Yes. A Hankel matrix can incorporate temporal measures by turning a time- or age-ordered sequence into overlapping temporal windows.
Suppose you measure a variable over time or age:
[
x_1, x_2, x_3, x_4, x_5, x_6
]
where each (x_i) could be gene expression, biomarker value, disease score, biological age estimate, etc.
A Hankel matrix with window length 3 is:
[
H =
\begin{bmatrix}
x_1 & x_2 & x_3 & x_4 \
x_2 & x_3 & x_4 & x_5 \
x_3 & x_4 & x_5 & x_6
\end{bmatrix}
]
Each column is a short temporal history:
[
\begin{bmatrix}
x_1\x_2\x_3
\end{bmatrix},
\quad
\begin{bmatrix}
x_2\x_3\x_4
\end{bmatrix},
\quad
\begin{bmatrix}
x_3\x_4\x_5
\end{bmatrix}
]
So the Hankel matrix converts a one-dimensional time series into many overlapping time-context samples.
1. Incorporating chronological time
Suppose measurements are collected at:
[
t_1, t_2, t_3, \ldots, t_n
]
and the observed values are:
[
x(t_1), x(t_2), x(t_3), \ldots, x(t_n)
]
Then the Hankel matrix is:
[
H_x =
\begin{bmatrix}
x(t_1) & x(t_2) & x(t_3) \
x(t_2) & x(t_3) & x(t_4) \
x(t_3) & x(t_4) & x(t_5)
\end{bmatrix}
]
This means each column represents a local temporal trajectory.
For example:
[
\text{Column 1} =
\begin{bmatrix}
x(t_1)\
x(t_2)\
x(t_3)
\end{bmatrix}
]
describes how the variable changes from (t_1) to (t_3).
This is useful for forecasting, trend detection, system identification, and trajectory modeling.
2. Incorporating age
Age can be treated similarly to time.
Suppose you measure a biomarker at ages:
[
20, 30, 40, 50, 60, 70
]
with corresponding values:
[
x_{20}, x_{30}, x_{40}, x_{50}, x_{60}, x_{70}
]
A Hankel matrix could be:
[
H =
\begin{bmatrix}
x_{20} & x_{30} & x_{40} & x_{50} \
x_{30} & x_{40} & x_{50} & x_{60} \
x_{40} & x_{50} & x_{60} & x_{70}
\end{bmatrix}
]
Each column captures a short aging trajectory:
[
\begin{bmatrix}
x_{20}\
x_{30}\
x_{40}
\end{bmatrix}
]
or
[
\begin{bmatrix}
x_{40}\
x_{50}\
x_{60}
\end{bmatrix}
]
This is useful for studying whether biological signals change gradually, accelerate, plateau, or shift at certain ages.
3. Adding age or time as explicit covariates
Sometimes the values alone are not enough. You may want the model to know the actual time or age.
For example, instead of only using:
[
\begin{bmatrix}
x_{20}\x_{30}\x_{40}
\end{bmatrix}
]
you can pair it with age:
[
\begin{bmatrix}
20 & x_{20}\
30 & x_{30}\
40 & x_{40}
\end{bmatrix}
]
or construct an augmented matrix:
[
\tilde{H} =
\begin{bmatrix}
x_{20} & x_{30} & x_{40} & x_{50} \
x_{30} & x_{40} & x_{50} & x_{60} \
x_{40} & x_{50} & x_{60} & x_{70} \
30 & 40 & 50 & 60
\end{bmatrix}
]
The last row gives the center age of each temporal window.
So each column now contains both the local biomarker history and the age context.
4. Multiple temporal variables: block Hankel matrix
For biological or health data, you often have many features:
[
x^{(1)}, x^{(2)}, x^{(3)}, \ldots, x^{(p)}
]
For example:
gene expression
methylation age
inflammation marker
glucose level
frailty score
treatment status
You can build one Hankel matrix for each variable and stack them:
[
H =
\begin{bmatrix}
H_{x^{(1)}} \
H_{x^{(2)}} \
H_{x^{(3)}} \
\vdots \
H_{x^{(p)}}
\end{bmatrix}
]
This is called a block Hankel matrix.
In aging research, this could represent a multi-omics aging trajectory:
[
H =
\begin{bmatrix}
H_{\text{gene expression}}\
H_{\text{methylation}}\
H_{\text{metabolomics}}\
H_{\text{clinical score}}\
H_{\text{age}}
\end{bmatrix}
]
Each column becomes a compact representation of a subject’s or cell population’s local aging state.
5. Handling irregular time or age intervals
Standard Hankel matrices assume evenly spaced observations, such as:
[
20, 30, 40, 50, 60
]
But real data may be irregular:
[
21, 27, 44, 53, 71
]
There are three common solutions.
Option A: Resample to a regular grid
Interpolate values to standard ages:
[
20, 30, 40, 50, 60, 70
]
Then build the Hankel matrix normally.
Option B: Include time gaps
Use both values and time intervals:
[
\Delta t_i = t_{i+1} - t_i
]
A column may look like:
[
\begin{bmatrix}
x(t_1)\
x(t_2)\
x(t_3)\
\Delta t_{1,2}\
\Delta t_{2,3}
\end{bmatrix}
]
This tells the model not only what changed, but also how much time passed.
Option C: Use age-aligned windows
Instead of using fixed row positions, define age lags:
[
a,\ a+10,\ a+20
]
For each subject or sample, construct windows around comparable age intervals.
This is useful when comparing aging trajectories across individuals.
6. Why this helps
A Hankel matrix allows you to move from isolated measurements to trajectory-aware features.
Instead of asking:
What is the gene expression at age 50?
you can ask:
What is the pattern of gene expression from age 30 to 50?
That is much more informative for aging, disease progression, and temporal biology.
In aging research, a Hankel matrix could support questions like:
Does a biomarker increase steadily with age?
Does it accelerate after middle age?
Are some genes early drivers and others late-response genes?
Are aging trajectories low-rank, meaning they can be summarized by a few dominant temporal patterns?
Can we predict future biological age from previous measurements?
Can we compare treated vs untreated aging trajectories, such as caloric restriction vs control?
Simple framing for your research context
For an aging-clock or longitudinal single-cell project, you could say:
We use a Hankel embedding to convert age-ordered molecular measurements into overlapping trajectory windows. Each column of the Hankel matrix represents a local aging segment, allowing downstream models to learn not only static molecular states but also temporal progression patterns. Chronological age, biological age, and time gaps can be added as covariates or incorporated through block Hankel matrices for multi-modal temporal modeling.
That is the core idea: Hankel matrices turn time or age into structured temporal context.
