PCA Manual in Python: FactoMineR Style​‌​‌​‌​​‍​‌​​​‌​‌‍​​‌‌‌​‌​‍​​‌‌​‌​​‍​‌‌​​‌​‌‍​​‌‌‌​​​‍​‌‌​​‌​​‍​​‌‌‌​​‌‍​‌‌​​​‌​‍​​‌‌​​​‌‍​‌‌​​​​‌‍​​‌‌​‌​‌‍​‌‌​​‌‌​‍​​‌‌​‌‌​‍​‌‌​​‌​‌‍​​‌‌​‌‌‌‍​‌‌​​​‌‌‍​​‌‌​​‌‌‍​‌‌​​‌​​‍​​‌‌‌​‌​‍​​‌‌​​‌​‍​​‌‌​​​​‍​​‌‌​​‌​‍​​‌‌​‌‌​‍​​‌‌​​​​‍​​‌‌​​‌​‍​​‌‌​​​‌‍​​‌‌​​‌‌‍​​‌‌‌​‌​‍​​‌‌​​‌​‍​‌‌​​​​‌‍​​‌‌​‌​​‍​‌‌​​​​‌‍​‌‌​​​‌‌‍​​‌‌​​​​‍​‌‌​​‌‌​‍​‌‌​​​‌‌¶

Replicating the Gold Standard of Multivariate Analysis¶

Certification and Reference Information¶

Original author/Reference: @TodoEconometria Professor: Juan Marcelo Gutierrez Miranda Methodology: Advanced Courses in Big Data, Data Science, AI Application Development & Applied Econometrics.​‌​‌​‌​​‍​‌​​​‌​‌‍​​‌‌‌​‌​‍​​‌‌​‌​​‍​‌‌​​‌​‌‍​​‌‌‌​​​‍​‌‌​​‌​​‍​​‌‌‌​​‌‍​‌‌​​​‌​‍​​‌‌​​​‌‍​‌‌​​​​‌‍​​‌‌​‌​‌‍​‌‌​​‌‌​‍​​‌‌​‌‌​‍​‌‌​​‌​‌‍​​‌‌​‌‌‌‍​‌‌​​​‌‌‍​​‌‌​​‌‌‍​‌‌​​‌​​‍​​‌‌‌​‌​‍​​‌‌​​‌​‍​​‌‌​​​​‍​​‌‌​​‌​‍​​‌‌​‌‌​‍​​‌‌​​​​‍​​‌‌​​‌​‍​​‌‌​​​‌‍​​‌‌​​‌‌‍​​‌‌‌​‌​‍​​‌‌​​‌​‍​‌‌​​​​‌‍​​‌‌​‌​​‍​‌‌​​​​‌‍​‌‌​​​‌‌‍​​‌‌​​​​‍​‌‌​​‌‌​‍​‌‌​​​‌‌ Certification Hash ID: 4e8d9b1a5f6e7c3d2b1a0f9e8d7c6b5a4f3e2d1c0b9a8f7e6d5c4b3a2f1e0d9c Repository: https://github.com/TodoEconometria/certificaciones

MAIN ACADEMIC REFERENCE:

• Husson, F., Le, S., & Pages, J. (2017). Exploratory Multivariate Analysis by Example Using R. CRC Press.
• FactoMineR Tutorial: http://factominer.free.fr/course/FactoTuto.html
• Le, S., Josse, J., & Husson, F. (2008). FactoMineR: An R Package for Multivariate Analysis. Journal of Statistical Software, 25(1), 1-18.
• Pedregosa, F., et al. (2011). Scikit-learn: Machine Learning in Python. JMLR 12, pp. 2825-2830.

1. Introduction: Why FactoMineR is the Gold Standard¶

1.1. The Problem of Multivariate Data¶

Imagine you have data from 41 athletes who competed in the decathlon (10 sporting events). Each athlete has 10 measurements: race times, jump distances, and throwing distances. The question is: How can we "see" patterns in 10 dimensions?

The human brain can only visualize 2 or 3 dimensions. Principal Component Analysis (PCA) is the mathematical tool that allows us to "compress" those 10 dimensions into 2, losing as little information as possible.

