User:
Mode analysis
Storyboard 
The correlation only determines similarity of the shape of the segment data without considering the absolute values.
It can be assumed that due to climate change and other mechanisms there is a possibility that the magnitudes of the parameters will vary over time.
For this reason, it is advisable to calculate how the magnitudes have varied between the reference segment and the one with the best correlation.
With this information, the historical data of the segment that will be used for the forecast can be scaled.
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Calculate coefficient $X_k$
Equation
To estimate the integral
| $ X_k = \displaystyle\frac{1}{ T } \displaystyle\int_{0}^{ T } x(t) e^{ i 2 \pi \nu_k t } dt$ |
you can discretize the function
| $ X_k = \displaystyle\frac{1}{ T } \displaystyle\sum_{ n =0}^{ N -1} x_n e^{ i 2 \pi \nu_k n \Delta t }$ |
ID:(14354, 0)
Coefficient in complex form
Equation
The coefficients of the Fourier transform
| $ x(t) = \displaystyle\sum_{k=-\infty}^{\infty}( a_k \cos 2 \pi \nu_k t + b_k \sin 2 \pi \nu_k t )$ |
can be regrouped as a complex number by defining
| $ X_k = a_k - i b_k $ |
ID:(14352, 0)
Magnitudes of the modes
Equation
If the complex coefficient is
| $ X_k = a_k - i b_k $ |
Its magnitude is defined as
| $ r_k = \sqrt{ a_k ^2 + b_k ^2}$ |
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Modes phase
Equation
If the complex coefficient is
| $ X_k = a_k - i b_k $ |
the phase can be calculated from
| $ \phi_k = \arctan\displaystyle\frac{ b_k }{ a_k }$ |
ID:(14356, 0)