我尝试复制类似于下方代码的工作，但当我使用这个链接中的数据https://raw.githubusercontent.com/plotly/datasets/master/api_docs/mt_bruno_elevation.csv时，出现了错误。我认为这是因为数据形状的问题，但我不确定具体如何修改它。
如果您能帮助我解决这个问题，将非常感激。
这是我的代码
from IPython.core.display import HTMLimport plotlyimport plotly.graph_objects as goimport noiseimport numpy as npimport matplotlibfrom mpl_toolkits.mplot3d import axes3d%matplotlib inlineimport pandas as pddata = = pd.read_csv('https://raw.githubusercontent.com/plotly/datasets/master/api_docs/mt_bruno_elevation.csv')z = dataimport numpy as npfrom numpy.lib.stride_tricks import as_strideddef sliding_window(arr, window_size): """ Construct a sliding window view of the array""" arr = np.asarray(arr) window_size = int(window_size) if arr.ndim != 2: raise ValueError("need 2-D input") if not (window_size > 0): raise ValueError("need a positive window size") shape = (arr.shape[0] - window_size + 1, arr.shape[1] - window_size + 1, window_size, window_size) if shape[0] <= 0: shape = (1, shape[1], arr.shape[0], shape[3]) if shape[1] <= 0: shape = (shape[0], 1, shape[2], arr.shape[1]) strides = (arr.shape[1]*arr.itemsize, arr.itemsize, arr.shape[1]*arr.itemsize, arr.itemsize) return as_strided(arr, shape=shape, strides=strides)def cell_neighbours(arr, i, j, d): """Return d-th neighbors of cell (i, j)""" w = sliding_window(arr, 2*d+1) ix = np.clip(i - d, 0, w.shape[0]-1) jx = np.clip(j - d, 0, w.shape[1]-1) i0 = max(0, i - d - ix) j0 = max(0, j - d - jx) i1 = w.shape[2] - max(0, d - i + ix) j1 = w.shape[3] - max(0, d - j + jx) return w[ix, jx][i0:i1,j0:j1].ravel()from dataclasses import dataclass@dataclassclass descent_step: """Class for storing each step taken in gradient descent""" value: float x_index: float y_index: floatdef gradient_descent_3d(array,x_start,y_start,steps=50,step_size=1,plot=False): # Initial point to start gradient descent at step = descent_step(array[y_start][x_start],x_start,y_start) # Store each step taken in gradient descent in a list step_history = [] step_history.append(step) # Plot 2D representation of array with startng point as a red marker if plot: matplotlib.pyplot.imshow(array,origin='lower',cmap='terrain') matplotlib.pyplot.plot(x_start,y_start,'ro') current_x = x_start current_y = y_start # Loop through specified number of steps of gradient descent to take for i in range(steps): prev_x = current_x prev_y = current_y # Extract array of neighbouring cells around current step location with size nominated neighbours=cell_neighbours(array,current_y,current_x,step_size) # Locate minimum in array (steepest slope from current point) next_step = neighbours.min() indices = np.where(array == next_step) # Update current point to now be the next point after stepping current_x, current_y = (indices[1][0],indices[0][0]) step = descent_step(array[current_y][current_x],current_x,current_y) step_history.append(step) # Plot each step taken as a black line to the current point nominated by a red marker if plot: matplotlib.pyplot.plot([prev_x,current_x],[prev_y,current_y],'k-') matplotlib.pyplot.plot(current_x,current_y,'ro') # If step is to the same location as previously, this infers convergence and end loop if prev_y == current_y and prev_x == current_x: print(f"Converged in {i} steps") break return next_step,step_historynp.random.seed(42)global_minimum = z.min()indices = np.where(z == global_minimum)print(f"Target: {global_minimum} @ {indices}")step_size = 0found_minimum = 99999# Random starting pointstart_x = np.random.randint(0,50)start_y = np.random.randint(0,50)# Increase step size until