This notebook was put together by Jake Vanderplas for UW's Astro 599 course. Source and license info is on GitHub.

Efficient Numerical Computing with Numpy

In this session, we'll discuss how to make your programs as efficient as possible, mainly by taking advantage of vectorization in NumPy.

Aside: the "Unladen Swallow"

In [2]:
from IPython.display import YouTubeVideo
YouTubeVideo("y2R3FvS4xr4")
Out[2]:

That video was the original reference behind the now-defunct unladen swallow project, which had the goal of making Python's C implementation faster.

Yes, Python is (unfortunately) a rather slow language.

Here is an example:

In [3]:
# A silly function implemented in Python

def func_python(N):
    d = 0.0
    for i in range(N):
        d += (i % 3 - 1) * i
    return d
In [4]:
# Use IPython timeit magic to time the execution
%timeit func_python(10000)
1000 loops, best of 3: 1.81 ms per loop

To compare to a compiled language, let's write the same function in fortran and use the f2py tool (included in NumPy) to compile it

In [5]:
%%file func_fortran.f

      subroutine func_fort(n, d)
           integer, intent(in) :: n
           double precision, intent(out) :: d
           integer :: i
           d = 0
           do i = 0, n - 1
                d = d + (mod(i, 3) - 1) * i
           end do
      end subroutine func_fort
Overwriting func_fortran.f

In [6]:
!f2py -c func_fortran.f -m func_fortran > /dev/null
In [7]:
from func_fortran import func_fort
%timeit func_fort(10000)
100000 loops, best of 3: 20.3 µs per loop

Fortran is about 100 times faster for this task!

Why is Python so slow?

We alluded to this yesterday, but languages tend to have a compromise between convenience and performance.

We like Python because our development time is generally more valuable than execution time. But sometimes speed can be an issue.

Strategies for making Python fast

  1. Use Numpy ufuncs to your advantage

  2. Use Numpy aggregates to your advantage

  3. Use Numpy broadcasting to your advantage

  4. Use Numpy slicing and masking to your advantage

  5. Use a tool like SWIG, cython or f2py to interface to compiled code.

Here we'll cover the first four, and leave the fifth strategy for a later session.

Strategy 1: Using UFuncs in Numpy

A ufunc in numpy is a Universal Function. This is a function which operates element-wise on an array. We've already seen examples of these in the various arithmetic operations:

In [8]:
import numpy as np
x = np.random.random(4)
print x
print x + 1  # add 1 to each element of x
[ 0.90839173  0.6614708   0.74607367  0.66114385]
[ 1.90839173  1.6614708   1.74607367  1.66114385]

In [9]:
print x * 2  # multiply each element of x by 2
[ 1.81678346  1.32294159  1.49214734  1.32228771]

In [10]:
print x * x  # multiply each element of x by itself
[ 0.82517553  0.43754362  0.55662592  0.43711119]

In [11]:
print x[1:] - x[:-1]
[-0.24692093  0.08460287 -0.08492982]

These are binary ufuncs: they take two arguments.

There are also many unary ufuncs:

In [12]:
-x
Out[12]:
array([-0.90839173, -0.6614708 , -0.74607367, -0.66114385])
In [13]:
np.sin(x)
Out[13]:
array([ 0.78851565,  0.61427811,  0.67876066,  0.61402009])

The Speed of Ufuncs

In [14]:
x = np.random.random(10000)
In [15]:
%%timeit
# compute element-wise x + 1 via a ufunc 
y = np.zeros_like(x)
y = x + 1
10000 loops, best of 3: 23.7 µs per loop

In [16]:
%%timeit
# compute element-wise x + 1 via a loop
y = np.zeros_like(x)
for i in range(len(x)):
    y[i] = x[i] + 1
100 loops, best of 3: 13.7 ms per loop

Why is NumPy so much faster?

Numpy UFuncs are faster than Python functions involving loops, because the looping happens in compiled code. This is only possible when types are known beforehand, which is why numpy arrays must be typed.

