NBSIR 73-189

A Comparison of Several Methods for

Forecasting U.S. Traffic Fatalities

Alexander R. Craw

Technical Analysis Division

National Bureau of Standards

Washington. D. C. 20234

May 1973

Final Report

Prepared for National Highway Traffic Safety Administration

Department of Transportation

Washington, D. C. 20590

NBSIR 73-189

A COMPARISON OF SEVERAL METHODS FOR

FORECASTING U.S. TRAFFIC FATALITIES

Alexander R. Craw

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

May 1973

Final Report

Prepared for National Highway Traffic Safety Administration

Department of Transportation

Washington, D. C. 20590

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

MATIOIMAL BUREAU OF STANDARDS, Richard W. Roberts. Director

SUMMARY

This is a second report to the Mathematical Analysis Division of the

National Highway Traffic Safety Adnjlnlstration (NHTSA) on the subject

of forecasting annual highway fatalities. This report concerns a comparison of several time series analysis programs based on exponential

smoothing and non-decompositlonal methods currently employed by NHTSA for

projecting the annual traffic fatalities for the entire U.S. Several

methods of data aggregation are studied.

It is found that there is some advantage in using lumped (pooled)

data for each region aggregated either quarterly or half yearly, and

using the Sum 6f Regional estimates to estimate the national value.

Also, there does not appear to be any great difference in the results

obtained using the non-decompositlonal methods and those obtained by

time-series analysis programs based on exponential smoothing methods.

Estimates for the 1972 and 1973 national traffic fatalities by a

variety of methods were made. For 1973 the estimates ranged from a low

of 54186 to a high of 55994, with a mean of 55055.
TABLE OF CONTENTS

Acknowledgements

Summary ~ . ^ -

List of Tables : List of Symbols and Formulas

1. Introduction and Problem Definition

2. Analyses Based on 12 Years and 6 Months of Data (12,6).

2.1 National Traffic Fatalities

2.2 A Look to Regions

3. Forecasts of Total U.S. Traffic Fatalities Using No Data

from Current Year

3.1 Modifications Needed

3.2 Extrapolations '--y'

4. Conclusions

Appendix: Comments on Forecasting Techniques and Computer

Programs
LIST OF TABLES

Table 1. Comparisons of Estimates by Data Features Rased on

Ag/S* (Fractional Error)

.

Estimates for Calendar 1972, Using 12 Years, 6

Months Data

Table 2.

Table 3.

Table 4.

Table 5,

Table 6.

Table 7.

Table 8.

Table 9.

Errors of Estimates of 1972 U.S. Traffic Fatalities

by Various Non-Decompositional Methods Using 12

Years, 6 Months of data.

Estimates of U.S. Traffic Fatalities for 1961-1972

Using a Difference Method: Yq = mo+(Y_i - m.^^)

Order of Region by Number of 1972 Traffic Fatalities

Tentative New Regions Based on Similarity of Seasonality Factors from TIMSR 4

Fractional Errors of Regional Estimates for 1972 by

Various Methods Based on 12 Years, 6 Months of Data.

Frequency of Firsts, Seconds and Third Best Procedures (Regions)

Computation of Forecasts of U.S. Traffic Fatalities

for 1973 by a Modified NHTSA Method:

<k)

o+ i _

VOL, = ratio of k average pairs

Computation of Estimates of U.S. Traffic Fatalities

for 1971, 1972 by a Modified NHTSA Method:

m.

Table 10. Computation of Estimates of U.S. Traffic Fatalities

for 1973, 1972 Using a Modified NHTSA Method:

"^o+l _ ^_.]c^-i+l

m. k"^m

= average of k ratios

-1

, iii
Table 11. Computation of Estimates of U.S. Traffic Fatalities

for 1969, 1970, 1971 Using a Modified NHTSA Method

y(^^ 1 Y O+l -Lri, -i + 1 = — = average ox k ratiios

Table 12. Computation of Estimates of U.S. Traffic Fatalities

for 1962-1973 Using a Modified Difference Method.

^o+l = Yo + (mo - m_i)

Table 13. Forecasts of U.S. Traffic Fatalities for 1973 Using

a Variety of Methods

Table 14. Estimates of U.S. Traffic Fatalities for 1972 Using

12 Years of Data

Table 15. Fractional Errors of Regional Estimates for 1972

• by Various Methods Based on 12 Years of Data.

