# RMSD - Root Mean Squared Deviations

Calculates the root mean squared deviations (aka., root mean squared error (RMSE)) function.

## Syntax

RMSD(X, Y, Ret_type)
X
is the original (eventual outcomes) time series sample data (a one dimensional array of cells (e.g. rows or columns)).
Y
is the forecast time series data (a one dimensional array of cells (e.g. rows or columns)).
Ret_type
is a switch to select the return output (1= RMSD (default), 2= NRMSD, 3= CV(RMSD)).
Order Description
1 RMSD (default)
2 Normalized RMSD (NRMSD)
3 Coefficient of Variation of the RMSD (CV(RMSD))

## Remarks

1. The RMSD is also known as root mean squared error (RMSE).
2. The RMSD compares differences between two data sets, neither of which is accepted as the "standard or actual."
3. The time series is homogeneous or equally spaced.
4. The two time-series must be identical in size.
5. The root mean squared errors (deviations) function is defined as follows:
$$\mathrm{RMSD}=\mathrm{RMSE}=\sqrt{\frac{\mathrm{SSE}}{N}}=\sqrt{\frac{\sum_{i=1}^N (x_i - \hat x_i)^2}{N}}$$
Where:
• $\{x_i\}$ is the actual observations time series
• $\{\hat x_i\}$ is the estimated or forecasted time series
• $N$ is the number of non-missing data points
6. The normalized root-mean deviation (or errors) function is defined as follows:
$$\mathrm{NRMSD}=\frac{\mathrm{RMSD}}{x_{\mathrm{max}}-x_{\mathrm{min}}}$$
Where:
• $x_{\mathrm{max}}$ is the maximum value in the first time series
• $x_{\mathrm{min}}$ is the minimum value in the first time series
7. The coefficient of variation of the RMSD $\mathrm{CV_{RMSD}}$ is defined as follows:
$$\mathrm{CV(RMSD)}=\frac{\mathrm{RMSD}}{\bar x}$$
Where:
• $\bar x$ is the mean of the first (observed) time series

## Examples

Example 1:

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A B C
Date Series1 Series2
1/1/2008 #N/A -2.61
1/2/2008 -2.83 -0.28
1/3/2008 -0.95 -0.90
1/4/2008 -0.88 -1.72
1/5/2008 1.21 1.92
1/6/2008 -1.67 -0.17
1/7/2008 0.83 -0.04
1/8/2008 -0.27 1.63
1/9/2008 1.36 -0.12
1/10/2008 -0.34 0.14
1/11/2008 0.48 -1.96
1/12/2008 -2.83 1.30
1/13/2008 -0.95 -2.51
1/14/2008 -0.88 -0.93
1/15/2008 1.21 0.39
1/16/2008 -1.67 -0.06
1/17/2008 -2.99 -1.29
1/18/2008 1.24 1.41
1/19/2008 0.64 2.37

Formula Description (Result)
=RMSD($B$1:$B$19,$C$1:$C$19,1) RMSD (1.689)
=NRMSD($B$1:$B$19,$C$1:$C$19,2) Normalized RMSD (1.689)
=CV(RMSD)($B$1:$B$19,$C$1:$C$19,3) Coefficient of Variation of the RMSD (1.689)