This site serves as the blog for the "collaborative" seminar on Biosurveillance being taught at the Department for Biomedical Informatics at the University of Pittsburgh. If you would like to participate in the seminar, please contact denver.h.dash@intel.com.
C. Lowry and D. Montgomery: A review of multivariate control charts (paper file uploaded in the google group)
This paper sounds weird and a lot of industrial engineering terminology for familiar things needs to deciphered first. This review is then a list of gotchas for reading the paper.
------------ The problem is monitoring p variables for excursions from normal behavior. The assumption is that we observe a sequence of normally (as in Gaussian) distributed p x 1 vectors X_1, X_2, ... with no autocorretion (iid). Also assumed known is their covariance matrix \Sigma.
------------ A central role in this paper is taken by what is perhaps better known in our community as the Mahalanobis distance. It basically measures the distance of a datapoint from the multivariate normal mean, in units that are monotonically related to the "variance" of the distribution:
M(x,\mu) = (x-\mu)^T \Sigma^{-1} (x-\mu)
Note that this is the "kernel" of the multivariate normal. Mahalanobis distance is the natural way to measure departure from a multivariate normal mean.
------------ Section 2 jumps right into saying what the control limits -- threshold -- without explaining what the central task is: to design a detector with a fixed false positive rate (FPR). The term false positive rate never actually appears. Instead, they tend to speak in terms of ARL - average run length, meaning the average length of time the process is "in control". "In control" and "out of control" mean normal and anomalous behavior as signaled by the detector, respectively. Thus rougly, FPR = 1/ARL.
------------ The Hotelling statistic is the Mahalanobis distance of $X_i$ from the process mean. It is the natural generalization of the univariate Student-t statistic. TODO: does the \chi^2 in (2) imply that the Hotelling statistic is $\chi^2$ distributed if mean and covariance are known? what about degrees of freedom?
------------ There are two phases of the statistical control process: -- Phase I, when we determine what the normal behavior is by computing the sample mean \mu and covariance \Sigma of normal datapoints and -- Phase II, where we monitor the process by computing the Hotelling statistic with the \mu and \Sigma computed in Phase I.
In industrial engineering, they like to take several samples of the variables at the same time. The number of these samples is $n$. They repeat the process $m$ times. Computationally, this is the same as taking $mn$ samples. The sampling process seems to have some practical motivation. It helps to realize this in reading Section 2.
Most of Section 2 is concerned with what exactly the distribution of the Hotelling statistic is in Phase II, depending on the number of samples taken in Phase I; and how should we choose the control limit to achieve the fixed false positive rate. Also discussed is how many Phase I samples one needs to approximate the exact distribution well with $\chi^2_{\alpha,p}$.
This whole section is basically a cautionary exposition for practicioners to use larger sample size or forgo the \chi^2 approximation. I suppose for a modern system we would build now, there is little concern about the Fisher distribution being somehow difficult to compute on the fly, if we don't have enough data to fit the \mu and \Sigma well.
------------ While the previous section was concerned with deviations from the mean, the short Section 3 gives two statistic for monitoring the process variability. Both are based on the intuition that the determinant of a matrix measures the volume of the space spanned by the rows of the matrix taken as vectors. The second one is simpler: |S| - the determinant of the sample covariance matrix, the first one is more involved and I lack the intuitive interpretation.
------------ Multivariate CUSUM
One interesting approach is to create a linear combination of the p monitored variables. This gives the optimal ARL in the direction of the anticipated shift (Healy 1987). This is probably the most relevant to biosurveillance, since we know which way e.g. an anthrax outbreak is going to send symptoms. One scheme forms a non-centrality measure (recent deviation)
that measures how far we are from previous estimate of deviation S_i = 0 if C_i < K = (S_{i-1} + X_i)(1-K/C_i)
So the deviation measure allows slow shifts (tolerance measured by K) and beyond that point, it will grow with the degree of recent deviation $C_i$. Note the estimate of deviation, S, looks a lot a like the data itself, scaled down a little bit. Detection occurs if S'\Sigma S > threshold.
Another MCUSUM procedure uses a moving average of data, Hotelling statistic on the moving average. The twist is that the signal that is explicitly attenuated with the length that it has been on.
------------ Multivariate EWMA. Really a straightforward extension of EWMA:
Z_i = R X_i + (I-R) Z_{i-1},
where R is a diagonal matrix. The detection signal is a Mahalanobis distance/Hotelling statistic on the moving average.
------------ The authors advise caution with using principal components projection to reduce the dimensionality of the space of the monitored variables and obtain independence between monitored variables. Their argument emphasizes the interpretability of the principal factor -- you want to know not only that something is wrong, but also which variable is going awry. I found the argument strange, it might often be a combination of variable that goes out of bounds, when all of the variables individually may be within their limits.
------------ Regression correction. Instead of viewing the data as coming from a Gaussian with dependencies between variables modeled with the off-diagonal entries of the covariance matrix, it has been proposed that we monitor the vector of regression residuals. We obtain the $j$-th element of the residual vector by regressing X^j_i, the $j$-th element of X_i, on all other values in $X_i$. Another option is to regress the values $X_i$ on some external values that are not the monitored covariates.
------------ Watch for autocorrelation. All methods presented are based on the (typically false) assumption that the X_1, X_2, ... are iid. The converse tends to be true in practice, with $X_i$ and $X_{i-1}$ exhibiting significant correlation.