1.2. Philosophy: FactoMineR vs. Scikit-Learn¶

This manual replicates the logic of FactoMineR (the gold standard in R) using Python.

The fundamental difference: Scikit-learn tells you "these are your compressed data". FactoMineR tells you "why your data compressed this way, which variables are responsible, and how reliable the representation of each point is".

2. BEFORE STARTING: Missing Data Treatment¶

2.1. The Most Common Error: "I'll Just Plug in the Mean"¶

One of the most frequent errors in multivariate analysis is imputing missing data with the mean without thinking. This practice, although common, can destroy the real structure of your data and lead you to erroneous conclusions.

GOLDEN RULE: Before imputing any missing value, you must understand WHY that data is missing. The imputation strategy depends on the missingness mechanism.

2.2. The Three Types of Missing Data (Rubin, 1976)¶

2.3. Why the Mean Can Destroy Your Analysis¶

Imagine this scenario in the decathlon:

Athlete     | 100m  | Long Jump   | Throwing
------------|-------|-------------|------------
Sprinter A  | 10.5s | 7.8m        | ???
Sprinter B  | 10.7s | 7.6m        | ???
Thrower C   | 11.8s | 6.2m        | 18.5m
Thrower D   | 12.0s | 6.0m        | 19.2m

Problem: The sprinters (A and B) did not complete the throwing event (due to injury).

If you impute with the throwing mean: - Throwing mean = (18.5 + 19.2) / 2 = 18.85m - You assign 18.85m to sprinters A and B

CATASTROPHIC CONSEQUENCE: - In the PCA, the sprinters appear as good throwers - The correlation circle will show that speed and throwing are positively correlated - THIS IS FALSE - in reality, sprinters tend to be worse throwers

The problem: The data is NOT missing at random (MCAR). It is missing because the sprinters were injured in the throwing event (probably due to lack of technique). This is a case of MAR or MNAR.

2.4. How to Diagnose the Type of Missing Data¶

Before imputing, perform these diagnostic tests:

Test 1: Visual Missingness Pattern¶

import missingno as msno
import matplotlib.pyplot as plt

# Visualize missing data pattern
msno.matrix(df)
plt.show()

# Visualize correlation between missing data
msno.heatmap(df)
plt.show()

Interpretation: - If the gaps are randomly dispersed: probably MCAR - If the gaps cluster in specific rows/columns: probably MAR or MNAR

Test 2: Little's Test (MCAR Test)¶

# Install: pip install pyampute
from pyampute.exploration import mcar_test

# Statistical test for MCAR
result = mcar_test(df)
print(f"p-value: {result}")
# If p > 0.05: MCAR not rejected (data MAY be random)
# If p < 0.05: MCAR rejected (data is NOT random)

Test 3: Compare Groups (Complete vs Incomplete Data)¶

# Create missing data indicator
df['has_na'] = df['throwing'].isna()

# Compare means of other variables
print(df.groupby('has_na')['100m'].mean())
print(df.groupby('has_na')['long_jump'].mean())

# If means are VERY different, the data is NOT MCAR

2.5. Imputation Strategies by Type¶

2.6. Alternatives to the Mean (Python Code)¶

Option 1: Drop Rows with NA (only if few and MCAR)¶

# Only if you have FEW missing data (<5%) and they are MCAR
df_clean = df.dropna()
print(f"Rows removed: {len(df) - len(df_clean)}")

Option 2: KNN Imputation (respects local structure)¶

from sklearn.impute import KNNImputer

# KNN finds the K most similar neighbors and averages their values
imputer = KNNImputer(n_neighbors=5)
df_imputed = pd.DataFrame(
    imputer.fit_transform(df[numeric_columns]),
    columns=numeric_columns
)

Advantage: A sprinter with NA in throwing will receive the value from other similar sprinters, not the global average.

Option 3: Multiple Imputation (MICE) - The Gold Standard¶

from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer

# MICE: Multiple Imputation by Chained Equations
imputer = IterativeImputer(max_iter=10, random_state=42)
df_imputed = pd.DataFrame(
    imputer.fit_transform(df[numeric_columns]),
    columns=numeric_columns
)

Advantage: Uses ALL variables to predict the missing value through iterative regressions.