IV
ACKNOWLEDGEMENTS

Particular thanks are due to Mr. William Obright, who made all of

the computer runs and carried the major burden of the computing load.

Mrs. Elizabeth Leyendecker passed along her knowledge of operating with

the GEXSMO and EXPSMOOTHING programs and graciously offered counsel

when snags appeared.

V
LIST OF SYMBOLS AND FORMULAS

S = the actual (observed) value of the time series datum (in this case fatalities). This may be a monthly, quarterly, halfyear of yearly value, depending on the context

S* = the actual value for a calendar year

S = an estimate of S

^, ^ below

A =S-S;ifA>0, the estimate is above the actual

A^ = the error in the forecast for the last 6 months of the

calendar year (Table 1)

A/S* = the mean absolute fractional error m = current total for specified months of fatalities from

o which calendar year forecasts are made. This may be for

3, 6 or 9 months. In this report a 6-month value is

customary.

m , = the corresponding total for specified months of fatalities -1

for i ^h previous year

Y__j^ = the total fatalities for the i^h previous year; i = 0, 1, 2,

Y. = an estimate of fatalities for year i 3 K = the number of years of data over which summation occurs

N = an NHTSA forecast method

(K) - an NHTSA forecast method applied to the Lumping mode using

K years of past data

(K) N2 = an NHTSA forecast method applied to the Sum of States mode

using K years of past data

H = monthly data

Q = quarterly data

H = half-year data

Y = yearly data

vi
L(M) = a lumping mode (for a region or U.S.) using monthly data.

Monthly data from individual states constituting a region

are added together to create a time-series for a region

SS(q) = a Sum of States mode consists of making individual analyses

and predictions for each state, then s\m the results over

the states constituting a region, then sum the results

of each of the regions in order to obtain a forecast for the

natural values; in this instance quarterly data are indicated

L = ^ [L(M) + L(Q) + L(H) + L(Y)]

= an average of liomping methods

= ^

[L(Q) + L(H) + L(Y)]

^ Regions = a forecast for the national value obtained by summing

the individual forecasts for each region

Entire US = a forecast for the national value obtained from one grand lumped time series for the nation

X = a leap year adjustment factor

vii
NHTSA FORECAST METHODS

;(K) yK

: = A —r:^ , a ratio or proportionality method

^1-1

A Difference Method:

Modified NHTSA Forecast Methods

n+i h -i

-i+1

y(K) yK

,,(k) 0+1 , ^1 -i ^. ^ ^ . ^ , M^ : = X — = ratio of K average pairs, K = 1, 1 m„ rK ^0

y(K) y

^(K) . ^0___ _ A vK -i+1

= average of K ratios, K = 1, 2. 2 ^1 m . ,

• K ^1 m ° " ' "

0 -1+1

Modified Difference Methods:

^0+1 - ^ = ^0 - ^-1 °^ W = ^0 ^^0 -

°^-i)

Variants of this are:

(Kj. ;(K) ^ _ 1 f""^ V (K) , K = 1 \ ^0+1 K ^ ^ ^^i -°^-i+l^ = ^0 - "^-1

1=0

4'' : 'ill -^0*k - --i.l' = ^0 ^ ^^^^

viii
Note that in any of the methods with a superscript (K) will yield

K estimates, say

yd) y(2) ;(K)

and that these may be averaged to produce still other estimates.

In table headings:

GEXS = GEXSMO time series analysis computer program

EXPS E EXPSMOOTHING time series analysis computer program

MAE = Mean absolute fractional error

ix

1.0 INTRODUCTION MD PROBLEM DEFINITIONS

In a previous study for the Mathematical Analysis Division

(mad) of the National Highway Traffic Safety Administration

(NHTSA), the Technical Analysis Division made a comparison of

forecasts for the current calendar year of national (entire U.S.)

highway fatalities. Those obtained by using readily available

computer routines for time series analyses based on exponential

smoothing procedure were compared with methods employed currently

by NHTSA. The methods were applied to estimate the current year

total of fatalities given an initial 3 months of fatality data,

then an initial 6 months of data and finally an initial 9 months

of data. That report concluded, in part, that there is no coercive

evidence to change their methods when operating on the national

highway fatalities time series.

