Computes trigonometric variables at different frequencies.
Arguments
- frequency
Frequency of the series, number of periods per year (12, 4, 3, 2...)
- start, length
First date (array with the first year and the first period, for instance
c(1980, 1)) and number of periods of the output variables. Can also be provided with thesargument- s
time series used to get the dates for the trading days variables. If supplied the parameters
frequency,startandlengthare ignored.- seasonal_frequency
the seasonal frequencies. By default the fundamental seasonal frequency and all the harmonics are used.
Details
Denote by \(P\) the value of frequency (= the period) and
\(f_1\), ..., \(f_n\) the frequencies provides by seasonal_frequency
(if seasonal_frequency = NULL then \(n=\lfloor P/2\rfloor\) and \(f_i\)=i).
trigonometric_variables returns a matrix of size \(length\times(2n)\).
For each date \(t\) associated to the period \(m\) (\(m\in[1,P]\)),
the columns \(2i\) and \(2i-1\) are equal to:
$$
\cos \left(
\frac{2 \pi}{P} \times m \times f_i
\right)
\text{ and }
\sin \left(
\frac{2 \pi}{P} \times m \times f_i
\right)
$$
Take for example the case when the first date (date) is a January, frequency = 12
(monthly time series), length = 12 and seasonal_frequency = NULL.
The first frequency, \(\lambda_1 = 2\pi /12\) represents the fundamental seasonal frequency and the
other frequencies (\(\lambda_2 = 2\pi /12 \times 2\), ..., \(\lambda_6 = 2\pi /12 \times 6\))
are the five harmonics. The output matrix will be equal to:
$$
\begin{pmatrix}
\cos(\lambda_1) & \sin (\lambda_1) & \cdots &
\cos(\lambda_6) & \sin (\lambda_6) \\
\cos(\lambda_1\times 2) & \sin (\lambda_1\times 2) & \cdots &
\cos(\lambda_6\times 2) & \sin (\lambda_6\times 2)\\
\vdots & \vdots & \cdots & \vdots & \vdots \\
\cos(\lambda_1\times 12) & \sin (\lambda_1\times 12) & \cdots &
\cos(\lambda_6\times 12) & \sin (\lambda_6\times 12)
\end{pmatrix}
$$
Examples
trigonometric_variables(
frequency = 12,
length = 480,
start = c(1990, 1),
seasonal_frequency = 3
)
#> Series 1 Series 2
#> Jan 1990 1.000000e+00 -4.311747e-14
#> Feb 1990 -8.335645e-15 1.000000e+00
#> Mar 1990 -1.000000e+00 -2.945342e-15
#> Apr 1990 -2.444961e-15 -1.000000e+00
#> May 1990 1.000000e+00 -7.835264e-15
#> Jun 1990 1.322557e-14 1.000000e+00
#> Jul 1990 -1.000000e+00 1.861587e-14
#> Aug 1990 -2.400617e-14 -1.000000e+00
#> Sep 1990 1.000000e+00 -2.939648e-14
#> Oct 1990 3.478678e-14 1.000000e+00
#> Nov 1990 -1.000000e+00 4.017708e-14
#> Dec 1990 1.127604e-14 -1.000000e+00
#> Jan 1991 1.000000e+00 5.885732e-15
#> Feb 1991 -4.954295e-16 1.000000e+00
