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A novel algorithm for detecting human circadian rhythms using a thoracic temperature sensor

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Abstract

Circadian rhythms undergo high perturbations due to cancer progression and worsening of metabolic diseases. This paper proposes an original method for detecting such perturbations using a novel thoracic temperature sensor. Such an infrared sensor records the skin temperature every five minutes, although some data might be missing. In this pilot study, five control subjects were evaluated over four days of recordings. In order to overcome the problem of missing data, first four different interpolation methods were compared. Using interpolation helps covering the gaps and extending the recordings frequency, subsequently prolonging sensor battery life. Afterwards, a Cosinor model was proposed to characterize circadian rhythms, and extract relevant parameters, with their confidence limits. A divergence study is then performed to detect changes in these parameters. The results are promising, supporting the enlargement of the sample size and warranting further assessment in cancer patients.
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A novel algorithm for detecting human circadian rhythms using a thoracic temperature sensor
Aly Chkeir*1, Farah Mourad-Chehade1, Jacques Beau4, Monique Maurice4, Sandra Komarzynski4, Francis Levi3, David J.
Hewson2, Jacques Duchêne1,
1Institut Charles Delaunay UMR CNRS 6281, Rosas, University of Technology of Troyes (UTT) 10000 Troyes, France
2University of Bedfordshire, Luton, United Kingdom
3Warwick University, United Kingdom
4INSERM, University of Paris, France
A R T I C L E I N F O
A B S T R A C T
Article history:
Received: 07 May, 2017
Accepted: 29 June, 2017
Online: 19 July, 2017
Circadian rhythms undergo high perturbations due to cancer progression and
worsening of metabolic diseases. This paper proposes an original method for detecting
such perturbations using a novel thoracic temperature sensor. Such an infrared sensor
records the skin temperature every five minutes, although some data might be missing.
In this pilot study, five control subjects were evaluated over four days of recordings. In
order to overcome the problem of missing data, first four different interpolation methods
were compared. Using interpolation helps covering the gaps and extending the
recordings frequency, subsequently prolonging sensor battery life. Afterwards, a
Cosinor model was proposed to characterize circadian rhythms, and extract relevant
parameters, with their confidence limits. A divergence study is then performed to detect
changes in these parameters. The results are promising, supporting the enlargement of
the sample size and warranting further assessment in cancer patients.
Keywords:
Human circadian rhythm
Interpolation
Cosinor
Divergence
Temperature Sensor
1. Introduction
Most biological processes follow circadian rhythms, with
the standard period of about 24 hours. These 24-hour rhythms
are driven by circadian clocks, which have been observed in
plants, fungi, cyanobacteria, and humans [1, 2]. In mammals,
each cell contains molecular clocks, whose coordination is
ensured by the suprachiasmatic nuclei in the hypothalamus
through the generation of rhythmic physiology. Such a
Circadian Timing System (CTS) regulates rhythmically cellular
metabolism and proliferation over the 24 hours.
The body temperature follows circadian rhythms, that are
tightly related to cancer development[3] [4]. Indeed, they play
an important role in the coordination of molecular circadian
clocks in various peripheral organs, such as lung, liver, kidney,
and intestine, and could have an effect on tumors through their
regulatory effects on Heat Shock Factor (HSF), Heat Shock
Proteins (HSPs), and Cold-Induced Proteins [4, 5]. Moreover,
it has been demonstrated that giving chemotherapy at an
accurate circadian timing improves tolerance up to fivefold and
to almost doubles antitumor efficacy, compared to constant
rates or wrongly timed administrations, in both rodent models
and cancer patients[6] [7]. In the same way, the destruction of
the suprachiasmatic nuclei suppressed any circadian rhythm in
body temperature in mice, which causes a 2-3 fold acceleration
of experimental cancer progression [8]. On the other hand, the
temperature circadian rhythms could be disrupted by anticancer
medications, and molecular clocks are impaired, as a function
of dose and circadian timing in mice [9, 10]. Therefore, the
circadian rhythm of body temperature is a useful biomarker of
CTS function[11] [12].
The relevance of monitoring core body temperature for CTS
assessment has been approved in previous works, however they
all suffer from a lack of non-invasive screening tools, which has
limited testing in cancer patients [13, 14]. To overcome such a
problem, one could use the skin temperature, which is
correlated to core temperature. However, changes in skin
temperature display an opposite pattern compared to that in core
body temperature, but with a similar circadian rhythm.
In our work, a non-invasive wearable sensor, Movisens®
GmbH, Karlsruhe, Germany, has been used. This sensor
monitors skin temperature for up to 200 hours, with the sensor
able to be worn at different positions such as the hip, wrist, or
chest. The temperature signals are collected at a relatively low
ASTESJ
ISSN: 2415-6698
*Corresponding Author: Aly Chkeir, Institut Charles Delaunay UMR CNRS
6281, Rosas, University of Technology of Troyes (UTT) 10000 Troyes, France
Email: aly.chkeir@utt.fr
Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 4, 105-110 (2017)
www.astesj.com
A. Chkeir et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 4, 105-110 (2017)
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sampling frequency with this sensor, that is, one point every
five minutes. Using such a low frequency is possible since
temperature varies slowly over time. Also, a minimal memory
is required and the battery life is extended, due to reduced
energy consumption for the measurement device. However,
having a low signal frequency produces more difficult
complications. Moreover, sensor malfunction or a subject not
wearing it for a period of time causes missing data in the
signals.
This paper proposes an original algorithm to analyze the
circadian rhythms in skin temperature signals. It first develops
an interpolation algorithm to set the signals to a higher
frequency such as one sample per minute, and to fill any gap in
the signals [15]. For this reason, a kernel-based machine
learning algorithm is presented and compared to other classic
interpolation methods. Afterwards, a rhythmometric modeling,
using Cosinor models, is proposed. It aims to extract parameters
such as the MESOR, the amplitude, the orthophase and the
bathyphase from interpolated signals. The paper then proposes
a divergence study over these features to detect changes caused
by chemotherapy. Five healthy subjects’ temperature signals
are involved in this study, as a prerequisite for subsequent
investigations involving a larger number in healthy controls and
cancer patients.
2. Subjects and Methods
2.1. Subjects and materiel
In this study, three female and two male control subjects,
aged 45.2 ±13.6 years, are considered. They were given a
detailed description of the objectives and requirements of the
study before the experiment, and they read and signed an
informed consent prior to testing. The infrared Movisens
(GmbH - move III) sensor was positioned on the thorax of the
subjects to monitor their skin temperature for four days. It
measures 5.0 x 3.6 x1.7 cm3, and weighs 32 g. The sensor is
composed of a tri-axial acceleration sensor (adxl345, Analog
Devices; range: ±8 g; sampling rate: 64 Hz; resolution: 12 bit)
embedded with a temperature sensor (MLX90615 high
resolution 16bit ADC; resolution of 0.02°C). The recorded data
is saved on a memory chip inside the sensor and transmitted to
a server via GPRS. A hypoallergenic patch has been used to
maintain the sensor in the upper right anterior thoracic area of
the subjects.
2.2. Interpolation of temperature signals
The temperature signals have a small frequency, with
occasional missing data. Let  denote the
temperature samples collected for a certain subject using the
sensor, with in minutes. The aim of interpolation is to
estimate a function
, that computes the temperature at any
time , and which verifies the following:


 (1)
Different algorithms for interpolation such as linear,
polynomial or cubic splines techniques exist in the literature
[15, 16]. In the linear interpolation, temperature is represented
by a straight line between any two consecutive collected
measurements. Having, for ,
(2)
with and 
.
This technique is easy to implement, but it yields several
functions, one per interval between two consecutive
measurements. Moreover, it is disadvantageous for large time-
intervals due to the non-specification of the linear estimation.
In the polynomial interpolation, temperature is represented by
a single polynomial function, which fits the measured data. By
using Lagrange polynomials, one obtains the following
function:




 (3)
where is the degree of the obtained polynomial and
. By taking close to , the obtained function
is specific, but with highly complex computations. Cubic
splines is the most commonly used interpolation technique. It
computes a set of piece-wise polynomial functions that
maximize the smoothness of the whole curve. The -th splines
function defined over the interval, for ,
is given as follows:
(4)
where:




  

where
is the derivative of the temperature model
. The
values of the coefficients are then computed iteratively as
shown in [2]. This technique is efficient, but proposes piece-
wise functions, that need iterative computations.
TABLE I. TYPICAL REPRODUCING KERNELS
General expression



This paper proposes a kernel-based regression approach that
generates a single function [17, 18]. A training database is first
constructed using the collected temperature measurements
. Then a model is computed using this
database, taking time as input and yielding temperature as
output. This model is defined using the kernel-based ridge
regression technique. The model is afterwards applied to other
times, where temperature values are unknown, for
interpolation. Consider a reproducing kernel defined from
to and denote its reproducing kernel Hilbert space.
Some commonly used reproducing kernels are given in Table I.
where the kernel parameters and are positive and is a
positive integer. Then the temperature model is defined by
minimizing the regularized mean quadratic error between the
model’s outputs and the measured data of the learning database:




(5)
where is a regularization parameter that controls the tradeoff
between the training error and the complexity of the solution
and
is the norm in the reproducing kernel Hilbert
space[19]. According to the Representer theorem, that all
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machine learning algorithms share, the minimization problem
could be reduced to a more computation-friendly problem.
Hence, the temperature model could be written as follows:

 (6)
where the coefficients are to be determined. Let denote
the column coefficients vector whose -th entry is . By
injecting the model expression in the minimization problem,
one obtains a dual optimization problem whose solution is
given by:
 (7)
where is a -by- matrix whose -th entry is given by
, is the -by- identity matrix and is the
temperature column vector, whose -th entry is given by
. Now that the model is defined for each temperature
signal, the temperature value at a given time is estimated by
The main advantage of the kernel-based approach remains in
the fact that a single-function model is obtained, unlike the
linear and cubic spline interpolations, where a piece-wise
expression is obtained. In the following, and for simplicity, the
notation is used for the estimated signal, having a value at
each minute, obtained after interpolation.
2.3. Detection of rhythmicity
The detection of rhythmicity is usually performed in the
frequency domain . The spectral analysis using the “Fourier
transform” is a well-known study to do this . In this
analysis, any signal, regardless of its shape and properties, can
be represented by a complex function of frequency that
highlights the frequencies that make it up. By applying the
inverse Fourier transform, the signal is then decomposed into
an infinite sum of sine and cosine functions of infinite
frequencies [22]. The signals could be deterministic such as
periodic/non-periodic or random such as stationary/non-
stationary. A similar analysis for periodic signals is the Fourier
series analysis, which represents a function as a sum of sine and
cosine functions of different frequencies.
In this paper, an algorithm based on Fourier analysis is first
proposed for frequency and harmonic detection. This algorithm
starts by estimating the fundamental frequency, the
fundamental amplitude and phase, and the harmonic amplitudes
and phases, to evaluate the periodogram. The term
periodogram was introduced by Schuster in December 1934
when Fourier analysis was used to estimate periodicity in
meteorological phenomena . The technique was evaluated
for the first time when inspecting circadian rhythms in the
1950s to measure circadian rhythms of mice after blinding .
The periodogram showed that periodic signals have a frequency
spectrum consisting of harmonics. For instance, if the time
domain repeats at f, the frequency spectrum will contain a first
harmonic at f, a second harmonic at 2f, a third harmonic at 3f,
and so forth. The first harmonic, which is the frequency at
which the time domain repeats itself, is called the fundamental
frequency, and has the highest amplitude. Periodograms and
spectral density were originally used in chronobiology in the
1960s [25].
In order to set the periodogram, a suitable window must be
applied to the signal, to reduce side-lobes. For the proposed
algorithm, a normalized Hanning window has been chosen [26]
since this window does not disturb the position of spectral peaks
in the spectral density, although the amplitude is decreased and
the peak is larger. Having the periodogram and thus the
fundamental and harmonic frequencies, the temperature signal
is then modeled, using the Fourier series as follows:

 (8)
where is the angular frequency i.e. 
, i.e. is the
fundamental period (duration of one cycle) and is the number
of the considered harmonics with the fundamental frequency.
With respect to circadian rhythms, the rhythm persists in
constant conditions with a period of around 24-hours, i.e.
 minutes. takes values from to infinity. The
higher is, the better the model
fits the observed value
. The parameters , and could be computed using
the computations of Fourier series. The main advantage of this
technique is that one is able to determine the exact fundamental
frequency, with its following ones, by analyzing the
periodogram, obtained with the Fourier transform. However, it
needs the data to be equidistant and to cover more than a single
cycle, otherwise the analysis would be erroneous.
One interesting method used for analyzing unequidistant
and time-limited observations is the single Cosinor procedure.
This study was developed to evaluate rhythmicity of un-
equidistant data series[27] [28], and is frequently used in the
analysis of biologic time series that have expected rhythms.
Cosinor uses the least squares method to fit a sum of sine
functions to a time series, as least squares procedures do not
have an equidistant data limitation. Practically, it considers the
Fourier series model of (8) with a precise fundamental
frequency, which corresponds to 24-hours, and number of
harmonics and then computes the parameters, and so
as to minimize the error between the signal and the model
. Let be the residual corresponding to the value ,
that is,
and consider the modeling error as the sum of the squared
residuals () for all the data, that is,





