We Keep Coding

Plot a 1D array of varying column size arrays in matplotlib

Given that, I have arrays of x & y, I am able to plot easy the points, like so:
a = np.arange(10).reshape(5, 2)
plt.scatter(a.T[0], a.T[1])
plt.xlabel("Interval")
plt.ylabel("Value")
plt.show()
But currently, I have a list of lists with varying size, like so:
a = [
[0, 1, 3],
[4, 2],
[1, 4, 7, 2],
[2],
[3, 4, 5, 6]
]
plt.scatter(a, list(range(len(a))))
plt.xlabel("Interval")
plt.ylabel("Value")
plt.show()
But, that gives an error: setting an array element with a sequence which surely will do when the dimensions aren't fixed and the sizes are not equal( (dimen) x != (dimen) y), but I am looking at getting something like that:
How can I obtain that kind of plot?

You could create the data to be plotted manually:
import numpy as np
import matplotlib.pyplot as plt
a = np.array([
[0, 1, 3],
[4, 2],
[1, 4, 7, 2],
[2],
[3, 4, 5, 6]
])
data = np.array([[x, y] for x, ys in enumerate(a) for y in ys])
plt.scatter(data[:, 0], data[:, 1], c='red')
plt.xlabel("Interval")
plt.ylabel("Value")
plt.show()
Output

You were almost close. You can just use a single for loop to plot individual lists as following. lst will be one sublist at a time and
[i]*len(lst) will generate the number of x-datapoints for that sublist.
for i, lst in enumerate(a):
plt.scatter([i]*len(lst), lst, color='r')
plt.xlabel("Interval")
plt.ylabel("Value")

Pose keypoints numpy averaging

I know you're supposed to give examples when you ask questions here, but I can't really think of anything that wouldn't involve pasting a massive project worth of code, so I'll just try to describe this as well as possible.
I'm working on a project that involves using keypoints generated by using OpenPose (after I've done some preprocessing on them to simplify everything, I come up with data formatted like this: [x0, y0, c0, x1, y1, c1...], where there are 18 points total, and the x's and y's represent their coordinates, while the c's represent confidence.) I want to take a nested list that has the keypoints for a single person listed in the above manner for each frame, and output a new nested list of lists, made up of the weighted average x's and y's (the weights would be the confidence values for each point) along with the average confidences by each second (instead of by frame), in the same format as above.
I have already converted the original list into a 3-dimensional list, with each second holding each of its frames, each of which holds its keypoint list. I know that I can write code myself to do all of this without using numpy.average(), but I was hoping that I wouldn't have to, because it quickly becomes confusing. Instead, I was wondering if there were a way I could iterate over each second, using said method, in a reasonably simple manner, and just append the resulting lists to a new list, like this:
out = []
for second in lst:
out.append(average(second, axis=1, weights=?, other params?))
Again, I'm sorry for not giving an example of some sort.

Maybe you could get some inspiration from this code:
import numpy as np
def pose_average(sequence):
x, y, c = sequence[0::3], sequence[1::3], sequence[2::3]
x_avg = np.average(x, weights=c)
y_avg = np.average(y, weights=c)
return x_avg, y_avg
sequence = [2, 4, 1, 5, 6, 3, 5, 2, 1]
pose_average(sequence)
>>> (4.4, 4.8)
For multiple sequences of grouped poses:
data = [[1, 2, 3, 2, 3, 4, 3, 4, 5], [1, 2, 3, 4, 5, 6, 7, 8, 9], [4, 1, 2, 5, 3, 3, 4, 1, 2]]
out = [ pose_average(seq) for seq in data ]
out
>>> [(2.1666666666666665, 3.1666666666666665),
(5.0, 6.0),
(4.428571428571429, 1.8571428571428572)]
Edit
By assuming that:
data is a list of sequence
a sequence is a list of grouped poses (for example grouped by seconds)
a pose is the coordinates of the joins positions: [x1, y1, c1, x2, y2, c2, ...]
the slightly modified code is now:
import numpy as np
data = [
[[1, 2, 3, 2, 3, 4, 3, 4, 5], [9, 2, 3, 4, 5, 6, 7, 8, 9], [4, 1, 2, 5, 3, 3, 4, 1, 2], [5, 3, 4, 1, 10, 6, 5, 0, 0]],
[[6, 9, 11, 0, 8, 6, 1, 5, 11], [3, 5, 4, 2, 0, 2, 0, 8, 8], [1, 5, 9, 5, 1, 0, 6, 6, 6]],
[[9, 4, 7, 0, 2, 1], [9, 4, 7, 0, 2, 1], [9, 4, 7, 0, 2, 1]]
]
def pose_average(sequence):
sequence = np.asarray(sequence)
x, y, c = sequence[:, 0::3], sequence[:, 1::3], sequence[:, 2::3]
x_avg = np.average(x, weights=c, axis=0)
y_avg = np.average(y, weights=c, axis=0)
return x_avg, y_avg
out = [ pose_average(seq) for seq in data ]
out
>>> [(array([4.83333333, 2.78947368, 5.375 ]),
array([2.16666667, 5.84210526, 5.875 ])),
(array([3.625, 0.5 , 1.88 ]), array([6.83333333, 6. , 6.2 ])),
(array([9., 0.]), array([4., 2.]))]
x_avg is now the list of x position averaged over the sequence for each point and weight by c.

