[Kelzang Wangdi] - Fab Futures - Data Science

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Math Score Distribution and Gaussian Fit

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
df = pd.read_csv("~/work/kelzang-wangdi/datasets/StudentsPerformance.csv") 
rating_map = df['parental level of education'].value_counts()
df["education_count"] = df['parental level of education'].map(rating_map)

math_score = df['math score'].dropna().values 
npts = len(math_score)
mean = np.mean(math_score)
stddev = np.std(math_score)

plt.figure(figsize=(10,6))

plt.hist(math_score, bins=max(npts//50, 5), density=True, alpha=0.6, color='gray', edgecolor='black')

plt.plot(math_score, 0*math_score, '|', ms=15, color='black')

# Gaussian fit
xi = np.linspace(mean - 3*stddev, mean + 3*stddev, 100)
yi = np.exp(-(xi - mean)**2 / (2*stddev**2)) / np.sqrt(2*np.pi*stddev**2)
plt.plot(xi, yi, 'b', lw=2, label='Gaussian Fit')

plt.xlabel("Math Score")
plt.ylabel("Density")
plt.title("Density Estimation: Gaussian Curve")
plt.legend()

plt.savefig("fig2.1.png", dpi=300, bbox_inches='tight', transparent=True)

plt.show()

Explanation

This density estimation histogram for "Math Score," showing the frequency distribution of observed data (grey bars) overlaid with a fitted Gaussian (Normal) curve (blue line). The data is roughly bell-shaped and unimodal, centered around a score near 75, suggesting that most scores cluster in the mid-to-high range. The blue curve serves as a model for the underlying probability distribution, indicating how well the observed scores follow a theoretical normal distribution.

Density Estimation

Showing K-Means, Voronoi Cluster, Elbow Plot

%matplotlib inline
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.spatial import Voronoi, voronoi_plot_2d
# -----------------------------------------------------------
# 1. LOAD YOUR DATA
# -----------------------------------------------------------
data = pd.read_csv("~/work/kelzang-wangdi/datasets/StudentsPerformance.csv") 
xcol = "reading score"  
ycol = "math score"  

# Extract x and y as NumPy arrays
x = data[xcol].dropna().values
y = data[ycol].dropna().values

# Make sure they have the same length
N = min(len(x), len(y))
x = x[:N]
y = y[:N]

# -----------------------------------------------------------
# 2. K-MEANS PARAMETERS
# -----------------------------------------------------------
nsteps = 1000
momentum = 0.   # not used, kept for compatibility

# -----------------------------------------------------------
# 3. K-MEANS FUNCTION
# -----------------------------------------------------------
def kmeans(x, y, momentum, nclusters):
    indices = np.random.uniform(low=0, high=len(x), size=nclusters).astype(int)
    mux = x[indices]
    muy = y[indices]

    for _ in range(nsteps):
        X = np.outer(x, np.ones(len(mux)))
        Y = np.outer(y, np.ones(len(muy)))
        Mux = np.outer(np.ones(len(x)), mux)
        Muy = np.outer(np.ones(len(x)), muy)

        distances = np.sqrt((X - Mux)**2 + (Y - Muy)**2)
        mins = np.argmin(distances, axis=1)

        for i in range(len(mux)):
            index = np.where(mins == i)
            if len(index[0]) > 0:
                mux[i] = np.mean(x[index])
                muy[i] = np.mean(y[index])

    distances = 0
    for i in range(len(mux)):
        index = np.where(mins == i)
        distances += np.sum(np.sqrt((x[index] - mux[i])**2 + (y[index] - muy[i])**2))

    return mux, muy, distances

# -----------------------------------------------------------
# 4. PLOTTING FUNCTIONS
# -----------------------------------------------------------
def plot_kmeans(x, y, mux, muy):
    fig, ax = plt.subplots()
    ax.plot(x, y, ".", alpha=0.5)
    ax.plot(mux, muy, "r.", markersize=20)
    ax.set_xlabel(xcol)
    ax.set_ylabel(ycol)
    ax.set_title(f"{len(mux)} clusters (centroids)")
    plt.show()

def plot_Voronoi(x, y, mux, muy):
    fig, ax = plt.subplots()
    ax.plot(x, y, ".", alpha=0.5)
    vor = Voronoi(np.stack((mux, muy), axis=1))
    voronoi_plot_2d(vor, ax=ax, show_points=True, show_vertices=False, point_size=20)
    ax.set_xlabel(xcol)
    ax.set_ylabel(ycol)
    ax.set_title(f"{len(mux)} clusters (Voronoi)")
    plt.show()

# -----------------------------------------------------------
# 5. RUN K-MEANS AND PLOT
# -----------------------------------------------------------
distances = []

for k in range(1, 6):
    mux, muy, d = kmeans(x, y, momentum, k)
    distances.append(d)
    
    if k <= 2:
        plot_kmeans(x, y, mux, muy)
    else:
        plot_Voronoi(x, y, mux, muy)

# -----------------------------------------------------------
# 6. ELBOW PLOT
# -----------------------------------------------------------
plt.figure()
plt.plot(range(1, 6), distances, "o-")
plt.xlabel("Number of clusters")
plt.ylabel("Total distance to clusters")
plt.title("Elbow Plot")
plt.xticks(range(1, 6))
plt.show()

K-Means Clustering (3 clusters Voronoi)

Explanation

This graph visualizes the distribution of data points (likely student scores, given the axis labels) and the cluster centers identified by the K-Means algorithm:Axes: The plot shows a relationship between two variables: the reading score on the x-axis and the math score on the y-axis. Each small, blue, translucent dot represents an individual data point (e.g., a student's score pair).Data Distribution: The data appears to be widely dispersed but shows a general positive correlation, meaning students with higher reading scores tend to have higher math scores.Cluster Centers (Centroids): The three large orange circles represent the centroids ($\mu$) or mean centers of the three clusters determined by the K-Means algorithm.One centroid is located in the lower-left of the scatterplot (low reading, low math scores).One centroid is in the center-middle (average scores).One centroid is in the upper-right (high reading, high math scores).Clustering Interpretation: The algorithm has successfully grouped the students into three distinct segments based on their scores, effectively categorizing them into groups like "low-achievers," "average-achievers," and "high-achievers" across both subjects.Voronoi Implication: The title indicates that the visualization is related to a Voronoi diagram. Although the Voronoi boundaries (which would partition the plane into regions around each centroid) are not explicitly drawn, the title confirms that the three orange points are the centers used to define the boundaries of the clusters, where every data point is assigned to the cluster whose centroid it is closest to.This plot is a classic result from an unsupervised machine learning task, where the goal is to identify inherent groupings within the data.