $X$ and $Y$.
• Prediction: estimates the function $f$ with the goal of making accurate $Y$ predictions for some unseen $X$.

We have recently used two highly popular, useful libraries, statsmodels and sklearn.

statsmodels is mostly focused on the inference task. It aims to make good estimates for $f()$ (via solving for our $\beta$'s), and it provides expansive details about its certainty. It provides lots of tools to discuss confidence, but isn't great at dealing with test sets.

sklearn is mostly focused on the prediction task. It aims to make a well-fit line to our input data $X$, so as to make good $Y$ predictions for some unseen inputs $X$. It provides a shallower analysis of our variables. In other words, sklearn is great at test sets and validations, but it can't really discuss uncertainty in the parameters or predictions.

• R-squared: An interpretable summary of how well the model did. 1 is perfect, 0 is a trivial baseline model based on the mean $y$ value, and negative is worse than the trivial model.
• F-statistic: A value testing whether we're likely to see these results (or even stronger ones) if none of the predictors actually mattered.
• Prob (F-statistic): The probability that we'd see these results (or even stronger ones) if none of the predictors actually mattered. If this probability is small then either A) some combination of predictors actually matters or B) something rather unlikely has happened
• coef: The estimate of each beta. This has several sub-components:
• std err: The amount we'd expect this value to wiggle if we re-did the data collection and re-ran our model. More data tends to make this wiggle smaller, but sometimes the collected data just isn't enough to pin down a particular value.
• t and P>|t|: similar to the F-statistic, these measure the probability of seeing coefficients this big (or even bigger) if the given variable didn't actually matter. Small probability doesn't necessarily mean the value matters
• [0.025 0.975]: Endpoints of the 95% confidence interval. This is a interval drawn in a clever way and which gives an idea of where the true beta value might plausibly live. (If you want to understand why "there's a 95% chance the true beta is in the interval" is wrong, start a chat with Will : )

## Part 2: Polynomial Regression, and Revisiting the Cab Data¶

Polynomial regression uses a linear model to estimate a non-linear function (i.e., a function with polynomial terms). For example:

$y = \beta_0 + \beta_1x_i + \beta_1x_i^{2}$

It is a linear model because we are still solving a linear equation (the linear aspect refers to the beta coefficients).

In [11]:
# read in the data, break into train and test
train_data, test_data = train_test_split(cab_df, test_size=.2, random_state=42)

Out[11]:
TimeMin PickupCount
0 860.0 33.0
1 17.0 75.0
2 486.0 13.0
3 300.0 5.0
4 385.0 10.0
In [12]:
cab_df.shape

Out[12]:
(1250, 2)
In [13]:
# do some data cleaning
X_train = train_data['TimeMin'].values.reshape(-1,1)/60 # transforms it to being hour-based
y_train = train_data['PickupCount'].values

X_test = test_data['TimeMin'].values.reshape(-1,1)/60 # hour-based
y_test = test_data['PickupCount'].values

def plot_cabs(cur_model, poly_transformer=None):

# build the x values for the prediction line
x_vals = np.arange(0,24,.1).reshape(-1,1)

# optionally use the passed-in transformer
if poly_transformer != None:
dm = poly_transformer.fit_transform(x_vals)
else:
dm = x_vals

# make the prediction at each x value
prediction = cur_model.predict(dm)

# plot the prediction line, and the test data
plt.plot(x_vals,prediction, color='k', label="Prediction")
plt.scatter(X_test, y_test, label="Test Data")

plt.ylabel("Number of Taxi Pickups")
plt.xlabel("Time of Day (Hours Past Midnight)")
plt.legend()
plt.show()

In [14]:
from sklearn.linear_model import LinearRegression
fitted_cab_model0 = LinearRegression().fit(X_train, y_train)
plot_cabs(fitted_cab_model0)

In [15]:
fitted_cab_model0.score(X_test, y_test)

Out[15]:
0.240661535615741
Exercise

Questions:

1. The above code uses sklearn. As more practice, and to help you stay versed in both libraries, perform the same task (fit a linear regression line) using statsmodels and report the $r^2$ score. Is it the same value as what sklearn reports, and is this the expected behavior?
In [16]:
### SOLUTION:

# augment the data with a column vector of 1's

# fit the model on the training data
OLSModel = OLS(train_data['PickupCount'].values, train_data_augmented).fit()

# get the prediction results
ols_predicted_pickups_test = OLSModel.predict(test_data_augmented)
r2_score_test = r2_score(test_data[['PickupCount']].values, ols_predicted_pickups_test)
print(r2_score_test)

0.240661535615741


We can see that there's still a lot of variation in cab pickups that's not being captured by a linear fit. Further, the linear fit is predicting massively more pickups at 11:59pm than at 12:00am. This is a bad property, and it's the conseqeuence of having a straight line with a non-zero slope. However, we can add columns to our data for $TimeMin^2$ and $TimeMin^3$ and so on, allowing a curvy polynomial line to hopefully fit the data better.

We'll be using sklearn's PolynomialFeatures() function to take some of the tedium out of building the expanded input data. In fact, if all we want is a formula like $y \approx \beta_0 + \beta_1 x + \beta_2 x^2 + ...$, it will directly return a new copy of the data in this format!

In [17]:
transformer_3 = PolynomialFeatures(3, include_bias=False)
expanded_train = transformer_3.fit_transform(X_train) # TRANSFORMS it to polynomial features
pd.DataFrame(expanded_train).describe() # notice that the columns now contain x, x^2, x^3 values

Out[17]:
0 1 2
count 1000.000000 1000.000000 1000.000000
mean 11.717217 182.833724 3234.000239
std 6.751751 167.225711 3801.801966
min 0.066667 0.004444 0.000296
25% 6.100000 37.210833 226.996222
50% 11.375000 129.390694 1471.820729
75% 17.437500 304.066458 5302.160684
max 23.966667 574.401111 13766.479963

A few notes on PolynomialFeatures:

• The interface is a bit strange. PolynomialFeatures is a 'transformer' in sklearn. We'll be using several transformers that learn a transformation on the training data, and then we will apply those transformations on future data. With PolynomialFeatures, the .fit() is pretty trivial, and we often fit and transform in one command, as seen above with .fit_transform().
• You rarely want to include_bias (a column of all 1's), since sklearn will add it automatically. Remember, when using statsmodels, you can just .add_constant() right before you fit the data.
• If you want polynomial features for a several different variables (i.e., multinomial regression), you should call .fit_transform() separately on each column and append all the results to a copy of the data (unless you also want interaction terms between the newly-created features). See np.concatenate() for joining arrays.
In [18]:
fitted_cab_model3 = LinearRegression().fit(expanded_train, y_train)
print("fitting expanded_train:", expanded_train)
plot_cabs(fitted_cab_model3, transformer_3)

fitting expanded_train: [[6.73333333e+00 4.53377778e+01 3.05274370e+02]
[2.18333333e+00 4.76694444e+00 1.04078287e+01]
[1.41666667e+00 2.00694444e+00 2.84317130e+00]
...
[1.96666667e+01 3.86777778e+02 7.60662963e+03]
[1.17333333e+01 1.37671111e+02 1.61534104e+03]
[1.42000000e+01 2.01640000e+02 2.86328800e+03]]

Exercise

Questions:

1. Calculate the polynomial model's $R^2$ performance on the test set.
2. Does the polynomial model improve on the purely linear model?
3. Make a residual plot for the polynomial model. What does this plot tell us about the model?
In [19]:
# ANSWER 1
expanded_test = transformer_3.fit_transform(X_test)
print("Test R-squared:", fitted_cab_model3.score(expanded_test, y_test))
# NOTE 1: unlike statsmodels' r2_score() function, sklearn has a .score() function
# NOTE 2: fit_transform() is a nifty function that transforms the data, then fits it

Test R-squared: 0.3341251257077902

In [20]:
# ANSWER 2: yes it does.

In [21]:
# ANSWER 3 (class discussion about the residuals)
x_matrix = transformer_3.fit_transform(X_train)

prediction = fitted_cab_model3.predict(x_matrix)
residual = y_train - prediction
plt.scatter(X_train, residual, label="Residual")
plt.axhline(0, color='k')

plt.title("Residuals for the Cubic Model")
plt.ylabel("Residual Number of Taxi Pickups")
plt.xlabel("Time of Day (Hours Past Midnight)")
plt.legend()

Out[21]:

#### Other features¶

Polynomial features are not the only constucted features that help fit the data. Because these data have a 24 hour cycle, we may want to build features that follow such a cycle. For example, $sin(24\frac{x}{2\pi})$, $sin(12\frac{x}{2\pi})$, $sin(8\frac{x}{2\pi})$. Other feature transformations are appropriate to other types of data. For instance certain feature transformations have been developed for geographical data.