#> Mar 1991 -1.000000e+00 4.894873e-15
#> Apr 1991 -1.028518e-14 -1.000000e+00
#> May 1991 1.000000e+00 -1.567548e-14
#> Jun 1991 2.106578e-14 1.000000e+00
#> Jul 1991 -1.000000e+00 2.645608e-14
#> Aug 1991 -3.184639e-14 -1.000000e+00
#> Sep 1991 1.000000e+00 -3.723669e-14
#> Oct 1991 4.262699e-14 1.000000e+00
#> Nov 1991 -1.000000e+00 -8.826123e-15
#> Dec 1991 3.435820e-15 -1.000000e+00
#> Jan 1992 1.000000e+00 -1.954483e-15
#> Feb 1992 7.344786e-15 1.000000e+00
#> Mar 1992 -1.000000e+00 1.273509e-14
#> Apr 1992 -1.812539e-14 -1.000000e+00
#> May 1992 1.000000e+00 -2.351569e-14
#> Jun 1992 2.890600e-14 1.000000e+00
#> Jul 1992 -1.000000e+00 3.429630e-14
#> Aug 1992 -3.968660e-14 -1.000000e+00
#> Sep 1992 1.000000e+00 1.176651e-14
#> Oct 1992 -6.376210e-15 1.000000e+00
#> Nov 1992 -1.000000e+00 -9.859072e-16
#> Dec 1992 -4.404396e-15 -1.000000e+00
#> Jan 1993 1.000000e+00 -9.794698e-15
#> Feb 1993 1.518500e-14 1.000000e+00
#> Mar 1993 -1.000000e+00 2.057530e-14
#> Apr 1993 -2.596561e-14 -1.000000e+00
#> May 1993 1.000000e+00 -3.135591e-14
#> Jun 1993 3.674621e-14 1.000000e+00
#> Jul 1993 -1.000000e+00 4.213652e-14
#> Aug 1993 9.316600e-15 -1.000000e+00
#> Sep 1993 1.000000e+00 3.926298e-15
#> Oct 1993 1.464005e-15 1.000000e+00
#> Nov 1993 -1.000000e+00 6.854308e-15
#> Dec 1993 -1.224461e-14 -1.000000e+00
#> Jan 1994 1.000000e+00 -1.763491e-14
#> Feb 1994 2.302522e-14 1.000000e+00
#> Mar 1994 -1.000000e+00 2.841552e-14
#> Apr 1994 -3.380582e-14 -1.000000e+00
#> May 1994 1.000000e+00 -3.919613e-14
#> Jun 1994 4.458643e-14 1.000000e+00
#> Jul 1994 -1.000000e+00 -6.866688e-15
#> Aug 1994 1.476385e-15 -1.000000e+00
#> Sep 1994 1.000000e+00 -3.913918e-15
#> Oct 1994 9.304221e-15 1.000000e+00
#> Nov 1994 -1.000000e+00 1.469452e-14
#> Dec 1994 -2.008483e-14 -1.000000e+00
#> Jan 1995 1.000000e+00 -2.547513e-14
#> Feb 1995 3.086543e-14 1.000000e+00
#> Mar 1995 -1.000000e+00 3.625574e-14
#> Apr 1995 -4.164604e-14 -1.000000e+00
#> May 1995 1.000000e+00 -4.703634e-14
#> Jun 1995 -4.416775e-15 1.000000e+00
#> Jul 1995 -1.000000e+00 9.735276e-16
#> Aug 1995 -6.363830e-15 -1.000000e+00
#> Sep 1995 1.000000e+00 -1.175413e-14
#> Oct 1995 1.714444e-14 1.000000e+00
#> Nov 1995 -1.000000e+00 2.253474e-14
#> Dec 1995 -2.792504e-14 -1.000000e+00
#> Jan 1996 1.000000e+00 -3.331534e-14
#> Feb 1996 3.870565e-14 1.000000e+00
#> Mar 1996 -1.000000e+00 4.409595e-14
#> Apr 1996 7.357166e-15 -1.000000e+00
#> May 1996 1.000000e+00 1.966863e-15
#> Jun 1996 3.423440e-15 1.000000e+00
#> Jul 1996 -1.000000e+00 8.813743e-15
#> Aug 1996 -1.420405e-14 -1.000000e+00
#> Sep 1996 1.000000e+00 -1.959435e-14