Summing array entries along a particular line, python

I have a 2D array, and would like to sum its entries along a particular line. It should basically be like numpy.sum(), not along a column or row but rather along a line (given by an equation).
I don't really know where to start from. There is this answer which uses a Radon transfer (though I haven't managed to porperly install the skimage package).
Is there any built-in function I can start from?

Here's what I've come up with:
array = [[1, 2, 3, 4, 5],
[1, 2, 3, 4, 5],
[1, 2, 3, 4, 5],
[1, 2, 3, 4, 5],
[1, 2, 3, 4, 5],
[1, 2, 3, 4, 5]]
def points_on_line(x0,y0, x1,y1):
dx = x1 - x0
dy = y1 - y0
D = 2*dy - dx
y = y0
for x in range(x0, x1):
yield (x,y)
if D > 0:
y = y + 1
D = D - 2*dx
D = D + 2*dy
print(sum([array[y][x] for x, y in points_on_line(0,0, 5, 4)]))
This uses Bresenheim's Line Algorithm to find the points lying on the line between two points. It's not perfect though, and won't return all the points that it touches. This should be a good jumping off point, though!

What is the purpose of meshgrid in Python / NumPy?

Can someone explain to me what is the purpose of meshgrid function in Numpy? I know it creates some kind of grid of coordinates for plotting, but I can't really see the direct benefit of it.
I am studying "Python Machine Learning" from Sebastian Raschka, and he is using it for plotting the decision borders. See input 11 here.
I have also tried this code from official documentation, but, again, the output doesn't really make sense to me.
x = np.arange(-5, 5, 1)
y = np.arange(-5, 5, 1)
xx, yy = np.meshgrid(x, y, sparse=True)
z = np.sin(xx**2 + yy**2) / (xx**2 + yy**2)
h = plt.contourf(x,y,z)
Please, if possible, also show me a lot of real-world examples.

The purpose of meshgrid is to create a rectangular grid out of an array of x values and an array of y values.
So, for example, if we want to create a grid where we have a point at each integer value between 0 and 4 in both the x and y directions. To create a rectangular grid, we need every combination of the x and y points.
This is going to be 25 points, right? So if we wanted to create an x and y array for all of these points, we could do the following.
x[0,0] = 0 y[0,0] = 0
x[0,1] = 1 y[0,1] = 0
x[0,2] = 2 y[0,2] = 0
x[0,3] = 3 y[0,3] = 0
x[0,4] = 4 y[0,4] = 0
x[1,0] = 0 y[1,0] = 1
x[1,1] = 1 y[1,1] = 1
...
x[4,3] = 3 y[4,3] = 4
x[4,4] = 4 y[4,4] = 4
This would result in the following x and y matrices, such that the pairing of the corresponding element in each matrix gives the x and y coordinates of a point in the grid.
x = 0 1 2 3 4 y = 0 0 0 0 0
0 1 2 3 4 1 1 1 1 1
0 1 2 3 4 2 2 2 2 2
0 1 2 3 4 3 3 3 3 3
0 1 2 3 4 4 4 4 4 4
We can then plot these to verify that they are a grid:
plt.plot(x,y, marker='.', color='k', linestyle='none')
Obviously, this gets very tedious especially for large ranges of x and y. Instead, meshgrid can actually generate this for us: all we have to specify are the unique x and y values.
xvalues = np.array([0, 1, 2, 3, 4]);
yvalues = np.array([0, 1, 2, 3, 4]);
Now, when we call meshgrid, we get the previous output automatically.
xx, yy = np.meshgrid(xvalues, yvalues)
plt.plot(xx, yy, marker='.', color='k', linestyle='none')
Creation of these rectangular grids is useful for a number of tasks. In the example that you have provided in your post, it is simply a way to sample a function (sin(x**2 + y**2) / (x**2 + y**2)) over a range of values for x and y.
Because this function has been sampled on a rectangular grid, the function can now be visualized as an "image".
Additionally, the result can now be passed to functions which expect data on rectangular grid (i.e. contourf)

