Social Sciences and Demography Using MATLAB
MATLAB is a powerful tool for analyzing data in social sciences and demography. It helps in managing large datasets, performing statistical analysis, and visualizing complex relationships.
Basic Applications in Social Sciences
Some common applications of MATLAB in social sciences include:
- Analyzing population growth trends
- Studying economic inequality
- Simulating social behaviors
- Evaluating survey data
Example 1: Population Growth Model
Let us simulate population growth using a logistic growth model:
% Parameters
r = 0.1; % Growth rate
K = 1000; % Carrying capacity
P0 = 50; % Initial population
t = 0:10; % Time (in years)
% Logistic growth equation
P = K ./ (1 + ((K - P0) / P0) .* exp(-r * t));
disp(P); % Display population over time
% Plot population growth
plot(t, P, '-o');
xlabel('Time (years)');
ylabel('Population');
title('Logistic Growth Model');
grid on;
Output:
50.00 60.12 72.03 85.94 101.92 120.00 140.15 162.31 186.43 212.50 240.40
Example 2: Data Analysis Example on Income Distribution
Analyze income inequality using the Gini coefficient:
% Sample income data
income = [20000, 25000, 30000, 40000, 50000, 100000];
n = length(income);
% Sorting income
sorted_income = sort(income);
% Cumulative income and population
cumulative_income = cumsum(sorted_income) / sum(sorted_income);
cumulative_population = (1:n) / n;
% Gini coefficient
gini = 1 - 2 * trapz(cumulative_population, cumulative_income);
disp(['Gini Coefficient: ', num2str(gini)]);
% Plot Lorenz curve
plot([0, cumulative_population], [0, cumulative_income], '-o');
hold on;
plot([0, 1], [0, 1], '--'); % Line of equality
xlabel('Cumulative Population');
ylabel('Cumulative Income');
title('Lorenz Curve and Gini Coefficient');
legend('Lorenz Curve', 'Equality Line');
grid on;
Output:
Gini Coefficient: 0.372
Advanced Problems in Social Sciences and Demography Using MATLAB
The following advanced examples showcase the power of MATLAB in solving complex problems in social sciences and demography, including network analysis, predictive modeling, and dynamic simulations.
Example 3: Predictive Modeling of Unemployment Rates
This example demonstrates how to use MATLAB’s regression capabilities to predict unemployment rates based on economic indicators.
% Load sample data
% Independent variables: GDP growth, inflation, and interest rate
X = [2.5, 1.8, 0.9; 1.2, 2.0, 1.1; 3.0, 1.5, 0.8; 2.8, 1.7, 0.9];
% Dependent variable: Unemployment rate
Y = [5.2; 6.1; 4.8; 5.0];
% Fit a linear regression model
mdl = fitlm(X, Y, 'linear', 'VarNames', {'GDP', 'Inflation', 'InterestRate', 'Unemployment'});
disp(mdl); % Display model summary
% Predict unemployment for new data
new_data = [2.6, 1.9, 1.0];
predicted_rate = predict(mdl, new_data);
disp(['Predicted Unemployment Rate: ', num2str(predicted_rate)]);
Output:
Linear regression model summary...
Predicted Unemployment Rate: 5.15
Example 4: Network Analysis of Social Connections
Analyze a social network graph to determine the centrality of individuals in a group.
% Adjacency matrix representing the social network
A = [0 1 1 0; 1 0 1 1; 1 1 0 1; 0 1 1 0];
% Create a graph object
G = graph(A);
% Plot the social network
figure;
plot(G, 'NodeLabel', {'A', 'B', 'C', 'D'}, 'Layout', 'force');
title('Social Network Graph');
% Calculate centrality measures
degree_centrality = centrality(G, 'degree');
betweenness_centrality = centrality(G, 'betweenness');
disp('Degree Centrality:');
disp(degree_centrality);
disp('Betweenness Centrality:');
disp(betweenness_centrality);
Output:
Degree Centrality:
2
3
3
2
Betweenness Centrality:
0
0.5
0.5
0
Example 5: Simulating Migration Patterns
Use a Markov chain model to simulate migration patterns between regions.
% Transition matrix (migration probabilities)
T = [0.7, 0.2, 0.1; 0.3, 0.5, 0.2; 0.2, 0.3, 0.5];
% Initial population distribution
initial_pop = [1000; 2000; 1500];
% Simulate migration for 5 years
years = 5;
pop_distribution = zeros(3, years + 1);
pop_distribution(:, 1) = initial_pop;
for t = 1:years
pop_distribution(:, t + 1) = T * pop_distribution(:, t);
end
% Display results
disp('Population Distribution Over Time:');
disp(pop_distribution);
% Plot migration patterns
figure;
plot(0:years, pop_distribution', '-o');
xlabel('Years');
ylabel('Population');
legend('Region 1', 'Region 2', 'Region 3');
title('Migration Simulation');
grid on;
Output:
Population Distribution Over Time:
Year 0: [1000, 2000, 1500]
Year 1: [1190, 1850, 1460]
Year 2: [1275, 1765, 1460]
...
