December 1, 2024

By: Bob Larson Pro Tour News

This is a terrific analysis of ranking vectors, which indicate whether a player is on his way up or down. Rafael Nadal is used as the primary example. This article may not be republished without permission of Bob Larson Tennis.

Being a journalist -- even a tennis journalist -- has its complications. When you get reader feedback, it's almost always negative ("so-and-so has a fine backhand, you !!!!"). So it's hard to know what people like.

But we did get one positive comment about our mention, some weeks ago, of rankings vectors -- that is, a ranking that tells you whether a player is on the rise or falling. An ordinary ranking is a scalar -- it measures what you've done over the last twelve months, and who cares if it all happened in the last month or ten months ago? A rankings vector tells you where you are and which direction you're going: "#30 and rising slowly." "#12 and staying steady." "#70 and falling fast."

But while such assessments are easy to make in rough outline (a player like Nadia Petrova is clearly rising; a player like Marat Safin obviously falling), a proper calculation is much more complicated. Let's take a specific case to show what we mean. Specifically, let's look at Rafael Nadal. And let's examine four data points: January 1, May 1, September 1, and year-end. His rankings at those times:

January 1: #235
May 1: #93
September 1: #45
Year-end (November 17): #47

We can graph this, to make things clearer; the big + signs represent Nadal's rankings on the four dates listed.:

300 - - - -
| : : :
|
| : : :
250 + - - -
+++ : : :
+
| : : :
200 - - - -
| : : :
|
| : : :
150 - - - -
| : : :
|
| + : :
100 - - +++ - : - :
| + : :
|
| : + +
50 - - - +++ - +++
| : + +
|
| : : :
0 +--|--|--|--|--|--|--|--|--|--|--|--|--|
JAN MAR MAY JUL SEP NOV

Obviously Nadal has improved fast. Obviously he is levelling off. So what exactly do we make of all this? Is his vector "up," based on his whole year, or "level," based on his last few months?

There is no actual answer; each answer is valid depending on how you define things. We have to define a model. There are many, many choices. And while you probably just want us to pick one, at least some explanation is probably required so you know what you're getting. You can skip this section and skip ahead to the section labelled The Results -- we are, perforce, going to talk math a little -- but ideally you should read it so you'll know what we're handing you. We'll try to make this as painless as possible.

It's important to understand that any method to assess ranking movement, like the rankings themselves, is based on the past. What we are doing is not predicting the future. The rankings don't predict the future. They describe what the player has already done. Ranking vectors describe the past, too, but in more detail: Not just how a player has done, but how he's been moving in the rankings. And, since there are many ways to do this (all equally valid, or equally invalid), we need to describe how.

To illustrate why we can't predict the future: if we just took the above information, and took the slope of Nadal's numbers from the start to the end of year one, we show that he started at #235 and ended at #47. In other words, he gained 188 spots during the year. Add on another year at that rate, and he'd end 2004 at -- negative #141.

Oops.

Clearly we can't just take starting and ending rankings and calculate from there. We want to say which players are on their way up, which are heading down, which are staying level. But players do even that in different ways. Some climb steadily. Some rise or fall in spurts. Nadal's curve looks, based on the above, like an exponential decay, settling down somewhere around #40. But it might be a parabola (heading back out to infinity), or he might just have hit a temporary rough patch. We don't know the answer, and even if we did, a model that fits Nadal probably wouldn't fit Andy Roddick's steady rise, and Alex Corretja's sudden fall and recent recovery, and everything else the players do.

Another approach we must reject is the usual one for "fitting" four points: a cubic equation, that is, an equation of the form

3 2 f(x) = ax + bx + cx + d.

This is even worse at predicting than the last one. If we arbitrarily set January 1, 2025 as date "0," make May 1 equal "1", September 1 be date "2," and the year-end be date "3," and apply all this to Nadal, then this gives us four equations, corresponding to x=0, 1, 2, and 3:

235 = d
93 = a + b + c + d
45 = 8a + 4b + 2c + d
47 = 27a + 9b + 3c + d

Which gives us values of a=-7.33333, b=69, c=-203.6666667. By which numbers, Nadal would be at #55 next May 1, at #25 next September 1 -- and at #-87 at the end of next year. Two years from now, he would have a negative ranking of #-1355! Cubic equations simply go out of control too fast. They are excellent for approximating within the period specified (e.g. the above formula says Nadal "should" have been #59 after Wimbledon, and he was in fact #61), but beyond the range is no good.