Courtesy of Microsoft Excel:

Actually the purpose of np.meshgrid is already mentioned in the documentation:
np.meshgrid
Return coordinate matrices from coordinate vectors.
Make N-D coordinate arrays for vectorized evaluations of N-D scalar/vector fields over N-D grids, given one-dimensional coordinate arrays x1, x2,..., xn.
So it's primary purpose is to create a coordinates matrices.
You probably just asked yourself:
Why do we need to create coordinate matrices?
The reason you need coordinate matrices with Python/NumPy is that there is no direct relation from coordinates to values, except when your coordinates start with zero and are purely positive integers. Then you can just use the indices of an array as the index.
However when that's not the case you somehow need to store coordinates alongside your data. That's where grids come in.
Suppose your data is:
1 2 1
2 5 2
1 2 1
However, each value represents a 3 x 2 kilometer area (horizontal x vertical). Suppose your origin is the upper left corner and you want arrays that represent the distance you could use:
import numpy as np
h, v = np.meshgrid(np.arange(3)*3, np.arange(3)*2)
where v is:
array([[0, 0, 0],
[2, 2, 2],
[4, 4, 4]])
and h:
array([[0, 3, 6],
[0, 3, 6],
[0, 3, 6]])
So if you have two indices, let's say x and y (that's why the return value of meshgrid is usually xx or xs instead of x in this case I chose h for horizontally!) then you can get the x coordinate of the point, the y coordinate of the point and the value at that point by using:
h[x, y] # horizontal coordinate
v[x, y] # vertical coordinate
data[x, y] # value
That makes it much easier to keep track of coordinates and (even more importantly) you can pass them to functions that need to know the coordinates.
A slightly longer explanation
However, np.meshgrid itself isn't often used directly, mostly one just uses one of similar objects np.mgrid or np.ogrid.
Here np.mgrid represents the sparse=False and np.ogrid the sparse=True case (I refer to the sparse argument of np.meshgrid). Note that there is a significant difference between
np.meshgrid and np.ogrid and np.mgrid: The first two returned values (if there are two or more) are reversed. Often this doesn't matter but you should give meaningful variable names depending on the context.
For example, in case of a 2D grid and matplotlib.pyplot.imshow it makes sense to name the first returned item of np.meshgrid x and the second one y while it's
the other way around for np.mgrid and np.ogrid.
np.ogrid and sparse grids
>>> import numpy as np
>>> yy, xx = np.ogrid[-5:6, -5:6]
>>> xx
array([[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]])
>>> yy
array([[-5],
[-4],
[-3],
[-2],
[-1],
[ 0],
[ 1],
[ 2],
[ 3],
[ 4],
[ 5]])
As already said the output is reversed when compared to np.meshgrid, that's why I unpacked it as yy, xx instead of xx, yy:
>>> xx, yy = np.meshgrid(np.arange(-5, 6), np.arange(-5, 6), sparse=True)
>>> xx
array([[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]])
>>> yy
array([[-5],
[-4],
[-3],
[-2],
[-1],
[ 0],
[ 1],
[ 2],
[ 3],
[ 4],
[ 5]])
This already looks like coordinates, specifically the x and y lines for 2D plots.
Visualized:
yy, xx = np.ogrid[-5:6, -5:6]
plt.figure()
plt.title('ogrid (sparse meshgrid)')
plt.grid()
plt.xticks(xx.ravel())
plt.yticks(yy.ravel())
plt.scatter(xx, np.zeros_like(xx), color="blue", marker="*")
plt.scatter(np.zeros_like(yy), yy, color="red", marker="x")
np.mgrid and dense/fleshed out grids
>>> yy, xx = np.mgrid[-5:6, -5:6]
>>> xx
array([[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]])
>>> yy
array([[-5, -5, -5, -5, -5, -5, -5, -5, -5, -5, -5],
[-4, -4, -4, -4, -4, -4, -4, -4, -4, -4, -4],
[-3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3],
[-2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2],
[-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],
[ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],
[ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4],
[ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]])
The same applies here: The output is reversed compared to np.meshgrid:
>>> xx, yy = np.meshgrid(np.arange(-5, 6), np.arange(-5, 6))
>>> xx
array([[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5],
[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]])
>>> yy
array([[-5, -5, -5, -5, -5, -5, -5, -5, -5, -5, -5],
[-4, -4, -4, -4, -4, -4, -4, -4, -4, -4, -4],
[-3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3],
[-2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2],
[-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],
[ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],
[ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4],
[ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]])
Unlike ogrid these arrays contain all xx and yy coordinates in the -5 <= xx <= 5; -5 <= yy <= 5 grid.
yy, xx = np.mgrid[-5:6, -5:6]
plt.figure()
plt.title('mgrid (dense meshgrid)')
plt.grid()
plt.xticks(xx[0])
plt.yticks(yy[:, 0])
plt.scatter(xx, yy, color="red", marker="x")
Functionality