The parameters , and are then obtained using least
squares by setting the derivatives of the  over each
parameter equal to zero. Since the temperature follows a
circardian rhythm, then the fundamental frequency corresponds
to 24h. Figure 1 shows an example of a Cosinor model obtained
with. The obtained model will subsequently be used to
compute significant rhythmometric parameters, as shown in the
following paragraph.
Once the temperature signals are modeled, using either
Fourier series or Cosinor, some features are extracted from their
sinusoidal representation, as will be shown in the following
paragraph.
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Figure 1 Circadian Rhythm Model
2.4. Features selection and divergence study
Once the temperature signal is decomposed into cumulative
sine functions, using Fourier series or Cosinor, the objective
here is to determine the sinusoidality of the data. This requires
the extraction of some features from the models by inspecting
their graph plotted against time, as shown in Figure 1. When the
model is composed of more than one trigonometric function,
that is, the fundamental period and some harmonics, four
features could be extracted [29]:
the MESOR (M), for “Midline Estimating Statistic Of
Rhythm”, that is the mean of the model,
the amplitude (A) that is defined as the half of the difference
between the maximum and the minimum of the model in
one fundamental period,
and, finally, the phases of the maximum and the minimum
of the composite model including harmonic terms, which
are called the orthophase ΦO and bathyphase ΦB
respectively.
Figure 1 illustrates these features. For a control subject,
these rhythmometric parameters vary slightly over time;
whereas chemotherapy could produce significant modification
in the circadian rhythm, which yields a divergence of one or
more rhythmometric parameters. In order to detect this
divergence, a sliding window algorithm is considered over the
temperature signal of a period of several days.
Then, for each window, the signal is modeled using the
cumulative sine functions, and the four rhythmometric
parameters are extracted. A statistical test, such as an exact
Fisher test, Wilcoxon test or another, is then applied to check
whether the values through the window diverge from their
previous values. This study is motivated by the perturbation of
the circadian rhythm due to chemotherapy, which induce a
divergence of the statistical distributions of the extracted
rhythmometric parameters.
3. Results
This section starts with the illustration of the effectiveness of
the interpolation techniques. To this end, the collected
temperature signals of two subjects out of the five are
considered. These two have a frequency of one sample per
minute over four days, whereas the remaining are measured with
a rate of one sample every five minutes. An example of a skin
temperature signal while wearing the IR sensor for four days, for
a control pattern with an expected pattern is shown in Figure 2.
Table II. Interpolation results
Interpolation
methods
Interpolation mean error ± SD
0
segment
removed
1
segment
removed
2
segments
removed
3
segments
removed
Linear
0.13±0.7
0.16±0.8
0.18±0.7
0.19±0.8
Cubic splines
0.15±0.7
0.17±0.8
0.19±0.8
0.25±0.9
Polynomial
0.80±1.4
0.80±1.4
0.81±1.4
0.81±1.4
Kernel-based
0.06±0.2
0.08±0.3
0.11±0.3
0.13±0.4
Figure 2 Skin temperature signal
Figure 3 Kernel-based interpolation signal (dashed line) and its true measured
one (straight line) for one subject over five hours, i.e. 300 minutes
In order to compare the interpolation approaches, the signals
of the two subjects were divided into segments of 1440 minutes,
i.e. one-day length, leading to eight segments. Then, these
segments are resampled by taking one point every five minutes,
leading to segments of 288 points. Thereafter, the linear,
polynomial, cubic spline and kernel-based interpolation
techniques were applied over these segments and the results are
compared to the original observed signal. For the kernel-based
interpolation, a Gaussian kernel was used, using a cross-
validation algorithm to select the optimal values of the
bandwidth and the regularization parameter according to the
learning data . The mean errors for the interpolations are
shown in Table II. These were computed by averaging the
absolute difference between the observed and the estimated
signals for the eight segments. In order to simulate the missing
data, a 30-min segment was subsequently removed from the one-
day length segment. This 30-min segment corresponds to six
consecutive points among the 288-point segments. Then, two
and three 30-min segments are removed. These segments were
randomly selected within each one-day length segment. Table II
shows the results for segments with no missing data (0 segment
removed), one 30-min segment removed, two segments
removed, and three segments removed. It is worth noting that
simulations are performed 50 times and errors are averaged over
all results, since the removed segments are selected randomly.
The table shows that the Kernel-based algorithm yields better
