How to Run PSO with Real Data

I want to use PSO to optimize a 3D problem using real data (lat, long, alt). How is this accomplished? I see in the examples and tutorials it is done against existing surface plots; Rosenbrock, Ackley, Griewank, etc.. How do I run it against real numbers?

R2020b

Answer 1

[Walter Roberson]

You use whatever formula is appropriate. For example

d=load('In_Flight_Data.mat') ;
lat=d.lat;
long=d.long;
alt=d.alt;
fun = @(x) sum((x(1)-lat).^2 + (x(2)-long).^2 + (x(3)-tan(alt)).^2);
pso(fun, 3)

[Timothy Turk]

Thank you, Walter. I was able to use the first four lines to build the matrix. That was one of my goals.

Answer 2

[John D'Errico]

You can't. That is, the optimal is just the optimal value at each point in your sample. You cannot do better than that, at least, not unless you are willing to build a model of the process.

So, what are you willing to do in terms of a model? What do you know? What are you willing to assume? If you cannot build a model, then you cannot do any form of optimization that will have some chance of being better than the best point from your data.

If your data is at all noisy, then don't expect to do well of course. But one option might be to use an interpolating spline surface model. Again, noise will totally destroy any spline fit, making it useless for this purpose.

[Timothy Turk]

My data is lat, long, and alt of spacecraft debris. Samples provided below. Is that what you would call noisy? I'm not sure what that means..

Lat, Long, Alt (Km)

  -42.748  92.546  540.752955 
  -77.248  -100.362  700.8914 
  57.159  41.664  758.619121 
  -39.979  24.617  688.364084 
  -81.627  3.396  732.495436 
  -65.705  -104.186  947.496077 
  33.811  46.185  493.514269 
  -67.838  81.76  704.196551 
  75.587  -59.466  684.452176 
  -54.777  100.233  689.940558 
  -12.685  133.573  771.593297 
  -34.102  117.206  744.037739 
  17.965  70.96  578.407638 
  -35.521  -50.495  773.915334 

[John D'Errico]

Sorry, but there is literally nothing i can suggest. Your data is seriously limited. 14 data points.

Is it noisy data? That would seem to be very true, OR you have a function that is seriously nasty.

xyz =
      -42.748       92.546       540.75
      -77.248      -100.36       700.89
       57.159       41.664       758.62
      -39.979       24.617       688.36
      -81.627        3.396        732.5
      -65.705      -104.19        947.5
       33.811       46.185       493.51
      -67.838        81.76        704.2
       75.587      -59.466       684.45
      -54.777       100.23       689.94
      -12.685       133.57       771.59
      -34.102       117.21       744.04
       17.965        70.96       578.41
      -35.521      -50.495       773.92

I see TWO data points that are close to each other, where Lat is -100 in both cases. What is the difference in altitude? I see 700 versus 947, for those two points. Likewise, other points that are quite close to each other have very different altitudes.

There is no way you would ever have a chance of fitting an interpolating spline that would be of any remote value.

As well, you seem to have no clue as to any model that might make sense, at least none that you have suggested.

Can I offer you good luck in this quest? Talk to Don Quixote, a fellow I heard of who likes these impossible dreams. :)

Ok, seriously, you need better data. Essentially, that means MORE data. You need to see replicates. How consistent is this surface in regard to replication? We don't know where the data comes from, or what it represents.

I'll plot a surface from your data, such as it can be...

I'm not sure what you can really learn from that.

Even with a bicubic surface of some ilk, the best estimate from this data of the minmimum is simply the lowest data point.

[minz,ind] = min(z)
minz =
       493.51
ind =
     7
>> xyz(ind,:)
ans =
       33.811       46.185       493.51

[Timothy Turk]

Thanks, John.

I have 303 data points. These data points represent a snapshot of the location of space debris from a specific collision orbiting the Earth at a given point in time. There are thousands more, but the space surveillance network can only catalog objects with a radar cross section over 10 cm.

The debris field converges at the north and south poles.

I am trying use those data points to identify the area of highest concentration of objects, radially, at the southern most part of the orbit, at or near, -86 lat, -179 long, 735 Km alt. Diameter of the search area is 240 Km.

It dawned on me whilst typing this, in order to replicate the scale of the problem, that is the actual number of objects, I should use a random number generator with the constraints in the paragraph above to generate thousands of random points.

I'm obviously VERY new to MATLAB and very much appreciate your help.

[Walter Roberson]

When you use PSO, you hope to get out a vector of values. What properties do you want for the vector of values?

Based upon what you have written, we could imagine that what you might maybe are hoping for, is a set of triples, (lat, long, alt), such that each triple represents the "center" of a region of high density. PSO cannot work with an unknown number of variables, so if you were looking for (say) 5 centers, you would run the PSO on 5*3 variables, getting out 5 groups of lat, long, alt. But any one run would not know in advance how many centers of high density there "really" are, so you might need to run with different numbers of centers to try to figure out the "best" collection of centers.

... If you just happen to be wanting what I describe above, then you might not need PSO at all. Instead you might want to use an algorithm such as kmeans or fuzzy c-means, but supplying a non-euclidean "distance" function.

