Hi Khairul

This is something I use to model seasonality in interrupted time series analyses. I got my approach from Article Time series analysis for psychological research: Examining a...

Here's the relevant text from Jebb's supplementary materials, which uses R code:

"Harmonic seasonal models.

A harmonic seasonal model is more complex than a model with an indicator per season, but will usually be more parameter efficient and provide a more realistic approximation. (This section was adapted from Cowpertwait & Metcalfe, 2009, pp. 103-104). First, a variable representing time should be created.

time = seq(from = 1, to = 90, by = 1)

Then, predictors with either a sine or cosine term should be created that represent the seasonal pattern. The formula for this values comes from Equation 9, where, for S seasons, there are S/2 s and c coefficients (here, six of each).

sin1 = sin(2 * pi * 1 * time/12)

cos1 = cos(2 * pi * 1 * time/12)

sin2 = sin(2 * pi * 2 * time/12)

cos2 = cos(2 * pi * 2 * time/12)

...

sin6 = sin(2 * pi * 6 * time/12)

cos6 = cos(2 * pi * 6 * time/12)

The series is then regressed on all of these terms.

model = gls(ts ~ time + sin1 + cos1 + sin2 + cos2 + sin3 + cos3 + sin4 + cos4 + sin5 + cos5 + sin6 + cos6)

The statistically significant coefficients can be identified through the summary function. summary(model) Only the statistically significant coefficients in the model should be retained. For our data, only five were kept, and the model was re-specified.

model = gls(ts ~ time + cos1 + sin2 + cos3 + sin3 + cos5)"

The important thing is to use pairs of sine and cosine terms. If you're using monthly data, it will be six sine and six cosine pairs; if quarterly, two sine and two cosine pairs. Similarly number at the end of the equation is just how many units are in your cycle. If months, the number is 12; if quarters, the number is 4.

The tutorial in Article Interrupted time series regression for the evaluation of pub...

A finite Fourier representation is adequate for a strict periodic seasonal component. This is a very strong assumption for a time series model. This is why softer forms like SARIMA models are often preferred.

Imdeed Spectral Analysis is frequently used în Time series analysis but SARIMA more versatile for the serie with Stochasic Trend!

Additionally, the Fourier series is the trigonometric sum of any periodic expression. However, since time series or financial data are non-periodic, a Fourier transform is applied to the sum of simple trigomometric waves. Thus, there is no loss of information with time-to-frequency conversions. This is an evaluation of the effect of variance for a given frequency with spectral representations.

Hi professors. I hope you are doing well. This is a mathematical article that proves a new recurrence relation that is fundamental for mathematics. The article proves also that four infinite series are equivalent. Hence, this article opens new opportunities to demonstrate and develop new mathematical findings and observations. This is the link: https://www.researchgate.net/publication/364651911_A_useful_new_equation_of_four_infinite_series_and_sums_by_using_a_new_demonstrated_recurrence_relation