It was pointed out by members of NHTSA^ that the cyclic and

trend properties of the highway fatality time series appear to

differ in the various regions and, perhaps, are idiosyncratic from

state to state. Thus it appeared possible that some important

information is lost when operating on the time series for the entire

U.S. To explore this possibility it was suggested that forecasts

for each region be made using an exponential smoothing method on

^U.S. Department of Commerce, National Bureau of Standards,

Time Series Forecasting of Highway Accident Fatalities . NBSIR-73-138

(19T3).

^Letter from R.J. Taylor, NHTSA , to Alexander Craw, NBS

January 29, 1973, Reference number NU3-31.

1
each region's time series and that these forecasts be summed, to

obtain a national forecast. These forecasts co\ild then be compared

with the forecasts on the national time series.

Unfortunately, the state (and therefore the regional) data

is available only for traffic fatalities. Non-traffic fatalities

are estimated on a national basis for inclusion into the national

time series for highway fatalities. Thus the summed regions

forecast must be compared with a forecast derived from the national

traffic fatalities time series.

At first, following the procedures of the previous study,

the analyses were performed using 12 years and 6 months of data

to obtain a forecast for the last 6 months of calendar year 1972.

From the known first 6 months totals and the forecast for the last

6 months forecasts were obtained for the highway traffic fatalities

for the entire calendar year 19T2.

Midway through the analyses, we were asked to provide forecasts

for the calendar year 1973 by each of the several methods under

review. This posed no problem as far as the methods using exponential

smoothing were concerned. However, because the NHTSA's procedures

required some data for the first several months of the calendar

year which were not then available, the current NHTSA methods

required slight modification. The details of these modifications

are discussed in section 3.
In the first study^ the analyses were based on the first

3 months of calendar year data, then the first 6 months of data,

and finally the first 9 months of data. In the new work, in order

to cut down on the amount of computing required, it was decided

to make a comprehensive comparisoh based on the first 6 months

data for a calendar year. We remark that in the previous work,

no systemmatic differences were noted between the results based on

3 and 9 months of data and those based on 6 months of data.

Also, rather than carrying out a large number of runs in the timesharing mode using a typewriter output, we decided to make the

comparisons using the GEXSMO time-series analysis routine and have

an additional partial comparison using the EXPSMOOTHING^ program

operated in a time-share mode but with high speed output from tape.

The notations adopted for the previous report will be used in

this report; any changes or additions will be noted explicitly.

(See List of Symbols and Formulas.)

^U.S. Department of Commerce, National Bureau of Standards,

Time Series Forecasting of Highway Accident Fatalities. NBSIR-T3-138

(1973).

^For discussion of these time series analysis programs see the

discussion and references given in NBSIR-T3-138 and the Appendix.

3
2.0 ANALYSES BASED ON 12 YEARS AND 6 MONTHS OF DATA (l2, 6)

Production runs were made using monthly (M) data, quarterly

(Q) data, half-year (H) data and yearly (Y) data for several levels

of aggregation and for two different modes of aggregation.

The first mode consists of making individual analyses and

predictions for each state, then sum the results over the states

constituting a region, then sum the results of each of the regions

in order to obtain a forecast for the national values. This mode

is denoted by SS(M), SS(Q), . . ;

The second mode of aggregation is called lumping and is denoted

by L(M), L(Q,) , .... In this mode, a time series for each region

was created by limping (adding) the data from each state in the

region. Forecasts were then obtained for each of the regions directly.

Finally, forecasts were obtained for the national valves by summing

the results of the regions. This latter is denoted by ^ Regions.

In the summary tables GEXS is used to denote the GEXSMO

program, EXPS, the EXPSMOOTHING program. N denotes an NHTSA

method. For each region (U.S.) the forecast for the current calendar

year from 6 months of data by a method N is independent of how the

aggregation for the region (U.S.) is made. However, the forecasts

for region (U.S.) by summing forecasts of states by a method N

does not necessarily equal the forecasts for region (U.S.) using

( K) ( K ) lumped data by method N. Thus, for a fixed K, NjJ^

^

=}= >

where K denotes the nimber of years of past data used in the N

method, and the subscripts refer to the lumping or Sum of States

mode, respectively. (See Appendix A for more detail.)

4
Table 1 contains a suimnary of the comparisons of forecasts

by data features based on the fractional error criterion, A/S*.