1 comment:
C. Lowry and D. Montgomery: A review of multivariate control charts
(paper file uploaded in the google group)
This paper sounds weird and a lot of industrial engineering
terminology for familiar things needs to deciphered first.
This review is then a list of gotchas for reading the paper.
------------
The problem is monitoring p variables for excursions from normal behavior.
The assumption is that we observe a sequence of normally (as in Gaussian)
distributed p x 1 vectors X_1, X_2, ... with no autocorretion (iid). Also
assumed known is their covariance matrix \Sigma.
------------
A central role in this paper is taken by what is perhaps better known
in our community as the Mahalanobis distance. It basically measures the
distance of a datapoint from the multivariate normal mean, in units
that are monotonically related to the "variance" of the distribution:
M(x,\mu) = (x-\mu)^T \Sigma^{-1} (x-\mu)
Note that this is the "kernel" of the multivariate normal. Mahalanobis
distance is the natural way to measure departure from a multivariate
normal mean.
------------
Section 2 jumps right into saying what the control limits -- threshold --
without explaining what the central task is: to design a detector
with a fixed false positive rate (FPR). The term false positive rate never
actually appears. Instead, they tend to speak in terms of ARL - average
run length, meaning the average length of time the process is "in control".
"In control" and "out of control" mean normal and anomalous behavior
as signaled by the detector, respectively. Thus rougly, FPR = 1/ARL.
------------
The Hotelling statistic is the Mahalanobis distance of $X_i$ from
the process mean. It is the natural generalization of the univariate
Student-t statistic.
TODO: does the \chi^2 in (2) imply that the Hotelling statistic is $\chi^2$
distributed if mean and covariance are known? what about degrees of freedom?
------------
There are two phases of the statistical control process:
-- Phase I, when we determine what the normal behavior is by computing
the sample mean \mu and covariance \Sigma of normal datapoints and
-- Phase II, where we monitor the process by computing the Hotelling statistic
with the \mu and \Sigma computed in Phase I.
In industrial engineering, they like to take several samples of
the variables at the same time. The number of these samples is $n$.
They repeat the process $m$ times. Computationally, this is the same
as taking $mn$ samples. The sampling process seems to have some
practical motivation. It helps to realize this in reading Section 2.
Most of Section 2 is concerned with what exactly the distribution
of the Hotelling statistic is in Phase II, depending on the number
of samples taken in Phase I; and how should we choose the control
limit to achieve the fixed false positive rate.
Also discussed is how many Phase I samples one needs to approximate
the exact distribution well with $\chi^2_{\alpha,p}$.
This whole section is basically a cautionary exposition for practicioners
to use larger sample size or forgo the \chi^2 approximation.
I suppose for a modern system we would build now, there is little concern
about the Fisher distribution being somehow difficult to compute on the fly,
if we don't have enough data to fit the \mu and \Sigma well.
------------
While the previous section was concerned with deviations from the mean,
the short Section 3 gives two statistic for monitoring the process variability.
Both are based on the intuition that the determinant of a matrix measures
the volume of the space spanned by the rows of the matrix taken as vectors.
The second one is simpler: |S| - the determinant of the sample covariance matrix,
the first one is more involved and I lack the intuitive interpretation.
------------
Multivariate CUSUM
One interesting approach is to create a linear combination
of the p monitored variables. This gives the optimal ARL
in the direction of the anticipated shift (Healy 1987).
This is probably the most relevant to biosurveillance, since we know
which way e.g. an anthrax outbreak is going to send symptoms.
One scheme forms a non-centrality measure (recent deviation)
C_i = {(S_{i-1} + X_i)' \Sigma (S_{i-1} + X_i)}^{1/2}
that measures how far we are from previous estimate of deviation
S_i = 0 if C_i < K
= (S_{i-1} + X_i)(1-K/C_i)
So the deviation measure allows slow shifts (tolerance measured by K)
and beyond that point, it will grow with the degree of recent deviation $C_i$.
Note the estimate of deviation, S, looks a lot a like the data itself,
scaled down a little bit. Detection occurs if S'\Sigma S > threshold.
Another MCUSUM procedure uses a moving average of data, Hotelling statistic
on the moving average. The twist is that the signal that is explicitly
attenuated with the length that it has been on.
------------
Multivariate EWMA. Really a straightforward extension of EWMA:
Z_i = R X_i + (I-R) Z_{i-1},
where R is a diagonal matrix. The detection signal is a Mahalanobis
distance/Hotelling statistic on the moving average.
------------
The authors advise caution with using principal components projection
to reduce the dimensionality of the space of the monitored variables
and obtain independence between monitored variables. Their argument
emphasizes the interpretability of the principal factor -- you want
to know not only that something is wrong, but also which variable is
going awry. I found the argument strange, it might often be a combination
of variable that goes out of bounds, when all of the variables
individually may be within their limits.
------------
Regression correction. Instead of viewing the data as coming from
a Gaussian with dependencies between variables modeled with the
off-diagonal entries of the covariance matrix, it has been proposed
that we monitor the vector of regression residuals. We obtain the
$j$-th element of the residual vector by regressing X^j_i, the
$j$-th element of X_i, on all other values in $X_i$.
Another option is to regress the values $X_i$ on some external values
that are not the monitored covariates.
------------
Watch for autocorrelation. All methods presented are based on the
(typically false) assumption that the X_1, X_2, ... are iid.
The converse tends to be true in practice, with $X_i$ and $X_{i-1}$
exhibiting significant correlation.
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