### Scaling Features¶

When using polynomials, we are explicitly trying to use the higher-order values for a given feature. However, sometimes these polynomial features can take on values that are drastically large, making it difficult for the system to learn an appropriate bias weight due to its large values and potentially large variance. To counter this, sometimes one may be interested in scaling the values for a given feature.

For our ongoing taxi-pickup example, using polynomial features improved our model. If we wished to scale the features, we could use sklearn's StandardScaler() function:

In [22]:
# SCALES THE EXPANDED/POLY TRANSFORMED DATA
# we don't need to convert to a pandas dataframe, but it can be useful for scaling select columns
train_copy = pd.DataFrame(expanded_train.copy())
test_copy = pd.DataFrame(expanded_test.copy())

# Fit the scaler on the training data
scaler = StandardScaler().fit(train_copy)

# Scale both the test and training data.
train_scaled = scaler.transform(expanded_train)
test_scaled = scaler.transform(expanded_test)

# we could optionally run a new regression model on this scaled data
fitted_scaled_cab = LinearRegression().fit(train_scaled, y_train)
fitted_scaled_cab.score(test_scaled, y_test)

Out[22]:
0.33412512570778274

## Part 3: Multiple regression and exploring the Football (aka soccer) data¶

Let's move on to a different dataset! The data imported below were scraped by Shubham Maurya and record various facts about players in the English Premier League. Our goal will be to fit models that predict the players' market value (what the player could earn when hired by a new team), as estimated by https://www.transfermarkt.us.

name: Name of the player
club: Club of the player
age : Age of the player
position : The usual position on the pitch
position_cat : 1 for attackers, 2 for midfielders, 3 for defenders, 4 for goalkeepers
market_value : As on www.transfermarkt.us.on July 20th, 2017
page_views : Average daily Wikipedia page views from September 1, 2016 to May 1, 2017
fpl_value : Value in Fantasy Premier League as on July 20th, 2017
fpl_sel : % of FPL players who have selected that player in their team
fpl_points : FPL points accumulated over the previous season
region: 1 for England, 2 for EU, 3 for Americas, 4 for Rest of World
nationality: Player's nationality
new_foreign: Whether a new signing from a different league, for 2017/18 (till 20th July)
age_cat: a categorical version of the Age feature
club_id: a numerical version of the Club feature
big_club: Whether one of the Top 6 clubs
new_signing: Whether a new signing for 2017/18 (till 20th July)

As always, we first import, verify, split, and explore the data.

## Part 3.1: Import and verification and grouping¶

In [29]:
league_df = pd.read_csv("league_data.txt")
print(league_df.dtypes)

# QUESTION: what would you guess is the mean age? mean salary?
league_df.head() # turns out, it's a lot

name             object
club             object
age               int64
position         object
position_cat      int64
market_value    float64
page_views        int64
fpl_value       float64
fpl_sel          object
fpl_points        int64
region          float64
nationality      object
new_foreign       int64
age_cat           int64
club_id           int64
big_club          int64
new_signing       int64
dtype: object

Out[29]:
name club age position position_cat market_value page_views fpl_value fpl_sel fpl_points region nationality new_foreign age_cat club_id big_club new_signing
0 Alexis Sanchez Arsenal 28 LW 1 65.0 4329 12.0 17.10% 264 3.0 Chile 0 4 1 1 0
1 Mesut Ozil Arsenal 28 AM 1 50.0 4395 9.5 5.60% 167 2.0 Germany 0 4 1 1 0
2 Petr Cech Arsenal 35 GK 4 7.0 1529 5.5 5.90% 134 2.0 Czech Republic 0 6 1 1 0
3 Theo Walcott Arsenal 28 RW 1 20.0 2393 7.5 1.50% 122 1.0 England 0 4 1 1 0
4 Laurent Koscielny Arsenal 31 CB 3 22.0 912 6.0 0.70% 121 2.0 France 0 4 1 1 0
In [30]:
league_df.shape