Option 4: Group-Based Imputation (when categories exist)¶

# If you know the data depends on a category
df['throwing'] = df.groupby('athlete_type')['throwing'].transform(
    lambda x: x.fillna(x.mean())
)

Example: Impute a sprinter's throwing with the mean of other sprinters, not the global mean.

2.7. Checklist BEFORE Performing PCA¶

Before running your PCA analysis, verify:

• How many NAs do I have? df.isna().sum() - If > 20% in a variable, consider removing it
• Where are the NAs? msno.matrix(df) - Look for patterns
• Are they random? Little's test or group comparison
• Which method should I use?
• MCAR + few NAs: Mean or drop rows
• MAR: KNN or MICE
• MNAR: Consult with a domain expert or sensitivity analysis

2.8. Practical Example: What to Do in the Decathlon¶

import pandas as pd
import numpy as np
from sklearn.impute import KNNImputer

# Load data
df = pd.read_csv('decathlon.csv')

# 1. Diagnosis
print("Missing data by column:")
print(df.isna().sum())

# 2. If there are NAs, check pattern
if df.isna().any().any():
    # Compare athletes with/without NAs
    df['has_na'] = df.isna().any(axis=1)
    print("\nMean comparison (with NA vs without NA):")
    print(df.groupby('has_na')[['100m', 'Long.jump']].mean())

    # If means are similar, probably MCAR -> we can impute
    # If means are very different, be careful -> use KNN or MICE

    # 3. Impute with KNN (respects structure)
    imputer = KNNImputer(n_neighbors=3)
    active_columns = ['100m', 'Long.jump', 'Shot.put', 'High.jump',
                        '400m', '110m.hurdle', 'Discus', 'Pole.vault',
                        'Javeline', '1500m']

    df[active_columns] = imputer.fit_transform(df[active_columns])
    print("\nImputation completed with KNN")

# 4. Now we CAN perform PCA
from prince import PCA
pca = PCA(n_components=5, rescale_with_std=True)
pca.fit(df[active_columns])

2.9. Summary: The 3-Step Rule¶

STEP 1: DIAGNOSE
   "Why is this data missing?"
   -> MCAR, MAR, or MNAR?

STEP 2: DECIDE
   "Can I impute without biasing?"
   -> If MNAR: DO NOT impute with mean
   -> If MAR: Use KNN or MICE
   -> If MCAR: Mean is acceptable

STEP 3: DOCUMENT
   "What did I do and why?"
   -> Report the % of NAs
   -> Report the method used
   -> Report whether results change

MORAL: Plugging in the mean "just because" is like filling a hole in a puzzle with a piece from another puzzle. It might fit, but the final picture will be distorted.

2.10. FUNDAMENTAL: Why 0 is Valid and You Should NOT Always Impute¶

2.10.1. The Core Problem: Confusing "Complete Database" with "Correct Database"¶

One of the most serious conceptual errors made by students (and many professionals) is thinking that:

FALSE: "A database must be complete to be analyzed"

This belief comes from programming (where NULL causes errors) and from machine learning (where many algorithms don't accept NaN). But in real data analysis, especially in surveys and Big Data, missing data IS INFORMATION, not a problem to "fix".

The Uncomfortable Truth:¶

2.10.2. The Fundamental Difference: 0 vs NaN vs "No Response"¶

Imagine a survey about income and expenses:

Analysis of Each Case:¶

Person A - The 0 is VALID: - cigarettes = 0 means "DOES NOT SMOKE", it is a real response - tobacco_expense = 0 is consistent with not smoking - IMPUTING these 0s with the mean would be CATASTROPHIC: you would turn a non-smoker into an average smoker

Person B - The NaN is INFORMATIVE: - cigarettes = NaN means "REFUSED TO ANSWER" - It could be because: - Smokes illegally (underage) - Embarrassed to admit smoking heavily - Simply doesn't want to share that information - IMPUTING with the mean assumes they are an average smoker, which may be FALSE

Person C - Complete Data: - Answered everything, no ambiguity

2.10.3. Sampling Concepts: Why Missing Data is Part of the Design¶

The Survey Context¶

When working with surveys or Big Data databases, missing data is a natural consequence of the collection process. According to sampling theory (see TodoEconometria - Sampling Strategies):

Simple Random Sampling: The selection of a representative sample is highly complex. For example, if we want to study people with type 2 diabetes, not everyone will answer every question, and that is expected and valid.

