# Boost Math - Statistics

## Statistics

The Statistics section of the Boost Math library cover a broad range of areas

### Univariate Statistics

    # Mean
    mean_boost(c(1, 2, 3, 4, 5))
    #> [1] 3
    # Variance
    variance(c(1, 2, 3, 4, 5))
    #> [1] 2
    # Sample Variance
    sample_variance(c(1, 2, 3, 4, 5))
    #> [1] 2.5
    # Mean and Sample Variance
    mean_and_sample_variance(c(1, 2, 3, 4, 5))
    #> [1] 3.0 2.5
    # Skewness
    skewness(c(1, 2, 3, 4, 5))
    #> [1] 0
    # Kurtosis
    kurtosis(c(1, 2, 3, 4, 5))
    #> [1] 1.7
    # Excess Kurtosis
    excess_kurtosis(c(1, 2, 3, 4, 5))
    #> [1] -1.3
    # First Four Moments
    first_four_moments(c(1, 2, 3, 4, 5))
    #> [1] 3.0 2.0 0.0 6.8
    # Median
    median_boost(c(1, 2, 3, 4, 5))
    #> [1] 3
    # Median Absolute Deviation
    median_absolute_deviation(c(1, 2, 3, 4, 5))
    #> [1] 1
    # Interquartile Range
    interquartile_range(c(1, 2, 3, 4, 5))
    #> [1] 3
    # Gini Coefficient
    gini_coefficient(c(1, 2, 3, 4, 5))
    #> [1] 0.2666667
    # Sample Gini Coefficient
    sample_gini_coefficient(c(1, 2, 3, 4, 5))
    #> [1] 0.3333333
    # Mode
    mode(c(1, 2, 2, 3, 4))
    #> [1] 2

### Bivariate Statistics

    # Covariance
    covariance(c(1, 2, 3), c(4, 5, 6))
    #> [1] 0.6666667
    # Means and Covariance
    means_and_covariance(c(1, 2, 3), c(4, 5, 6))
    #> [1] 2.0000000 5.0000000 0.6666667
    # Correlation Coefficient
    correlation_coefficient(c(1, 2, 3), c(4, 5, 6))
    #> [1] 1

### Signal Statistics

    # Absolute Gini Coefficient
    absolute_gini_coefficient(c(1, 2, 3, 4, 5))
    #> [1] 0.2666667
    # Sample Absolute Gini Coefficient
    sample_absolute_gini_coefficient(c(1, 2, 3, 4, 5))
    #> [1] 0.3333333
    # Hoyer Sparsity
    hoyer_sparsity(c(1, 0, 0, 2, 3))
    #> [1] 0.5117037

    noisy_signal <- c(1.1, 2.1, 3.1)
    # Oracle SNR
    oracle_snr(c(1, 2, 3), c(1.1, 2.1, 3.1))
    #> [1] 466.6667
    # Oracle SNR in dB
    oracle_snr_db(c(1, 2, 3), c(1.1, 2.1, 3.1))
    #> [1] 26.69007
    # M2M4 SNR Estimator in dB
    m2m4_snr_estimator(c(1.1, 2.1, 3.1), 3, 2)
    #> [1] NaN
    # M2M4 SNR Estimator in dB
    m2m4_snr_estimator_db(c(1.1, 2.1, 3.1), 3, 2)
    #> [1] NaN

### Anderson-Darling Test

    # Anderson-Darling test for normality
    anderson_darling_normality_statistic(c(1, 2, 3, 4, 5), 0, 1)
    #> [1] 19.49684

### T-Tests

    # One Sample T-Test with parameters
    one_sample_t_test_params(sample_mean = 2, sample_variance = 1, num_samples = 30, assumed_mean = 0)
    #> [1] 1.095445e+01 8.021287e-12
    # One Sample T-Test
    one_sample_t_test(c(1, 2, 3, 4, 5), assumed_mean = 0)
    #> [1] 4.2426407 0.0132356
    # Two Sample T-Test
    two_sample_t_test(c(1, 2, 3), c(4, 5, 6))
    #> [1] -3.67423461  0.02131164
    # Paired Samples T-Test
    paired_samples_t_test(c(1, 2, 3), c(4, 5, 6))
    #> [1] -Inf    0

### Z-Tests

    # One Sample Z-Test with parameters
    one_sample_z_test_params(sample_mean = 2, sample_variance = 1, num_samples = 30, assumed_mean = 0)
    #> [1] 1.095445e+01 8.021287e-12
    # One Sample Z-Test
    one_sample_z_test(c(1, 2, 3, 4, 5), assumed_mean = 0)
    #> [1] 4.2426407 0.0132356
    # Two Sample ZTest
    two_sample_z_test(c(1, 2, 3), c(4, 5, 6))
    #> [1] -3.67423461  0.02131164

### Runs Tests

    # Runs Above and Below Threshold
    runs_above_and_below_threshold(c(1, 2, 3, 4, 5), threshold = 3)
    #> [1] -1.2247449  0.2206714
    #' # Runs Above and Below Median
    runs_above_and_below_median(c(1, 2, 3, 4, 5))
    #> [1] -1.2247449  0.2206714

### Ljung-Box Tests

    # Ljung-Box test for autocorrelation
    ljung_box(c(1, 2, 3, 4, 5), lags = 2, fit_dof = 0)
    #> [1] 1.5166667 0.4684465

### Linear Regression

    x <- c(1, 2, 3, 4, 5)
    y <- c(2, 3, 5, 7, 11)
    # Simple Ordinary Least Squares
    simple_ordinary_least_squares(x, y)
    #> [1] -1.0  2.2
    # Simple Ordinary Least Squares with R-squared
    simple_ordinary_least_squares_with_R_squared(x, y)
    #> [1] -1.0000000  2.2000000  0.9453125