Exponential family statistical distributions, including the well-known normal, binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distributions are stable through the entire duration of the data collection process. A failure of stability leads to nonsmoothness and nonlinearity in the physical processes that result in the data. In this paper, we propose testing for stability of parameters of exponential family distributions and generalized linear models. A rejection of the hypothesis of stable parameters leads to change detection. We derive the related likelihood ratio test statistic. We compare the performance of this test statistic to the popular normal distributional assumption dependent cumulative sum (Gaussian CUSUM) statistic in change detection problems. We study Atlantic tropical storms using the techniques developed here, so to understand whether the nature of these tropical storms has remained stable over the last few decades.

One important way in which nonlinear structures may be present in data related to many physical and natural phenomena is by structural breaks and changes. Generally, elicitation of the time and nature of such breaks with statistical guarantees involves change detection techniques like the cumulative sum (CUSUM) or the exponentially weighted moving average (EWMA).

The standard framework for applying such change detection techniques requires assuming
that the order in which the sampled observations arrive
is known, with the question of interest being whether the data generating process
has remained stable over time. The observations are
assumed to follow a known Gaussian distribution, and are monitored for a potential change
to a different, but still known, Gaussian distribution. Statistical guarantees are typically expressed
in terms of expected run length, i.e., how long it takes on average
for a true change to be detected,
when there is a control for the expected length of time before false signaling occurs.
These normality-based sequential monitoring
and stability detection techniques originated from industrial process
control (

Note that in many modern applications, the assumption of normality is not tenable. In this paper, we discuss change detection in a general exponential family, and in regression models including generalized linear models like logistic regression and log-linear regression. We present several mathematical results concerning the different kinds of CUSUM statistics that may result, depending on the probabilistic structure under consideration, and whether certain parameters are estimated or assumed to be known. A natural question here is on the performance of the normality-based CUSUM statistic, when the probability models do not satisfy the Gaussian assumptions. We study this issue, and present mathematical results, simulation studies, and discussions about when and how the Gaussian CUSUM may yield high quality results. Finally, we discuss properties of Atlantic tropical storms, and use the techniques developed in the rest of this paper to study structural changes in the fundamental physical properties for which we have data records for such storms.

In order to generalize the scope of statistical change detection tools, in this paper
we propose a variant of the sequential industrial monitoring framework,
by considering the stability of the
data generation process as a problem of detecting the time of the distributional
change; in other words, we conduct a hypothesis test, and under the null hypothesis, the data generation
process remains stable through the entire sampling time

Simulation studies show that in most situations the
EF CUSUM method performs better than Gaussian CUSUM. The
EF CUSUM has a shorter average run length, smaller
variation of run length and shorter maximum run length compared with
Gaussian CUSUM. Moreover, smaller shifts can be detected more quickly
by EF CUSUM than by Gaussian CUSUM, which is a big advantage
of using EF CUSUM. Under some circumstances
the Gaussian CUSUM approximates the EF CUSUM
well, we discuss this issue below.
It is also important to note that whether the change point

We also extend our study to that of parameter
change in the generalized linear model. In this context,

Our case study for illustrating our instability and change detection techniques is
based on Atlantic tropical storm data. There are several studies in recent times
on whether, and how, the properties of these storms have changed with climate
change;
see for example

Exponential family CUSUM: binomial, exponential, gamma and multivariate normal distributions.

Section

CUSUM statistic for normal distribution: the first row is more general with both mean and variance change. The remaining three rows are special cases of the first one.

In this section we provide a partial list of techniques for change detection.
As mentioned earlier,
some of these originated in industrial quality context, and related
methods include Shewhart charts
(

The CUSUM technique has been extended to better suit
practical needs, including

Some researchers have treated special
cases in the EF CUSUM family, including

Let the data be the random sample

We assume that

Here

Note that the time ordering of the observations is not an integral part to our methodology. Also, multiple
change points may be allowed. For the former, we would assume that there is some permutation of the data,
say

In our first result below, we obtain the test statistic for the hypothesis test described above.
We adopt the convention that

Let

The likelihood ratio test statistic for testing the null hypothesis

We omit the proof of this and several other theorems in the interest of brevity.

In general, the distribution of the test statistic

The critical value

Note that the test statistic

The necessity part: if

The sufficiency part:
If

In the very special case where

The statistic

Under these assumptions, the
difference between the normality-based CUSUM

The CUSUM for multivariate normal
distribution is somewhat more complicated and therefore we divide
this problem into
the following cases based on the nature of the variance–covariance matrix. In all the
cases listed below, the test statistic is

Assume rank

Based on discussion of case 2, our CUSUM statistic is based on

In this section, we consider data of the form

We now illustrate that the results presented above extend to the case where
the parameters are unknown. For simplicity of presentation, we omit the scaling function

It may be noted, however, that the above test statistic can suffer from extremely low
power, depending on the values of

The above test statistic can be obtained from the profile likelihood (for null
and alternative), when

There can be several other results relating to stability detection with estimated
parameters, under various assumptions and technical conditions, which we will address
in future work. We conclude this section with a result on stability detection when
parameters are estimated in a generalized linear model.