The Three Real Scenarios:¶

Scenario 1: Control Questions (Filters)

In well-designed surveys, there are questions that intentionally generate NaN in other variables:

Q1: "Do you own a car?"
    [ ] Yes  -> Go to Q2
    [ ] No   -> Skip to Q5

Q2: "What brand is your car?" (Only if answered Yes in Q1)
    _____________

Result in the database:

Question: Should we impute Mary's car brand?

Answer: NO. The NaN is correct. Mary doesn't have a car, therefore she cannot have a brand. Imputing "Toyota" (the mean/mode) would be absurd.

Scenario 2: Non-Response Due to Sensitivity

Some questions are sensitive:

Q10: "What is your monthly income?"
     [ ] Less than $1000
     [ ] $1000 - $3000
     [ ] $3000 - $5000
     [ ] More than $5000
     [ ] Prefer not to answer

Result:

Question: Should we impute Ana's income with the median?

Answer: IT DEPENDS on the missingness type: - If Ana is wealthy and doesn't want to disclose it: MNAR (data is missing because it's high) - If Ana randomly chose "prefer not to answer": MCAR (random)

Imputing with the median in the MNAR case will introduce downward bias (we assume that the wealthy who don't respond have average income).

Scenario 3: Big Data with Sensors

In IoT databases, sensors, or logs:

Temperature Sensor:
Time  | Temperature
------|------------
10:00 | 25.3°C
10:05 | 25.5°C
10:10 | NaN  (sensor failed)
10:15 | 26.1°C

Question: Should we impute 10:10 with the mean of 10:05 and 10:15?

Answer: IT DEPENDS: - If the sensor fails randomly: MCAR -> Linear imputation is reasonable - If the sensor fails when it's very hot (overheats): MNAR -> Imputing with the mean will underestimate the actual temperature

2.10.4. The Conceptual Error: "I Need Complete Data to Analyze"¶

Myth 1: "Algorithms don't accept NaN"¶

Reality: - Many algorithms DO accept NaN: decision trees, Random Forest, XGBoost - Those that don't accept NaN (linear regression, classic PCA) have robust versions: - PCA with prince accepts NA - Regression with statsmodels has missing='drop'

Myth 2: "More data = Better model"¶

Reality: - Incorrect data (poorly imputed) are worse than fewer correct data - Example:

# Option A: Impute with mean (BAD if MNAR)
df['income'].fillna(df['income'].mean())  # 1000 rows, but with bias

# Option B: Drop rows with NA (GOOD if few)
df.dropna(subset=['income'])  # 800 rows, but without bias

Which is better? It depends on the % of NAs: - If NA < 5%: Dropping is safe - If NA > 20%: Investigate why they are missing before deciding

2.10.5. Practical Case: Consumer Habits Survey¶

Imagine this database:

Correct Analysis:¶

Person A: - Cigarettes_Day = 0 is VALID (doesn't smoke) - Tobacco_Expense = 0 is VALID (consistent) - DO NOT IMPUTE

Person B: - Drinks_Week = 0 is VALID (doesn't drink) - DO NOT IMPUTE

Person C: - Cigarettes_Day = NaN is PROBLEMATIC - Options: 1. Drop row (if it's the only one with NA) 2. Impute with KNN (find similar smokers) 3. Leave as NA and use algorithms that support it

INCORRECT Analysis (Common Among Beginners):¶

# ERROR: Impute EVERYTHING with the mean
df.fillna(df.mean())

Consequence: - Person A (non-smoker) now has Cigarettes_Day = 5 (mean of B and C) - Person D (non-smoker, non-drinker) now smokes and drinks - The database is DESTROYED

2.10.6. Golden Rules for Students¶

Rule 1: Before Touching an NA, Ask Yourself:¶

1. Why is this data missing?
   - Filter question? -> NA is CORRECT, DO NOT impute
   - Non-response? -> Investigate type (MCAR/MAR/MNAR)
   - Technical error? -> Could be MCAR

