source: anuga_core/source/anuga/fit_interpolate/fit.py @ 4214

Last change on this file since 4214 was 4214, checked in by duncan, 18 years ago

spacing
File size: 18.9 KB
Line 
1"""Least squares fitting.
2
3   Implements a penalised least-squares fit.
4
5   The penalty term (or smoothing term) is controlled by the smoothing
6   parameter alpha.
7   With a value of alpha=0, the fit function will attempt
8   to interpolate as closely as possible in the least-squares sense.
9   With values alpha > 0, a certain amount of smoothing will be applied.
10   A positive alpha is essential in cases where there are too few
11   data points.
12   A negative alpha is not allowed.
13   A typical value of alpha is 1.0e-6
14
15
16   Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou
17   Geoscience Australia, 2004.
18
19   TO DO
20   * test geo_ref, geo_spatial
21
22   IDEAS
23   * (DSG-) Change the interface of fit, so a domain object can
24      be passed in. (I don't know if this is feasible). If could
25      save time/memory.
26"""
27import types
28
29from Numeric import zeros, Float, ArrayType,take
30
31from anuga.caching import cache           
32from anuga.geospatial_data.geospatial_data import Geospatial_data, \
33     ensure_absolute
34from anuga.fit_interpolate.general_fit_interpolate import FitInterpolate
35from anuga.utilities.sparse import Sparse, Sparse_CSR
36from anuga.utilities.polygon import in_and_outside_polygon
37from anuga.fit_interpolate.search_functions import search_tree_of_vertices
38from anuga.utilities.cg_solve import conjugate_gradient
39from anuga.utilities.numerical_tools import ensure_numeric, gradient
40
41import exceptions
42class ToFewPointsError(exceptions.Exception): pass
43class VertsWithNoTrianglesError(exceptions.Exception): pass
44
45DEFAULT_ALPHA = 0.001
46
47
48class Fit(FitInterpolate):
49   
50    def __init__(self,
51                 vertex_coordinates,
52                 triangles,
53                 mesh_origin=None,
54                 alpha = None,
55                 verbose=False,
56                 max_vertices_per_cell=30):
57
58
59        """
60        Fit data at points to the vertices of a mesh.
61
62        Inputs:
63
64          vertex_coordinates: List of coordinate pairs [xi, eta] of
65              points constituting a mesh (or an m x 2 Numeric array or
66              a geospatial object)
67              Points may appear multiple times
68              (e.g. if vertices have discontinuities)
69
70          triangles: List of 3-tuples (or a Numeric array) of
71              integers representing indices of all vertices in the mesh.
72
73          mesh_origin: A geo_reference object or 3-tuples consisting of
74              UTM zone, easting and northing.
75              If specified vertex coordinates are assumed to be
76              relative to their respective origins.
77
78          max_vertices_per_cell: Number of vertices in a quad tree cell
79          at which the cell is split into 4.
80
81          Note: Don't supply a vertex coords as a geospatial object and
82              a mesh origin, since geospatial has its own mesh origin.
83
84
85        Usage,
86        To use this in a blocking way, call  build_fit_subset, with z info,
87        and then fit, with no point coord, z info.
88       
89        """
90        # Initialise variabels
91
92        if alpha is None:
93
94            self.alpha = DEFAULT_ALPHA
95        else:   
96            self.alpha = alpha
97        FitInterpolate.__init__(self,
98                 vertex_coordinates,
99                 triangles,
100                 mesh_origin,
101                 verbose,
102                 max_vertices_per_cell)
103       
104        m = self.mesh.number_of_nodes # Nbr of basis functions (vertices)
105       
106        self.AtA = None
107        self.Atz = None
108
109        self.point_count = 0
110        if self.alpha <> 0:
111            if verbose: print 'Building smoothing matrix'
112            self._build_smoothing_matrix_D()
113           
114    def _build_coefficient_matrix_B(self,
115                                  verbose = False):
116        """
117        Build final coefficient matrix
118
119        Precon
120        If alpha is not zero, matrix D has been built
121        Matrix Ata has been built
122        """
123
124        if self.alpha <> 0:
125            #if verbose: print 'Building smoothing matrix'
126            #self._build_smoothing_matrix_D()
127            self.B = self.AtA + self.alpha*self.D
