The goal of this first notebook is to explore multiple regression and feature engineering with existing graphlab functions.

In this notebook you will use data on house sales in King County to predict prices using multiple regression. You will:

- Use SFrames to do some feature engineering
- Use built-in graphlab functions to compute the regression weights (coefficients/parameters)
- Given the regression weights, predictors and outcome write a function to compute the Residual Sum of Squares
- Look at coefficients and interpret their meanings
- Evaluate multiple models via RSS

```
In [1]:
```import graphlab

```
In [2]:
```sales = graphlab.SFrame('kc_house_data.gl/')

```
```

```
In [3]:
```train_data,test_data = sales.random_split(.8,seed=0)

Recall we can use the following code to learn a multiple regression model predicting 'price' based on the following features: example_features = ['sqft_living', 'bedrooms', 'bathrooms'] on training data with the following code:

(Aside: We set validation_set = None to ensure that the results are always the same)

```
In [4]:
```example_features = ['sqft_living', 'bedrooms', 'bathrooms']
example_model = graphlab.linear_regression.create(train_data, target = 'price', features = example_features,
validation_set = None)

```
```

```
In [5]:
```example_weight_summary = example_model.get("coefficients")
print example_weight_summary

```
```

In the gradient descent notebook we use numpy to do our regression. In this book we will use existing graphlab create functions to analyze multiple regressions.

Recall that once a model is built we can use the .predict() function to find the predicted values for data we pass. For example using the example model above:

```
In [6]:
```example_predictions = example_model.predict(train_data)
print example_predictions[0] # should be 271789.505878

```
```

```
In [10]:
```def get_residual_sum_of_squares(model, data, outcome):
# First get the predictions
predictions = model.predict(data)
# Then compute the residuals/errors
residuals = predictions - outcome
# Then square and add them up
RSS = (residuals * residuals).sum()
return(RSS)

Test your function by computing the RSS on TEST data for the example model:

```
In [11]:
```rss_example_train = get_residual_sum_of_squares(example_model, test_data, test_data['price'])
print rss_example_train # should be 2.7376153833e+14

```
```

```
In [25]:
```from math import log

Next create the following 4 new features as column in both TEST and TRAIN data:

- bedrooms_squared = bedrooms*bedrooms
- bed_bath_rooms = bedrooms*bathrooms
- log_sqft_living = log(sqft_living)
- lat_plus_long = lat + long As an example here's the first one:

```
In [23]:
```train_data['bedrooms_squared'] = train_data['bedrooms'].apply(lambda x: x**2)
test_data['bedrooms_squared'] = test_data['bedrooms'].apply(lambda x: x**2)

```
In [27]:
```# create the remaining 3 features in both TEST and TRAIN data
train_data['bed_bath_rooms'] = train_data.apply(lambda x: x['bedrooms'] * x['bathrooms'])
train_data['log_sqft_living'] = train_data['sqft_living'].apply(lambda x: log(x))
train_data['lat_plus_long'] = train_data.apply(lambda x: x['lat'] + x['long'])
test_data['bed_bath_rooms'] = test_data.apply(lambda x: x['bedrooms'] * x['bathrooms'])
test_data['log_sqft_living'] = test_data['sqft_living'].apply(lambda x: log(x))
test_data['lat_plus_long'] = test_data.apply(lambda x: x['lat'] + x['long'])

- Squaring bedrooms will increase the separation between not many bedrooms (e.g. 1) and lots of bedrooms (e.g. 4) since 1^2 = 1 but 4^2 = 16. Consequently this feature will mostly affect houses with many bedrooms.
- bedrooms times bathrooms gives what's called an "interaction" feature. It is large when
*both*of them are large. - Taking the log of squarefeet has the effect of bringing large values closer together and spreading out small values.
- Adding latitude to longitude is totally non-sensical but we will do it anyway (you'll see why)

```
In [ ]:
```

```
In [38]:
```print test_data['bedrooms_squared'].mean()
print test_data['bed_bath_rooms'].mean()
print test_data['log_sqft_living'].mean()
print test_data['lat_plus_long'].mean()

```
```

Now we will learn the weights for three (nested) models for predicting house prices. The first model will have the fewest features the second model will add one more feature and the third will add a few more:

- Model 1: squarefeet, # bedrooms, # bathrooms, latitude & longitude
- Model 2: add bedrooms*bathrooms
- Model 3: Add log squarefeet, bedrooms squared, and the (nonsensical) latitude + longitude

```
In [31]:
```model_1_features = ['sqft_living', 'bedrooms', 'bathrooms', 'lat', 'long']
model_2_features = model_1_features + ['bed_bath_rooms']
model_3_features = model_2_features + ['bedrooms_squared', 'log_sqft_living', 'lat_plus_long']

```
In [33]:
```# Learn the three models: (don't forget to set validation_set = None)
model_1 = graphlab.linear_regression.create(train_data, target = 'price', features = model_1_features, validation_set = None)
model_2 = graphlab.linear_regression.create(train_data, target = 'price', features = model_2_features, validation_set = None)
model_3 = graphlab.linear_regression.create(train_data, target = 'price', features = model_3_features, validation_set = None)

```
```

```
In [36]:
```# Examine/extract each model's coefficients:
print 'model 1'
print model_1.get("coefficients")
print 'model 2'
print model_2.get("coefficients")
print 'model 3'
print model_3.get("coefficients")

```
```

**Quiz Question: What is the sign (positive or negative) for the coefficient/weight for 'bathrooms' in model 1?**

**Quiz Question: What is the sign (positive or negative) for the coefficient/weight for 'bathrooms' in model 2?**

Think about what this means.

```
In [41]:
```# Compute the RSS on TRAINING data for each of the three models and record the values:
print get_residual_sum_of_squares(model_1, train_data, train_data['price'])
print get_residual_sum_of_squares(model_2, train_data, train_data['price'])
print get_residual_sum_of_squares(model_3, train_data, train_data['price'])

```
```

**Quiz Question: Which model (1, 2 or 3) has lowest RSS on TRAINING Data?** Is this what you expected?

Now compute the RSS on on TEST data for each of the three models.

```
In [42]:
```# Compute the RSS on TESTING data for each of the three models and record the values:
print get_residual_sum_of_squares(model_1, test_data, test_data['price'])
print get_residual_sum_of_squares(model_2, test_data, test_data['price'])
print get_residual_sum_of_squares(model_3, test_data, test_data['price'])

```
```

**Quiz Question: Which model (1, 2 or 3) has lowest RSS on TESTING Data?** Is this what you expected?Think about the features that were added to each model from the previous.