1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140

/// Estimate the arithmetic mean, the variance and the skewness of a sequence of
/// numbers ("population").
///
/// This can be used to estimate the standard error of the mean.
#[derive(Debug, Clone)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct Skewness {
    /// Estimator of mean and variance.
    avg: MeanWithError,
    /// Intermediate sum of cubes for calculating the skewness.
    sum_3: f64,
}

impl Skewness {
    /// Create a new skewness estimator.
    #[inline]
    pub fn new() -> Skewness {
        Skewness {
            avg: MeanWithError::new(),
            sum_3: 0.,
        }
    }

    /// Increment the sample size.
    ///
    /// This does not update anything else.
    #[inline]
    fn increment(&mut self) {
        self.avg.increment();
    }

    /// Add an observation given an already calculated difference from the mean
    /// divided by the number of samples, assuming the inner count of the sample
    /// size was already updated.
    ///
    /// This is useful for avoiding unnecessary divisions in the inner loop.
    #[inline]
    fn add_inner(&mut self, delta: f64, delta_n: f64) {
        // This algorithm was suggested by Terriberry.
        //
        // See https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance.
        let n = f64::approx_from(self.len()).unwrap();
        let term = delta * delta_n * (n - 1.);
        self.sum_3 += term * delta_n * (n - 2.)
            - 3.*delta_n * self.avg.sum_2;
        self.avg.add_inner(delta_n);
    }

    /// Determine whether the sample is empty.
    #[inline]
    pub fn is_empty(&self) -> bool {
        self.avg.is_empty()
    }

    /// Estimate the mean of the population.
    ///
    /// Returns 0 for an empty sample.
    #[inline]
    pub fn mean(&self) -> f64 {
        self.avg.mean()
    }

    /// Return the sample size.
    #[inline]
    pub fn len(&self) -> u64 {
        self.avg.len()
    }

    /// Calculate the sample variance.
    ///
    /// This is an unbiased estimator of the variance of the population.
    #[inline]
    pub fn sample_variance(&self) -> f64 {
        self.avg.sample_variance()
    }

    /// Calculate the population variance of the sample.
    ///
    /// This is a biased estimator of the variance of the population.
    #[inline]
    pub fn population_variance(&self) -> f64 {
        self.avg.population_variance()
    }

    /// Estimate the standard error of the mean of the population.
    #[inline]
    pub fn error_mean(&self) -> f64 {
        self.avg.error()
    }

    /// Estimate the skewness of the population.
    #[inline]
    pub fn skewness(&self) -> f64 {
        if self.sum_3 == 0. {
            return 0.;
        }
        let n = f64::approx_from(self.len()).unwrap();
        let sum_2 = self.avg.sum_2;
        debug_assert_ne!(sum_2, 0.);
        n.sqrt() * self.sum_3 / (sum_2*sum_2*sum_2).sqrt()
    }
}

impl Default for Skewness {
    fn default() -> Skewness {
        Skewness::new()
    }
}

impl Estimate for Skewness {
    #[inline]
    fn add(&mut self, x: f64) {
        let delta = x - self.mean();
        self.increment();
        let n = f64::approx_from(self.len()).unwrap();
        self.add_inner(delta, delta/n);
    }

    #[inline]
    fn estimate(&self) -> f64 {
        self.skewness()
    }
}

impl Merge for Skewness {
    #[inline]
    fn merge(&mut self, other: &Skewness) {
        let len_self = f64::approx_from(self.len()).unwrap();
        let len_other = f64::approx_from(other.len()).unwrap();
        let len_total = len_self + len_other;
        let delta = other.mean() - self.mean();
        let delta_n = delta / len_total;
        self.sum_3 += other.sum_3
            + delta*delta_n*delta_n * len_self*len_other*(len_self - len_other)
            + 3.*delta_n * (len_self * other.avg.sum_2 - len_other * self.avg.sum_2);
        self.avg.merge(&other.avg);
    }
}

impl_from_iterator!(Skewness);