#> Oct 1996 2.498465e-14 1.000000e+00
#> Nov 1996 -1.000000e+00 3.037495e-14
#> Dec 1996 -3.576526e-14 -1.000000e+00
#> Jan 1997 1.000000e+00 -4.115556e-14
#> Feb 1997 4.654586e-14 1.000000e+00
#> Mar 1997 -1.000000e+00 5.193617e-14
#> Apr 1997 -4.830499e-16 -1.000000e+00
#> May 1997 1.000000e+00 -6.271677e-14
#> Jun 1997 1.126366e-14 1.000000e+00
#> Jul 1997 -1.000000e+00 7.349738e-14
#> Aug 1997 -2.204426e-14 -1.000000e+00
#> Sep 1997 1.000000e+00 2.940885e-14
#> Oct 1997 3.282487e-14 1.000000e+00
#> Nov 1997 -1.000000e+00 -1.862825e-14
#> Dec 1997 -4.360547e-14 -1.000000e+00
#> Jan 1998 1.000000e+00 7.847643e-15
#> Feb 1998 5.438608e-14 1.000000e+00
#> Mar 1998 -1.000000e+00 2.932963e-15
#> Apr 1998 -6.516668e-14 -1.000000e+00
#> May 1998 1.000000e+00 -1.371357e-14
#> Jun 1998 7.594729e-14 1.000000e+00
#> Jul 1998 -1.000000e+00 2.449417e-14
#> Aug 1998 2.695894e-14 -1.000000e+00
#> Sep 1998 1.000000e+00 -3.527478e-14
#> Oct 1998 -1.617834e-14 1.000000e+00
#> Nov 1998 -1.000000e+00 4.605539e-14
#> Dec 1998 5.397731e-15 -1.000000e+00
#> Jan 1999 1.000000e+00 -5.683599e-14
#> Feb 1999 5.382875e-15 1.000000e+00
#> Mar 1999 -1.000000e+00 6.761660e-14
#> Apr 1999 -1.616348e-14 -1.000000e+00
#> May 1999 1.000000e+00 -7.839720e-14
#> Jun 1999 2.694409e-14 1.000000e+00
#> Jul 1999 -1.000000e+00 -2.450903e-14
#> Aug 1999 -3.772469e-14 -1.000000e+00
#> Sep 1999 1.000000e+00 1.372842e-14
#> Oct 1999 4.850530e-14 1.000000e+00
#> Nov 1999 -1.000000e+00 -2.947818e-15
#> Dec 1999 -5.928590e-14 -1.000000e+00
#> Jan 2000 1.000000e+00 -7.832788e-15
#> Feb 2000 7.006651e-14 1.000000e+00
#> Mar 2000 -1.000000e+00 1.861339e-14
#> Apr 2000 3.283972e-14 -1.000000e+00
#> May 2000 1.000000e+00 -2.939400e-14
#> Jun 2000 -2.205912e-14 1.000000e+00
#> Jul 2000 -1.000000e+00 4.017460e-14
#> Aug 2000 1.127851e-14 -1.000000e+00
#> Sep 2000 1.000000e+00 -5.095521e-14
#> Oct 2000 -4.979055e-16 1.000000e+00
#> Nov 2000 -1.000000e+00 6.173582e-14
#> Dec 2000 -1.028270e-14 -1.000000e+00
#> Jan 2001 1.000000e+00 -7.251642e-14
#> Feb 2001 2.106331e-14 1.000000e+00
#> Mar 2001 -1.000000e+00 -3.038981e-14
#> Apr 2001 -3.184391e-14 -1.000000e+00
#> May 2001 1.000000e+00 1.960920e-14
#> Jun 2001 4.262452e-14 1.000000e+00
#> Jul 2001 -1.000000e+00 -8.828599e-15
#> Aug 2001 -5.340512e-14 -1.000000e+00
#> Sep 2001 1.000000e+00 -1.952007e-15
#> Oct 2001 6.418573e-14 1.000000e+00
#> Nov 2001 -1.000000e+00 1.273261e-14
#> Dec 2001 -7.496633e-14 -1.000000e+00
#> Jan 2002 1.000000e+00 -2.351322e-14
#> Feb 2002 -2.793990e-14 1.000000e+00
#> Mar 2002 -1.000000e+00 3.429382e-14