2. Is it a 0 or a NaN?
   - 0 = Valid response ("I don't have", "I don't do")
   - NaN = Absence of response

3. What happens if I impute incorrectly?
   - Bias in results
   - Erroneous conclusions
   - Invalid model

Rule 2: Decision Workflow¶

STEP 1: Count NAs
   df.isna().sum()
   -> If < 5%: Consider dropping rows
   -> If > 20%: Investigate pattern

STEP 2: Visualize pattern
   import missingno as msno
   msno.matrix(df)
   -> Random? -> MCAR
   -> Clustered? -> MAR/MNAR

STEP 3: Statistical test
   from pyampute.exploration import mcar_test
   mcar_test(df)
   -> p > 0.05: MCAR (you can impute)
   -> p < 0.05: MAR/MNAR (be careful)

STEP 4: Decide strategy
   - MCAR + few NAs: Mean or drop
   - MAR: KNN or MICE
   - MNAR: DO NOT impute or use specific model
   - Filter question: LEAVE as NA

Rule 3: ALWAYS Document¶

# GOOD PRACTICE
print("="*50)
print("MISSING DATA REPORT")
print("="*50)
print(f"Total observations: {len(df)}")
print(f"Variables with NA: {df.isna().any().sum()}")
print(f"\nDetail by variable:")
print(df.isna().sum())
print(f"\nPercentage of NAs:")
print((df.isna().sum() / len(df) * 100).round(2))
print("\nStrategy applied:")
print("- Variables with valid 0: ['Cigarettes_Day', 'Drinks_Week']")
print("- Variables imputed with KNN: ['Income']")
print("- Rows removed: 3 (0.5% of total)")

2.10.7. Full Example: Correct vs Incorrect Code¶

INCORRECT Code (Beginner):¶

import pandas as pd

# Load data
df = pd.read_csv('survey.csv')

# ERROR: Impute EVERYTHING with the mean
df = df.fillna(df.mean())

# ERROR: Don't verify what happened
print("Done, complete database!")

Problems: 1. Doesn't distinguish between valid 0 and NaN 2. Doesn't verify missingness type 3. Doesn't document what was done 4. May have destroyed the real structure

CORRECT Code (Professional):¶

import pandas as pd
import numpy as np
from sklearn.impute import KNNImputer
import missingno as msno
import matplotlib.pyplot as plt

# 1. Load and explore
df = pd.read_csv('survey.csv')
print("INITIAL DIAGNOSIS")
print("="*50)
print(df.info())
print("\nMissing data:")
print(df.isna().sum())

# 2. Visualize pattern
msno.matrix(df)
plt.title("Missing Data Pattern")
plt.show()

# 3. Separate variables with valid 0 from variables with NA
# Variables where 0 is valid (do not impute)
vars_with_valid_zero = ['Cigarettes_Day', 'Tobacco_Expense', 'Drinks_Week']

# Variables where NA must be treated
vars_to_impute = ['Income', 'Age', 'Work_Hours']

# 4. Verify consistency (0 in dependent variables)
# Example: If doesn't smoke, tobacco expense must be 0
mask_non_smoker = df['Smokes'] == 'No'
assert (df.loc[mask_non_smoker, 'Cigarettes_Day'] == 0).all(), \
    "ERROR: There are non-smokers with cigarettes > 0"

# 5. Impute only numeric variables that need it
if df[vars_to_impute].isna().any().any():
    print("\nApplying KNN Imputer to:", vars_to_impute)
    imputer = KNNImputer(n_neighbors=5)
    df[vars_to_impute] = imputer.fit_transform(df[vars_to_impute])
    print("Imputation completed")

# 6. Document
print("\n" + "="*50)
print("FINAL REPORT")
print("="*50)
print(f"Total rows: {len(df)}")
print(f"Variables with valid 0 (NOT imputed): {vars_with_valid_zero}")
print(f"Variables imputed with KNN: {vars_to_impute}")
print(f"Remaining NAs: {df.isna().sum().sum()}")

2.10.8. Summary: The Data Analyst's Philosophy¶

"It's not your job to fill gaps. It's your job to UNDERSTAND why there are gaps."

