W. Edwards Deming and Frederick F. Stephan. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. The Annals of Mathematical Statistics, 11(4):427--444, 1940.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the case of updating based on single marginal probability distribution [KK66] Formula 1 in Section 4.2. In the case of several marginal probability distributions, the proposed soft evidential update method corresponds to the iterative proportional fitting procedure (IPFP) of Deming and Stephan [DS40]. Note that in Section 3 we have shown how updating based on either a probability assignment to an arbitrary logical function, or a probability assignment to a probabilistic function can be transformed to updating based on a marginal probability distribution. Therefore the correspondence of soft ....

....ffl The marginals from the U.S. Census Bureau agent are respected. ffl Q 2 (X 1 ; X 2 ) is the distribution that is closest to the distribution P (X 1 ; X 2 ) As a measure of distance among distribution, Kullback Leibler divergence could be used. As early as 1940, W.E. Deming and F.F. Stephen [DS40] proposed IPFP for solving such a problem. We propose that our agent does the same. One may argue that the task of this example is rather a model learning or model revision task than an evidential update task. In the context of AEBNs, it is more appropriate to consider this an evidential update ....

W.E. Deming and F.F. Stephan. On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics, 11:427, 1940.

....on the specified constraints and derive the estimation with minimal human intervention. 4 Algorithmic Solution In this section, we propose the use of an algorithmic approach for the solution of problem 1. The technique is called Iterative Proportional Fitting (IPF) and was introduced in [DS40] It is an iterative algorithm that converges to the maximum entropy solution. IPF has the following properties [BFH75] 7 1. it always converges monotonically to the required unique maximum entropy estimation, given a number of marginals; 2. a stopping rule may be used that ensures accuracy ....

W. E. Deming and F. F. Stephan. On a Least Square Adjustment of a Sampled Frequency Table When the Expected Marginal Totals Are Known. Annals of Mathematical Statistics, 11:427--444, 1940.

....The component wise conditional expectations of the OD traOEc, given the link traOEc, estimated parameters, and the positivity constraints on the OD traOEc, are used as the initial estimates of the OD traOEc. The linear equation y = Ax is enforced via the iterative proportional tting algorithm (cf. [42, 43]) to obtain the nal estimates of the OD traOEc. The positivity and the linear constraints are very important nal steps to get reliable estimates of the OD traOEc, in addition to the implicit regularization introduced by the iid statistical model. To smooth the parameter estimates, a random walk ....

W. E. Deming and F. F. Stephen. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist., 11:427444.

....are formed using more than one variable, but only the marginal population totals are known. It shares the same idea with poststratification as to match the sample with what is known about the target population. Iterative proportional fitting (IPF) is simply iteration of the raking procedure (Deming and Stephan, 1940). Although these methods have the same intuitive interpretation, they have different statistical properties, with potential implications for the estimation of standard errors (see, Gelman and Carlin, 2000, for some simple examples) 2.2 Notation: cell weights and unit weights We have found it ....

Deming, W. E., and Stephan, F. F., 1940. On a least squares adjustment of a sampled frequency table when the expected marginal tables are known. The Annuals of Mathematical Statistics, 11, 427-444.

....procedure, 0 denotes transpose, and g is a link function. Since the quantities to be predicted are positive, generalized linear models which lead to multplicative models are appropriate) 2. 1 Fitting algorithm The estimation procedure is inspired by iterative proportional fitting (IPF; see Deming and Stephan, 1940), a technique for estimating cell values in a multi way table of probabilities when marginal totals are known and a sample from the population is available. IPF alternates (in the two way case) between rescaling the rows so that the cells sum to the correct row totals, and performing the same ....

W. E. Deming and F. F. Stephan, "On a least-squares adjustment of a sampled frequency table when the expected marginal totals are known," Annals of Mathematical Statistics, vol. 11, pp. 427--444, 1940.

....the fraction of the observed count to the expected counts in marginal tables determined by the interaction term. The IPS algorithm for log linear models was described in detail in Darroch and Ratcliff (1972) and is also known as the iterative proportional fitting or the Deming Stephan algorithm (Deming Stephan, 1940). It is, however, not necessary to store a table of probabilities corresponding to classifying according to the set of factors of the irreducible component. By representing the distributions of the irreducible components in an economical way the space requirements can be reduced (Jirousek, 1991 ....

Deming, W.E. & Stephan, F.F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, Annals of Mathematical Statistics 11: 427-444.

....as follows: 1. Obtain initial values (fi (0) f (0) ff (0) One can use (fi (0) f (0) resulting from a regression assuming independence and ff (0) 0. 2. Use a modified version of the iterative proportional fitting algorithm (IPF) which was originally introduced by Deming and Stephan (1940) and has also been used by Heagerty and Zeger (1996) and others, to obtain the joint probabilities in the bivariate marginal tables. That is, get the current estimates of the local odds ratios, k) ist , from the current estimate ff (k) and construct bivariate tables having this odds ratios. ....

DEMING, W.E., STEPHAN, F.F. (1940). On a Least Squares Adjustment of Sampled Frequency Table when the Expected Marginal Totals are Known, Annals of Mathematical Statistics, 11, 427-444.

....are achieved. This procedure increases efficiency of the estimates, because each cell has a larger weight, but increases potential bias because of the additional independence assumptions. Some organizations go beyond this to adjust for two way classifications, using iterative proportional fitting (Deming and Stephan 1940). This method derives marginal totals for specific population characteristics from Census Bureau reports, and then cycles through a series of variable combinations, weighting the data to match these targets. During iteration through this list of variable combinations, the sample distribution ....

Deming, W. E. and F. F. Stephan. 1940. "On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals Are Known." Annals of Mathematical Statistics 11: 427--444.