#> Apr 2002 1.715929e-14 -1.000000e+00
#> May 2002 1.000000e+00 -4.507443e-14
#> Jun 2002 -6.378686e-15 1.000000e+00
#> Jul 2002 -1.000000e+00 5.585504e-14
#> Aug 2002 -4.401920e-15 -1.000000e+00
#> Sep 2002 1.000000e+00 -6.663564e-14
#> Oct 2002 1.518253e-14 1.000000e+00
#> Nov 2002 -1.000000e+00 7.741625e-14
#> Dec 2002 -2.596313e-14 -1.000000e+00
#> Jan 2003 1.000000e+00 2.548998e-14
#> Feb 2003 3.674374e-14 1.000000e+00
#> Mar 2003 -1.000000e+00 -1.470938e-14
#> Apr 2003 -4.752434e-14 -1.000000e+00
#> May 2003 1.000000e+00 3.928773e-15
#> Jun 2003 5.830495e-14 1.000000e+00
#> Jul 2003 -1.000000e+00 6.851832e-15
#> Aug 2003 -6.908555e-14 -1.000000e+00
#> Sep 2003 1.000000e+00 -1.763244e-14
#> Oct 2003 7.986616e-14 1.000000e+00
#> Nov 2003 -1.000000e+00 2.841304e-14
#> Dec 2003 2.304007e-14 -1.000000e+00
#> Jan 2004 1.000000e+00 -3.919365e-14
#> Feb 2004 -1.225947e-14 1.000000e+00
#> Mar 2004 -1.000000e+00 4.997426e-14
#> Apr 2004 1.478861e-15 -1.000000e+00
#> May 2004 1.000000e+00 -6.075486e-14
#> Jun 2004 9.301745e-15 1.000000e+00
#> Jul 2004 -1.000000e+00 7.153547e-14
#> Aug 2004 -2.008235e-14 -1.000000e+00
#> Sep 2004 1.000000e+00 3.137077e-14
#> Oct 2004 3.086296e-14 1.000000e+00
#> Nov 2004 -1.000000e+00 -2.059016e-14
#> Dec 2004 -4.164356e-14 -1.000000e+00
#> Jan 2005 1.000000e+00 9.809554e-15
#> Feb 2005 5.242417e-14 1.000000e+00
#> Mar 2005 -1.000000e+00 9.710517e-16
#> Apr 2005 -6.320477e-14 -1.000000e+00
#> May 2005 1.000000e+00 -1.175166e-14
#> Jun 2005 7.398538e-14 1.000000e+00
#> Jul 2005 -1.000000e+00 2.253226e-14
#> Aug 2005 2.892085e-14 -1.000000e+00
#> Sep 2005 1.000000e+00 -3.331287e-14
#> Oct 2005 -1.814025e-14 1.000000e+00
#> Nov 2005 -1.000000e+00 4.409347e-14
#> Dec 2005 7.359641e-15 -1.000000e+00
#> Jan 2006 1.000000e+00 -5.487408e-14
#> Feb 2006 3.420964e-15 1.000000e+00
#> Mar 2006 -1.000000e+00 6.565469e-14
#> Apr 2006 -1.420157e-14 -1.000000e+00
#> May 2006 1.000000e+00 -7.643529e-14
#> Jun 2006 2.498218e-14 1.000000e+00
#> Jul 2006 -1.000000e+00 -2.647094e-14
#> Aug 2006 -3.576278e-14 -1.000000e+00
#> Sep 2006 1.000000e+00 1.569033e-14
#> Oct 2006 4.654339e-14 1.000000e+00
#> Nov 2006 -1.000000e+00 -4.909729e-15
#> Dec 2006 -5.732399e-14 -1.000000e+00
#> Jan 2007 1.000000e+00 -5.870877e-15
#> Feb 2007 6.810460e-14 1.000000e+00
#> Mar 2007 -1.000000e+00 1.665148e-14
#> Apr 2007 -7.888520e-14 -1.000000e+00
#> May 2007 1.000000e+00 -2.743209e-14
#> Jun 2007 -2.402103e-14 1.000000e+00
#> Jul 2007 -1.000000e+00 3.821269e-14
#> Aug 2007 1.324042e-14 -1.000000e+00
#> Sep 2007 1.000000e+00 -4.899330e-14
#> Oct 2007 -2.459816e-15 1.000000e+00
