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58 Table 9. Balancing load on a loop. (a) Splitting Data into Entering and Exiting Trips Combined Entering Trip Exiting Trip Stop Off On Load Off On Load Off On Load Inherited 0 0 0 Before loop 6 42 36 6 42 36 A 10 5 31 10 26 5 5 B 10 5 26 10 16 5 10 C 10 5 21 10 6 5 15 D 10 5 16 10 -4 5 20 After loop 18 0 -2 18 0 2 Total 64 62 46 42 18 20 (b) Correcting Entering and Exiting Trips Entering Trip Exiting Trip Stop Off On Load Off On Load Inherited 0 0 Before loop 6 44 38 0 A 10 28 5 5 B 9 19 5 10 C 10 9 4 14 D 9 0 5 19 After loop 19 0 0 Total 44 44 19 19 (c) Splitting Trips at Terminal (Stop) C Entering Trip Exiting Trip Stop Off On Load Off On Load Inherited 0 Before loop 6 44 38 A 10 5 33 B 9 5 29 C (off only) 10 19 Bequeathed / inherited 19 19 C (on only) 4 23 D 9 5 19 After loop 19 0 0 Total 35 54 28 9 The researchers are not aware of efforts in APC data pro- Besides achieving on-off balance, corrections aim to pre- cessing to identify and exclude operator movements; clearly, vent negative loads. The algorithms the researchers have seen there is a need for such algorithms, especially those that work consider only departing load. A more stringent feasibility test well in the presence of possible counting error. looks also at through load. For example, suppose a bus arrives at a stop with a load of 2 passengers; on-off counts then indi- cate that 6 passengers got off and 5 got on. The departing load 8.3 Trip-Level Balancing Methods is calculated to be 1, a positive number that raises no alarm. Comparing on and off totals gives APCs a built-in error However, something about these figures is not right; how check. Large errors can simply be screened out; that is part of could 6 passengers get off when only 2 were on the bus? This the advantage of the large sample size that comes with auto- discrepancy is clear when one calculates that through load mated data collection. Counts with smaller imbalances will be based on these figures is -4. Negative through loads can occur accepted, leaving the question of how counts should be cor- if passengers (or the bus operator) get on and off at the same rected in order to balance ons and offs. stops (e.g., passenger steps on the bus, finds out it is the wrong

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59 bus, and steps off); but that is a rather rare occurrence, except tions to the largest counts, the assumption being that count- perhaps at terminals. To account for an occasional balker or ing errors are more likely to have occurred there. an operator getting off and back on, correction algorithms Another systematic method uses proportional corrections. can set -1 as the lower limit on through load, something illus- One simply multiplies all the ons in a sub-block by the cor- trated in the following example. Offs occurring at terminals rection factor f = T on /Ton, and all offs by the correction factor well after the bus has arrived and discharged its load are likely f = T off / Toff. This method has two desirable properties: first, it to be balkers, and processing algorithms may explicitly seek leaves the alighting and boarding centroids unchanged, leav- to identify and deal with them. ing average passenger trip length unaffected; second, it calls for greater corrections where counts were greater, which is 8.3.1 Selecting Target On/Off Total consistent with the notion that the bigger the count, the more likely an error. In a sub-block, boundary conditions require that the differ- A rounding procedure is illustrated in the example that fol- ence between total ons and offs equal the difference between lows. To round ons, first generate the profile of cumulative bequeathed and inherited passengers. If the totals do not bal- adjusted ons; round the cumulative profile; and then gener- ance, should total ons be adjusted to match total offs, or vice ate rounded ons by stop as the difference between successive versa, or should the correction be shared between the two? cumulative ons. As mentioned earlier, this rounding proce- If the counting system has known