It's not only limited to 2D, these functions work for arbitrary dimensions (well, there is a maximum number of arguments given to function in Python and a maximum number of dimensions that NumPy allows):
>>> x1, x2, x3, x4 = np.ogrid[:3, 1:4, 2:5, 3:6]
>>> for i, x in enumerate([x1, x2, x3, x4]):
... print('x{}'.format(i+1))
... print(repr(x))
x1
array([[[[0]]],
[[[1]]],
[[[2]]]])
x2
array([[[[1]],
[[2]],
[[3]]]])
x3
array([[[[2],
[3],
[4]]]])
x4
array([[[[3, 4, 5]]]])
>>> # equivalent meshgrid output, note how the first two arguments are reversed and the unpacking
>>> x2, x1, x3, x4 = np.meshgrid(np.arange(1,4), np.arange(3), np.arange(2, 5), np.arange(3, 6), sparse=True)
>>> for i, x in enumerate([x1, x2, x3, x4]):
... print('x{}'.format(i+1))
... print(repr(x))
# Identical output so it's omitted here.
Even if these also work for 1D there are two (much more common) 1D grid creation functions:
np.arange
np.linspace
Besides the start and stop argument it also supports the step argument (even complex steps that represent the number of steps):
>>> x1, x2 = np.mgrid[1:10:2, 1:10:4j]
>>> x1 # The dimension with the explicit step width of 2
array([[1., 1., 1., 1.],
[3., 3., 3., 3.],
[5., 5., 5., 5.],
[7., 7., 7., 7.],
[9., 9., 9., 9.]])
>>> x2 # The dimension with the "number of steps"
array([[ 1., 4., 7., 10.],
[ 1., 4., 7., 10.],
[ 1., 4., 7., 10.],
[ 1., 4., 7., 10.],
[ 1., 4., 7., 10.]])
Applications
You specifically asked about the purpose and in fact, these grids are extremely useful if you need a coordinate system.
For example if you have a NumPy function that calculates the distance in two dimensions:
def distance_2d(x_point, y_point, x, y):
return np.hypot(x-x_point, y-y_point)
And you want to know the distance of each point:
>>> ys, xs = np.ogrid[-5:5, -5:5]
>>> distances = distance_2d(1, 2, xs, ys) # distance to point (1, 2)
>>> distances
array([[9.21954446, 8.60232527, 8.06225775, 7.61577311, 7.28010989,
7.07106781, 7. , 7.07106781, 7.28010989, 7.61577311],
[8.48528137, 7.81024968, 7.21110255, 6.70820393, 6.32455532,
6.08276253, 6. , 6.08276253, 6.32455532, 6.70820393],
[7.81024968, 7.07106781, 6.40312424, 5.83095189, 5.38516481,
5.09901951, 5. , 5.09901951, 5.38516481, 5.83095189],
[7.21110255, 6.40312424, 5.65685425, 5. , 4.47213595,
4.12310563, 4. , 4.12310563, 4.47213595, 5. ],
[6.70820393, 5.83095189, 5. , 4.24264069, 3.60555128,
3.16227766, 3. , 3.16227766, 3.60555128, 4.24264069],
[6.32455532, 5.38516481, 4.47213595, 3.60555128, 2.82842712,
2.23606798, 2. , 2.23606798, 2.82842712, 3.60555128],
[6.08276253, 5.09901951, 4.12310563, 3.16227766, 2.23606798,
1.41421356, 1. , 1.41421356, 2.23606798, 3.16227766],
[6. , 5. , 4. , 3. , 2. ,
1. , 0. , 1. , 2. , 3. ],
[6.08276253, 5.09901951, 4.12310563, 3.16227766, 2.23606798,
1.41421356, 1. , 1.41421356, 2.23606798, 3.16227766],
[6.32455532, 5.38516481, 4.47213595, 3.60555128, 2.82842712,
2.23606798, 2. , 2.23606798, 2.82842712, 3.60555128]])
The output would be identical if one passed in a dense grid instead of an open grid. NumPys broadcasting makes it possible!
Let's visualize the result:
plt.figure()
plt.title('distance to point (1, 2)')
plt.imshow(distances, origin='lower', interpolation="none")
plt.xticks(np.arange(xs.shape[1]), xs.ravel()) # need to set the ticks manually
plt.yticks(np.arange(ys.shape[0]), ys.ravel())
plt.colorbar()
And this is also when NumPys mgrid and ogrid become very convenient because it allows you to easily change the resolution of your grids:
ys, xs = np.ogrid[-5:5:200j, -5:5:200j]
# otherwise same code as above
However, since imshow doesn't support x and y inputs one has to change the ticks by hand. It would be really convenient if it would accept the x and y coordinates, right?
It's easy to write functions with NumPy that deal naturally with grids. Furthermore, there are several functions in NumPy, SciPy, matplotlib that expect you to pass in the grid.
I like images so let's explore matplotlib.pyplot.contour:
ys, xs = np.mgrid[-5:5:200j, -5:5:200j]
density = np.sin(ys)-np.cos(xs)
plt.figure()
plt.contour(xs, ys, density)
Note how the coordinates are already correctly set! That wouldn't be the case if you just passed in the density.
Or to give another fun example using astropy models (this time I don't care much about the coordinates, I just use them to create some grid):
from astropy.modeling import models
z = np.zeros((100, 100))
y, x = np.mgrid[0:100, 0:100]
for _ in range(10):
g2d = models.Gaussian2D(amplitude=100,
x_mean=np.random.randint(0, 100),
y_mean=np.random.randint(0, 100),
x_stddev=3,
y_stddev=3)
z += g2d(x, y)
a2d = models.AiryDisk2D(amplitude=70,
x_0=np.random.randint(0, 100),
y_0=np.random.randint(0, 100),
radius=5)
z += a2d(x, y)
Although that's just "for the looks" several functions related to functional models and fitting (for example scipy.interpolate.interp2d,
scipy.interpolate.griddata even show examples using np.mgrid) in Scipy, etc. require grids. Most of these work with open grids and dense grids, however some only work with one of them.