Here Ag is the error in the forecast for the last 6 months of the

calendar year and S* denotes the observed value of the fatalities

for the calendar year. The forecasts in Table 1 are for the calendar

year 1972, based on 12 years, 6 months of data (12, 6). The (signed)

fractional error for each region is shown for each mode aiad method

of aggregation.

For the forecasts produced by the GEXSMO program:

1) the best forecasts for regions is by L(Q);

2) the best forecasts for U.S. is for ^ Regions by L(Q);

3) ^ Regions forecasts are better than the forecasts for the

entire U.S. (based on one grand lumping).

k) In general, for regions the mean absolute fractional error

(MAE) for limped forecasts is less than the MA.E for forecasts

by Sum of States mode. The exception is SS(M).

5) In general, for ^ Regions, the error for forecasts by

the lumping mode is less than that for forecasts by Sum

of States mode. The exception is SS(M).

Guided by these results we decided to make some computer runs

using a different computer time-series analysis program, namely

EXPSMOOTHING . Rather than run a complete duplication of the GEXSMO

runs, it was decided to operate in the lumped (L) mode only and

to use only quarterly, half year and yearly data. The results of

these computations are shown in Table 1 also.

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Of the EXPSMOOTHING methods, the best forecasts were obtained

using half-year data, L(H) . For this mode, L(H), the Regions

forecast was better than the L(H) forecast for the Entire U.S.,

although both were very good. For both quarterly and yearly data,

however, the Entire U.S. estimates were better than the ^ Regions

estimates. However, the relatively poorer forecasts of ^ Regions

in this case of L(Q) is due primarily to a poor estimate for

one region. Region 8. Very poor estimates were obtained were

obtained for Regions 2, 3 and 8 using L(Y). We remark that in

general Region 8 was poorly forecast by all methods.

Using the NHTSA method and K=12 (see Appendix) the forecasts

for J Regions! by lumping were better than those produced by the

]J

Regions]

Sum of States mode (see Table 2), and the ^ Regions forecast by

lumping was better than that for the Entire U.S.

Applying the NHTSA methods to the Entire U.S., it was possible

to produce 12 different forecasts, depending on whether 1, 2,

12 years of past data was employed. These ranged in value from

5522U to 55930 with a mean of 55593 for the U.S. traffic fatalities

The corresponding fractional absolute errors ranged from O.OO61

to 0.0190 with a mean of 0.0128. The values of K corresponding to

8
these forecasts vere K=h and K=ll with the K-value corresponding

to the mean between 6 and T. (Table 2}

The NHTSA methods^ ma^^ be called ratio methods, since they

may be vritten y(K) rk

22 ^ ^ ^i=l -i

m„

0 ) . ^m . ^1=1 -1

Forecasts were also made by a difference method

If m^ is the sum of data for the first 6 months of the current

year, then Y ^

- m_-j^ is the sum of the data for the 1st 6 months

of the previous year. Res-ults for 1972 by this method were excellent,

the fractional absolute error being 0.0005. This appears to be

a fortuitous result. When the method was applied to previous years

a wide range of values resulted. (see Table 3)

^George Suzuki points out that the current MTSA methods utilizes

the so-called "non-seasonal" or "non-decompositional" procedure.

This is in contrast to time-series analysis which generally goes

through a decomposition process to isolate the trend. In this

decomposition process , the seasonal and other regular variations

are identified. The reconstitution of a future time series, i.e.,

extrapolation, is performed by extrapolating the trend and then

imposing the corresponding cyclical variations. Both the GEXSMO

and EXPSMOOTHING time-series programs do the latter. (See Appendix.)

9
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11
2.2 A Look to Regions

On the basis of the forecasts for 1972 using 12 years

and 6 months of data, it appeared that some improvement could be •

obtained in estimating national traffic fatalities, if the estimates

for each region could be improved. There also appears to be some

legitimate merit in making acc\irate estimates for the regions in

their own right. j

Two places to look for possible improvements are (a) those

regions which contribute the largest numbers to the national total

and (b) those regions which were forecast poorly.