Out[30]:
(461, 17)
In [31]:
league_df.describe()

Out[31]:
age position_cat market_value page_views fpl_value fpl_points region new_foreign age_cat club_id big_club new_signing
count 461.000000 461.000000 461.000000 461.000000 461.000000 461.000000 460.000000 461.000000 461.000000 461.000000 461.000000 461.000000
mean 26.804772 2.180043 11.012039 763.776573 5.447939 57.314534 1.993478 0.034707 3.206074 10.334056 0.303688 0.145336
std 3.961892 1.000061 12.257403 931.805757 1.346695 53.113811 0.957689 0.183236 1.279795 5.726475 0.460349 0.352822
min 17.000000 1.000000 0.050000 3.000000 4.000000 0.000000 1.000000 0.000000 1.000000 1.000000 0.000000 0.000000
25% 24.000000 1.000000 3.000000 220.000000 4.500000 5.000000 1.000000 0.000000 2.000000 6.000000 0.000000 0.000000
50% 27.000000 2.000000 7.000000 460.000000 5.000000 51.000000 2.000000 0.000000 3.000000 10.000000 0.000000 0.000000
75% 30.000000 3.000000 15.000000 896.000000 5.500000 94.000000 2.000000 0.000000 4.000000 15.000000 1.000000 0.000000
max 38.000000 4.000000 75.000000 7664.000000 12.500000 264.000000 4.000000 1.000000 6.000000 20.000000 1.000000 1.000000

### (Stratified) train/test split¶

We want to make sure that the training and test data have appropriate representation of each region; it would be bad for the training data to entirely miss a region. This is especially important because some regions are rather rare.

Exercise

Questions:

1. Use the train_test_split() function, while (a) ensuring the test size is 20% of the data, and; (2) using 'stratify' argument to split the data (look up documentation online), keeping equal representation of each region. This doesn't work by default, correct? What is the issue?
2. Deal with the issue you encountered above. Hint: you may find numpy's .isnan() and panda's .dropna() functions useful!
3. How did you deal with the error generated by train_test_split? How did you justify your action?
In [32]:
### SOLUTION:
try:

# Doesn't work: a value is missing
train_data, test_data = train_test_split(league_df, test_size = 0.2, stratify=league_df['region'])
except:

# Count the missing lines and drop them
missing_rows = np.isnan(league_df['region'])
print("Uh oh, {} lines missing data! Dropping them".format(np.sum(missing_rows)))
league_df = league_df.dropna(subset=['region'])
train_data, test_data = train_test_split(league_df, test_size = 0.2, stratify=league_df['region'])

Uh oh, 1 lines missing data! Dropping them

In [33]:
train_data.shape, test_data.shape

Out[33]:
((368, 17), (92, 17))

Now that we won't be peeking at the test set, let's explore and look for patterns! We'll introduce a number of useful pandas and numpy functions along the way.

### Groupby¶

Pandas' .groupby() function is a wonderful tool for data analysis. It allows us to analyze each of several subgroups.

Many times, .groupby() is combined with .agg() to get a summary statistic for each subgroup. For instance: What is the average market value, median page views, and maximum fpl for each player position?

In [34]:
train_data.groupby('position').agg({
'market_value': np.mean,
'page_views': np.median,
'fpl_points': np.max
})

Out[34]:
market_value page_views fpl_points
position
AM 29.134615 1413.0 218
CB 8.402308 273.0 178
CF 14.799020 841.0 224
CM 8.190217 360.0 225
DM 11.716667 513.5 131
GK 6.025000 419.0 141
LB 7.833333 288.0 177
LM 4.142857 347.0 99
LW 11.427778 467.0 264
RB 7.291667 234.0 142
RM 13.125000 1042.5 105
RW 11.312500 504.0 162
SS 12.666667 1161.0 180
In [35]:
train_data.position.unique()

Out[35]:
array(['LM', 'GK', 'DM', 'RB', 'CB', 'LW', 'CF', 'RW', 'CM', 'AM', 'LB',
'RM', 'SS'], dtype=object)
In [36]:
train_data.groupby(['big_club', 'position']).agg({
'market_value': np.mean,
'page_views': np.mean,
'fpl_points': np.mean
})