convergence on global minimumwhile found_minimum != global_minimum: step_size += 1 found_minimum,steps = gradient_descent_3d(z,start_x,start_y,step_size=step_size,plot=False)print(f"Optimal step size {step_size}")found_minimum,steps = gradient_descent_3d(z,start_x,start_y,step_size=step_size,plot=True)print(f"Steps: {steps}")def multiDimenDist(point1,point2): #find the difference between the two points, its really the same as below deltaVals = [point2[dimension]-point1[dimension] for dimension in range(len(point1))] runningSquared = 0 #because the pythagarom theorm works for any dimension we can just use that for coOrd in deltaVals: runningSquared += coOrd**2 return runningSquared**(1/2)def findVec(point1,point2,unitSphere = False): #setting unitSphere to True will make the vector scaled down to a sphere with a radius one, instead of it's orginal length finalVector = [0 for coOrd in point1] for dimension, coOrd in enumerate(point1): #finding total differnce for that co-ordinate(x,y,z...) deltaCoOrd = point2[dimension]-coOrd #adding total difference finalVector[dimension] = deltaCoOrd if unitSphere: totalDist = multiDimenDist(point1,point2) unitVector =[] for dimen in finalVector: unitVector.append( dimen/totalDist) return unitVector else: return finalVectordef generate_3d_plot(step_history): # Initialise empty lists for markers step_markers_x = [] step_markers_y = [] step_markers_z = [] step_markers_u = [] step_markers_v = [] step_markers_w = [] for index, step in enumerate(step_history): step_markers_x.append(step.x_index) step_markers_y.append(step.y_index) step_markers_z.append(step.value) # If we haven't reached the final step, calculate the vector between the current step and the next step if index < len(steps)-1: vec1 = [step.x_index,step.y_index,step.value] vec2 = [steps[index+1].x_index,steps[index+1].y_index,steps[index+1].value] result_vector = findVec(vec1,vec2) step_markers_u.append(result_vector[0]) step_markers_v.append(result_vector[1]) step_markers_w.append(result_vector[2]) else: step_markers_u.append(0.1) step_markers_v.append(0.1) step_markers_w.append(0.1) # Include cones at each marker to show direction of step, scatter3d is to show the red line between points and surface for the terrain fig = go.Figure(data=[ go.Cone( x=step_markers_x, y=step_markers_y, z=step_markers_z, u=step_markers_u, v=step_markers_v, w=step_markers_w, sizemode="absolute", sizeref=2, anchor='tail'), go.Scatter3d( x=step_markers_x, y=step_markers_y, z=step_markers_z, mode='lines', line=dict( color='red', width=2 )), go.Surface(colorscale=terrain,z=world,opacity=0.5)]) # Z axis is limited to the extent of the terrain array fig.update_layout( title='Gradient Descent Steps', scene = dict(zaxis = dict(range=[world.min(),world.max()],),),) return fig # Generate 3D plot from previous random starting locationfig = generate_3d_plot(steps)HTML(plotly.offline.plot(fig, filename='random_starting_point_3d_gradient_descent.html',include_plotlyjs='cdn'))回答：
错误发生的原因是found_minimum是一个int类型，而global_minimum是一个Series类型。我认为您参考的教程假设数据是以numpy的array形式加载的，但这点并未明确说明。
因此，z = data.to_numpy()解决了一个问题，同时也暴露了另一个问题，即教程中的数据集是50×50，而您的数据集是25×25。仅仅更改随机起始点的限制似乎是一个诱人的解决方案，但实际上效果并不好。这个数据集对于这种梯度下降实现来说太小了，无法适当收敛。
为了解决这个问题，我调整了您的数据集，制造了一个50×50的数据集：
data_arr = data.to_numpy()double_arr = np.append(data_arr, 1.5*data_arr + 50, axis=0)quad_arr = np.append(double_arr, 1.5*double_arr + 50, axis=1)在整个代码中使用这个quad_arr，并将plotly的颜色比例更新为go.Surface(colorscale=Earth)，结果如下：
完整的、可复制的代码如下：
