Decomposition¶

Decomposition is the process of using an orthogonal transformation to convert a set of observations of possibly-correlated variables (with numerical values) into a set of values of linearly-uncorrelated variables called principal components.

Since some algorithms are sensitive to correlated predictors, it can be a good idea to use the PCA (Principal Component Analysis: Decomposition Technique) before applying the algorithm. Since some algorithms are also sensitive to the number of predictors, we’ll have to be picky with which variables we include.

To demonstrate data decomposition in VerticaPy, we’ll use the well-known iris dataset.

from verticapy.datasets import load_iris

iris = load_iris()
iris.head(100)

	123 SepalLengthCm Numeric(5,2) 100%	...	123 PetalWidthCm Numeric(5,2) 100%	Abc Species Varchar(30) 100%
1	4.6	...	0.2	Iris-setosa
2	4.7	...	0.2	Iris-setosa
3	4.7	...	0.2	Iris-setosa
4	4.8	...	0.1	Iris-setosa
5	4.8	...	0.2	Iris-setosa
6	4.8	...	0.2	Iris-setosa
7	4.9	...	0.2	Iris-setosa
8	4.9	...	0.1	Iris-setosa
9	4.9	...	0.1	Iris-setosa
10	4.9	...	0.1	Iris-setosa
11	5.0	...	1.0	Iris-versicolor
12	5.0	...	0.2	Iris-setosa
13	5.1	...	0.2	Iris-setosa
14	5.4	...	1.5	Iris-versicolor
15	5.4	...	0.4	Iris-setosa
16	5.4	...	0.4	Iris-setosa
17	5.5	...	1.0	Iris-versicolor
18	5.5	...	1.1	Iris-versicolor
19	5.6	...	1.3	Iris-versicolor
20	5.7	...	1.2	Iris-versicolor
21	5.7	...	0.4	Iris-setosa
22	5.8	...	2.4	Iris-virginica
23	5.9	...	1.8	Iris-versicolor
24	6.1	...	1.4	Iris-versicolor
25	6.1	...	1.8	Iris-virginica
26	6.3	...	1.5	Iris-versicolor
27	6.3	...	1.6	Iris-versicolor
28	6.3	...	2.5	Iris-virginica
29	6.4	...	1.3	Iris-versicolor
30	6.5	...	1.8	Iris-virginica
31	6.5	...	2.2	Iris-virginica
32	6.7	...	1.7	Iris-versicolor
33	6.8	...	1.4	Iris-versicolor
34	6.8	...	2.3	Iris-virginica
35	7.0	...	1.4	Iris-versicolor
36	7.1	...	2.1	Iris-virginica
37	7.7	...	2.2	Iris-virginica
38	4.4	...	0.2	Iris-setosa
39	4.5	...	0.3	Iris-setosa
40	4.8	...	0.2	Iris-setosa
41	5.0	...	1.0	Iris-versicolor
42	5.1	...	0.5	Iris-setosa
43	5.1	...	0.2	Iris-setosa
44	5.2	...	1.4	Iris-versicolor
45	5.2	...	0.2	Iris-setosa
46	5.2	...	0.1	Iris-setosa
47	5.4	...	0.4	Iris-setosa
48	5.5	...	0.2	Iris-setosa
49	5.6	...	1.3	Iris-versicolor
50	5.8	...	1.2	Iris-versicolor
51	5.8	...	1.9	Iris-virginica
52	5.8	...	1.9	Iris-virginica
53	5.9	...	1.5	Iris-versicolor
54	5.9	...	1.8	Iris-virginica
55	6.0	...	1.6	Iris-versicolor
56	6.0	...	1.5	Iris-versicolor
57	6.1	...	1.2	Iris-versicolor
58	6.2	...	1.8	Iris-virginica
59	6.2	...	1.3	Iris-versicolor
60	6.3	...	1.3	Iris-versicolor
61	6.3	...	1.8	Iris-virginica
62	6.4	...	2.3	Iris-virginica
63	6.5	...	1.5	Iris-versicolor
64	6.5	...	2.0	Iris-virginica
65	6.5	...	2.0	Iris-virginica
66	6.6	...	1.3	Iris-versicolor
67	6.6	...	1.4	Iris-versicolor
68	6.7	...	1.4	Iris-versicolor
69	6.7	...	1.5	Iris-versicolor
70	6.9	...	1.5	Iris-versicolor
71	6.9	...	2.1	Iris-virginica
72	6.9	...	2.3	Iris-virginica
73	7.2	...	1.6	Iris-virginica
74	7.2	...	1.8	Iris-virginica
75	7.3	...	1.8	Iris-virginica
76	7.7	...	2.3	Iris-virginica
77	3.3	...	7.8	Iris-setosa
78	3.3	...	7.8	Iris-setosa
79	3.3	...	7.8	Iris-setosa
80	3.3	...	7.8	Iris-setosa
81	3.3	...	7.8	Iris-setosa
82	3.3	...	7.8	Iris-setosa
83	3.3	...	7.8	Iris-setosa
84	3.3	...	7.8	Iris-setosa
85	3.3	...	7.8	Iris-setosa
86	3.3	...	7.8	Iris-setosa
87	3.3	...	7.8	Iris-setosa
88	3.3	...	7.8	Iris-setosa
89	3.3	...	7.8	Iris-setosa
90	3.3	...	7.8	Iris-setosa
91	3.3	...	7.8	Iris-setosa
92	3.3	...	7.8	Iris-setosa
93	3.3	...	7.8	Iris-setosa
94	3.3	...	7.8	Iris-setosa
95	3.3	...	7.8	Iris-setosa
96	3.3	...	7.8	Iris-setosa
97	3.3	...	7.8	Iris-setosa
98	3.3	...	7.8	Iris-setosa
99	3.3	...	7.8	Iris-setosa
100	3.3	...	7.8	Iris-setosa

