Hi,
I am currently utilizing pysparks `GeneralizedLinearRegression`, and have a
question when I try to include a categorical:categorical interaction in my
model.
I have tried searching stackoverflow and exisitngs mail, but with no luck. Let
me know if I have missed anything 😊.
My question boils down to, if the `Interaction` functionality between
categorical variables is usable in combination with GeneralizedLinearRegression
( and generally functions using IWLS)? Since I am unsure of whether the design
matrix generated from the feature column considers collinearity.
Please see code and further comments below:
The following code utilizes the RFormula transformation to visualize the model
matrix:
```
from pyspark.ml.feature import RFormula
from pyspark.sql.functions import lit
# Create sample data with all combinations
data = [("N", "N"), ("Y", "N"), ("N", "Y"), ("Y", "Y")]
df = spark.createDataFrame(data, ["x1", "x2"])
df = df.withColumn("y", lit(1.0))
formula = RFormula(formula="y ~ x1+x2+x1:x2", featuresCol="features")
transformed = formula.fit(df).transform(df)
# Display
transformed.select("x1", "x2", "features").show(truncate=False)
``
Returning
+---+---+-------------------------+
|x1 |x2 |features |
+---+---+-------------------------+
|N |N |[1.0,1.0,1.0,0.0,0.0,0.0]|
|Y |N |(6,[1,4],[1.0,1.0]) |
|N |Y |(6,[0,3],[1.0,1.0]) |
|Y |Y |(6,[5],[1.0]) |
+---+---+-------------------------+
This will introduce a 4x6 (7 with intercept) dimensioned matrix:
Perfect Multicollinearity - The interaction columns are perfectly determined by
the main effects. For example, the (N,N) interaction is exactly where x1="N"
AND x2="N".
Rank Deficiency - The design matrix is rank deficient. For a two-factor
interaction model, we should need only 4 parameters (intercept + 3
coefficients), but PySpark creates 6.
Causing converging problems when using the IWLS algorithm.
--------------------
If we look a the same example using patsy’s dmatrix implementation:
```
import pandas as pd
from patsy import dmatrix
data = pd.DataFrame({
'x1': ['N', 'Y', 'N', 'Y'],
'x2': ['N', 'N', 'Y', 'Y'],
'y': [1.0, 1.0, 1.0, 1.0]
})
formula_matrix = dmatrix("x1 + x2 + x1:x2", data, return_type="dataframe")
print(formula_matrix)
```
Patsy formula matrix:
Intercept x1[T.Y] x2[T.Y] x1[T.Y]:x2[T.Y]
0 1.0 0.0 0.0 0.0
1 1.0 1.0 0.0 0.0
2 1.0 0.0 1.0 0.0
3 1.0 1.0 1.0 1.0
We get the correct model matrix setup.
Venlig hilsen | Best Regards
Emil Hofman | Aktuar
T +45 51 55 86 80 | M +45 51 55 86 80
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