results with less estimation error in all cases, which is expected
due its malleability and its adaptation to the curve, even under
non-linear conditions.
The measured temperature signal for a typical subject for a
five-hour period is shown in Figure 3. The figure also shows the
interpolated signal obtained with the kernel-based interpolation.
01000 2000 3000 4000 5000 6000
34
35
36
37
38
Time (min)
Temperature (°C)
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www.astesj.com 109
The plot shows that the computed signal is close to the measured
one, with a smooth curve produced.
Then, the objective is to illustrate the rhythmometric modeling
techniques, which use the Fourier series or Cosinor. To this end,
we consider the 5 temperature signals over a 4-day period,
having passed the interpolation phase, i.e. the signals have no
gaps, with a rate of one sample per minute. We start by applying
the Fourier series, taking the fundamental period to be equal to
24 hours, i.e. 1440 minutes, followed by 3 harmonics, i.e. 12
hours, 8 hours and 6 hours. We consider that the modeling error
is the average of the absolute differences between the 4-day
temperature signals and their modeled ones for the 5 signals. In
this case, the modeling error of the Fourier series technique with
a 24-hour period is equal to 0.77. We then apply the Fourier
transform to the signals to obtain their periodograms. These
computations showed that the local maxima of the spectral
energy are not obtained exactly at the frequencies 1/24, 1/12, 1/8
and 1/6, but very close to them.
Figure 4 Periodogram generated by the Fourier analysis of the time series
Figure 4 shows a periodogram obtained for a certain subject,
where the fundamental frequency is equal to 1/23.5. Let be
the fundamental period obtained by taking the maximal point of
the periodogram, i.e.  is very close to 24 hours. If we take
 and the following three local maxima, and perform the
inverse Fourier transform, the modeling error decreases to 0.73,
which was expected since this way more information is
considered in the modeling. Having the fundamental period not
equal to 24h is related to the fact that only four cycles (4 days)
are considered, which is not enough for Fourier analysis.
Afterwards, by applying Cosinor computations, while taking the
fundamental period equal to 24h, and considering the following
3 harmonics, the modeling error is the least, being equal to 0.53.
This shows the power of such a method with limited-duration
signals.
Figure 5 Temperature modeled signals using the Fourier series with a 24h
period in thin dashed line, the Fourier transform with a 23.5h period in thin
straight line and the Cosinor in thick dashed line, and the original signal in thick
straight line.
Figure 5 shows the modeled signals in a thin dashed line for the
Fourier series using a 24h period, a thin straight line for the
Fourier transform using a  23.5h period and a thick
dashed line for the Cosinor analysis. It also shows the original
signal in thick straight line, for a one day period going from 5
a.m. till 5 am. the following day. The curves show that the
obtained signals follow the original one with a small modeling
error. The rhythmometric parameters, i.e. orthophase and
bathyphase, are then computed. Table III shows their mean
values in degrees with the standard deviations over the 5 signals.
Here the parameters are extracted from the rhythmometric
models obtained for the whole 4-days signals. As expected,
considering the fundamental period  from the periodogram,
modeling leads to a difference in the characteristics equivalent
to 30 minutes, whereas both Fourier series and Cosinor lead to
close values. This study shows that the Cosinor computations have
promising results, to be validated with more signals later on.
4. Discussion
This paper proposed a longitudinal evaluation of skin
temperature measurement, collected using a thoracic sensor.
The collected signals have a sampling frequency of one point
every five minutes, with a length of four days for control
subjects. At a first phase, we performed interpolation, using a
kernel-based machine learning technique, to resample the
signals to a higher frequency of one point every minute and to
fill in the gaps in the measured signals. This step is crucial for
the following data processing techniques. A rhythmometric
study is then followed up on resampled signals, using either a
Fourier representation or Cosinor. A Fourier analysis with the
Fourier transform and the Fourier series is first proposed to
model the signals, then a Cosinor model is constructed using
Fourier analysis and least squares computations. The advantage
of Cosinor remains in its robustness against un-equidistant and
low-duration data, which is not the case for Fourier analysis.
Rhythmometric parameters such as MESOR, amplitude,
orthophase and bathyphase, are then extracted using the
obtained model. For a given temperature signal of a given
subject, such computations are performed for each sliding
window of the signal, leading to a set of random variables that
are the rhythmometric parameters. Then a divergence study is