Out[36]:
market_value page_views fpl_points
big_club position
0 AM 14.791667 590.333333 60.000000
CB 4.584694 280.979592 40.734694
CF 8.576389 873.055556 52.138889
CM 5.668919 345.783784 40.135135
DM 7.875000 463.850000 47.750000
GK 4.011364 379.181818 50.045455
LB 4.657895 225.684211 44.736842
LM 4.000000 356.666667 45.166667
LW 6.738095 400.238095 50.285714
RB 4.511364 277.227273 51.045455
RM 6.250000 351.000000 2.500000
RW 8.489583 587.416667 56.541667
SS 7.600000 2165.400000 58.400000
1 AM 41.428571 2590.714286 152.714286
CB 20.093750 1085.625000 69.062500
CF 29.733333 2536.533333 89.400000
CM 18.555556 1634.222222 89.444444
DM 19.400000 1414.200000 69.600000
GK 9.716667 704.333333 50.833333
LB 15.375000 875.625000 76.750000
LM 5.000000 936.000000 26.000000
LW 27.841667 2153.833333 111.166667
RB 14.937500 987.375000 81.000000
RM 20.000000 2028.000000 94.000000
RW 28.250000 1415.750000 78.000000
SS 38.000000 2196.000000 180.000000
Exercise

Question:

1. Notice that the .groupby() function above takes a list of two column names. Does the order matter? What happens if we switch the two so that 'position' is listed before 'big_club'?
In [37]:
### SOLUTION:
train_data.groupby(['position', 'big_club']).agg({
'market_value': np.mean,
'page_views': np.mean,
'fpl_points': np.mean
})

# in this case, our values are the same, as we are not aggregating anything differently;
# however, our view / grouping is merely different. visually, it often makes most sense to
# group such that the left-most (earlier) groupings have fewer distinct options than
# the ones to the right of it, but it all depends on what you're trying to discern.

Out[37]:
market_value page_views fpl_points
position big_club
AM 0 14.791667 590.333333 60.000000
1 41.428571 2590.714286 152.714286
CB 0 4.584694 280.979592 40.734694
1 20.093750 1085.625000 69.062500
CF 0 8.576389 873.055556 52.138889
1 29.733333 2536.533333 89.400000
CM 0 5.668919 345.783784 40.135135
1 18.555556 1634.222222 89.444444
DM 0 7.875000 463.850000 47.750000
1 19.400000 1414.200000 69.600000
GK 0 4.011364 379.181818 50.045455
1 9.716667 704.333333 50.833333
LB 0 4.657895 225.684211 44.736842
1 15.375000 875.625000 76.750000
LM 0 4.000000 356.666667 45.166667
1 5.000000 936.000000 26.000000
LW 0 6.738095 400.238095 50.285714
1 27.841667 2153.833333 111.166667
RB 0 4.511364 277.227273 51.045455
1 14.937500 987.375000 81.000000
RM 0 6.250000 351.000000 2.500000
1 20.000000 2028.000000 94.000000
RW 0 8.489583 587.416667 56.541667
1 28.250000 1415.750000 78.000000
SS 0 7.600000 2165.400000 58.400000
1 38.000000 2196.000000 180.000000

## Part 3.2: Linear regression on the football data¶

This section of the lab focuses on fitting a model to the football (soccer) data and interpreting the model results. The model we'll use is

$$\text{market_value} \approx \beta_0 + \beta_1\text{fpl_points} + \beta_2\text{age} + \beta_3\text{age}^2 + \beta_4log_2\left(\text{page_views}\right) + \beta_5\text{new_signing} +\beta_6\text{big_club} + \beta_7\text{position_cat}$$

We're including a 2nd degree polynomial in age because we expect pay to increase as a player gains experience, but then decrease as they continue aging. We're taking the log of page views because they have such a large, skewed range and the transformed variable will have fewer outliers that could bias the line. We choose the base of the log to be 2 just to make interpretation cleaner.