Over 50 percent of the national traffic fatalities for 1972

were contributed by Regions U, 5 and 6. (See Table 4). Regions 9

and 3 contribute another 20 percent of the total. If one were to

look for improvements in forecasting national traffic fatalities,

these are the regions that should be examined for homogeneity of

data characteristics. "'

If Regions U, 5 and 6 were forecast accurately, by say the

GEXSMO program using the lumped mode and the other Regions remained

unchanged, then the total error in the forecast could be reduced

from 60k to 70 (out of 5^889) if monthly data were used. The reduction

in error would be from U78 to -110, from 993 to 23^ and from -hlj

to -101 if quarterly, half-year or yearly data were employed,

respectively. These might be looked upon as upper bounds, it being

presumptious to assume that the forecast could be made with no

error. However, if the accuracy of the forecasts for these three

12
Table 4. Order of Region by Number of 1972 Traffic Fatalities

1972 Percent

Order Region Fatalities of Total Cum %

1 4 11998 21.86 21.86

2 5 10363 18.88 40.74

3 6 6862 12.50 53.24

4 9 6189 11.28 64.52

5 3 5177 9.43 73.95

6 2 4508 8.21 82.16

7 7 3477 6.33 88.49

8 8 2182 3.98 92.47

9 1 2143 3.90 96.37

10 10 1990 3.62 99.99

Total 54889

13
regions alone were improved by a factor of 2, i.e. from slightly

over 2 percent average error to slightly over 1 percent average

error, then the total error in the national forecast would have

been reduced to 137, l8U, 663, -259 by the four data aggregation

M, Q, H and Y, respectively, using the GEXSMO program.

Before any calculations were made, the NHTSA time-series

program TIMSRU output was examined by state for each region. The

seasonality factors were scanned to determine if a particular region

could be broken down into sub-regions that had similar seasonality

factor patterns. The results of this exercise are given in Table 5-

Using these definitions of sub-regions as a guideline, calculations

of regional fatalities were made using the GEXSMO program with liimped

quarterly, L(Q), data for the sub-regions U.2, 5.1 and 5-2. Since

subregion h.l consisted of the one state (Florida), this calculation

was at hand in the SS(Q) set of calculations. Adding the results

of fatalities for regions U.l and k.2 the new estimate of fatalities

for Region k became 12126, an overforecast by 83 fatalities with

fractional error A/S* = .0178 in both cases. These are an improvement

from A = 272 to A = I83 when comparison was made to the SS(Q)

mode for Region 5- .

.

No recalculations were made for Region 6, the region with the

third highest fatality total for 1972, because there did not appear

to be any sub-regions distinguishable from the entire region on the

basis of seasonality factors.

14
Table 5. Tentative New Regions Based on Similarity of Seasonality Factors

from TIMSR4

New

Region* Region States

4 4.1 Florida

4.2 Others: Alabama, Georgia, Kentucky, Mississippi,

North Carolina, South Carolina, Tennessee

5 5.1 Minnesota, Wisconsin

5.2 Illinois, Indiana, Ohio, Michigan

6 6.0 No change; Arkansas, Louisiana, New Mexico, Oklahoma,

Texas

9 9.1 California, Arizona

9.2, Hawaii

9.3 Nevada

3 3.1 District of Columbia

3.2 Others: Delaware, Maryland, Pennsylvania, Virginia,

West Virginia

2 2.0 No change: New Jersey, New York, Puerto Rico**

7 7.0 No change: Iowa, Kansas, Missouri, Nebraska

8 8.0 No change: Colorado, Montana, North Dakota,

South Dakota, Utah, Wyoming

1 1.1 Maine, Vermont

1.2 Rhode Island

1.3 Connecticut, Massachusetts, New Hampshire

10 10.1 Alaska

10.2 Idaho

10.3 Oregon, Washington

* In order of decreasing magnitude of traffic fatalities for 1972.

** Puerto Rico not included in this study.

I

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Region 9» the fourth highest in 1972, was the best forecast

region by both the GEXSMO and EXPONENTIAL programs. There did

not appear to be any great chance of picking up much improvement

in forecast in this region.

Table 6 gives a comparison of forecasts for 1972 by region by

each of the methods using 12 years and 6 months of data. The comparison

are on the basis of the fractional error (A/S*). In this table,

which contains the data of Tables 1 and 2 but is presented to

emphasize regions, entries are marked B to indicate the "best"

mode of forecasting each regions fatalities. "Best" mode is that

for which the absolute value of the fraction error is smallest.