#> Nov 2007 -1.000000e+00 5.977391e-14
#> Dec 2007 -8.320789e-15 -1.000000e+00
#> Jan 2008 1.000000e+00 -7.055451e-14
#> Feb 2008 1.910140e-14 1.000000e+00
#> Mar 2008 -1.000000e+00 8.133512e-14
#> Apr 2008 -2.988200e-14 -1.000000e+00
#> May 2008 1.000000e+00 2.157112e-14
#> Jun 2008 4.066261e-14 1.000000e+00
#> Jul 2008 -1.000000e+00 -1.079051e-14
#> Aug 2008 -5.144321e-14 -1.000000e+00
#> Sep 2008 1.000000e+00 9.903690e-18
#> Oct 2008 6.222382e-14 1.000000e+00
#> Nov 2008 -1.000000e+00 1.077070e-14
#> Dec 2008 -7.300442e-14 -1.000000e+00
#> Jan 2009 1.000000e+00 -2.155131e-14
#> Feb 2009 8.378503e-14 1.000000e+00
#> Mar 2009 -1.000000e+00 3.233191e-14
#> Apr 2009 1.912120e-14 -1.000000e+00
#> May 2009 1.000000e+00 -4.311252e-14
#> Jun 2009 -8.340597e-15 1.000000e+00
#> Jul 2009 -1.000000e+00 5.389312e-14
#> Aug 2009 -2.440009e-15 -1.000000e+00
#> Sep 2009 1.000000e+00 -6.467373e-14
#> Oct 2009 1.322061e-14 1.000000e+00
#> Nov 2009 -1.000000e+00 7.545434e-14
#> Dec 2009 -2.400122e-14 -1.000000e+00
#> Jan 2010 1.000000e+00 -8.623494e-14
#> Feb 2010 3.478183e-14 1.000000e+00
#> Mar 2010 -1.000000e+00 -1.667129e-14
#> Apr 2010 -4.556243e-14 -1.000000e+00
#> May 2010 1.000000e+00 5.890684e-15
#> Jun 2010 5.634304e-14 1.000000e+00
#> Jul 2010 -1.000000e+00 4.889921e-15
#> Aug 2010 -6.712364e-14 -1.000000e+00
#> Sep 2010 1.000000e+00 -1.567053e-14
#> Oct 2010 7.790425e-14 1.000000e+00
#> Nov 2010 -1.000000e+00 2.645113e-14
#> Dec 2010 2.500198e-14 -1.000000e+00
#> Jan 2011 1.000000e+00 -3.723174e-14
#> Feb 2011 -1.422138e-14 1.000000e+00
#> Mar 2011 -1.000000e+00 4.801234e-14
#> Apr 2011 3.440772e-15 -1.000000e+00
#> May 2011 1.000000e+00 -5.879295e-14
#> Jun 2011 7.339834e-15 1.000000e+00
#> Jul 2011 -1.000000e+00 6.957356e-14
#> Aug 2011 -1.812044e-14 -1.000000e+00
#> Sep 2011 1.000000e+00 -8.035416e-14
#> Oct 2011 2.890105e-14 1.000000e+00
#> Nov 2011 -1.000000e+00 -2.255207e-14
#> Dec 2011 -3.968165e-14 -1.000000e+00
#> Jan 2012 1.000000e+00 1.177146e-14
#> Feb 2012 5.046226e-14 1.000000e+00
#> Mar 2012 -1.000000e+00 -9.908591e-16
#> Apr 2012 -6.124286e-14 -1.000000e+00
#> May 2012 1.000000e+00 -9.789747e-15
#> Jun 2012 7.202347e-14 1.000000e+00
#> Jul 2012 -1.000000e+00 2.057035e-14
#> Aug 2012 -8.280407e-14 -1.000000e+00
#> Sep 2012 1.000000e+00 -3.135096e-14
#> Oct 2012 -2.010216e-14 1.000000e+00
#> Nov 2012 -1.000000e+00 4.213156e-14
#> Dec 2012 9.321552e-15 -1.000000e+00
#> Jan 2013 1.000000e+00 -5.291217e-14
#> Feb 2013 1.459053e-15 1.000000e+00
#> Mar 2013 -1.000000e+00 6.369278e-14
#> Apr 2013 -1.223966e-14 -1.000000e+00