Other Available Ufuncs

In [17]:
x = np.random.random(5)
print x
print np.minimum(x, 0.5)
print np.maximum(x, 0.5)
[ 0.03731972  0.94058919  0.79808336  0.45500933  0.03967752]
[ 0.03731972  0.5         0.5         0.45500933  0.03967752]
[ 0.5         0.94058919  0.79808336  0.5         0.5       ]

In [18]:
# contrast this behavior with that of min() and max()
print np.min(x)
print np.max(x)
0.0373197161493
0.940589193882

In [19]:
%matplotlib inline
# On older IPython versions, use %pylab inline
import matplotlib.pyplot as plt
In [20]:
x = np.linspace(0, 10, 1000)
plt.plot(x, np.sin(x))
Out[20]:
[<matplotlib.lines.Line2D at 0x1060badd0>]
In [21]:
from scipy.special import gammaln
plt.plot(x, gammaln(x))
Out[21]:
[<matplotlib.lines.Line2D at 0x106353610>]

Some interesting properties of UFuncs

UFuncs have some methods built-in, which allow for some very interesting, flexible, and fast operations:

In [22]:
x = np.arange(5)
y = np.arange(1, 6)
np.add(x, y)
Out[22]:
array([1, 3, 5, 7, 9])
In [23]:
np.add.accumulate(x)
Out[23]:
array([ 0,  1,  3,  6, 10])
In [24]:
np.multiply.accumulate(x)
Out[24]:
array([0, 0, 0, 0, 0])
In [25]:
np.multiply.accumulate(y)
Out[25]:
array([  1,   2,   6,  24, 120])
In [26]:
np.add.identity
Out[26]:
0
In [27]:
np.multiply.identity
Out[27]:
1
In [28]:
np.add.outer(x, y)
Out[28]:
array([[1, 2, 3, 4, 5],
       [2, 3, 4, 5, 6],
       [3, 4, 5, 6, 7],
       [4, 5, 6, 7, 8],
       [5, 6, 7, 8, 9]])
In [29]:
# make a times-table
x = np.arange(1, 13)
print np.multiply.outer(x, x)
[[  1   2   3   4   5   6   7   8   9  10  11  12]
 [  2   4   6   8  10  12  14  16  18  20  22  24]
 [  3   6   9  12  15  18  21  24  27  30  33  36]
 [  4   8  12  16  20  24  28  32  36  40  44  48]
 [  5  10  15  20  25  30  35  40  45  50  55  60]
 [  6  12  18  24  30  36  42  48  54  60  66  72]
 [  7  14  21  28  35  42  49  56  63  70  77  84]
 [  8  16  24  32  40  48  56  64  72  80  88  96]
 [  9  18  27  36  45  54  63  72  81  90  99 108]
 [ 10  20  30  40  50  60  70  80  90 100 110 120]
 [ 11  22  33  44  55  66  77  88  99 110 121 132]
 [ 12  24  36  48  60  72  84  96 108 120 132 144]]

Ufunc mini-exercises

Each of the following functions take an array as input, and return an array as output. They are implemented using loops, which is not very efficient.

  1. For each function, implement a fast version which uses ufuncs to calculate the result more efficiently. Double-check that you get the same result for several different arrays.

  2. use the %timeit magic to time the execution of the two implementations for a large array (say, 1000 elements).

In [30]:
# 1. computing the element-wise sine + cosine

from math import sin, cos
def slow_sincos(x):
    """x is a 1-dimensional array"""
    y = np.zeros_like(x)
    for i in range(len(x)):
        y[i] = sin(x[i]) + cos(x[i])
    return y

x = np.random.random(5)
print slow_sincos(x)
[ 1.41379464  1.24311597  1.39711558  1.1674103   1.41168674]

In [31]:
# write a fast_sincos function
In [32]:
# 2. computing the difference between adjacent squares

def slow_sqdiff(x):
    """x is a 1-dimensional array"""
    y = np.zeros(len(x) - 1)
    for i in range(len(y)):
        y[i] = x[i + 1] ** 2 - x[i] ** 2
    return y

x = np.random.random(5)
print slow_sqdiff(x)
[-0.11965629 -0.25739054  0.01840098  0.2762583 ]

In [33]:
# write a fast_sqdiff function
In [34]:
# 3. computing the outer-product of each consecutive pair

def slow_pairprod(x):
    """x is a 1-dimensional array"""
    if len(x) % 2 != 0:
        raise ValueError("length of x must be even")
    N = len(x) / 2
    y = np.zeros((N, N))
    for i in range(N):
        for j in range(N):
            y[i, j] = x[2 * i] * x[2 * j + 1]
    return y

x = np.arange(1, 9)
print slow_pairprod(x)
[[  2.   4.   6.   8.]
 [  6.  12.  18.  24.]
 [ 10.  20.  30.  40.]
 [ 14.  28.  42.  56.]]

In [35]:
# write a fast_pairprod function

Strategy 2. Using Numpy Aggregates

Aggregates are functions over arrays which return smaller arrays.