On this basis, for the GEXSMO program forecasts, the best mode of

forecasting is L(M) for Region 1, L(H) for Region 2, L(Q) for

Region 3, SS(M) for Region U, etc. The most frequent "best" mode

is L(Q) with a frequency of 3 out of 10. The mode with the lowest

mean absolute fractional error is also L(Q). The modes with no

best regional forecasts are SS(Q) and SS(H). Region 8 had the worst

"best" forecast with A/S* for this region being -.0197- On the

basis of mean absolute fractional error the region that had best

forecasts is Region 9*

The ranking order for each of the regions is also given in

Table ih. The symbol, h , means that (in this case Region l)

was the hth best forecast region; 10 indicates the worst forecast

on the average. This dubious distinction went to Region 8 for all

three methods: GEXSMO, EXPSMOOTHING and KHTSA methods.

17
While 8 modes of forecasting were done by GEXSMO, only three

modes vere run using the EXPSMOOTHING program: L(Q) , L(H) and

L(Y). On the basis of frequency of best regional forecast, L(H)

had a frequency of 6, L(Q) had 3 and L(Y) had 1.

Results for the two possible NHTSA methods, by sums of states method

(Ng) and the lumping method (N-j^) were fairly evenly divided:

6 to 5 in favor of N^. - . .

The frequency of firsts , seconds and third "best" regional

forecasts by procedure is given in Table 7

^^-^ . . . TABLE 7

Frequency of First, Second and Third Best Procediire (Regions)

GEXS EXPS N

First 5 3 2

Second h i+ 2

Third 1 3 6

On the basis of this sample, if one were interested in accuracy

of regional forecasts, it would appear that one of the time series

programs should be preferred over the current N methods

18
The second way to improve forecasts would be to improve the

accuracy of the worst forecast regions, provided of course that

the improvement would be large enough to seriously affect the national

values. Since the current national traffic fatalities is near 55,000,

a change of 550 would affect a change of approximately one percent in

the national forecast, 275 would affect a change of approximately 0.5

percent and 110 would effect a change of approximately 0,2 percent. In

our notation the fractional error would be ,01, .005 and .002 respectively.

Region 8 was the worst forecast region by each of the procedures.

The total fatalities for 1972 was 2182 or approximately 4 percent of the

national value. The best forecast by any method is close to 2 percent

or Z 44 fatalities. But this is less than 0.1 of a percent of the national

value, so we abandoned any effort to make special progress on this region.

We remark in passing that Region 8 was underforecast by all of the GEXSMO

and EXPSMOOTHONG programs, and that Region 7 was underforecast by all

the GEXSMO programs.

The second worst forecast region is Region 2 which contributes about ,

8 percent of the national total. However, this region was forecast very

well by GEXSMO using L(H) and by EXPSMOOTHING using L(H) and by both N

methods. Table 5 indicates there is no need to break the region into

subregions. Also this case has already been studied by the sum of

states mode since the region consists of only 2 states:"'' New York and

New Jersey.

"'"Puerto Rico was not included in this study.

19
3.0 FORECASTS OF TOTAL U.S. TRAFFIC FATALITIES USING NO DATA FROM

CURRENT YEAR

3. 1 Modifications Needed

To provide forecasts for the calendar year 1973, two things were

necessary: (1) the need to modify the NHTSA method slightly since data

for the current calendar year are not available and (2) to obtain some

forecasts for previous years using these methods and forecasts by the

exponential smoothing methods using no current year data. The latter is

needed to get some feel for the error involved the 1973 forecasts.

To obtain forecasts for this year based on last years data two

modifications of the NHTSA method were used. The first was

- ^(K) lyK yK

0+1 ^ K / 1 -i+1 ^ ^1 -i+1 ^

0 K^l "^-1 ^1 "^-i

The right side of this equation represents the ratio of average pairs.

For 1973, K = 12, while for 1972, K = 11. The second method was

0+1

- —}1 -i+1 ^. ^ T ^ = average of K ratios, K = 1, . . . . m^ K '—1 m

^

Each of these methods generates K estimates

;(1) (2) (K)

0+1' 0+1' ' •

* '0+1

and one can take the mean of these K estimates to provide other

estimates. .

Using difference techniques we obtained

^0+1 - "^0 = ^0 - "^-1 °^ ^0+1 = ^0 ("^0 - "-1^

20
A simple variant on this is

Y(K) - m = i 5; (Y

0+1 0 ^ i=o i

m K = 1,

Another is

K-1

Y(K) = Y

0+1 '0 i=0

mi.,^)=Y +m„-m , -1+1 0 0 -1

K = 1,

Results of employing some of these modifications to obtain forecasts

for the current year and previous years are presented in Tables 8-12.