applied to detect any perturbation of the rhythm. In fact,
chemotherapy can alter the circadian rhythm of the patients,
therefore it is expected in that case to observe a divergence of
the distributions of the rhythmometric parameters between the
time span on chemotherapy and that before or after it. To detect
such divergence, we can separate each parameter values into
two sets (on or off chemotherapy), then apply a statistical test
to detect the divergence between their distributions. This study
is promising since chemotherapy-induced perturbations have
been documented for over 15 anticancer drugs in experimental
models, and for several drug combination protocols in cancer
patients [6 14 15].
The kernel-based interpolation phase was validated using
different cases of missing data in the signals, and by comparing
them to well-known interpolation methods such as linear,
polynomial and cubic splines. A comparison between the
rhythmometric modeling techniques is then conducted,
showing that Cosinor analysis leads to fewer modeling errors.
Having only five signals with a 4-day duration is not enough to
draw conclusions, however the results are promising for future
works. Thus a larger study involving more healthy controls and
5 7 9 11 13 15 17 19 21 23 1 3
34
35
36
37
38
Temperature(°C)
Time (hours)
Table III. Rhythmometric parameters
Mean (±SD)
Fourier
series
(24h)
Fourier
transform
()
Cosinor
model
(24h)
Orthophase
96.9±0.6
104.1±1
97.8±0.5
Bathyphase
27.58±0.8
38.8±0.8
28.4±0.5
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cancer patients is planned to further determine the relevance of
the methodology developed here and the minimum number of
harmonics. It should be possible to evaluate the effect of
harmonics such as 12h, 8h, and 6h, and even to use only specific
harmonics of varying frequency in order to determine the exact
moment when the circadian rhythm is modified. Advanced
processing such as the goodness of fit and the rhythm detection
will also be applied on the signals using the F-test. Moreover,
an evaluation of the IR sensor is also to be done, to verify if the
observed temperature with such a sensor remains a mirror of the
central temperature.
5. Acknowledgments
The authors would like to thank all clinicians, clinical
research associates and technicians for their involvement in the
device design and the protocol definition, set-up and
monitoring. This work was supported by the Champagne-
Ardenne Regional Council; PICADO project (Projet Innovant
pour le Changement d’Ampleur de la Domomédecine).
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Geophysics and cosmical physics owe to Sir Arthur Schuster the fundamental discovery that research on periodicities, cycles, and related phenomena requires, besides purely analytical calculations, such as harmonic analysis or Fourier series, essentially statistical considerations based on the theory of probability. His first paper on this subject, which appeared in this Journal, gives the outline of a theory which was later applied to various meteorological, terrestrial-magnetic, and cosmical periodicities. These classical papers, the study of which should be part of the training of every geophysicist, are: (1) On the investigation of hidden periodicities with application to a supposed 26-day period of meteorological phenomena (Terr. Mag., 3, 13-41, 1898); (2) The periodogram of magnetic declination as obtained from the records of the Greenwich Observatory during the years 1871-1895 (Cambridge, Trans. Phil. Soc., 18, 107-135, 1899); (3) On the spectrum of an irregular disturbance (Phil. Mag., 5, 344-346, 1903); (4) The periodogram and its optical analogy (Proc. R. Soc., A, 77, 136-140, 1906); (5) On the periodicity of sunspots (Phil. Trans. R. Soc., A, 206, 69-100, 1906); and (6) On the periodicity of sunspots (Proc. R. Soc., A, 85, 50-53, 1911). Abstract Geophysics and cosmical physics owe to Sir Arthur Schuster the fundamental discovery that research on periodicities, cycles, and related phenomena requires, besides purely analytical calculations, such as harmonic analysis or Fourier series, essentially statistical considerations based on the theory of probability. His first paper on this subject, which appeared in this Journal, gives the outline of a theory which was later applied to various meteorological, terrestrial-magnetic, and cosmical periodicities. These classical papers, the study of which should be part of the training of every geophysicist, are: (1) On the investigation of hidden periodicities with application to a supposed 26-day period of meteorological phenomena (Terr. Mag., 3, 13-41, 1898); (2) The periodogram of magnetic declination as obtained from the records of the Greenwich Observatory during the years 1871-1895 (Cambridge, Trans. Phil. Soc., 18, 107-135, 1899); (3) On the spectrum of an irregular disturbance (Phil. Mag., 5, 344-346, 1903); (4) The periodogram and its optical analogy (Proc. R. Soc., A, 77, 136-140, 1906); (5) On the periodicity of sunspots (Phil. Trans. R. Soc., A, 206, 69-100, 1906); and (6) On the periodicity of sunspots (Proc. R. Soc., A, 85, 50-53, 1911).