Fundamental Principles:¶

1. Missing data is information, not an error
2. 0 is a valid response, not missing data
3. NaN means "unknown", and sometimes "unknown" is the correct answer
4. Imputing incorrectly is worse than not imputing
5. Document EVERYTHING you do with NAs

Final Checklist Before Performing PCA (or Any Analysis):¶

• Have I identified all variables with valid 0?
• Have I verified that the 0s are consistent with other variables?
• Have I diagnosed the missingness type (MCAR/MAR/MNAR)?
• Have I decided on a justified strategy for each variable?
• Have I documented how many NAs there were and what I did with them?
• Have I verified that the results make sense after imputation?

3. The Dataset: Decathlon - Context and Research Questions¶

3.1. The Real Scenario: What Problem Are We Trying to Solve?¶

Imagine you are the technical director of an athletics federation. You have data from 41 athletes who competed in the decathlon (10 events) in two different competitions: the Olympic Games and the Decastar Meeting.

You face several business/research questions:

KEY: We don't apply PCA "just because" or "because it's trendy". We apply it because we have specific questions that this technique can answer efficiently and validly.

3.2. Why PCA is the Right Tool (and Not Another)¶

The Mathematical Problem¶

We have 10 numerical variables (the events). Visualizing 10 dimensions is impossible. But:

• We don't want to lose important information
• We want to see patterns (groups of athletes, relationships between events)
• We want to reduce complexity without destroying the structure

Why NOT Other Techniques?¶

Why USE PCA?¶

3.3. The Sporting Context of the Decathlon¶

The decathlon is known as "the king of athletics events" because it evaluates the most complete athlete. It consists of 10 events over 2 days:

Day 1¶

Day 2¶

Prior hypothesis (before PCA): We expect to find that events group by "skill type": speed (100m, 400m, hurdles), strength (shot put, discus, javelin), and technique (pole vault, high jump). PCA will tell us whether this hypothesis is correct.

3.4. The Two Competitions: Why Does It Matter?¶

Key question: Do Olympic athletes have a different performance profile? Or are they simply "better at everything"?

• If they are "better at everything", they will be in the same direction as Decastar athletes but farther from the origin.
• If they have a different profile, they will be in a different zone of the map.

This question is answered by using Competition as a supplementary variable: it doesn't participate in the PCA calculation, but we project it to see if the groups separate.

3.5. Data Structure¶

3.6. Key Concept: Active vs. Supplementary Variables¶

This is one of the most powerful concepts in FactoMineR:

Active Variables: Participate in the mathematical calculation of principal components. They are the ones that "build" the axes. In our case, the 10 sporting events.

Supplementary Variables: Do NOT participate in the calculation, but are projected onto the factorial space to see how they relate to the found dimensions.

Why is it useful? Imagine you want to know whether the type of competition (Olympic Games vs Decastar) influences the athletes' performance profile. If you include "Competition" as an active variable, you contaminate the analysis (you introduce a categorical variable in a numerical calculation). But if you use it as a supplementary variable, you can see if the groups separate without having forced that separation.

3.7. Other Classic PCA Datasets and Their Contexts¶

So you understand that PCA is applied in many domains, here are other classic datasets:

Common pattern: In all cases, we have many numerical variables and want to discover latent structure (hidden dimensions that explain the variation).

3.8. Checklist: Before Applying PCA, Ask Yourself...¶

• Do I have a clear question? "I want to explore patterns" is valid. "I want to predict Y" is not for PCA.
• Are my variables numerical? PCA requires quantitative variables (or convertible to numerical).
• Do I have enough observations? Rule of thumb: at least 5-10 observations per variable.
• Are the variables on comparable scales? If not, I must standardize (Prince does this by default).
• Have I handled missing data? (See section 2)
• Do I have supplementary variables to validate? Very useful for interpreting results.

4. Analysis Results: Chart Panel¶

Below is the complete chart panel generated by the 02_PCA_FactoMineR_style.py script:

5. Detailed Interpretation of Each Chart¶

4.1. SCREE PLOT¶

Location: Upper left panel

What it shows: The variance explained by each principal component.

Reading the Results:¶

Interpretation:¶

1. The gray dashed line (Threshold = 10%) represents the expected contribution if all variables were independent (100% / 10 variables = 10%). Dimensions above this line contribute more information than the average.