In each of these tables m refers to the lumped data for the last 6 months

of the year and Y refers to the yearly total.

Forecasts for 1973, 1972, and 1971 using the method of the ratio

of K average pairs are presented in Tables 8 and 9. For 1973, the

low value is obtained for K = 3, the high value for K = 11 and the K

corresponding to the mean value is between 6 and 7- For 1972, the

the low value is obtained for K = 2, the high for K = 10 and the K

corresponding to the mean value is between 5 and 6. For 1971, the values

are K = 1, K = 9, and between 7 and 8. For both 1971 and 1972 the

observed value is located in the same interval as the mean value of the

K forecasts.

Similar results were obtained using the method of the average of K

ratios. See Tables 10 and 11.

Results by the simple difference method (modified) for the years

1962-1973 are presented in Table 12.

To reduce the amount of computation the exponential smoothing

programs were run only in the lumped mode. Forecasts for 1973 are

presented in Table 13, while corresponding forecasts and errors for

1972 are given in Tables ik and 15 . In addition to forecasting for

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these four methods was taken. This Is denoted by L, where

L = i4[L(M) + L(Q) + L(H) + L(Y)].

Using GEXSMO, the best estimate for the national traffic fatalities

for 1972 was obtained using L(H), and identical estimates were obtained

for E Regions and Entire U.S. by L(H). However, all of the forecasts

for 1972 by Z Regions and Entire U.S. had percent error less than 1 percent, with the exception being Entire U.S. by L(Y). Excluding this

outlier of -.0315, all the GEXSMO estimates were in the interval

[.0019 to .0089]

.

Using EXPSMOOTHING the best estimate for the national traffic

fatalities for 1972 was obtained using Z Regions and L(Q). (Tables 14,

15)

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for 1972 range from 54121 to 55870 with mean average fractional errors

of .0046 and .0053. (Tables 14, 15)

A grand average of the 25 estimates for 1972 not all Independent,

slightly underforecast the national traffic fatality value by -265

(-.0048). (Tables 14, 15).

28
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30
h . 0 CONCLUSIONS

Based on the computations using 12 years and 6 months of data to

estimate the national highway traffic fatalities for the last 6 months of

1972 (and thus the total calendar year traffic fatalities), there

appears to be some advantage in using lumped data for each region

aggregated either quarterly or half yearly, and using the Sma. of Regions

Method to estimate the national value.

There does not appear to be any great difference in the results

obtained by using the non-decompositional methods and those obtained by

time-series analysis programs based on exponential smoothing methods.

Qnly a few of the vast number of variations possible by non-decompositional

methods were tried, and the results of these compared favorably with those

done using exponential smoothing. Of the K forecasts possible with the

NHTSA methods (or slight modifications) good results were obtained for

national estimates by averaging over the K forecasts.

There does not appear to be any great gain to be had in creating new

sub-regions of existing regions on the basis of homogeneity of seasonality

factors, although some benefit was evidenced. Although not investigated

here, possibly a redefinition of the regions by homogeneity of data

properties might lead to better forecasts for the nation.

Several regions (notably Regions 8 and 2) were consistently forecast

poorly by all methods employed. However, the forecasts for Region 2 were

(12) fl?) in error by less than one percent by the NHTSA methods N^^ ' and N^ .

By contrast. Region 9 was forecast well by all methods tried.

31
On the basis of the sample taken, if one were interested in accuracy

of regional forecasts, it would appear that one of the time-series programs

should be preferred over the current N methods.

To forecast national highway traffic fatalities for 1973 for

non-decompositional methods used by NHTSA had to be modified slightly

since no data for the early months of 1973 were at hand. Using these

modified methods and the two exponential smoothing methods programmed

to yield yearly rather than half-year forecasts, 25 estimates, not

necessarily independent, were made for calendar year 1973- These

ranged from a low of 5^186 to a high of 5599^, with mean of 55055.

The corresponding estimates by the same methods for 1972 ranged from a

low of 53166 to a high of 55537 with a mean of '^h62k. Observed national

traffic fatalities for 1972 were 5^889.