error patterns, they can dure can be applied to force counts in the database to be inte- be taken into account in selecting the target. Let gers, or it can be applied when generating reports. , T T on off = target (corrected) sub-block ons and offs total Ton, Toff = block level raw totals for ons and offs 8.3.3 Correcting Negative Loads M = bequeathedinherited passengers for the sub- After a sub-block is balanced, calculated loads may be neg- block (given) ative at one or more points along a route. Most commonly kon, koff = external adjustment factors for ons and offs checked is departing load; however, a stronger test is for s2on , soff = variance of sub-block errors for on and off totals 2 through load (departing load minus ons). There should be con = 1/s 2 on , coff = 1/soff 2 thresholds for negative departing load and negative through The external adjustment factors are correction factors an load beyond which a trip should be rejected. agency might have that account for known systematic bias; Trips not rejected should be adjusted to eliminate negative for example, kon = 1.03 implies that ons have a systematic loads. One option, illustrated in this section, is to allow undercount of 3%. through loads of -1 to account for an operator exiting and re- From information theory, the best (least variance) esti- entering an otherwise empty bus, or a passenger boarding and mates are given by then alighting an empty bus (e.g., upon discovering it was the wrong bus). c on konTon + c off koff Toff + Mc off If there are multiple points with negative load, the point of Ton = (3) ) c on + c off greatest violation should be corrected first, because its cor- rection is likely to cure negative loads elsewhere. The point of = T T off on - M (4) correction becomes a new sub-block boundary, dividing the sub-block into two new sub-blocks which are then balanced, When M = 0, this expression is a weighted average of adjusted making the procedure recursive. on and off totals, using relative certainty as weights. For A balancing example is given in Table 10. Original counts example, if con/coff = 3 and a sub-block had 4 excess ons, the [Table 10(a) and (b)] have 36 ons and 34 offs. First, target on correction would be to reduce ons by 1 and increase offs by 3. and off totals are calculated to be 35, then ons and offs are Even if on counts are known to be more accurate than off adjusted and rounded. counts, or vice versa, the target is more accurately estimated A check for negative loads in Table 10(c) finds the greatest using information from both sets of counts, rather than violation at Stop 5. Through load there is -4, but because ignoring the weaker set. through loads of -1 are allowed, the violation is -3. The trip is split at Stop 5 into two new sub-blocks, which are balanced in Table 10(d) through (k). Stop 5's offs belong to the early 8.3.2 Distributing Corrections sub-block (the one ending at Stop 5), and its ons to the late One simple way to distribute corrections that has already sub-block. been mentioned is to put them at the end (or start) of the sub- The early sub-block [Table 10(d) and (e)] begins with an block. The method used in TriTAPT is to make the correc- imbalance of -3 (25 ons, 29 offs, and a target difference of -1).

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60 Table 10. Balancing initial load and correcting negative load. (a) Balance Ons Input Cumulative Scaled Rounded Balanced Stop Ons Ons Cumulative Cumulative Ons 1 12 12 11.67 12 12 2 8 20 19.44 19 7 3 6 26 25.28 25 6 4 0 26 25.28 25 0 5 2 28 27.22 27 2 6 5 33 32.08 32 5 7 2 35 34.03 34 2 8 0 35 34.03 34 0 9 1 36 35.00 35 1 10 36 35.00 35 0 Total 36 35 Target 35 f = 0.972 (b) Balance Offs Input Cumulative Scaled Rounded Balanced Stop Offs Offs Cumulative Cumulative Offs 1 0 0.00 0 0 2 2 2 2.06 2 2 3 4 6 6.18 6 4 4 10 16 16.47 16 10 5 12 28 28.82 29 13 6 0 28 28.82 29 0 7 1 29 29.85 30 1 8 0 29 29.85 30 0 9 3 32 32.94 33 3 10 2 34 35.00 35 2 Total 34 35 Target 35 f = 1.029 (c) Check for negative load Stop Off On Thru Load Dep Load Violation Comment 1 0 12 0 12 2 2 7 10 17 3 4 6 13 19 4 10 0 9 9 5 13 2 -4 -2 -3 split here 6 0 5 -2 3 -1 7 1 2 2 4 8 0 0 4 4 9 3 1 1 2 10 2 0 0 0 (continued on next page)