from IPython.core.display import HTMLimport plotlyimport plotly.graph_objects as goimport noiseimport numpy as npimport matplotlibfrom mpl_toolkits.mplot3d import axes3d%matplotlib inlineimport pandas as pddata = pd.read_csv('https://raw.githubusercontent.com/plotly/datasets/master/api_docs/mt_bruno_elevation.csv')data_arr = data.to_numpy()double_arr = np.append(data_arr, 1.5*data_arr + 50, axis=0)quad_arr = np.append(double_arr, 1.5*double_arr + 50, axis=1)z = quad_arrmatplotlib.pyplot.imshow(z,origin='lower',cmap='terrain')# Find maximum value index in numpy arrayindices = np.where(z == z.max())max_z_x_location, max_z_y_location = (indices[1][0],indices[0][0])matplotlib.pyplot.plot(max_z_x_location,max_z_y_location,'ro',markersize=15)# Find minimum value index in numpy arrayindices = np.where(z == z.min())min_z_x_location, min_z_y_location = (indices[1][0],indices[0][0])matplotlib.pyplot.plot(min_z_x_location,min_z_y_location,'yo',markersize=15)import numpy as npfrom numpy.lib.stride_tricks import as_strideddef sliding_window(arr, window_size): """ Construct a sliding window view of the array""" arr = np.asarray(arr) window_size = int(window_size) if arr.ndim != 2: raise ValueError("need 2-D input") if not (window_size > 0): raise ValueError("need a positive window size") shape = (arr.shape[0] - window_size + 1, arr.shape[1] - window_size + 1, window_size, window_size) if shape[0] <= 0: shape = (1, shape[1], arr.shape[0], shape[3]) if shape[1] <= 0: shape = (shape[0], 1, shape[2], arr.shape[1]) strides = (arr.shape[1]*arr.itemsize, arr.itemsize, arr.shape[1]*arr.itemsize, arr.itemsize) return as_strided(arr, shape=shape, strides=strides)def cell_neighbours(arr, i, j, d): """Return d-th neighbors of cell (i, j)""" w = sliding_window(arr, 2*d+1) ix = np.clip(i - d, 0, w.shape[0]-1) jx = np.clip(j - d, 0, w.shape[1]-1) i0 = max(0, i - d - ix) j0 = max(0, j - d - jx) i1 = w.shape[2] - max(0, d - i + ix) j1 = w.shape[3] - max(0, d - j + jx) return w[ix, jx][i0:i1,j0:j1].ravel()from dataclasses import dataclass@dataclassclass descent_step: """Class for storing each step taken in gradient descent""" value: float x_index: float y_index: floatdef gradient_descent_3d(array,x_start,y_start,steps=50,step_size=1,plot=False): # Initial point to start gradient descent at step = descent_step(array[y_start][x_start],x_start,y_start) # Store each step taken in gradient descent in a list step_history = [] step_history.append(step) # Plot 2D representation of array with startng point as a red marker if plot: matplotlib.pyplot.imshow(array,origin='lower',cmap='terrain') matplotlib.pyplot.plot(x_start,y_start,'ro') current_x = x_start current_y = y_start # Loop through specified number of steps of gradient descent to take for i in range(steps): prev_x = current_x prev_y = current_y # Extract array of neighbouring cells around current step location with size nominated neighbours=cell_neighbours(array,current_y,current_x,step_size) # Locate minimum in array (steepest slope from current point) next_step = neighbours.min() indices = np.where(array == next_step) # Update current point to now be the next point after stepping current_x, current_y = (indices[1][0],indices[0][0]) step = descent_step(array[current_y][current_x],current_x,current_y) step_history.append(step) # Plot each step taken as a black line to the current point nominated by a red marker if plot: matplotlib.pyplot.plot([prev_x,current_x],[prev_y,current_y],'k-') matplotlib.pyplot.plot(current_x,current_y,'ro') # If step is to the same location as previously, this infers convergence