Spectral Analysis Functions

Spectral Transfer Function using Welch's Method

SPECTRAL_TRANSFER_FUNCTION_WELCH

Overview

SPECTRAL_TRANSFER_FUNCTION_WELCH estimates the frequency-domain relationship between two time series using Welch's averaged periodogram method. It can reveal which repeating cycles are shared, how strongly the series are related at each cycle length, and whether one series tends to lead or lag the other.

Welch's method divides the data into overlapping segments, estimates the spectrum for each segment, and averages the results. This reduces noise and generally produces a more stable estimate than a single-periodogram calculation.

Syntax

result = SPECTRAL_TRANSFER_FUNCTION_WELCH(series1, series2)
extract(result, 0)

Parameters

Parameter	Description
`series1`	The input or explanatory time series.
`series2`	The output or response time series.

The two series should normally use the same observation frequency and cover a compatible date range.

Example

str = SPECTRAL_TRANSFER_FUNCTION_WELCH(logdiff(U6Rate), logdiff(TCU))
extract(str, 0)

This example examines how changes in the U-6 unemployment rate are related to changes in industrial capacity utilization across different business-cycle frequencies.

What the Analysis Can Show

Coherence: the strength of the relationship at each frequency.
Gain: the size of the response in one series relative to the other.
Phase: the lead-lag relationship between the two series.
Dominant periods: the cycle lengths where the relationship is strongest.
Power spectra: the frequencies containing the greatest variation in each series.

Typical Uses

Business-cycle analysis
Leading-indicator research
Financial-market relationships
Economic and industrial systems
Signal and time-series analysis

Interpretation note: Two series can have a weak overall correlation and still share a very strong relationship at a specific cycle length, such as 24, 36, or 48 months.

Spectral Transfer Matrix

SPECTRAL_TRANSFER_MATRIX

Overview

SPECTRAL_TRANSFER_MATRIX performs pairwise spectral analysis across a list of time series and presents the selected result as a matrix. It is designed for rapid exploratory analysis when the user wants to compare several variables without running each pair separately.

Syntax

SPECTRAL_TRANSFER_MATRIX(seriesList, statistic)

Parameters

Parameter	Description
`seriesList`	A list containing two or more time series.
`statistic`	The spectral result to display for every pair of series.

Supported Statistics

Statistic	Description
`PEAK`	The maximum coherence found between each pair of series.
`AVG`	The average coherence across the analyzed frequencies.
`BEST_PERIOD`	The cycle length at which the strongest relationship occurs.

Example

l = logDiff(List(DGS10, SP500, VIXCLS, VXNCLS))
spectral_transfer_matrix(l, PEAK)

This example calculates the maximum spectral coherence between the 10-year Treasury yield, the S&P 500, the VIX, and the Nasdaq Volatility Index.

Interpreting the Matrix

High PEAK: the two series share at least one strong common cycle.
High AVG: the relationship persists across a broad range of frequencies.
BEST_PERIOD: identifies the cycle length where the pair is most closely related.

Typical Uses

Scanning groups of economic indicators
Comparing financial assets or volatility measures
Finding candidate leading indicators
Identifying synchronized business cycles
Selecting series for more detailed analysis

Exploratory command: The matrix is best used to identify interesting pairs. A full Welch spectral transfer analysis can then be run on those pairs to inspect coherence, gain, phase, and cycle structure in greater detail.

Recommended Workflow

Use SPECTRAL_TRANSFER_MATRIX to scan a group of series and identify strong or unusual relationships.
Use SPECTRAL_TRANSFER_FUNCTION_WELCH on a selected pair for a complete, frequency-by-frequency analysis.