Create random points distributed as cities?

Question

How can you create 1000 random points with a distribution similar to the distribution of cities and cities, for example. Ohio?
I'm afraid I can’t pinpoint "distributed as cities"; uniformly distributed centers + small Gaussian clouds are easy, but ad hoc.
Added: should there be a family of 2d distributions with a clustering parameter that can change in accordance with a given set of points?

Answer 1

Perhaps you can take a look at Walter Cristall's Theory of Central Places . I suppose there must be some kind of generator somewhere, or you can make your own.

Answer 2

Start with a model of water bodies in your target area (or make it up if it is in an imaginary place), then group the cities near the river joints, along the lakes, river interchanges. Then create imaginary highways connecting these large cities. Now sprinkle some intermediate cities along these highways at a reasonable distance, preferring to be close to motorway junctions. Now sprinkle some small cities through empty spaces.

Answer 3

java new Random().nextGaussian(). java, :

synchronized public double nextGaussian() {
    // See Knuth, ACP, Section 3.4.1 Algorithm C.
    if (haveNextNextGaussian) {
        haveNextNextGaussian = false;
        return nextNextGaussian;
    } else {
        double v1, v2, s;
        do {
            v1 = 2 * nextDouble() - 1; // between -1 and 1
            v2 = 2 * nextDouble() - 1; // between -1 and 1
            s = v1 * v1 + v2 * v2;
        } while (s >= 1 || s == 0);
        double multiplier = StrictMath.sqrt(-2 * StrictMath.log(s)/s);
        nextNextGaussian = v2 * multiplier;
        haveNextNextGaussian = true;
        return v1 * multiplier;
    }
}

30000

x = r.nextGaussian() * rad/4 + rad;
y = r.nextGaussian() * rad/4 + rad;

:

Answer 4

.

: , , , , .

:
) , " " "clusternumbers (A), " clusternumbers (B) ".
N = 100, 1000 fcluster , ncluster = 25,

N 100 ncluster 25: 22 + 3  r 117
sizes: av 4     10   9   8   7   6   6   5   5   4   4   4 ...
radii: av 117  202 198 140 134  64  62  28 197 144 148 132 ...

N 1000 cluster 25: 22 + 3  r 197
sizes: av 45  144 139 130  85  84  69  63  43  38  33  30  ...
radii: av 197  213 279 118 146 282 154 245 212 243 226 235 ...

b) 2 3 .
:

def randomclusters( N, ncluster=25,  radius=1, box=box ):
    """ -> N 2d points: Gaussian clusters, Poisson cluster sizes """
    pts = []
    lam = eval( str( N // ncluster ))
    clustersize = lambda: np.random.poisson(lam - 1) + 1
        # poisson 2:  14  27  27  18   9   4  %
        # poisson 3:   5  15  22  22  17  10  %
    while len(pts) < N:
        u = uniformrandom2(box)
        csize = clustersize()
        if csize == 1:
            pts.append( u )
        else:
            pts.extend( inbox( gauss2( u, radius, csize )))
    return pts[:N]


    # Utility functions --

import scipy.cluster.hierarchy as hier

def fcluster( pts, ncluster, method="average", criterion="maxclust" ):
    """ -> (pts, Y pdist, Z linkage, T fcluster, clusterlists)
        ncluster = n1 + n2 + ... (including n1 singletons)
        av cluster size = len(pts) / ncluster
    """
        # Clustering is pretty fast:
        # sort pdist, then like Kruskal MST, O( N^2 ln N )
        # Many metrics and parameters are possible; these satisfice.
    pts = np.asarray(pts)
    Y = scipy.spatial.distance.pdist( pts )  # N*(N-1)/2
    Z = hier.linkage( Y, method )  # N-1, like mst
    T = hier.fcluster( Z, ncluster, criterion=criterion )
    clusters = clusterlists(T)
    return (pts, Y, Z, T, clusters)

def clusterlists(T):
    """ T = hier.fcluster( Z, t ) e.g. [a b a b c a]
        -> [ [0 2 5] [1 3] ] sorted by len, no singletons [4]
    """
    clists = [ [] for j in range( max(T) + 1 )]
    for j, c in enumerate(T):
        clists[c].append( j )
    clists.sort( key=len, reverse=True )
    n1 = np.searchsorted(  map( len, clists )[::-1], 2 )
    return clists[:-n1]

def radius( x ):
    """ rms |x - xmid| """
    return np.sqrt( np.mean( np.var( x, axis=0 )))
        # * 100  # 1 degree lat/long ~ 70 .. 111 km