32
APPENDIX

COMMENTS ON FORECASTING TECHNIQUES AND COMPUTER PROGRAMS

Time series analysis generally goes through a decomposition process

to isolate the trend. In this process, the seasonal and other regular

variations are identified. The reconstitution of a future time series,

i.e., extrapolation, is performed by extrapolating the trend and then

imposing the corresponding cyclical variations. Since the NHTSA need is

for the annual total, the requirement is less severe since the annual

total can be projected without direct consideration of seasonal or other

cyclical variations.

The current NHTSA procedures utilizes the "non-seasonal" or

"non-decompositional" procedure, but does this in a very straightforward

way with one assumption and variants of the assumption. The ass\imption

NHTSA uses is one of proportionality, that is, that this year's fatalities

is proportional to last years' fatalities as the cumulative total up to

this time is to the corresponding cumulative total last year. The

variants have to do with using the average of the last k years of

cumulative totals instead of just last year's. Further variations not

now among the NHTSA methods can be tried on this, such as various

weighted averaging ( experiment ial smoothing) of the last k years just

like the weighted schemes in time series analysis.

A different basic ass\amption would be one of additivity instead of

proportionality. The simplest procedure under this assumption is to

take the fatalities over the latest one year period. In your notation,

= m^ + - m_-|^), where the second term is the fatalities last

year over the period complementary to mg. Many variations can be

33
imposed on this, such as various weighting schemes on the prior years'

data. This additive scheme is more conservative, but may be most suitable

"When the third quarter data are in. For the first quarter data, this

procedxare may be too conservative "when there is a significant trend.

Clearly, it is possible to take combinations of the additive and

proportional schemes. All of these procedures would be very simple

methods . . .

A further possibility is to combine either of the "non-decompositional"

approach with that of time series analysis, principally the trend analysis portion. If the trend projection is imposed on the procedures with

the additive assumption the result will be somewhat kin to the procedures using time series analysis. If imposed on the proportionality

procedure, the concept will be quite different as might be the results.

Then there is the wide gamut of procedures that utilize other data

in addition to the fatalities. Thus, related data such as auto registration, number of teenaged drivers, etc., that might be used as additional

predictors would be utilized. This procedure could soon extend outside of

the "simple" procedures, but would certainly be the appropriate approach

to investigate if the extrapolation extends much beyond the current

annual total.

These (edited) comments have been graciously furnished by Dr. George

Suziiki of the Technical Analysis Division, National Bureau of Standards.

As indicated in the main report and the list of symbols and formulae,

several of these methods have been tried, but an exhaustive study was not

made.

34
FORM NBS-114A (1-71) U.S. DEPT. OF CCMM. 1. PUBLICATION OR REPORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA j^BSIR 73-189

3. Recipient's Accession No.

4. TITLE AND SUB 11 1' LE A Comparison of Several Methods for Forecasting

U.S. Traffic Fatalities

5. Publication Date

6. Performing Organization Code

7. AUTHORCS) „ „ Alexander R. Craw

g. Performing Organization

NBSIR 73-189

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF CO.MMERCE

WASHINGTON, D.C. 20234

10. Project/Task/'Work Unit No. 4 O ± 4 4 y b

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

National Highway Traffic Safety Administration

Department of Transportation

Washington, D.C, 20590

13. Type of Report & Period

Covered

Final Report

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) This is a second report to the Mathematical Analysis Division of the National Highway

Traffic "ifety Administration (NHTSA) on the subject of forecasting annual highway

fatalities. This report concerns a comparison of several time series analysis programs,

based on exponential smoothing and non-decompositional methods currently employed by

NHTSA for projecting the annual traffic fatalities for the entire U.S. Several methods

of data aggregation are studied.

It is found that there is some advantage in using lumped (pooled) data for each regidn

aggregated either quarterly or half yearly, and using the Sum of Regional estimates

to estimate the national value,

Also, there does not appear to be any great difference in the results obtained using

the non-decompositional methods and those obtained by time-series analysis programs

based on exponential smoothing methods.

Estimates for the 1972 and 1973 national traffic fatalities by a variety of methods were made. For 1973 the estimates ranged from a low of 54186 to a high of 55994,

with a mean of 55055.

17. KEY UORDS (.Alphabetical order, separated by semicolons)

Exponential smoothing; forecasting; time-series; traffic fatalities.

18. AVAILABILITY STATEMENT

[3 UNLIMITED.

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF P.^GES

20. SECURITY CLASS

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