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# Pearson Correlation Coefficient

## Overview

The Pearson correlation coefficient measures the linear correlation between two variables. The Pearson correlation coefficient between two nodes in graph is calculated by using N properties of node to form two N-dimensional vectors.

## Basic Concept

### Vector

Vector is one of the basic concepts in Advanced Mathematics, vectors in low dimensional spaces are relatively easy to understand and express. The following diagram shows the relationship between vectors A, B and coordinate axes in 2- and 3-dimensional spaces respectively, as well as the angle `θ` between them:

When comparing two nodes in graph, N properties of node are used to form the two N-dimensional vectors.

### Pearson Correlation Coefficient

The range of Pearson correlation coefficient values is [-1,1]; let `r` to denote the Pearson correlation coefficient, then:

• `r > 0` indicates positive correlation, i.e. as one variable becomes larger, the other variable becomes larger;
• `r < 0` indicates negative correlation, i.e. as one variable becomes larger, the other variable becomes smaller;
• `r = 1` or `r = -1` indicates that two variables can be described by a linear equation, i.e. them fall on the same line;
• `r = 0` indicates that there is no linear correlation (but may exist some other correlations).

For two variables X= (x1, x2, ..., xn) and Y = (y1, y2, ..., yn) , Pearson correlation coefficient (r) is defined as the ratio of the covariance of them and the product of their standard deviations:

## Special Case

### Isolated Node, Disconnected Graph

Theoretically, the calculation of Pearson Correlation Coefficient between two nodes does not depend on the existence of edges in the graph. Regardless of whether the two nodes to be calculated are isolated nodes or whether they are in the same connected component, it does not affect the calculation of their Pearson Correlation Coefficient.

### Self-loop Edge

The calculation of Pearson Correlation Coefficient has nothing to do with edges.

### Directed Edge

The calculation of Pearson Correlation Coefficient has nothing to do with edges.

## Command and Configuration

• Command: `algo(similarity)`
• Configurations for the parameter `params()`:
Name
Type
Default
Specification
Description
ids / uuids []`_id` / []`_uuid` / Mandatory IDs or UUIDs of the first set of nodes to be calculated
ids2 / uuids2 []`_id` / []`_uuid` / Optional IDs or UUIDs of the second set of nodes to be calculated
type string cosine jaccard / overlap / cosine / pearson / euclideanDistance / euclidean Measurement of the similarity:
jaccard: Jaccard Similarity
overlap: Overlap Similarity
cosine: Cosine Similarity
pearson: Pearson Correlation Coefficient
euclideanDistance: Euclidean Distance
euclidean: Normalized Euclidean Distance
node_schema_property []`@<schema>?.<property>` / Numeric node property; LTE needed; schema can be either carried or not When `type` is cosine / pearson / euclideanDistance / euclidean, must specify two or more node properties to form the vector; when `type` is jaccard / overlap, this parameter is invalid
limit int -1 >=-1 Number of results to return; return all results if sets to -1
top_limit int -1 >=-1 Only available in the selection mode, limit the length of selection results (`top_list`) of each node, return the full `top_list` if sets to -1

## Calculation Mode

This algorithm has two calculation modes:

1. Pairing mode: when two sets of valid nodes are configured, pair each node in the first set with each node in the second set (Cartesian product), similarities are calculated for all node pairs.
2. Selection mode: when only one set (the first) of valid nodes are configured, for each node in the set, calculate its similarities with all other nodes in the graph, return the results if the similarity > 0, order the results the descending similarity.

## Examples

### Example Graph

The example graph has product1, product2, product3 and product4 (UUIDs are 1, 2, 3 and 4 in order; edges are ignored), product node has properties price, weight, weight and height:

#### 1. File Writeback

Calculation Mode
Configuration
Data in Each Row
Pairing mode filename `node1`,`node2`,`similarity`
Selection mode filename `node`,`top_list`

Example: Calculate Pearson correlation coefficient between product UUID = 1 and products UUID = 2,3,4 through properties price, weight, width and height, write the algorithm results back to file

``````algo(similarity).params({
uuids: [1],
uuids2: [2,3,4],
node_schema_property: [price,weight,width,height],
type: "pearson"
}).write({
file:{
filename: "pearson"
}
})
``````

Results: File pearson

``````product1,product2,0.998785
product1,product3,0.474384
product1,product4,0.210494
``````

Example: Calculate Pearson correlation coefficient between products UUID = 1,2,3,4 and all other products in the graph respectively through properties price, weight, width and height, write the algorithm results back to file

``````algo(similarity).params({
uuids: [1,2,3,4],
node_schema_property: [price,weight,width,height],
type: "pearson"
}).write({
file:{
filename: "list"
}
})
``````

Results: File list

``````product1,product2:0.998785;product3:0.474384;product4:0.210494;
product2,product1:0.998785;product3:0.507838;product4:0.253573;
product3,product2:0.507838;product1:0.474384;product4:0.474021;
product4,product3:0.474021;product2:0.253573;product1:0.210494;
``````

#### 2. Property Writeback

Not supported by this algorithm.

#### 3. Statistics Writeback

This algorithm has no statistics.

### Direct Return

Calculation Mode
Alias Ordinal
Type Description Column Name
Pairing mode 0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
Selection mode 0 []perNode Node and its selection results `node`, `top_list`

Example: Calculate Pearson correlation coefficient between product UUID = 1 and products UUID = 2,3,4 through properties price, weight, width and height, order results in the ascending similarity

``````algo(similarity).params({
uuids: [1],
uuids2: [2,3,4],
node_schema_property: [price,weight,width,height],
type: "pearson"
}) as p
return p order by p.similarity
``````

Results:

node1 node2 similarity
1 4 0.210494150169583
1 3 0.474383803132863
1 2 0.998785121601255

Example: Select the product with the highest Pearson correlation coefficient with products UUID = 1,2 respectively through properties price, weight, width and height,

``````algo(similarity).params({
uuids: [1,2],
type: "pearson",
node_schema_property: [price,weight,width,height],
top_limit: 1
}) as top
``````

Results:

node top_list
1 2:0.998785,
2 1:0.998785,

### Streaming Return

Calculation Mode
Alias Ordinal
Type Description Column Name
Pairing mode 0 []perNodePair Node pair and its similarity `node1`, `node2`, `similarity`
Selection mode 0 []perNode Node and its selection results `node`, `top_list`

Example: Calculate Pearson correlation coefficient between product UUID = 3 and products UUID = 1,2,4 through properties price, weight, width and height, only return results that have similariy above 0.5

``````algo(similarity).params({
uuids: [3],
uuids2: [1,2,4],
node_schema_property: [price,weight,width,height],
type: "pearson"
}).stream() as p
where p.similarity > 0.5
return p
``````

Results:

node1 node2 similarity
3 2 0.50783775659896

Example: Select the product with the highest Pearson correlation coefficient with products UUID = 1,3 respectively

``````algo(similarity).params({
uuids: [1,3],
node_schema_property: [price,weight,width,height],
type: "pearson",
top_limit: 1
}).stream() as top