So we aren't even going to try to predict mathematically. Any method we use will fit some players and not others. We're going to do something very simple instead. We're going to take just the four points specified -- January 1, May 1, September 1, year-end. We're going to calculate three slopes: January 1 to year-end, May 1 to year-end, and September 1 to year-end. And then we're going to average those three numbers. This means that recent results have more weight, but not overwhelming weight. We then divide by the final ranking to normalize the results.

To put this in symbols, if J is the January rankings, M the May ranking, S the September ranking, and Y the year-end ranking, then our vector V is defined by

(J-Y)+(M-Y)+(S-Y) (J+M+S-3Y)
V = ----------------- = ----------
3Y 3Y

Taking Nadal as an example, his one-year slope is 235-47 = 188. His eight-month slope is 93-47 = 46. But his four-month slope is 45-47 = -2. In other words, his ranking rose quickly over the year, less quickly over eight months, and actually fell over the last part of the year. Adding all these up, we get an average of 232/3 = 77.3 That means he's going up, and fairly quickly.

But we have to measure that against where he started. If you're #2, and gain one spot, it's huge. If you're #99, and gain one spot, it's just noise. What matters is how much you move with respect to your ranking. Hence we divide by the player's final ranking. In Nadal's case, e.g., that means we divide 77.3 by his current #47 ranking. That gives us a ratio of 1.65. And that is our "vector." Nadal's ranking is #47. His direction is upward at a rate of 1.65. That's a dimensionless number; it's not 1.65 "per month" or some such thing. It's just 1.65, relative to the values posted by other players.

The Results

The results of the above calculations are, obviously, positive or negative numbers, with positive numbers being good (a player who is rising); the higher this number, the better a player is doing. A number larger than one (for that matter, a number larger than about .4) means a fast-rising player. Negative numbers indicate players who are, on the whole, falling; a number smaller than -0.4 means a guy who is in a fair bit of trouble.

To keep this list within some sort of bounds, we will look only at players who either ended up in the Top 100 at year-end or were in the Top 80 at some time this year, and who were actually active during the year (that still leaves us with 119 players). We're also going to "clamp" the rankings: Any player ranked below #200 in a given ranking period will have his ranking treated as #200.

That gives us this list, in ranking order. Remember, positive numbers in the "Vector" column are good, negative bad, and the larger the positive number, the better.

Note how many of the final Top Ten have positive numbers, and large ones.