We present below a sketch of the proof of the above result.

The likelihood function under the alternative hypothesis is

Furthermore, in the generalized linear model case, the parametric bootstrap is a viable way
of approximating the distribution of

In this section, we discuss a simulation study on the
change of parameter(s) for binomial, exponential, gamma, and Poisson
distributions, and compare the EF CUSUM statistic with the
Gaussian CUSUM statistic, under the constraint that the mean and the
standard deviation of both distributions are equal.
Based on the exponential family density

The simulation procedure can be described as follows:
First, we control false alarms by carefully choosing

From the simulation results in
Fig.

We also discover that

We now discuss a case study of Atlantic tropical storms, for which we use HURDAT (hurricane database) data from the US National Hurricane Center. For each storm, the following information is recorded: date and time, tropical storm identity, tropical storm name, position in latitude and longitude, maximum sustained winds in knots, and central pressure in millibars.

We present our results from three studies on Atlantic tropical storms here. Each of these studies are carried out on two data sets: a longer series from 1851 to 2008 and a shorter series from 1951 to 2008. The expectation–maximization algorithm was used for missing data segments in the longer series when required, this problem does not arise in the shorter series.

Performance comparison: EF CUSUM with Gaussian CUSUM. Dot-dash, dashed, and solid line stand for mean, median, and standard deviation. The top panel describes run length comparison from Binomial(15,0.95) to Binomial(15,0.90), the middle panel describes run length comparison from Poisson(3) to Poisson(3.1), the bottom panel describes run length comparison from Gamma(1,2) to Gamma(1.5,1.5). Due to length limitation of the graphs, we do not include the MAX line here.

First, we consider the problem of TDS
for the yearly number of
tropical storms between 1851 and 2008.
This yearly data is modeled as
Poisson(

In view of the fact that the data from the nineteenth century and the first
half of the twentieth century may not be entirely reliable, we repeated the
above analysis
on detecting change for the
Atlantic tropical storms from year 1951 to 2008.
We assume that the potential change could only occur after 1970.
For detecting potential change
Poisson(

The observed data and a Poisson fit for the number of tropical storms between 1951 and 2008.

In both of these analyses, our results are not particularly sensitive to the
choice of the initial segment when no change is assumed to occur (i.e., until
1900 and 1970, respectively, in the first and second analysis described above). We also
verified that the assumption that the number of tropical storms in
a given year follows a Poisson distribution is reasonable. For example,
a goodness-of-fit

The second study has two parts. For the data
from 1851 to 2008, we
model the maximum sustained winds
and maximum central pressure as

In a variation of the second study, we consider
maximum sustained wind
speed and minimum central pressure as

Atlantic tropical storm data from 1851 to 2008 are used to
detect any mean change in tropical storm characteristics. Here

Atlantic tropical storm data from 1951 to 2008 are used to
detect any mean change in tropical storm characteristics. Here

The results are summarized in Tables

A moving average estimate of the average number of tropical storms between 1951 and 2008.

In the third study,
we consider the relationship between
the number of tropical storms

For the 1851–2008 data, we take the first 50 observations,
and get

For the 1951–2008 data, we take the first 20 observations,
and get

The EF CUSUM generally performs better than the Gaussian CUSUM. In practice, in situations where the data do not follow normal distribution, we should consider the appropriate distribution for modeling the data and choose the corresponding CUSUM statistic to effectively detect the change in parameter(s) if there is any. Further details for the mathematical proofs, simulation studies, and our analysis of Atlantic tropical storms record are available from the authors.

In general, optimality results for our proposed methods should follow along the lines similar to
those established by

The presence of temporal dependence is typically not problematic; furthermore, our likelihood-based schemes generalize easily to standard time series frameworks, although additional mathematical technicalities cannot be avoided. In addition, cases where the observations are not temporally ordered, or when there are multiple break points, need suitable generalizations and mathematical treatment. Note that there is a relationship between the number of structural breaks in the distribution of a data sequence, the size of such breaks, and the probabilities of true/false inference from hypothesis testing. Establishing the limits of our proposed methodology along these lines is work to be realized in the future.

It should be noted that the methodology discussed here may fail under several different scenarios. For example, when parameters of the distributions are unknown, there will be no reasonable way of obtaining the null or alternative distribution consistently if there are too few observations before or after any change point. This also suggests that the proposed method may not be able to adapt to situations where there are many change points, or when one or more changes in the parameters asymptotes to zero quickly. Although we consider exponential family distributions here which lends itself to several standard statistical techniques, our proposed tests may require modifications if other distributions are involved, and a parametric bootstrap is not guaranteed to produce consistent distributional approximations.

This research is partially supported by the National Science Foundation under grant nos. IIS-1029711 and SES-0851705, and by grants from the Institute on the Environment (IonE), and College of Liberal Arts, U. Minnesota. Edited by: D. Wang Reviewed by: three anonymous referees