Exercise

Questions:

1. Build the data and fit this model to it. How good is the overall model?
2. Interpret the regression model. What is the meaning of the coefficient for:
• age and age$^2$
• $log_2($page_views$)$
• big_club
3. What should a player do in order to improve their market value? How many page views should a player go get to increase their market value by 10?
In [38]:
# Q1: we'll do most of it for you ...
y_train = train_data['market_value']
y_test = test_data['market_value']
def build_football_data(df):
x_matrix = df[['fpl_points','age','new_signing','big_club','position_cat']].copy()
x_matrix['log_views'] = np.log2(df['page_views'])

# CREATES THE AGE SQUARED COLUMN
x_matrix['age_squared'] = df['age']**2

# OPTIONALLY WRITE CODE to adjust the ordering of the columns, just so that it corresponds with the equation above
x_matrix = x_matrix[['fpl_points','age','age_squared','log_views','new_signing','big_club','position_cat']]

return x_matrix

# use build_football_data() to transform both the train_data and test_data
train_transformed = build_football_data(train_data)
test_transformed = build_football_data(test_data)

fitted_model_1 = OLS(endog= y_train, exog=train_transformed, hasconst=True).fit()
fitted_model_1.summary()

# WRITE CODE TO RUN r2_score(), then answer the above question about the overall goodness of the model
r2_score(y_test, fitted_model_1.predict(test_transformed))

# The model is reasonably good. We're capturing about 64%-69% of the variation in market values,
# and the test set confirms that we're not overfitting too badly.

Out[38]:
0.6513108642702654
In [39]:
# Q2: let's use the age coefficients to show the effect of age has on one's market value;
# we can get the age and age^2 coefficients via:
agecoef = fitted_model_1.params.age
age2coef = fitted_model_1.params.age_squared

# let's set our x-axis (corresponding to age) to be a wide range from -100 to 100,
# just to see a grand picture of the function
x_vals = np.linspace(-100,100,1000)
y_vals = agecoef*x_vals +age2coef*x_vals**2

# WRITE CODE TO PLOT x_vals vs y_vals
plt.plot(x_vals, y_vals)
plt.title("Effect of Age")
plt.xlabel("Age")
plt.ylabel("Contribution to Predicted Market Value")
plt.show()

# Q2A: WHAT HAPPENS IF WE USED ONLY AGE (not AGE^2) in our model (what's the r2?); make the same plot of age vs market value
# Q2B: WHAT HAPPENS IF WE USED ONLY AGE^2 (not age) in our model (what's the r2?); make the same plot of age^2 vs market value
# Q2C: PLOT page views vs market value

# SOLUTION
page_view_coef = fitted_model_1.params.log_views
x_vals = np.linspace(0,15)
y_vals = page_view_coef*x_vals
plt.plot(x_vals, y_vals)
plt.title("Effect of Page Views")
plt.xlabel("Page Views")
plt.ylabel("Contribution to Predicted Market Value")
plt.show()

# 3- Linear regression on non-experimental data can't determine causation, so we can't prove that
# a given relationship runs in the direction we might think. For instance, doing whatever it
# takes to get more page views probably doesn't meaningfully increase market value; it's likely
# the causation runs in the other direction and great players get more views. Even so, we can use
# page views to help us tell who is a great player and thus likely to be paid well.


### Part 3.3: Turning Categorical Variables into multiple binary variables¶

Of course, we have an error in how we've included player position. Even though the variable is numeric (1,2,3,4) and the model runs without issue, the value we're getting back is garbage. The interpretation, such as it is, is that there is an equal effect of moving from position category 1 to 2, from 2 to 3, and from 3 to 4, and that this effect is probably between -0.5 to -1 (depending on your run).

In reality, we don't expect moving from one position category to another to be equivalent, nor for a move from category 1 to category 3 to be twice as important as a move from category 1 to category 2. We need to introduce better features to model this variable.

We'll use pd.get_dummies to do the work for us.

In [40]:
train_design_recoded = pd.get_dummies(train_transformed, columns=['position_cat'], drop_first=True)
test_design_recoded = pd.get_dummies(test_transformed, columns=['position_cat'], drop_first=True)


Out[40]:
const fpl_points age age_squared log_views new_signing big_club position_cat_2 position_cat_3 position_cat_4
279 1.0 26 32 1024 9.870365 0 1 1 0 0
52 1.0 0 32 1024 6.686501 0 0 0 0 1
312 1.0 80 25 625 8.894818 0 0 1 0 0
57 1.0 0 36 1296 7.748193 0 0 0 0 0
151 1.0 31 27 729 7.857981 0 0 0 1 0

We've removed the original position_cat column and created three new ones.