Rank.Name...............Jan1.May1.Sep1..End..Vector
..1..Roddick, Andy........10....6....4....1....5.67
..2..Federer, Roger........6....5....2....2....1.17
..3..Ferrero, Juan Carl....4....3....3....3....0.11
..4..Agassi, Andre.........2....1....1....4...-0.67
..5..Coria, Guillermo.....45...16....5....5....3.40
..6..Schuettler, Rainer...33...13....8....6....2.00
..7..Moya, Carlos..........5....4....7....7...-0.24
..8..Nalbandian, David....12...12...13....8....0.54
..9..Philippoussis, Mar...80...62...20....9....5.00
.10..Grosjean, Sebastie...17...14....9...10....0.33
.11..Srichaphan, Parado...16...10...11...11....0.12
.12..Massu, Nicolas.......56...89...42...12....4.19
.13..Novak, Jiri...........7....9...10...13...-0.33
.14..El Aynaoui, Younes...22...17...21...14....0.43
.15..Henman, Tim...........8...31...33...15....0.60
.16..Kuerten, Gustavo.....37...15...14...16....0.38
.17..Hewitt, Lleyton.......1....2....6...17...-0.82
.18..Schalken, Sjeng......20...11...12...18...-0.20
.19..Verkerk, Martin......86...68...16...19....1.98
.20..Fish, Mardy..........84...53...26...20....1.72
.21..Robredo, Tommy.......30...20...17...21....0.06
.22..Mantilla, Felix......55...47...22...22....0.88
.23..Mirnyi, Max..........43...33...18...23....0.36
.24..Calleri, Agustin.....50...28...19...24....0.35
.25..Costa, Albert.........9....8...24...25...-0.45
.26..Ferreira, Wayne......39...27...23...26....0.14
.27..Zabaleta, Mariano....53...36...27...27....0.43
.28..Lopez, Feliciano.....62...56...29...28....0.75
.29..Spadea,Vincent.......67...32...31...29....0.49
.30..Ginepri, Robby......100...49...40...30....1.10
.31..Bjorkman, Jonas......48...71...44...31....0.75
.32..Clement, Arnaud......38...37...32...32....0.11
.33..Dent, Taylor.........57...39...73...33....0.71
.34..Gaudio, Gaston.......21...29...30...34...-0.22
.35..Gonzalez, Fernando...18...21...15...35...-0.49
.36..Nieminen, Jarkko.....40...35...39...36....0.06
.37..Blake, James.........28...23...35...37...-0.23
.38..Chela, Juan Ignaci...23...34...41...38...-0.14
.39..Sargsian, Sargis....103...75...52...39....0.97
.40..Kucera, Karol........83...50...37...40....0.42
.41..Kafelnikov, Yevgen...27...25...28...41...-0.35
.42..Ljubicic, Ivan.......49...46...43...42....0.10
.43..Youzhny, Mikhail.....32...30...36...43...-0.24
.44..Davydenko, Nikolay...81...41...38...44....0.21
.45..Saretta, Flavio......92...80...49...45....0.64
.46..Stepanek, Radek......63...51...46...46....0.16
.47..Nadal,Rafael........200...93...45...47....1.40
.48..Volandri,Filippo....154...92...47...48....1.03
.49..Rochus, Olivier......64...60...48...49....0.17
.50..Sanchez, David.......60...48...55...50....0.09
.51..Gambill, Jan-Micha...42...40...50...51...-0.14
.52..Arazi, Hicham........90...96...65...52....0.61
.53..Lee, Hyung-Taik......87...54...69...53....0.32
.54..Koubek, Stefan.......54...63...68...54....0.14
.55..Martin, Alberto......61...65...63...55....0.15
.56..Malisse, Xavier......25...26...67...56...-0.30
.57..Lapentti, Nicolas....29...42...57...57...-0.25
.58..Kiefer, Nicolas......72...73...62...58....0.19
.59..Soderling, Robin....176..158...99...59....1.45
.60..Sluiter,Raemon.......71...66...54...60....0.06
.61..Hrbaty, Dominik......51...58...53...61...-0.11
.62..Santoro, Fabrice.....35...52...58...62...-0.22
.63..Carraz,Gregory......138..130..100...63....0.95
.64..Ancic, Mario.........89...77...74...64....0.25
.65..Beck, Karol.........125...74...71...65....0.38
.66..Vicente, Fernando....58...57...61...66...-0.11
.67..Horna, Luis..........85...78...76...67....0.19
.68..Sanguinetti, David...46...84...86...68....0.06
.69..Martin, Todd.........47...95...93...69....0.14
.70..Hanescu,Victor......170..147...84...70....0.91
.71..Ferrer, David........59...59...59...71...-0.17
.72..Pavel, Andrei........26...38..103...72...-0.23
.73..Labadze, Irakli......93..104..105...73....0.38
.74..Karlovic, Ivo.......175..200..109...74....1.18
.75..van Lottum,John.....115..102...83...75....0.33
.76..Burgsmuller, Lars....76..100...85...76....0.14
.77..Safin, Marat..........3....7...25...77...-0.85
.78..Dupuis, Anthony......70...86...64...78...-0.06
.79..Melzer, Jurgen.......91...88...81...79....0.10
.80..Ramirez Hidalgo,Ru..145..146...70...80....0.50
.81..Montanes, Albert.....78...91...87...81....0.05
.82..Moodie,Wesley.......200..153...95...82....0.82
.83..Mathieu, Paul-Henr...36...44...56...83...-0.45
.84..Vahaly, Brian.......102...72...78...84....0.00
.85..Rochus,Christophe...146..134...89...85....0.45
.86..Portas, Albert.......88...85...88...86....0.01
.87..Saulnier, Cyril.....161..124..101...87....0.48
.88..Behrend, Tomas......142..101...79...88....0.22
.89..Ascione,Thierry.....200..200..116...89....0.93
.90..Andreev,Igor........200..200..122...90....0.93
.91..Hernandez,Oscar.....200..195..113...91....0.86
.92..Popp,Alexander......117..163..107...92....0.40
.93..Gasquet,Richard.....166..114..124...93....0.45
.95..Enqvist, Thomas......44...90..143...95...-0.03
.96..Lisnard,Jean-Rene...104...99...94...96....0.03
.97..Bogomolov Jr., Ale..198..122..117...97....0.50
.98..Tursonov,Dmitry.....200..200..174...98....0.95
.99..Blanco,Galo.........133..126..110...99....0.24
100..Corretja, Alex.......19...18..102..100...-0.54
101..Acasuso, Jose........41...45...75..101...-0.47
104..Mutis, Olivier.......94...79...80..104...-0.19
105..Rios, Marcelo........24...43...51..105...-0.63
108..Boutter, Julien......77...69...96..108...-0.25
112..Escude, Nicolas......34...61...60..112...-0.54
114..Carlsen, Kenneth.....65...67...72..114...-0.40
116..Arthurs, Wayne.......52...55..118..116...-0.35
118..Rusedski, Greg.......31...70...90..118...-0.46
119..Vinciguerra, Andre..181..109...92..119....0.07
120..Squillari, Franco....79...94...97..120...-0.25
123..Norman, Magnus......107...82...77..123...-0.28
127..Voltchkov, Vladimi...73...64...66..127...-0.47
129..Kratochvil, Michel...69...81..150..129...-0.22
154..Voinea, Adrian.......68...87..135..154...-0.37
210..Sa, Andre............66...83..155..200...-0.49
213..Meligeni, Fernando...74..107..153..200...-0.44
262..Ulihrach, Bohdan.....75..108..188..200...-0.38
274..Canas, Guillermo.....15...22..200..200...-0.60
434..Pless, Kristian......82..105..133..200...-0.47