Notice that all the predictors are well-correlated with each other.

iris.corr()

Let’s compute the PCA of the different elements.

from verticapy.machine_learning.vertica import PCA

model = PCA()

model.fit(
    iris,
    [
        "PetalLengthCm",
        "SepalWidthCm",
        "SepalLengthCm",
        "PetalWidthCm",
    ],
)



=======
columns
=======
index|    name     |  mean  |   sd   
-----+-------------+--------+--------
  1  |petallengthcm| 5.29520| 2.65046
  2  |sepalwidthcm | 3.67240| 0.83213
  3  |sepallengthcm| 5.02600| 1.23158
  4  |petalwidthcm | 2.63920| 2.66236


===============
singular_values
===============
index| value  |explained_variance|accumulated_explained_variance
-----+--------+------------------+------------------------------
  1  | 3.10953|      0.59239     |            0.59239           
  2  | 2.35110|      0.33866     |            0.93104           
  3  | 0.99380|      0.06051     |            0.99155           
  4  | 0.37133|      0.00845     |            1.00000           


====================
principal_components
====================
index|  PC1   |  PC2   |  PC3   |  PC4   
-----+--------+--------+--------+--------
  1  | 0.62676| 0.76262| 0.09839|-0.12608
  2  | 0.20444| 0.03229|-0.42096| 0.88315
  3  |-0.25445| 0.17577| 0.83700| 0.45143
  4  | 0.70755|-0.62167| 0.33548| 0.01884


========
counters
========
   counter_name   |counter_value
------------------+-------------
accepted_row_count|     250     
rejected_row_count|      0      
 iteration_count  |      1      


===========
call_string
===========
SELECT PCA('"public"."_verticapy_tmp_pca_v_mldb_43ed918e97bc11efa8720242ac120002_"', '"public"."_verticapy_tmp_view_v_mldb_443e79fa97bc11efa8720242ac120002_"', '"PetalLengthCm", "SepalWidthCm", "SepalLengthCm", "PetalWidthCm"'
USING PARAMETERS scale=false);

Let’s compute the correlation matrix of the result of the PCA.

model.transform().corr()

Notice that the predictors are now independant and combined together and they have the exact same amount of information than the previous variables. Let’s look at the accumulated explained variance of the PCA components.

model.explained_variance_
Out[4]: array([0.59238864, 0.33865565, 0.06050821, 0.00844749])

Most of the information is in the first two components with more than 97.7% of explained variance. We can export this result to a vDataFrame.

model.transform(n_components = 2)

	Abc Species Varchar(30) 100%	...	123 col1 Float(22) 100%	123 col2 Float(22) 100%
1	Iris-setosa	...	-1.15198969119672	0.27622003481473
2	Iris-setosa	...	-1.24742330877637	0.392295855356348
3	Iris-setosa	...	-1.32375724011031	0.445025462456261
4	Iris-setosa	...	-1.32183515786365	0.541843247223654
5	Iris-setosa	...	-1.28152511107279	0.518058857623143
6	Iris-setosa	...	-1.29652722712352	0.580475646448235
7	Iris-setosa	...	-1.18840369447931	0.555939181390084
8	Iris-setosa	...	-1.26415979574863	0.638911299240627
9	Iris-setosa	...	-1.26415979574863	0.638911299240627
10	Iris-setosa	...	-1.26415979574863	0.638911299240627
11	Iris-versicolor	...	-1.18624026043726	0.446220218801964
12	Iris-setosa	...	-1.06939651708109	0.662694415132234
13	Iris-setosa	...	-0.960831867410113	0.724609395749264
14	Iris-versicolor	...	-0.743987167322463	0.673959286317241
15	Iris-setosa	...	-0.677181455381914	0.843411194948978
16	Iris-setosa	...	-0.52407247568726	0.824403426424333
17	Iris-versicolor	...	-0.954194559795171	0.901067700886617
18	Iris-versicolor	...	-0.908883807654076	0.856477715011076
19	Iris-versicolor	...	-0.745143708179378	0.888400452127056
20	Iris-versicolor	...	-0.691891221349984	1.03651661116969