Mean value, variance, mean square value, mean variance calculation and their physical meaning in digital signal processing

1 mean value

The mean value represents the size of the DC component in the signal, expressed in E (x). For a Gaussian white noise signal, its mean value is 0, so it has only the AC component.

2 square of the mean value

The square of the mean, denoted by {E (x)}^2, represents the power of the DC component in the signal.

3 mean square value

The mean square value represents the mean value of the signal squared, denoted by E (x^2). The mean square value represents the average power of the signal. Average power of signal = signal AC component Power + signal DC component power

For example: x, Y, Z 3 to find the mean square value. Mean square value = (square of square +y of x squared +z)/3


RMS (root mean square), the square root of the mean squared value of 4
5 mean Variance

Mean variance (mean square error), expressed in MSE. The mean variance is the average of the sum of the squares of the distance from the true value of each data, that is, the mean square sum of the error, the formula is close to the variance, its root is called the RMS error, and the RMS error is close to the standard deviation form. Mean variance is sometimes considered equal to variance

6 root mean square error

The RMS error is indicated by Rmse (root mean square error). It is the square root of the square and the number of observations n ratio of the observed value and the truth deviation, in the actual measurement, the number of observations n is always limited, the truth value can only be replaced by the most reliable (best) values. The root-square error is very sensitive to the large or small errors in a set of measurements, so the RMS error can reflect The RMS error is sometimes considered a standard deviation.

6 Variance

Variance is expressed in variance or deviation or var. Variance describes the fluctuation range of the signal, indicating the strength of the AC component in the signal, i.e. the average power of the AC signal.

or expressed in a formula as

Note that the above is divided by the n-1, only so the variance estimated by the sample value is unbiased, that is, the expectation of the above is the variance of x. But some places are also useful to divide by N to represent variance, except that the result is not unbiased estimation of variance, the mathematical expectation of the result is not the variance of x, but the X-variance.

7 Standard deviation

Standard deviation (Deviation) is expressed in σ, and sometimes the standard deviation can be called the RMS error RMSE. The standard deviation is the average distance from the average deviation of the data, which is the square root of the squared and average deviation, and the standard deviation can reflect the degree of dispersion of a data set by σ.

The standard deviation σ, which reflects the degree to which the measured data deviates from the true value, the smaller the σ, indicates the higher the accuracy of the measurement, so σ can be used as a criterion for measuring the accuracy of this measurement process.

with the variance why use standard deviation. What is the advantage of the standard deviation analogy difference?

Because the variance is inconsistent with the dimension of the data we are dealing with, although it can well describe the degree of deviation of the data from the mean, the processing result is not in accordance with our intuitive thinking.
For example: There are 60 students in a class, the average score is 70 points, the standard deviation is 9, the variance is 81, the results obey the normal distribution, then we can not intuitively determine the variance of the class students and the mean deviation of how many points, through the standard deviation we are very intuitive to get student results distributed in [61,79] The probability of a range is 0.6826, which is approximately equal to the 34.2%*2 in the image below.

Summary:

(1) In general, the mean variance, RMS error and variance, the standard deviation can not be equated, although their formulas are similar. We need to differentiate them from the relationship between real and mean values.

(2) for variance and standard deviation, they reflect the relationship between the data sequence and the mean value.

(3) for mean and root mean square errors, they reflect the relationship between the data sequence and the real value.