.... A general solution to this problem is to model the responses conditional on the poststratification variables (see Little, 1993) For example, the standard approach to adjusting for several demographic variables is to rake across one way or two way margins (i.e. iterative proportional fitting, Deming and Stephan, 1940), which essentially corresponds to poststratification on the complete multi way table, but with a model of the responses, conditional on the demographic variables, that sets higher level interactions to zero. Methods based on smoothing weights can also be viewed as poststratification, with ....

Deming, W., and Stephan, F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal tables are known. Annals of Mathematical Statistics 11, 427--444.

....Republic. As concerns the probabilistic framework 3 no direct method of solving the marginal problem is known but there exists an asymptotic method. Using the collection of prescribed less dimensional measures one can define by means of the so called iterative proportional fitting procedure [3] a sequence of multidimensional probability measures which is proved in [2] to converge iff there exists a joint measure having the prescribed measures as marginals. The limit measure then has the prescribed marginals and minimizes I divergence within the class of such joint measures. ....

....ae N and is a possibility measure over N , then its marginal on S is a possibility measure S over S defined as follows ( N j ) S (A) A Theta XNnS ) for ; 6= A ae XS ; 6= S 6= N . One of the most popular approaches for dealing with uncertainty in AI is Dempster Shafer theory [3, 13]. Knowledge can be described here in several equivalent ways (belief or respectively commonality or plausibility function) we chose the concept of basic probability assignment. Definition 5 (basic probability assignment) A basic probability assignment (BPA) over N is a real function m : exp XN ....

Deming, W.E., Stephan, F.F.: On a least square adjustment of a sampled frequency table when expected marginal totals are known. Ann. Math. Statis. 11 (1940) 427-- 444.

....is bounded above by zero, the procedure converges. At convergence, the score equations are satisfied, and the ij s and p are consistent. This algorithm is similar in flavour to the Iterative Proportional Scaling (IPS) procedure used in log linear models. IPS has a long history, dating back to Deming Stephan (1940). Bishop, Fienberg Holland (1975) give a modern treatment and many references. The resulting classification rule is d(x) argmax i [ p i (x) 6) 3 Pairwise threshold optimization As pointed out by Friedman (1996) approaching the classification problem in a pairwise fashion allows one to ....

Deming, W. & Stephan, F. (1940), `On a least squares adjustment of a sampled frequency table when the expected marginal totals are known', Ann. Math.

....model. For further details, see [Agresti, 1990] 40.2.3 The Iterative Estimation Procedure For some loglinear models, it is possible to obtain closed forms for the expected cell counts. For more complicated models, the iterative proportional fitting algorithm for hierarchical loglinear models [Deming and Stephan, 1940] can be used. Briefly, this procedure works as follows. The interaction terms in the loglinear models represent constraints on the estimated expected marginal totals. Each of these marginal constraints translates into an adjustment scaling factor for the cell entries. The iterative procedure has ....

Deming, W. E. and Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statis, (11):427--444.

....is bounded above by zero, the procedure converges. At convergence, the score equations are satisfied, and the ij s and p are consistent. This algorithm is similar in flavour to the Iterative Proportional Scaling (IPS) procedure used in log linear models. IPS has a long history, dating back to Deming Stephan (1940). Bishop, Fienberg Holland (1975) give a modern treatment and many references. The resulting classification rule is d(x) argmax i [p i (x) 6) 3 Properties of the solution The weights n ij in (4) can improve the efficiency of the estimates a little, but do not have much effect unless the ....

Deming, W. & Stephan, F. (1940), `On a least squares adjustment of a sampled frequency table when the expected marginal totals are known', Ann. Math. Statist. pp. 427--444.

....is bounded above by zero, the procedure converges. At convergence, the score equations are satisfied, and the ij s and p are consistent. This algorithm is similar in flavour to the Iterative Proportional Scaling (IPS) procedure used in log linear models. IPS has a long history, dating back to Deming Stephan (1940). Bishop, Fienberg Holland (1975) give a modern treatment and many references. The resulting classification rule is d(x) argmax i [p i (x) 6) 3 Properties of the solution The weights n ij in (4) can improve the efficiency of the estimates a little, but do not have much effect unless the ....

Deming, W. & Stephan, F. (1940), `On a least squares adjustment of a sampled frequency table when the expected marginal totals are known', Ann. Math. Statist. pp. 427--444.

....of the given marginal probabilities. In general, however, an iterative method will be required to obtain the maximum likelihood (maximum entropy) estimates for scaling factors to be applied to the marginals. The most common such method is iterative proportional scaling, generally attributed to [DS40], which is guaranteed to converge to a unique solution whenever the marginal arrays have all positive elements and are consistent with each other. One drawback of the standard iterative proportional scaling algorithm is that it requires storage and computation over the complete estimated ....

W. Deming and F. Stephan. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals Math. Stat., 11:427--444, 1940]

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W. Edwards Deming and Frederick F. Stephan. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. The Annals of Mathematical Statistics, 11(4):427--444, 1940.

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W. E. Deming and F. F. Stephan. On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist., 11:427-444, 1940.

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Deming, W. and Stephan, F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics, 11:427--444.

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W.E. Deming and F.F. Stephan. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics, pages 427--444, 1940.

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W. E. Deming and F. F. Stephan, On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, Annals Mathematical Statistics, 11 (1940), pp. 427--444.

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W.E. Deming and F.F. Stephan, On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 (1940), pp. 427--444.

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Deming, W. and Stephan, F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics, 11:427--444.

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W.E. Deming and F.F. Stephan, On a least square adjustment of a sampled frequency table when the expected marginal totals are known, Ann. Math. Stat. 11 (1940), pp. 427-444.

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