Searching the list for our ten biggest upward movers (or, correctly, our ten players with the largest upward vectors; they are not our biggest movers in terms of places gained), they are:

Rank.Name...............Jan1.May1.Sep1..End..Vector
..1..Roddick, Andy........10....6....4....1....5.67
..9..Philippoussis, Mar...80...62...20....9....5.00
.12..Massu, Nicolas.......56...89...42...12....4.19
..5..Coria, Guillermo.....45...16....5....5....3.40
..6..Schuettler, Rainer...33...13....8....6....2.00
.19..Verkerk, Martin......86...68...16...19....1.98
.20..Fish, Mardy..........84...53...26...20....1.72
.59..Soderling, Robin....176..158...99...59....1.45
.47..Nadal,Rafael........200...93...45...47....1.40
.74..Karlovic, Ivo.......175..200..109...74....1.18

Obviously Roddick can't move above #1, but he can strengthen his position -- and, indeed, that seems fairly likely, given how little he has to defend between the Australian Open and Paris. A lot of those other guys, like Schuettler and Philippoussis, are probably at the ends of their runs also. But Soderling and Nadal and perhaps even Fish still have some room to rise, and Karlovic, who finally will get to play regularly on the ATP, has a big opportunity.

Our ten biggest ground-losers are:

Rank.Name...............Jan1.May1.Sep1..End..Vector
127..Voltchkov, Vladimi...73...64...66..127...-0.47
101..Acasuso, Jose........41...45...75..101...-0.47
.35..Gonzalez, Fernando...18...21...15...35...-0.49
210..Sa, Andre............66...83..155..200...-0.49
100..Corretja, Alex.......19...18..102..100...-0.54
112..Escude, Nicolas......34...61...60..112...-0.54
274..Canas, Guillermo.....15...22..200..200...-0.60
105..Rios, Marcelo........24...43...51..105...-0.63
..4..Agassi, Andre.........2....1....1....4...-0.67
.17..Hewitt, Lleyton.......1....2....6...17...-0.82
.77..Safin, Marat..........3....7...25...77...-0.85

There are, of course, a lot of injuries in there -- Safin, Canas, Escude; we'd expect them to spring back. The others -- well, we'll just have to wait and see.