【Advanced】MATLAB Statistical Analysis

# 1. Introduction to MATLAB Statistical Analysis**

The MATLAB Statistics Toolbox provides a comprehensive collection of functions for performing a wide variety of statistical analyses, including descriptive statistics, inferential statistics, and regression analysis. These functions empower researchers and data analysts to process and analyze data with ease and efficiency, thereby uncovering valuable insights.

# 2. Statistical Data Analysis in MATLAB

### 2.1 Descriptive Statistical Analysis

#### 2***

***mon measures include:

- **Mean:** The sum of all data divided by the count of data points. The mean represents the typical value of the data.
- **Median:** The data value in the middle when the data is ordered from smallest to largest. The median is unaffected by extreme values.
- **Mode:** The most frequently occurring data. There may be multiple modes.

Functions in MATLAB to compute measures of central tendency:

% Calculating mean
mean_value = mean(data);

% Calculating median
median_value = median(data);

% Calculating mode
mode_value = mode(data);

#### 2.1.2 ***

***mon measures include:

- **Variance:** The sum of squared deviations from the mean, divided by the count of data points minus one. The larger the variance, the more spread out the data.
- **Standard Deviation:** The square root of the variance. Standard deviation represents the typical deviation from the mean.
- **Coefficient of Variation:** The ratio of the standard deviation to the mean. The coefficient of variation represents the relative spread of the data.

Functions in MATLAB to compute measures of dispersion:

% Calculating variance
variance_value = var(data);

% Calculating standard deviation
standard_deviation = std(data);

% Calculating coefficient of variation
coefficient_of_variation = std(data) / mean(data);

### 2.2 Inferential Statistical Analysis

#### 2.2.1 Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether a given data set supports a particular hypothesis. The process includes:

1. Stating the null hypothesis (H0) and the alternative hypothesis (H1).
2. Determining the significance level (α).
3. Calculating the test statistic.
4. Making decisions based on the test statistic and the significance level.

Functions in MATLAB for performing hypothesis tests:

% T-test
[h, p, ci, stats] = ttest(data1, data2);

% One-way ANOVA
[p, table, stats] = anova1(data, groups);

% Chi-square test
[p, chi2, df] = chi2test(data);

#### 2.2.2 Confidence Intervals

A confidence interval is a statistical method used to estimate the true value of a parameter. It consists of a lower bound and an upper bound, and the probability that the true value lies within the confidence interval is high at a given confidence level.

Functions in MATLAB for calculating confidence intervals:

% Calculating confidence interval for mean
[lower_bound, upper_bound] = meanci(data, confidence_level);

% Calculating confidence interval for proportions
[lower_bound, upper_bound] = binofit(successes, trials, confidence_level);

# 3. Regression Analysis in MATLAB**

### 3.1 Linear Regression Models

#### 3.1.1 Model Fitting and Evaluation

Linear regression models are statistical models used for predicting continuous target variables. They assume a linear relationship between the target variable and the predictor variables. In MATLAB, the `fitlm` function is used to fit linear regression models.

% Data preparation
x = [1, 2, 3, 4, 5];
y = [2, 4, 6, 8, 10];

% Model fitting
model = fitlm(x, y);

% Model evaluation
disp('Model summary:');
disp(model);

The `fitlm` function returns a model object containing detailed information about the fitted model, including regression coefficients, R-squared values, and p-values.

#### 3.1.2 Model Prediction and Diagnostics

After fitting the model, predictions can be made for new data using the `predict` function.

% New data prediction
new_x = 6;
prediction = predict(model, new_x);

The `predict` function returns the predicted value and can be subjected to further analysis and visualization as needed.

To diagnose the goodness-of-fit of the model, residual analysis can be performed. Residuals are the differences between observed and predicted values.

% Residual analysis
residuals = model.Residuals.Raw;
plot(x, residuals, 'o');
xlabel('Independent variable');
ylabel('Residuals');

A residual plot can reveal trends and patterns in the model fit. If residuals are randomly scattered around 0, it indicates a good model fit.

### 3.2 Nonlinear Regression Models

#### 3.2.1 Model Selection and Fitting

Nonlinear regression models are use