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Table 10. (Continued). (d) Balance Ons, Early Subblock (g) Balance Ons, Late Subblock (j) Balance Counts Input Cum Scaled Round'd Bal'c'd Input Cum Scaled Round'd Bal'c'd Thru Dep Stop Stop Stop Offs Ons Ons Ons Cum Cum Ons Ons Ons Cum Cum Ons Load Load 1 12 12 12.96 13 13 5 2 2 1.60 2 2 1 13 0 13 2 7 19 20.52 21 8 6 5 7 5.60 6 4 2 2 8 11 19 3 6 25 27.00 27 6 7 2 9 7.20 7 1 3 4 6 15 21 4 0 25 27.00 27 0 8 0 9 7.20 7 0 4 9 0 12 12 5 25 27.00 27 0 9 1 10 8.00 8 1 5 13 2 -1 1 Total 25 f= 27 10 10 8.00 8 0 6 0 4 1 5 Target 27 1.080 Total 10 f= 8 7 1 1 4 5 Target 8 0.800 8 0 0 5 5 9 4 1 1 2 (e) Balance Offs, Early Subblock 10 2 0 (h) Balance Offs, Late Subblock Input Cum Scaled Round'd Bal'c'd Total 35 35 Stop Offs Offs Cum Cum Offs Input Cum Scaled Round'd Bal'c'd Stop 1 0 0.00 0 0 Offs Offs Cum Cum Offs (k) Differences (Balanced - Original) 2 2 2 1.93 2 2 5 0 0.00 0 0 3 4 6 5.79 6 4 6 0 0 0.00 0 0 Thru Dep Stop Offs Ons 4 10 16 15.45 15 9 7 1 1 1.17 1 1 Load Load 5 13 29 28.00 28 13 8 0 1 1.17 1 0 1 0 1 0 1 Total 29 f= 28 9 3 4 4.67 5 4 2 0 0 1 1 Target 28 0.966 10 2 6 7.00 7 2 3 0 0 1 1 Total 6 f= 7 4 -1 0 2 2 Target 7 1.167 5 1 0 1 1 (f) Check for Neg Load, Early Subblock 6 0 -1 1 0 Thru Dep 7 0 -1 0 -1 Stop Offs Ons Violation (i) Check for Neg Load, Late Subblock Load Load 8 0 0 -1 -1 1 0 13 0 13 Thru Dep 9 1 0 -2 -2 Stop Offs Ons Violation 2 2 8 11 19 Load Load 10 0 -2 -2 3 4 6 15 21 5 0 2 -1 1 Total 1 -1 4 9 0 12 12 6 0 4 1 5 5 13 -1 7 1 1 4 5 8 0 0 5 5 9 4 1 1 2 10 2 0 0 0

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62 Target ons is calculated to be 27, target offs to 28. Corrections helps preserve the integrity of the data, is useful for investi- are distributed proportionally and rounded as before. Note gations, and can be used for testing new balancing methods. that in balancing the late sub-block in Table 10(i), through Can records with invalid counts be retained? There are load at Stop 5 is initialized to -1, the correction target. likely to be many trips for which count data is judged invalid, Both segments, after balancing, have no negative load vio- but for which time and location data, useful for analyzing lations, so the correction procedure ends. operations measures such as schedule adherence, is not. The reverse can also be. This concern is met by including flags in the database for "counts invalid" and "times invalid." 8.3.4 Other Count Correction Issues Should corrected counts be forced to be integers? Most Independent of the balancing procedure, several questions users prefer that corrected counts be integers. The appear- related to databases and corrected counts arise: ance of fractional counts in a database tends to raise ques- tions and emphasizes that the raw counts were deemed Should the database store corrected counts, or should incorrect. On the other hand, it is quite possible to store corrections be made on the fly? In the APC analysis sys- fractional counts and simply round analysis results. tems studied, corrections are made on the fly. However, Should the database store load, or should it be derived those systems used simple algorithms, not accounting for from counts? Passenger load can be derived on the fly from inherited passengers or loops. With more complex cor- counts starting at the beginning of a trip, provided there rection algorithms, it seems preferable to make correc- are records for inherited passengers. However, storing cor- tions during entry processing, storing the corrected rected load can be a way of compensating for not storing counts in the database. corrected on/off counts or inherited passengers, as is the Should raw counts be stored as well as corrected counts? practice at Tri-Met. Tri-Met uses uncorrected counts in If corrected counts are stored, agencies and vendors both analyses of boardings, while for analyses involving load it still want raw counts retained as well. Storing raw counts uses balanced load estimates.