2. The first 4 dimensions have eigenvalues greater than 1 (Kaiser Rule), suggesting retaining 4 components.

3. The red curve (cumulative) shows that with 5 dimensions we capture 65.5% of the total variability. This is relatively low, indicating that the decathlon is a multidimensional sport where no single "factor" explains all performance.

Sporting Insight: Unlike sports such as swimming (where speed explains almost everything), the decathlon requires multiple independent skills: speed, strength, endurance, technique. That's why no dimension clearly dominates.

4.2. CORRELATION CIRCLE (The Most Important Chart)¶

Location: Upper center panel

This is the most important chart in FactoMineR. It shows how variables relate to each other and to the axes.

How to Read the Circle:¶

Interpretation of Our Results:¶

Group 1 - Lower Left Quadrant (Speed/Agility): - Long.jump (long jump): Long arrow to the left - 400m: Similar direction - 110m.hurdle: Similar direction

Interpretation: These events are positively correlated with each other. An athlete good at long jump tends to be good at 400m and hurdles. This makes sense: they all require explosive speed and leg power.

Group 2 - Upper Left Quadrant (Endurance): - 1500m: Arrow toward upper-left - Shot.put (shot put): Upward

Interpretation: The 1500m (endurance) and shot put appear in different directions from the speed group. This suggests that endurance is a distinct skill from explosive speed.

Group 3 - Lower Right Quadrant: - Pole.vault (pole vault): Long arrow downward - Discus: Similar direction

Interpretation: These technical events form their own group.

Special Variable - Long.jump: - It is the longest arrow toward the left - Correlation with Dim.1: -0.74 (very high)

Conclusion: Long jump is the variable that best defines the first dimension. An athlete with a high score on Dim.1 (right side of the map) tends to have a worse long jump.

4.3. INDIVIDUAL MAP¶

Location: Upper right panel

Shows where each athlete is located in the space of the first two dimensions, colored by the supplementary variable Competition.

Reading the Chart:¶

• Blue points: Athletes from the Decastar competition
• Red points: Athletes from the Olympic Games (OlympicG)

Interpretation:¶

1. General spread: Athletes are widely distributed across the plane, without forming very compact groups. This confirms that there is diversity of profiles in the decathlon.

2. Separation by competition: Observe that red points (Olympic) tend to be more dispersed and some are in extreme positions (e.g., BERNARD at the far right). Olympic athletes show greater variability in their profiles.

3. Notable athletes:

4. BERNARD (far right): Very distinctive profile, different from the average
5. Zsivoczky (below): Another atypical profile

6. BOURGUIGNON (upper left): Different profile

7. Combined interpretation with the circle:

8. Athletes on the left of the map tend to be good at Long.jump, 400m, 110m.hurdle (speed events)
9. Athletes above tend to be good at Shot.put, 1500m (strength/endurance)

4.4. VARIABLE CONTRIBUTIONS TO DIM.1¶

Location: Lower left panel

This chart answers the question: Which variables "built" the first axis?

Results:¶

Interpretation:¶

• The dashed red line marks the expected contribution threshold (10% = 100%/10 variables)
• The red bars (above threshold) are the variables that truly define that dimension
• The blue bars (below threshold) contribute less than expected

Conclusion for Dim.1: This dimension is dominated by Long.jump (32.9%), followed by 1500m (19.9%) and 400m (12.0%). This suggests that Dim.1 represents a continuum between explosive athletes (good at long jump, 400m) vs endurance athletes (1500m).

4.5. VARIABLE CONTRIBUTIONS TO DIM.2¶

Location: Lower center panel

Results:¶

Interpretation:¶

Conclusion for Dim.2: This dimension is dominated by Shot.put (34.1%) and Pole.vault (30.7%). It represents a contrast between throwing athletes (brute strength) vs technical athletes (pole vault).

Important note: Observe that Shot.put and Pole.vault point in opposite directions in the correlation circle. This means: - Athletes above on the map: Good at shot put - Athletes below on the map: Good at pole vault

4.6. BIPLOT (Individuals + Variables)¶

Location: Lower right panel

The biplot combines the individual map with variable arrows, allowing you to directly see which variables characterize which athletes.