Question: are the debris measurements taken "shortly" after a collision, or are they taken after the debris has had time to move notably in orbit? If so, then because objects at different altitudes move at different velocities (unless they are being affected by atmospheric drag forces that are limiting the velocity), a piece that might be half way around the world might be best deemed to be part of a particular cluster. Do you have velocity measurements as well as position measurements?

[Timothy Turk]

Thanks, Walter.

Ideally, there would only be one center.

I'll investigate the algorithms you mentioned.

The debris along with all other objects orbiting Earth larger that 10 cm cross section are continuosly cataloged by the space surveillance network. I do believe I can pull the velocity of each object from the Systems Tool Kit (STK), but don't have access to it right now. I will check tonight and respond. And, yes, agreed, velocities at different altitudes are different; lower altitudes have higher velocities.

[Walter Roberson]

When you use PSO, you hope to get out a vector of values. What properties do you want for the vector of values?

[Timothy Turk]

The vector I would desire is a the location of the highest density of debris in the debris field as it passes closest to the South Pole.

This is a sample of the data I can pull from STK. This only a snapshot of one piece of debris. There are 303 pieces of debris and I would use one minute increments for each piece over one orbit ~ 90 minutes.

Satellite-IRIDIUM_33_DEB_33773: LLA Position

Time (UTCG) Lat (deg) Lon (deg) Alt (km) Lat Rate (deg/sec) Lon Rate (deg/sec) Alt Rate (km/sec)

----------------------- --------- --------- ---------- ------------------ ------------------ -----------------

1 Mar 2019 18:00:00.000 -24.381 -47.011 766.760719 0.060052 0.000350 -0.028500

1 Mar 2019 18:01:00.000 -20.776 -46.997 765.107684 0.060107 0.000123 -0.026564

1 Mar 2019 18:02:00.000 -17.168 -46.996 763.578772 0.060156 -0.000057 -0.024369

1 Mar 2019 18:03:00.000 -13.558 -47.003 762.188564 0.060198 -0.000195 -0.021947

1 Mar 2019 18:04:00.000 -9.945 -47.018 760.949686 0.060233 -0.000296 -0.019331

1 Mar 2019 18:05:00.000 -6.330 -47.038 759.872637 0.060261 -0.000364 -0.016559

1 Mar 2019 18:06:00.000 -2.714 -47.061 758.965646 0.060282 -0.000401 -0.013669

1 Mar 2019 18:07:00.000 0.904 -47.086 758.234571 0.060296 -0.000408 -0.010701

1 Mar 2019 18:08:00.000 4.522 -47.110 757.682831 0.060302 -0.000385 -0.007697

1 Mar 2019 18:09:00.000 8.140 -47.131 757.311380 0.060300 -0.000331 -0.004697

1 Mar 2019 18:10:00.000 11.758 -47.149 757.118724 0.060291 -0.000245 -0.001742

1 Mar 2019 18:11:00.000 15.375 -47.160 757.100970 0.060275 -0.000124 0.001129

1 Mar 2019 18:12:00.000 18.990 -47.163 757.251922 0.060252 0.000036 0.003877

1 Mar 2019 18:13:00.000 22.605 -47.155 757.563209 0.060222 0.000242 0.006469

1 Mar 2019 18:14:00.000 26.217 -47.133 758.024451 0.060185 0.000500 0.008873

1 Mar 2019 18:15:00.000 29.827 -47.093 758.623449 0.060142 0.000823 0.011059

1 Mar 2019 18:16:00.000 33.434 -47.032 759.346408 0.060092 0.001223 0.013003

1 Mar 2019 18:17:00.000 37.038 -46.945 760.178185 0.060037 0.001722 0.014685

1 Mar 2019 18:18:00.000 40.638 -46.823 761.102548 0.059974 0.002346 0.016088

1 Mar 2019 18:19:00.000 44.235 -46.660 762.102452 0.059905 0.003136 0.017202

1 Mar 2019 18:20:00.000 47.827 -46.443 763.160329 0.059828 0.004146 0.018021

1 Mar 2019 18:21:00.000 51.414 -46.156 764.258371 0.059742 0.005461 0.018542

1 Mar 2019 18:22:00.000 54.995 -45.779 765.378823 0.059642 0.007208 0.018769

1 Mar 2019 18:23:00.000 58.570 -45.278 766.504260 0.059523 0.009593 0.018709

1 Mar 2019 18:24:00.000 62.138 -44.608 767.617864 0.059375 0.012955 0.018376

1 Mar 2019 18:25:00.000 65.694 -43.693 768.703681 0.059179 0.017897 0.017785

1 Mar 2019 18:26:00.000 69.237 -42.407 769.746856 0.058900 0.025565 0.016957

1 Mar 2019 18:27:00.000 72.759 -40.524 770.733859 0.058459 0.038350 0.015916

1 Mar 2019 18:28:00.000 76.246 -37.595 771.652675 0.057673 0.061881 0.014688

1 Mar 2019 18:29:00.000 79.663 -32.591 772.492973 0.056011 0.111760 0.013302