128        else:
129            self.B = self.AtA
130
131        #Convert self.B matrix to CSR format for faster matrix vector
132        self.B = Sparse_CSR(self.B)
133
134    def _build_smoothing_matrix_D(self):
135        """Build m x m smoothing matrix, where
136        m is the number of basis functions phi_k (one per vertex)
137
138        The smoothing matrix is defined as
139
140        D = D1 + D2
141
142        where
143
144        [D1]_{k,l} = \int_\Omega
145           \frac{\partial \phi_k}{\partial x}
146           \frac{\partial \phi_l}{\partial x}\,
147           dx dy
148
149        [D2]_{k,l} = \int_\Omega
150           \frac{\partial \phi_k}{\partial y}
151           \frac{\partial \phi_l}{\partial y}\,
152           dx dy
153
154
155        The derivatives \frac{\partial \phi_k}{\partial x},
156        \frac{\partial \phi_k}{\partial x} for a particular triangle
157        are obtained by computing the gradient a_k, b_k for basis function k
158        """
159       
160        #FIXME: algorithm might be optimised by computing local 9x9
161        #"element stiffness matrices:
162
163        m = self.mesh.number_of_nodes # Nbr of basis functions (1/vertex)
164
165        self.D = Sparse(m,m)
166
167        #For each triangle compute contributions to D = D1+D2
168        for i in range(len(self.mesh)):
169
170            #Get area
171            area = self.mesh.areas[i]
172
173            #Get global vertex indices
174            v0 = self.mesh.triangles[i,0]
175            v1 = self.mesh.triangles[i,1]
176            v2 = self.mesh.triangles[i,2]
177
178            #Get the three vertex_points
179            xi0 = self.mesh.get_vertex_coordinate(i, 0)
180            xi1 = self.mesh.get_vertex_coordinate(i, 1)
181            xi2 = self.mesh.get_vertex_coordinate(i, 2)
182
183            #Compute gradients for each vertex
184            a0, b0 = gradient(xi0[0], xi0[1], xi1[0], xi1[1], xi2[0], xi2[1],
185                              1, 0, 0)
186
187            a1, b1 = gradient(xi0[0], xi0[1], xi1[0], xi1[1], xi2[0], xi2[1],
188                              0, 1, 0)
189
190            a2, b2 = gradient(xi0[0], xi0[1], xi1[0], xi1[1], xi2[0], xi2[1],
191                              0, 0, 1)
192
193            #Compute diagonal contributions
194            self.D[v0,v0] += (a0*a0 + b0*b0)*area
195            self.D[v1,v1] += (a1*a1 + b1*b1)*area
196            self.D[v2,v2] += (a2*a2 + b2*b2)*area
197
198            #Compute contributions for basis functions sharing edges
199            e01 = (a0*a1 + b0*b1)*area
200            self.D[v0,v1] += e01
201            self.D[v1,v0] += e01
202
203            e12 = (a1*a2 + b1*b2)*area
204            self.D[v1,v2] += e12
205            self.D[v2,v1] += e12
206
207            e20 = (a2*a0 + b2*b0)*area
208            self.D[v2,v0] += e20
209            self.D[v0,v2] += e20
210
211
212    def get_D(self):
213        return self.D.todense()
214
215
216    def _build_matrix_AtA_Atz(self,
217                              point_coordinates,
218                              z,
219                              verbose = False):
220        """Build:
221        AtA  m x m  interpolation matrix, and,
222        Atz  m x a  interpolation matrix where,
223        m is the number of basis functions phi_k (one per vertex)
224        a is the number of data attributes
225
226        This algorithm uses a quad tree data structure for fast binning of
227        data points.
228
229        If Ata is None, the matrices AtA and Atz are created.
230
231        This function can be called again and again, with sub-sets of
232        the point coordinates.  Call fit to get the results.
233       
234        Preconditions
235        z and points are numeric
236        Point_coordindates and mesh vertices have the same origin.
237
238        The number of attributes of the data points does not change
239        """
240        #Build n x m interpolation matrix
241
242        if self.AtA == None:
243            # AtA and Atz need to be initialised.
244            m = self.mesh.number_of_nodes
245            if len(z.shape) > 1:
246                att_num = z.shape[1]
247                self.Atz = zeros((m,att_num), Float)
248            else:
249                att_num = 1
250                self.Atz = zeros((m,), Float)
251            assert z.shape[0] == point_coordinates.shape[0] 
252
253            self.AtA = Sparse(m,m)
254            # The memory damage has been done by now.