How to Read It:¶

1. Mentally project each athlete onto each variable arrow
2. If the projection falls in the direction of the arrow: the athlete has a high value in that variable
3. If the projection falls in the opposite direction: the athlete has a low value in that variable

Interpretation Example:¶

• BERNARD (far right):
• Projection onto Long.jump (left): opposite = poor long jump

• Projection onto Shot.put (up): perpendicular = average

• Athletes upper-left:

• Good at 1500m and Shot.put
• Endurance and strength profile

6. Advanced FactoMineR Metrics¶

5.1. Quality of Representation (\(\cos^2\))¶

The \(\cos^2\) measures how well represented a point is in the factorial plane.

Interpretation:¶

Results for Athletes (Top 5 Best Represented):¶

Practical application: If you are going to interpret a specific athlete's position, first check their \(\cos^2\). If it is low (< 0.5), their position on the map may be misleading.

5.2. Contributions¶

Contributions indicate how much each individual/variable contributes to the construction of an axis.

For Variables:¶

We already saw that Long.jump contributes 32.9% to Dim.1. This means that if we removed long jump from the analysis, the first dimension would change drastically.

For Individuals:¶

Interpretation: BERNARD and Warners are "influential" athletes who "pull" the axis toward their positions. They are athletes with extreme profiles.

7. R/Python Equivalence Table¶

For those coming from R who want to replicate FactoMineR analyses:

8. Decathlon Analysis Conclusions¶

7.1. Main Findings¶

1. The decathlon is multidimensional: No dimension explains more than 17% of the variance. At least 4 components are needed to capture 56% of the information.

2. Dimension 1 - Speed/Agility vs Endurance:

3. Dominated by: Long.jump (32.9%), 1500m (19.9%), 400m (12.0%)

4. Separates explosive athletes from endurance athletes

5. Dimension 2 - Strength vs Technique:

6. Dominated by: Shot.put (34.1%), Pole.vault (30.7%)

7. Separates throwing athletes from technical athletes

8. Supplementary variable (Competition):

9. Olympic athletes show greater variability in their profiles
10. There is no clear separation between competitions, suggesting that the level of competition does not determine the profile type

7.2. Practical Implications¶

• For coaches: Identifying which quadrant an athlete falls in helps design personalized training programs
• For analysts: Contributions reveal which variables are most important for characterizing overall performance

9. Generated Files¶

10. Proposed Exercise for the Student¶

Comprehension Questions:¶

1. If an athlete has a negative coordinate on Dim.1, what can you infer about their long jump?

2. Why does Discus have an almost null contribution (0.1%) to Dim.1?

3. If an athlete's \(\cos^2\) in the Dim1-Dim2 plane is 0.30, should we trust their position on the map? Why?

4. Looking at the biplot, what type of profile does athlete YURKOV (above on the map) have?

Answers:¶

1. They have a good long jump (Long.jump's correlation with Dim.1 is negative: -0.74)

2. Because Discus variability is mainly represented in Dim.3 (contribution 39.4%), not in Dim.1

3. We should not trust it much. Only 30% of their variability is captured in that plane. The remaining 70% is in other dimensions.

4. YURKOV has high values in the variables that point upward (Shot.put, 1500m, High.jump): a strength and endurance profile.

11. Academic References¶

Main Tutorial Reference¶

FactoMineR Tutorial Husson, F., & Josse, J. http://factominer.free.fr/course/FactoTuto.html "This course presents the main methods used in Exploratory Data Analysis with the FactoMineR package."

Original FactoMineR Article¶

Le, S., Josse, J., & Husson, F. (2008). FactoMineR: An R Package for Multivariate Analysis. Journal of Statistical Software, 25(1), 1-18. DOI: 10.18637/jss.v025.i01

Reference Book¶

Husson, F., Le, S., & Pages, J. (2017). Exploratory Multivariate Analysis by Example Using R. 2^nd Edition. CRC Press. ISBN: 978-1138196346

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Institutional Information¶

Author/Reference: @TodoEconometria Professor: Juan Marcelo Gutierrez Miranda Area: Big Data, Data Science & Applied Econometrics.

Certification Hash ID: 4e8d9b1a5f6e7c3d2b1a0f9e8d7c6b5a4f3e2d1c0b9a8f7e6d5c4b3a2f1e0d9c