Suppose you have a function:
def sinus2d(x, y):
return np.sin(x) + np.sin(y)
and you want, for example, to see what it looks like in the range 0 to 2*pi. How would you do it? There np.meshgrid comes in:
xx, yy = np.meshgrid(np.linspace(0,2*np.pi,100), np.linspace(0,2*np.pi,100))
z = sinus2d(xx, yy) # Create the image on this grid
and such a plot would look like:
import matplotlib.pyplot as plt
plt.imshow(z, origin='lower', interpolation='none')
plt.show()
So np.meshgrid is just a convenience. In principle the same could be done by:
z2 = sinus2d(np.linspace(0,2*np.pi,100)[:,None], np.linspace(0,2*np.pi,100)[None,:])
but there you need to be aware of your dimensions (suppose you have more than two ...) and the right broadcasting. np.meshgrid does all of this for you.
Also meshgrid allows you to delete coordinates together with the data if you, for example, want to do an interpolation but exclude certain values:
condition = z>0.6
z_new = z[condition] # This will make your array 1D
so how would you do the interpolation now? You can give x and y to an interpolation function like scipy.interpolate.interp2d so you need a way to know which coordinates were deleted:
x_new = xx[condition]
y_new = yy[condition]
and then you can still interpolate with the "right" coordinates (try it without the meshgrid and you will have a lot of extra code):
from scipy.interpolate import interp2d
interpolated = interp2d(x_new, y_new, z_new)
and the original meshgrid allows you to get the interpolation on the original grid again:
interpolated_grid = interpolated(xx[0], yy[:, 0]).reshape(xx.shape)
These are just some examples where I used the meshgrid there might be a lot more.