and end loop if prev_y == current_y and prev_x == current_x: print(f"Converged in {i} steps") break return next_step,step_historynp.random.seed(42)global_minimum = z.min()indices = np.where(z == global_minimum)print(f"Target: {global_minimum} @ {indices}")step_size = 0found_minimum = 99999# Random starting pointstart_x = np.random.randint(0,50)start_y = np.random.randint(0,50)# Increase step size until convergence on global minimumprint('==========================')print(found_minimum)print(global_minimum)print('==========================')while found_minimum != global_minimum: step_size += 1 try: found_minimum,steps = gradient_descent_3d(z,start_x,start_y,step_size=step_size,plot=True) except ValueError: passprint(f"Optimal step size {step_size}")found_minimum,steps = gradient_descent_3d(z,start_x,start_y,step_size=step_size,plot=True)print(f"Steps: {steps}")def multiDimenDist(point1,point2): #find the difference between the two points, its really the same as below deltaVals = [point2[dimension]-point1[dimension] for dimension in range(len(point1))] runningSquared = 0 #because the pythagarom theorm works for any dimension we can just use that for coOrd in deltaVals: runningSquared += coOrd**2 return runningSquared**(1/2)def findVec(point1,point2,unitSphere = False): #setting unitSphere to True will make the vector scaled down to a sphere with a radius one, instead of it's orginal length finalVector = [0 for coOrd in point1] for dimension, coOrd in enumerate(point1): #finding total differnce for that co-ordinate(x,y,z...) deltaCoOrd = point2[dimension]-coOrd #adding total difference finalVector[dimension] = deltaCoOrd if unitSphere: totalDist = multiDimenDist(point1,point2) unitVector =[] for dimen in finalVector: unitVector.append( dimen/totalDist) return unitVector else: return finalVectordef generate_3d_plot(step_history): # Initialise empty lists for markers step_markers_x = [] step_markers_y = [] step_markers_z = [] step_markers_u = [] step_markers_v = [] step_markers_w = [] for index, step in enumerate(step_history): step_markers_x.append(step.x_index) step_markers_y.append(step.y_index) step_markers_z.append(step.value) # If we haven't reached the final step, calculate the vector between the current step and the next step if index < len(steps)-1: vec1 = [step.x_index,step.y_index,step.value] vec2 = [steps[index+1].x_index,steps[index+1].y_index,steps[index+1].value] result_vector = findVec(vec1,vec2) step_markers_u.append(result_vector[0]) step_markers_v.append(result_vector[1]) step_markers_w.append(result_vector[2]) else: step_markers_u.append(0.1) step_markers_v.append(0.1) step_markers_w.append(0.1) # Include cones at each marker to show direction of step, scatter3d is to show the red line between points and surface for the terrain fig = go.Figure(data=[ go.Cone( x=step_markers_x, y=step_markers_y, z=step_markers_z, u=step_markers_u, v=step_markers_v, w=step_markers_w, sizemode="absolute", sizeref=2, anchor='tail'), go.Scatter3d( x=step_markers_x, y=step_markers_y, z=step_markers_z, mode='lines', line=dict( color='red', width=2 )), go.Surface(colorscale='Earth', z=quad_arr,opacity=0.5)]) # Z axis is limited to the extent of the terrain array fig.update_layout( title='Gradient Descent Steps', scene = dict(zaxis = dict(range=[quad_arr.min(),quad_arr.max()],),),) return fig # Generate 3D plot from previous random starting locationfig = generate_3d_plot(steps)HTML(plotly.offline.plot(fig, filename='random_starting_point_3d_gradient_descent.html',include_plotlyjs='cdn'))