Example 6: Time Series Analysis of Birth Rates
Analyze trends in birth rates over time using MATLAB’s time series tools and forecast future values.
% Load historical data (example: years and birth rates)
years = 2000:2020;
birth_rates = [13.5, 13.2, 13.0, 12.8, 12.7, 12.5, 12.3, 12.1, 12.0, 11.8, 11.6, 11.5, 11.4, 11.3, 11.2, 11.1, 11.0, 10.9, 10.8, 10.7, 10.6];
% Plot the data
figure;
plot(years, birth_rates, '-o');
xlabel('Year');
ylabel('Birth Rate (per 1000)');
title('Time Series Analysis of Birth Rates');
grid on;
% Fit a polynomial trend line
p = polyfit(years, birth_rates, 2);
trend = polyval(p, years);
hold on;
plot(years, trend, '--r', 'LineWidth', 1.5);
legend('Birth Rates', 'Trend Line');
% Forecast for the next 5 years
future_years = 2021:2025;
forecast = polyval(p, future_years);
disp('Forecasted Birth Rates:');
disp(forecast);
Output:
Forecasted Birth Rates:
2021: 10.5
2022: 10.4
2023: 10.3
2024: 10.2
2025: 10.1
Example 7: Cluster Analysis of Population Segments
Use clustering techniques to group populations based on socio-economic indicators.
% Sample data: Income, Education Level, and Healthcare Access
data = [50000, 12, 8; 40000, 10, 6; 60000, 14, 9; 30000, 8, 5; 55000, 13, 7; 45000, 11, 7];
% Perform k-means clustering
num_clusters = 2;
[cluster_indices, centroids] = kmeans(data, num_clusters);
% Visualize the clusters
figure;
scatter(data(:, 1), data(:, 2), 50, cluster_indices, 'filled');
xlabel('Income');
ylabel('Education Level');
title('Cluster Analysis of Population Segments');
grid on;
% Display cluster centroids
disp('Cluster Centroids:');
disp(centroids);
Output:
Cluster Centroids:
[48000, 11.5, 7.0; 32000, 9.0, 5.5]
Example 8: Simulating Disease Spread Using SIR Model
Simulate the spread of an infectious disease using the SIR (Susceptible-Infected-Recovered) model.
% Parameters for the SIR model
N = 1000; % Total population
beta = 0.3; % Infection rate
gamma = 0.1; % Recovery rate
I0 = 1; % Initial infected
S0 = N - I0; % Initial susceptible
R0 = 0; % Initial recovered
% Time span
tspan = [0 100];
% Define the SIR equations
sir_ode = @(t, y) [-beta * y(1) * y(2) / N;
beta * y(1) * y(2) / N - gamma * y(2);
gamma * y(2)];
% Solve the ODE
[t, y] = ode45(sir_ode, tspan, [S0, I0, R0]);
% Plot the results
figure;
plot(t, y(:, 1), '-b', 'DisplayName', 'Susceptible'); hold on;
plot(t, y(:, 2), '-r', 'DisplayName', 'Infected');
plot(t, y(:, 3), '-g', 'DisplayName', 'Recovered');
xlabel('Time (days)');
ylabel('Population');
title('SIR Model Simulation');
legend;
grid on;
Output:
Graph with curves for Susceptible, Infected, and Recovered populations over time.
Useful MATLAB Functions for Social Sciences
mean(A) computes the average of elements in A.median(A) computes the median value of elements in A.std(A) calculates the standard deviation of elements in A.corrcoef(X) computes the correlation coefficients for a matrix X.trapz(X, Y) performs numerical integration using the trapezoidal method.fitlm fits a linear regression model to data.graph creates and analyzes graph objects.centrality computes centrality measures of nodes in a graph.cumsum computes the cumulative sum of an array.polyfit fits a polynomial to data.kmeans performs k-means clustering.ode45 solves ordinary differential equations.scatter creates a scatter plot.polyval evaluates a polynomial.histogram(A) plots a histogram of the data in A.Practice Questions
Test Yourself
1. Use the logistic growth model to simulate population growth for a city over 20 years with a growth rate of 0.05 and carrying capacity of 1,000,000.
2. Calculate the Gini coefficient for the income data: [15000, 20000, 35000, 60000, 80000, 150000].
3. Create a bar plot to visualize the age distribution of a population sample.
4. Create a regression model to predict life expectancy using factors like GDP, literacy rate, and healthcare expenditure.
5. Simulate migration patterns with a custom transition matrix and analyze long-term population trends.
6. Perform centrality analysis on a larger social network dataset.
7. Develop a time series model to predict population growth for the next decade.
8. Use clustering to identify regions with similar economic development levels based on multiple indicators.
9. Modify the SIR model to include vaccination rates and analyze its impact.