Short answer
The purpose of meshgrid is to help replace slow Python loops by faster vectorized operations available in NumPy library. meshgrid role is to prepare 2D arrays required by the vectorized operation.
Basic example showing the principle
Let's say we have two sequences of values,
a = [2,7,9,20]
b = [1,6,7,9] ​
and we want to perform an operation on each possible pair of values, one taken from the first list, one taken from the second list. We also want to store the result. For example, let's say we want to get the sum of the values for each possible pair.
Slow and laborious method
c = []
for i in range(len(b)):
row = []
for j in range(len(a)):
row.append (a[j] + b[i])
c.append (row)
print (c)
Result:
[[3, 8, 10, 21],
[8, 13, 15, 26],
[9, 14, 16, 27],
[11, 16, 18, 29]]
Fast and easy method
i,j = np.meshgrid (a,b)
c = i + j
print (c)
Result:
[[ 3 8 10 21]
[ 8 13 15 26]
[ 9 14 16 27]
[11 16 18 29]]
You can see from this basic illustration how the explicit slow Python loops have been replaced by hidden faster C loops in Numpy library. This principle is widely used for 3D operations, included colored pixel maps. The common example is a 3D plot.
Common use: 3D plot
x = np.arange(-4, 4, 0.25)
y = np.arange(-4, 4, 0.25)
X, Y = np.meshgrid(x, y)
R = np.sqrt(X**2 + Y**2)
Z = np.sin(R)
(Borrowed from this site)
meshgrid is used to create pairs of coordinates between -4 and +4 with .25 increments in each direction X and Y. Each pair is then used to find R, and Z from it. This way of preparing "a grid" of coordinates is frequently used in plotting 3D surfaces, or coloring 2D surfaces.
Meshgrid under the hood
The two arrays prepared by meshgrid are:
(array([[ 2, 7, 9, 20],
[ 2, 7, 9, 20],
[ 2, 7, 9, 20],
[ 2, 7, 9, 20]]),
array([[1, 1, 1, 1],
[6, 6, 6, 6],
[7, 7, 7, 7],
[9, 9, 9, 9]]))
These arrays are created by repeating the values provided, either horizontally or vertically. The two arrays are shape compatible for a vector operation.
Origin
numpy.meshgrid comes from MATLAB, like many other NumPy functions. So you can also study the examples from MATLAB to see meshgrid in use, the code for the 3D plotting looks the same in MATLAB.

meshgrid helps in creating a rectangular grid from two 1-D arrays of all pairs of points from the two arrays.
x = np.array([0, 1, 2, 3, 4])
y = np.array([0, 1, 2, 3, 4])
Now, if you have defined a function f(x,y) and you wanna apply this function to all the possible combination of points from the arrays 'x' and 'y', then you can do this:
f(*np.meshgrid(x, y))
Say, if your function just produces the product of two elements, then this is how a cartesian product can be achieved, efficiently for large arrays.
Referred from here