#> May 2013 1.000000e+00 -7.447338e-14
#> Jun 2013 2.302026e-14 1.000000e+00
#> Jul 2013 -1.000000e+00 8.525399e-14
#> Aug 2013 -3.380087e-14 -1.000000e+00
#> Sep 2013 1.000000e+00 1.765225e-14
#> Oct 2013 4.458148e-14 1.000000e+00
#> Nov 2013 -1.000000e+00 -6.871640e-15
#> Dec 2013 -5.536208e-14 -1.000000e+00
#> Jan 2014 1.000000e+00 -3.908966e-15
#> Feb 2014 6.614269e-14 1.000000e+00
#> Mar 2014 -1.000000e+00 1.468957e-14
#> Apr 2014 -7.692329e-14 -1.000000e+00
#> May 2014 1.000000e+00 -2.547018e-14
#> Jun 2014 8.770390e-14 1.000000e+00
#> Jul 2014 -1.000000e+00 3.625078e-14
#> Aug 2014 1.520233e-14 -1.000000e+00
#> Sep 2014 1.000000e+00 -4.703139e-14
#> Oct 2014 -4.421727e-15 1.000000e+00
#> Nov 2014 -1.000000e+00 5.781199e-14
#> Dec 2014 -6.358879e-15 -1.000000e+00
#> Jan 2015 1.000000e+00 -6.859260e-14
#> Feb 2015 1.713948e-14 1.000000e+00
#> Mar 2015 -1.000000e+00 7.937321e-14
#> Apr 2015 -2.792009e-14 -1.000000e+00
#> May 2015 1.000000e+00 2.353303e-14
#> Jun 2015 3.870070e-14 1.000000e+00
#> Jul 2015 -1.000000e+00 -1.275242e-14
#> Aug 2015 -4.948130e-14 -1.000000e+00
#> Sep 2015 1.000000e+00 1.971814e-15
#> Oct 2015 6.026191e-14 1.000000e+00
#> Nov 2015 -1.000000e+00 8.808791e-15
#> Dec 2015 -7.104251e-14 -1.000000e+00
#> Jan 2016 1.000000e+00 -1.958940e-14
#> Feb 2016 8.182312e-14 1.000000e+00
#> Mar 2016 -1.000000e+00 3.037000e-14
#> Apr 2016 2.108311e-14 -1.000000e+00
#> May 2016 1.000000e+00 -4.115061e-14
#> Jun 2016 -1.030251e-14 1.000000e+00
#> Jul 2016 -1.000000e+00 5.193121e-14
#> Aug 2016 -4.780981e-16 -1.000000e+00
#> Sep 2016 1.000000e+00 -6.271182e-14
#> Oct 2016 1.125870e-14 1.000000e+00
#> Nov 2016 -1.000000e+00 7.349243e-14
#> Dec 2016 -2.203931e-14 -1.000000e+00
#> Jan 2017 1.000000e+00 -8.427303e-14
#> Feb 2017 3.281992e-14 1.000000e+00
#> Mar 2017 -1.000000e+00 -1.863320e-14
#> Apr 2017 -4.360052e-14 -1.000000e+00
#> May 2017 1.000000e+00 7.852595e-15
#> Jun 2017 5.438113e-14 1.000000e+00
#> Jul 2017 -1.000000e+00 2.928011e-15
#> Aug 2017 -6.516173e-14 -1.000000e+00
#> Sep 2017 1.000000e+00 -1.370862e-14
#> Oct 2017 7.594234e-14 1.000000e+00
#> Nov 2017 -1.000000e+00 2.448922e-14
#> Dec 2017 -8.672294e-14 -1.000000e+00
#> Jan 2018 1.000000e+00 -3.526983e-14
#> Feb 2018 -1.618329e-14 1.000000e+00
#> Mar 2018 -1.000000e+00 4.605043e-14
#> Apr 2018 5.402682e-15 -1.000000e+00
#> May 2018 1.000000e+00 -5.683104e-14
#> Jun 2018 5.377923e-15 1.000000e+00
#> Jul 2018 -1.000000e+00 6.761164e-14
#> Aug 2018 -1.615853e-14 -1.000000e+00
#> Sep 2018 1.000000e+00 -7.839225e-14
#> Oct 2018 2.693913e-14 1.000000e+00
#> Nov 2018 -1.000000e+00 8.917286e-14