255           
256        self.point_count += point_coordinates.shape[0]
257        #print "_build_matrix_AtA_Atz - self.point_count", self.point_count
258        if verbose: print 'Getting indices inside mesh boundary'
259        #print 'point_coordinates.shape', point_coordinates.shape         
260        #print 'self.mesh.get_boundary_polygon()',\
261        #      self.mesh.get_boundary_polygon()
262
263        inside_poly_indices, outside_poly_indices  = \
264                     in_and_outside_polygon(point_coordinates,
265                                            self.mesh.get_boundary_polygon(),
266                                            closed = True, verbose = verbose)
267        #print "self.inside_poly_indices",self.inside_poly_indices
268        #print "self.outside_poly_indices",self.outside_poly_indices
269
270       
271        n = len(inside_poly_indices)
272        if verbose: print 'Building fitting matrix from %d points' %n       
273        #Compute matrix elements for points inside the mesh
274        for k, i in enumerate(inside_poly_indices):
275            #For each data_coordinate point
276            if verbose and k%((n+10)/10)==0: print 'Doing %d of %d' %(k, n)
277            x = point_coordinates[i]
278            element_found, sigma0, sigma1, sigma2, k = \
279                           search_tree_of_vertices(self.root, self.mesh, x)
280           
281            if element_found is True:
282                j0 = self.mesh.triangles[k,0] #Global vertex id for sigma0
283                j1 = self.mesh.triangles[k,1] #Global vertex id for sigma1
284                j2 = self.mesh.triangles[k,2] #Global vertex id for sigma2
285
286                sigmas = {j0:sigma0, j1:sigma1, j2:sigma2}
287                js     = [j0,j1,j2]
288
289                for j in js:
290                    self.Atz[j] +=  sigmas[j]*z[i]
291                    #print "self.Atz building", self.Atz
292                    #print "self.Atz[j]", self.Atz[j]
293                    #print " sigmas[j]", sigmas[j]
294                    #print "z[i]",z[i]
295                    #print "result", sigmas[j]*z[i]
296                   
297                    for k in js:
298                        self.AtA[j,k] += sigmas[j]*sigmas[k]
299            else:
300                msg = 'Could not find triangle for point', x
301                raise Exception(msg)
302   
303       
304    def fit(self, point_coordinates_or_filename=None, z=None,
305            verbose=False,
306            point_origin=None,
307            attribute_name=None,
308            max_read_lines=500):
309        """Fit a smooth surface to given 1d array of data points z.
310
311        The smooth surface is computed at each vertex in the underlying
312        mesh using the formula given in the module doc string.
313
314        Inputs:
315        point_coordinates: The co-ordinates of the data points.
316              List of coordinate pairs [x, y] of
317              data points or an nx2 Numeric array or a Geospatial_data object
318          z: Single 1d vector or array of data at the point_coordinates.
319         
320        """
321        # use blocking to load in the point info
322        if type(point_coordinates_or_filename) == types.StringType:
323            msg = "Don't set a point origin when reading from a file"
324            assert point_origin is None, msg
325            filename = point_coordinates_or_filename
326            for i,geo_block in enumerate(Geospatial_data(filename,
327                                              max_read_lines=max_read_lines,
328                                              load_file_now=False,
329                                              verbose=verbose)):
330                if verbose is True and 0 == i%200: # round every 5 minutes
331                    print 'Block %i' %i
332                # build the array
333                points = geo_block.get_data_points(absolute=True)
334                z = geo_block.get_attributes(attribute_name=attribute_name)
335                self.build_fit_subset(points, z)
336            point_coordinates = None
337        else:
338            point_coordinates =  point_coordinates_or_filename
339           
340        if point_coordinates is None:
341            assert self.AtA <> None
342            assert self.Atz <> None
343            #FIXME (DSG) - do  a message
344        else:
345            point_coordinates = ensure_absolute(point_coordinates,
346                                                geo_reference=point_origin)
347            #if isinstance(point_coordinates,Geospatial_data) and z is None:
348            # z will come from the geo-ref
349            self.build_fit_subset(point_coordinates, z, verbose)
350
351        #Check sanity
352        m = self.mesh.number_of_nodes # Nbr of basis functions (1/vertex)
353        n = self.point_count
354        if n<m and self.alpha == 0.0:
355            msg = 'ERROR (least_squares): Too few data points\n'
356            msg += 'There are only %d data points and alpha == 0. ' %n
357            msg += 'Need at least %d\n' %m
358            msg += 'Alternatively, set smoothing parameter alpha to a small '
359            msg += 'positive value,\ne.g. 1.0e-3.'
360            raise ToFewPointsError(msg)
361
362        self._build_coefficient_matrix_B(verbose)
363        loners = self.mesh.get_lone_vertices()
364        # FIXME  - make this as error message.
365        # test with
366        # Not_yet_test_smooth_att_to_mesh_with_excess_verts.
