均值与方差计算

by luoshi006

均值方差 的数值计算方法。

在阅读 px4 代码时遇到了求取均值与方差的代码实现,比较优美:

https://github.com/PX4/ecl/blob/master/validation/data_validator.cpp#L88-L114

for (unsigned i = 0; i < dimensions; i++) {
        if (_time_last == 0) {
            _mean[i] = 0;
            _lp[i] = val[i];
            _M2[i] = 0;
        } else {
            float lp_val = val[i] - _lp[i];

            float delta_val = lp_val - _mean[i];
            _mean[i] += delta_val / _event_count;
            _M2[i] += delta_val * (lp_val - _mean[i]);
            _rms[i] = sqrtf(_M2[i] / (_event_count - 1));

            if (fabsf(_value[i] - val[i]) < 0.000001f) {
                _value_equal_count++;
            } else {
                _value_equal_count = 0;
            }
        }

        _vibe[i] = _vibe[i] * 0.99f + 0.01f * fabsf(val[i] - _lp[i]);

        // XXX replace with better filter, make it auto-tune to update rate
        _lp[i] = _lp[i] * 0.99f + 0.01f * val[i];

        _value[i] = val[i];
    }


对其分析如下:

以下公式中,约定 vjvj 表示样本中第 jj 个值。

平均值

meann=v1+v2+...+vnn=meann1(n1)+vnn=meann1+vnmeann1n (1)(2)(3)(1)meann=v1+v2+...+vnn(2)=meann−1∗(n−1)+vnn(3)=meann−1+vn−meann−1n 

即:

//此处代码为 PX4 validator 代码:
float delta_val = lp_val - _mean[i];
_mean[i] += delta_val / _event_count;

样本方差

RMS=(vnmeann)2n1RMS=∑(vn−meann)2n−1

方差在统计中,分母为 n1n−1 ,是为了保证估计的无偏性。

//此处代码为 PX4 validator 代码:
_M2[i] += delta_val * (lp_val - _mean[i]);
_rms[i] = sqrtf(_M2[i] / (_event_count - 1));

此处,_M2[i] 的计算公式如下:

M2=(vnmn1)(vnmn)=(vnmn1)(vnmn1vnmn1n)=(n1n)(vnmn1)2(4)(5)(6)(4)M2=(vn−mn−1)(vn−mn)(5)=(vn−mn−1)(vn−mn−1−vn−mn−1n)(6)=(n−1n)(vn−mn−1)2

时间仓促,此公式未查到出处,怀疑与下一行分母的 n1n−1 有关。

存疑,待查证。

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