#> Dec 2018 -3.771974e-14 -1.000000e+00
#> Jan 2019 1.000000e+00 1.373338e-14
#> Feb 2019 4.850035e-14 1.000000e+00
#> Mar 2019 -1.000000e+00 -2.952770e-15
#> Apr 2019 -5.928095e-14 -1.000000e+00
#> May 2019 1.000000e+00 -7.827836e-15
#> Jun 2019 7.006156e-14 1.000000e+00
#> Jul 2019 -1.000000e+00 1.860844e-14
#> Aug 2019 -8.084216e-14 -1.000000e+00
#> Sep 2019 1.000000e+00 -2.938905e-14
#> Oct 2019 9.162277e-14 1.000000e+00
#> Nov 2019 -1.000000e+00 4.016965e-14
#> Dec 2019 1.128346e-14 -1.000000e+00
#> Jan 2020 1.000000e+00 -5.095026e-14
#> Feb 2020 -5.028573e-16 1.000000e+00
#> Mar 2020 -1.000000e+00 6.173086e-14
#> Apr 2020 -1.027775e-14 -1.000000e+00
#> May 2020 1.000000e+00 -7.251147e-14
#> Jun 2020 2.105835e-14 1.000000e+00
#> Jul 2020 -1.000000e+00 8.329208e-14
#> Aug 2020 -3.183896e-14 -1.000000e+00
#> Sep 2020 1.000000e+00 -9.407268e-14
#> Oct 2020 4.261957e-14 1.000000e+00
#> Nov 2020 -1.000000e+00 -8.833550e-15
#> Dec 2020 -5.340017e-14 -1.000000e+00
#> Jan 2021 1.000000e+00 -1.947055e-15
#> Feb 2021 6.418078e-14 1.000000e+00
#> Mar 2021 -1.000000e+00 1.272766e-14
#> Apr 2021 -7.496138e-14 -1.000000e+00
#> May 2021 1.000000e+00 -2.350827e-14
#> Jun 2021 8.574199e-14 1.000000e+00
#> Jul 2021 -1.000000e+00 3.428887e-14
#> Aug 2021 1.716424e-14 -1.000000e+00
#> Sep 2021 1.000000e+00 -4.506948e-14
#> Oct 2021 -6.383638e-15 1.000000e+00
#> Nov 2021 -1.000000e+00 5.585008e-14
#> Dec 2021 -4.396968e-15 -1.000000e+00
#> Jan 2022 1.000000e+00 -6.663069e-14
#> Feb 2022 1.517757e-14 1.000000e+00
#> Mar 2022 -1.000000e+00 7.741130e-14
#> Apr 2022 -2.595818e-14 -1.000000e+00
#> May 2022 1.000000e+00 -8.819190e-14
#> Jun 2022 3.673878e-14 1.000000e+00
#> Jul 2022 -1.000000e+00 -1.471433e-14
#> Aug 2022 -4.751939e-14 -1.000000e+00
#> Sep 2022 1.000000e+00 3.933725e-15
#> Oct 2022 5.830000e-14 1.000000e+00
#> Nov 2022 -1.000000e+00 6.846880e-15
#> Dec 2022 -6.908060e-14 -1.000000e+00
#> Jan 2023 1.000000e+00 -1.762749e-14
#> Feb 2023 7.986121e-14 1.000000e+00
#> Mar 2023 -1.000000e+00 2.840809e-14
#> Apr 2023 -9.064181e-14 -1.000000e+00
#> May 2023 1.000000e+00 -3.918870e-14
#> Jun 2023 -1.226442e-14 1.000000e+00
#> Jul 2023 -1.000000e+00 4.996930e-14
#> Aug 2023 1.483813e-15 -1.000000e+00
#> Sep 2023 1.000000e+00 -6.074991e-14
#> Oct 2023 9.296793e-15 1.000000e+00
#> Nov 2023 -1.000000e+00 7.153051e-14
#> Dec 2023 -2.007740e-14 -1.000000e+00
#> Jan 2024 1.000000e+00 -8.231112e-14
#> Feb 2024 3.085800e-14 1.000000e+00
#> Mar 2024 -1.000000e+00 9.309173e-14
#> Apr 2024 -4.163861e-14 -1.000000e+00
#> May 2024 1.000000e+00 -1.038723e-13