367        if len(loners)>0:
368            msg = 'WARNING: (least_squares): \nVertices with no triangles\n'
369            msg += 'All vertices should be part of a triangle.\n'
370            msg += 'In the future this will be inforced.\n'
371            msg += 'The following vertices are not part of a triangle;\n'
372            msg += str(loners)
373            print msg
374            #raise VertsWithNoTrianglesError(msg)
375       
376       
377        return conjugate_gradient(self.B, self.Atz, self.Atz,
378                                  imax=2*len(self.Atz) )
379
380       
381    def build_fit_subset(self, point_coordinates, z=None, attribute_name=None,
382                              verbose=False):
383        """Fit a smooth surface to given 1d array of data points z.
384
385        The smooth surface is computed at each vertex in the underlying
386        mesh using the formula given in the module doc string.
387
388        Inputs:
389        point_coordinates: The co-ordinates of the data points.
390              List of coordinate pairs [x, y] of
391              data points or an nx2 Numeric array or a Geospatial_data object
392        z: Single 1d vector or array of data at the point_coordinates.
393        attribute_name: Used to get the z values from the
394              geospatial object if no attribute_name is specified,
395              it's a bit of a lucky dip as to what attributes you get.
396              If there is only one attribute it will be that one.
397
398        """
399
400        #FIXME(DSG-DSG): Check that the vert and point coords
401        #have the same zone.
402        if isinstance(point_coordinates,Geospatial_data):
403            point_coordinates = point_coordinates.get_data_points( \
404                absolute = True)
405       
406        #Convert input to Numeric arrays
407        if z is not None:
408            z = ensure_numeric(z, Float)
409        else:
410            msg = 'z not specified'
411            assert isinstance(point_coordinates,Geospatial_data), msg
412            z = point_coordinates.get_attributes(attribute_name)
413           
414        point_coordinates = ensure_numeric(point_coordinates, Float)
415
416        self._build_matrix_AtA_Atz(point_coordinates, z, verbose)
417
418
419############################################################################
420
421def fit_to_mesh(vertex_coordinates,
422                triangles,
423                point_coordinates, # this can also be a .csv/.txt file name
424                point_attributes=None,
425                alpha=DEFAULT_ALPHA,
426                verbose=False,
427                acceptable_overshoot=1.01,
428                mesh_origin=None,
429                data_origin=None,
430                max_read_lines=None,
431                attribute_name=None,
432                use_cache = False):
433    """
434    Fit a smooth surface to a triangulation,
435    given data points with attributes.
436
437
438        Inputs:
439        vertex_coordinates: List of coordinate pairs [xi, eta] of
440              points constituting a mesh (or an m x 2 Numeric array or
441              a geospatial object)
442              Points may appear multiple times
443              (e.g. if vertices have discontinuities)
444
445          triangles: List of 3-tuples (or a Numeric array) of
446          integers representing indices of all vertices in the mesh.
447
448          point_coordinates: List of coordinate pairs [x, y] of data points
449          (or an nx2 Numeric array)
450
451          alpha: Smoothing parameter.
452
453          acceptable overshoot: controls the allowed factor by which fitted values
454          may exceed the value of input data. The lower limit is defined
455          as min(z) - acceptable_overshoot*delta z and upper limit
456          as max(z) + acceptable_overshoot*delta z
457
458          mesh_origin: A geo_reference object or 3-tuples consisting of
459              UTM zone, easting and northing.
460              If specified vertex coordinates are assumed to be
461              relative to their respective origins.
462         
463
464          point_attributes: Vector or array of data at the
465                            point_coordinates.
466
467    """
468    #Since this is a wrapper for fit, lets handle the geo_spatial att's
469    if use_cache is True:
470        interp = cache(_fit,
471                       (vertex_coordinates,
472                        triangles),
473                       {'verbose': verbose,
474                        'mesh_origin': mesh_origin,
475                        'alpha':alpha},
476                       verbose = verbose)       
477       
478    else:
479        interp = Fit(vertex_coordinates,
480                     triangles,
481                     verbose=verbose,
482                     mesh_origin=mesh_origin,
483                     alpha=alpha)
484       
485    vertex_attributes = interp.fit(point_coordinates,
486                                   point_attributes,
487                                   point_origin=data_origin,
488                                   max_read_lines=max_read_lines,
489                                   attribute_name=attribute_name,
490                                   verbose=verbose)
491
492       
493    # Add the value checking stuff that's in least squares.
494    # Maybe this stuff should get pushed down into Fit.
495    # at least be a method of Fit.
496    # Or intigrate it into the fit method, saving teh max and min's
497    # as att's.
498   
499    return vertex_attributes
500
501def _fit(*args, **kwargs):
502    """Private function for use with caching. Reason is that classes
503    may change their byte code between runs which is annoying.
504    """
505   
506    return Fit(*args, **kwargs)
507
Note: See TracBrowser for help on using the repository browser.