Based on these 12 events, the resulting equation provided below is accurate within ¡ 0。1 (one tenth) of a second。

CVR Elapsed Time + 63732。8 5 FDR SRN

The time correlation from FDR SRN to Eastern Standard Time was provided by the Aircraft Performance Specialist:

Eastern Standard Time 5 FDR SRN–9701。119 (where [EST] is expressed as seconds after midnight)。 (Brazy, 2009, p。 2)。

The CVR to FDR time alignment problem is a specific instance of the general problem of time series analysis。 Time series analysis is a method for analyzing time series data that can be pided into two segments, one of which is time-domain。 Time-domain methods include  convolution,

auto-correlation, and cross-correlation analysis。 Cross- correlation is a mathematical process of determining how much function f must be shifted along the x-axis to be identical to function g and is a necessary step in a broad variety of applications, from forensics to chemistry (Fischer, Roth, & Buhmann, 2007)。

Gregor (2006) characterizes the CVR to FDR time alignment problem as having the following components: (a) knowing which FDR event corresponds to which CVR event; and (b) optimizing the offset between two series of events。 For the purpose of discussion, the first step will be referred to as pattern matching, and the second step as alignment optimization。 Each step is discussed in  turn。

Pattern Matching

Gregor introduces the pattern matching problem as one of maximization of a cross-correlation between the CVR and FDR series of events (Gregor, 2006, p。  6)

Z(n) 5 Sk FDR(k ) * CVR(k + n)

where k represents an index to each of the FDR events。 The cross-correlation essentially tries every combination of FDR event and CVR event, finding a maximization of the two alignment areas。 In this case, each FDR(k) and CVR(k) is a binary value representing a one second sampling interval, however the equation can be computationally applied to real numbers。 Cross-correlation can also be found in matrix operations by the name of convolution (MathWorks, 2011) as well as in the cross join operator of structured query languages (Microsoft, 2011) or as a Cartesian Product。

Linear programming (LP) can be applied to pattern matching。 For example, in the case of searches being issued to multiple Internet search engines, LP was used to determine which document had the maximum relevancy across all search engine results。 Amin and Emrouznejad (2010) outlined an LP model with an objective function of

max Sj (lkjvj)

where l represents the kth document being returned as the jth ranked place and v is the weighting being sought for the jth place。 The associated constraints are

Sj  (lijvj) ,51 (i51,…,r)

vj-vj+1 。5 e (j51,…,l–1)

vl 。5 e

where the first constraint bounds the relevancy of each document。 The second and third constraints provide that the weights assigned to document j as the progression continues from document j to document  j+1。

Insight into the pattern matching problem can also be drawn from the methods used in time series analysis and forecasting。 Patterns in time series data are known to follow particular patterns, such as random, trend, seasonal, cyclical。 Anderson et al。 (2011) show how data can be curve-fit into various time series models for the purpose of forecasting;  providing  examples  of  how  this  is  done in

Microsoft Excel and LINGO。 Time series forecasting may be adaptable to CVR/FDR pattern matching by curve- fitting the CVR data series to the  FDR。

Alignment Optimization

The alignment optimization step may be a simple offset,

C, given by

tcvr 5 tfdr + C

This optimization is applicable when the timebase rate of the CVR and FDR are the same (Gregor, 2006, p。 5)。 If the timebase rate is different between the two samplings, such as may occur in legacy tape based units or where solid state units have flaws, a first order linear offset is appropriate, represented by