#> Jun 2024 -6.126762e-14 1.000000e+00
#> Jul 2024 -1.000000e+00 9.660999e-16
#> Aug 2024 -6.319982e-14 -1.000000e+00
#> Sep 2024 1.000000e+00 -1.254335e-13
#> Oct 2024 -3.970641e-14 1.000000e+00
#> Nov 2024 -1.000000e+00 2.252731e-14
#> Dec 2024 -8.476103e-14 -1.000000e+00
#> Jan 2025 1.000000e+00 -1.469948e-13
#> Feb 2025 -1.814520e-14 1.000000e+00
#> Mar 2025 -1.000000e+00 4.408852e-14
#> Apr 2025 -1.063222e-13 -1.000000e+00
#> May 2025 1.000000e+00 5.881771e-14
#> Jun 2025 3.416012e-15 1.000000e+00
#> Jul 2025 -1.000000e+00 6.564973e-14
#> Aug 2025 -1.278835e-13 -1.000000e+00
#> Sep 2025 1.000000e+00 3.725650e-14
#> Oct 2025 2.497722e-14 1.000000e+00
#> Nov 2025 -1.000000e+00 8.721095e-14
#> Dec 2025 -1.494447e-13 -1.000000e+00
#> Jan 2026 1.000000e+00 1.569529e-14
#> Feb 2026 4.653844e-14 1.000000e+00
#> Mar 2026 -1.000000e+00 1.087722e-13
#> Apr 2026 5.636780e-14 -1.000000e+00
#> May 2026 1.000000e+00 -5.865925e-15
#> Jun 2026 6.809965e-14 1.000000e+00
#> Jul 2026 -1.000000e+00 1.303334e-13
#> Aug 2026 3.480659e-14 -1.000000e+00
#> Sep 2026 1.000000e+00 -2.742714e-14
#> Oct 2026 8.966086e-14 1.000000e+00
#> Nov 2026 -1.000000e+00 1.518946e-13
#> Dec 2026 1.324537e-14 -1.000000e+00
#> Jan 2027 1.000000e+00 -4.898835e-14
#> Feb 2027 1.112221e-13 1.000000e+00
#> Mar 2027 -1.000000e+00 -5.391788e-14
#> Apr 2027 -8.315838e-15 -1.000000e+00
#> May 2027 1.000000e+00 -7.054956e-14
#> Jun 2027 1.327833e-13 1.000000e+00
#> Jul 2027 -1.000000e+00 -3.235667e-14
#> Aug 2027 -2.987705e-14 -1.000000e+00
#> Sep 2027 1.000000e+00 -9.211077e-14
#> Oct 2027 1.543445e-13 1.000000e+00
#> Nov 2027 -1.000000e+00 -1.079546e-14
#> Dec 2027 -5.143826e-14 -1.000000e+00
#> Jan 2028 1.000000e+00 -1.136720e-13
#> Feb 2028 -5.146797e-14 1.000000e+00
#> Mar 2028 -1.000000e+00 1.076575e-14
#> Apr 2028 -7.299947e-14 -1.000000e+00
#> May 2028 1.000000e+00 -1.352332e-13
#> Jun 2028 -2.990676e-14 1.000000e+00
#> Jul 2028 -1.000000e+00 3.232696e-14
#> Aug 2028 -9.456068e-14 -1.000000e+00
#> Sep 2028 1.000000e+00 -1.567944e-13
#> Oct 2028 -8.345549e-15 1.000000e+00
#> Nov 2028 -1.000000e+00 5.388817e-14
#> Dec 2028 -1.161219e-13 -1.000000e+00
#> Jan 2029 1.000000e+00 4.901806e-14
#> Feb 2029 1.321566e-14 1.000000e+00
#> Mar 2029 -1.000000e+00 7.544938e-14
#> Apr 2029 -1.376831e-13 -1.000000e+00
#> May 2029 1.000000e+00 2.745685e-14
#> Jun 2029 3.477687e-14 1.000000e+00
#> Jul 2029 -1.000000e+00 9.701060e-14
#> Aug 2029 6.812936e-14 -1.000000e+00
#> Sep 2029 1.000000e+00 5.895636e-15
#> Oct 2029 5.633809e-14 1.000000e+00
#> Nov 2029 -1.000000e+00 1.185718e-13
#> Dec 2029 4.656815e-14 -1.000000e+00