Basic Idea
Given possible x values, xs, (think of them as the tick-marks on the x-axis of a plot) and possible y values, ys, meshgrid generates the corresponding set of (x, y) grid points---analogous to set((x, y) for x in xs for y in yx). For example, if xs=[1,2,3] and ys=[4,5,6], we'd get the set of coordinates {(1,4), (2,4), (3,4), (1,5), (2,5), (3,5), (1,6), (2,6), (3,6)}.
Form of the Return Value
However, the representation that meshgrid returns is different from the above expression in two ways:
First, meshgrid lays out the grid points in a 2d array: rows correspond to different y-values, columns correspond to different x-values---as in list(list((x, y) for x in xs) for y in ys), which would give the following array:
[[(1,4), (2,4), (3,4)],
[(1,5), (2,5), (3,5)],
[(1,6), (2,6), (3,6)]]
Second, meshgrid returns the x and y coordinates separately (i.e. in two different numpy 2d arrays):
xcoords, ycoords = (
array([[1, 2, 3],
[1, 2, 3],
[1, 2, 3]]),
array([[4, 4, 4],
[5, 5, 5],
[6, 6, 6]]))
# same thing using np.meshgrid:
xcoords, ycoords = np.meshgrid([1,2,3], [4,5,6])
# same thing without meshgrid:
xcoords = np.array([xs] * len(ys)
ycoords = np.array([ys] * len(xs)).T
Note, np.meshgrid can also generate grids for higher dimensions. Given xs, ys, and zs, you'd get back xcoords, ycoords, zcoords as 3d arrays. meshgrid also supports reverse ordering of the dimensions as well as sparse representation of the result.
Applications
Why would we want this form of output?
Apply a function at every point on a grid:
One motivation is that binary operators like (+, -, *, /, **) are overloaded for numpy arrays as elementwise operations. This means that if I have a function def f(x, y): return (x - y) ** 2 that works on two scalars, I can also apply it on two numpy arrays to get an array of elementwise results: e.g. f(xcoords, ycoords) or f(*np.meshgrid(xs, ys)) gives the following on the above example:
array([[ 9, 4, 1],
[16, 9, 4],
[25, 16, 9]])
Higher dimensional outer product: I'm not sure how efficient this is, but you can get high-dimensional outer products this way: np.prod(np.meshgrid([1,2,3], [1,2], [1,2,3,4]), axis=0).
Contour plots in matplotlib: I came across meshgrid when investigating drawing contour plots with matplotlib for plotting decision boundaries. For this, you generate a grid with meshgrid, evaluate the function at each grid point (e.g. as shown above), and then pass the xcoords, ycoords, and computed f-values (i.e. zcoords) into the contourf function.

Behind the scenes:
import numpy as np
def meshgrid(x , y):
XX = []
YY = []
for colm in range(len(y)):
XX.append([])
YY.append([])
for row in range(len(x)):
XX[colm].append(x[row])
YY[colm].append(y[colm])
return np.asarray(XX), np.asarray(YY)
Lets take dataset of #Sarsaparilla's answer as example:
y = [7, 6, 5]
x = [1, 2, 3, 4]
xx, yy = meshgrid(x , y)
and it outputs:
>>> xx
array([[1, 2, 3, 4],
[1, 2, 3, 4],
[1, 2, 3, 4]])
>>> yy
array([[7, 7, 7, 7],
[6, 6, 6, 6],
[5, 5, 5, 5]])

matplotlib plotting a dot every nth x

I'm trying to plot two things:
The first is this:
plotted_values = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
I then want to plot a dot every nth value in the plotted values list. Say n is 5:
nth_value_plot = [5, 10]
I want to plot the nth_value_plot values such that the 5 and 10 share the same x, y coordinate as the 5 and 10 in the plotted_values. Plotting as such plots the nth_value_plot with the x coordinates 0 and 1 as expected:
plt.plot(plotted_values)
plt.plot(nth_value_plot, "o")
plt.show()
How do I correctly plot this as described above?
EDIT:
The final plot coordinates (x, y) should be:
plotted_values = [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]]
nth_value_plot = [[4, 5], [9, 10]]
The current plot from the code above has plotted values with these coordinates and nth_value_plot with the coordinates:
nth_value_plot = [[0, 5], [1, 10]]
Thanks

Assuming your x values are the same as your y values:
v = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
n=5
plt.scatter(v[::n],v[::n])

You must specify the x coordinates:
Replace
plt.plot(nth_value_plot, "o")
with
n = 5
plt.plot([x + n - 1 for x in range(0, len(plotted_values), n)], nth_value_plot, "o")

Within the plot command there is a 'markevery' command that permits you to set the number of points to skip within the data array.