#### Why only three new columns?¶

Why does pandas give us the option to drop the first category?

Exercise

Questions:

1. If we're fitting a model without a constant, should we have three dummy columns or four dummy columns?
2. Fit a model on the new, recoded data, then interpret the coefficient of position_cat_2.
In [41]:
### SOLUTION:
resu = OLS(y_train, train_design_recoded).fit()
resu.summary()
print("r2:", r2_score(y_test, resu.predict(test_design_recoded)))
print("position_cat_2 coef:", resu.params.position_cat_2)
train_design_recoded.shape, y_train.shape

r2: 0.6468080878333718
position_cat_2 coef: -1.7978920532876277

Out[41]:
((368, 10), (368,))

SOLUTION:

1. If our model does not have a constant, we must include all four dummy variable columns. If we drop one, we're not modeling any effect of being in that category, and effectively assuming the dropped category's effect is 0.
2. Being in position 2 (instead of position 1) has an impact between -1.54 and +2.38 on a player's market value. Since we're using an intercept, the dropped category becomes the baseline and the effect of any dummy variable is the effect of being in that category instead of the baseline category.

## Part 4: A nice trick for forward-backwards¶

XOR (operator ^) is a logical operation that only returns true when input differ. We can use it to implement forward-or-backwards selection when we want to keep track of whet predictors are "left" from a given list of predictors.

The set analog is "symmetric difference". From the python docs:

s.symmetric_difference(t) s ^ t new set with elements in either s or t but not both

In [42]:
set() ^ set([1,2,3])

Out[42]:
{1, 2, 3}
In [43]:
set([1]) ^ set([1,2,3])

Out[43]:
{2, 3}
In [44]:
set([1, 2]) ^ set([1,2,3])

Out[44]:
{3}
Exercise

Outline a step-forwards algorithm which uses this idea

SOLUTION:

Start with no predictors in a set, selected_predictors. Then the "xor" will give the set of all predictors. Go through them 1-by -1, seeing which has the highest score/ OR lowestaic/bic. Add this predictor to the selected_predictors.

Now repeat. The xor will eliminate this predictor from the remaining predictors. In the next iteration we will pick the next predictor which when combined with the first one gibes the lowest aic/bic of all 2-predictor models.

We repeat. We finally chose the best bic model from the 1 -predictor models, 2-predictor models, 3-predictor models and so on...

## BONUS EXERCISE:¶

We have provided a spreadsheet of Boston housing prices (data/boston_housing.csv). The 14 columns are as follows:

1. CRIM: per capita crime rate by town
2. ZN: proportion of residential land zoned for lots over 25,000 sq.ft.
3. INDUS: proportion of non-retail business acres per town
4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
5. NOX: nitric oxides concentration (parts per 10 million)
6. RM: average number of rooms per dwelling
7. AGE: proportion of owner-occupied units built prior to 1940
8. DIS: weighted distances to ﬁve Boston employment centers
10. TAX: full-value property-tax rate per \$10,000 11. PTRATIO: pupil-teacher ratio by town 12. B: 1000(Bk−0.63)2 where Bk is the proportion of blacks by town 13. LSTAT: % lower status of the population 14. MEDV: Median value of owner-occupied homes in$1000s We can see that the input attributes have a mixture of units

There are 450 observations.

Exercise

Using the above file, try your best to predict housing prices. (the 14th column) We have provided a test set data/boston_housing_test.csv but refrain from looking at the file or evaluating on it until you have finalized and trained a model.

1. Load in the data. It is tab-delimited. Quickly look at a summary of the data to familiarize yourself with it and ensure nothing is too egregious.
2. Use a previously-discussed function to automatically partition the data into a training and validation (aka development) set. It is up to you to choose how large these two portions should be.
3. Train a basic model on just a subset of the features. What is the performance on the validation set?
4. Train a basic model on all of the features. What is the performance on the validation set?
5. Toy with the model until you feel your results are reasonably good.
6. Perform cross-validation with said model, and measure the average performance. Are the results what you expected? Were the average results better or worse than that from your original 1 validation set?
7. Experiment with other models, and for each, perform 10-fold cross-validation. Which model yields the best average performance? Select this as your final model.
8. Use this model to evaulate your performance on the testing set. What is your performance (MSE)? Is this what you expected?
In [0]:


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