Numpy has several built-in

In [36]:
# 10 x 10 array drawn from a standard normal
x = np.random.randn(10, 10)
In [37]:
print x.mean()
0.00323045925586

In [38]:
print np.mean(x)
0.00323045925586

In [39]:
print x.std()
1.0801379167

In [40]:
print x.var()
1.1666979191

In [41]:
print x.sum()
0.323045925586

In [42]:
print x.prod()
-5.72610473992e-23

In [43]:
print np.median(x)
0.00389134929792

In [44]:
print np.percentile(x, 50)
0.00389134929792

In [45]:
print np.percentile(x, (25, 75))
[-0.78582974895964608, 0.7374780554465159]

Aggregates along certain dimensions

In [46]:
x = np.random.rand(3, 5)
print x
[[ 0.20461757  0.63148584  0.87816143  0.60442118  0.79228768]
 [ 0.40654761  0.97038926  0.96468488  0.27046867  0.17623957]
 [ 0.08734478  0.38353736  0.16706956  0.17744179  0.14886722]]

In [47]:
print x.sum(0)  # sum along rows
[ 0.69850996  1.98541246  2.00991587  1.05233165  1.11739447]

In [48]:
print x.sum(1)  # sum along columns
[ 3.11097371  2.78832999  0.96426071]

In [49]:
print np.median(x, 1)
[ 0.63148584  0.40654761  0.16706956]

In [50]:
print np.mean(x, 1)
[ 0.62219474  0.557666    0.19285214]

Binary ufuncs as aggregates

Any binary ufunc (a ufunc taking two arguments) can be turned into an aggregate using the reduce() method:

In [51]:
np.sum(x, 1)
Out[51]:
array([ 3.11097371,  2.78832999,  0.96426071])
In [52]:
np.add.reduce(x, 1)
Out[52]:
array([ 3.11097371,  2.78832999,  0.96426071])
In [53]:
np.prod(x, 1)
Out[53]:
array([ 0.05433798,  0.01814108,  0.00014784])
In [54]:
np.multiply.reduce(x, 1)
Out[54]:
array([ 0.05433798,  0.01814108,  0.00014784])
In [55]:
np.divide.reduce(x, 1)
Out[55]:
array([  0.77051726,   9.11086414,  51.60324064])

A caution: for reduce methods, the default axis is 0:

In [56]:
np.add.reduce(x)
Out[56]:
array([ 0.69850996,  1.98541246,  2.00991587,  1.05233165,  1.11739447])
In [57]:
np.sum(x)
Out[57]:
6.8635644110229004

A quick efficiency note:

Beware the built-in Python aggregates!

Python has a min, max, and sum aggregate built-in. These are much more general than the versions in NumPy:

In [58]:
x = np.random.random(10000)

%timeit np.sum(x)
%timeit sum(x)
100000 loops, best of 3: 16.4 µs per loop
100 loops, best of 3: 4 ms per loop

Dynamic type-checking is slow.

Make sure to use Numpy's sum, min, and max.

Aggregate Mini-exercises

Take the following functions, and convert them into an efficient form using aggregates. Each function expects a 1-dimensional array as input. Double-check that your function returns the same result as the original

In [59]:
def slow_cubesum(x):
    """x is a 1D array"""
    result = 0
    for i in range(len(x)):
        result += x[i] ** 3
    return result

x = np.random.random(100)
print slow_cubesum(x)
28.3126613072

In [60]:
# implement fast_cubesum
In [61]:
def slow_rms(x):
    """x is a 1D array"""
    m = np.mean(x)
    rms = 0
    for i in range(len(x)):
        rms += (x[i] - m) ** 2
    rms /= len(x)
    return np.sqrt(rms)

x = np.random.random(100)
print slow_rms(x)
0.298246500479

In [62]:
# implement fast_rms

Now we return to our silly function from the beginning of this section. Can you implement a fast version using ufuncs and aggregates?

In [63]:
def slow_sillyfunc(N):
    """N is an integer"""
    d = 0.0
    for i in range(N):
        d += (i % 3 - 1) * i
    return d

print slow_sillyfunc(100)
-33.0

In [64]:
# Implement fast_sillyfunc using ufuncs & aggragates

Strategy 3: Using Numpy Broadcasting

We've taken a look at broadcasting previously. But it's important enough that we'll review it quickly here:

(image source)

Broadcasting rules:

  1. If the two arrays differ in their number of dimensions, the shape of the array with fewer dimensions is padded with ones on its leading (left) side.

  2. If the shape of the two arrays does not match in any dimension, the array with shape equal to 1 in that dimension is stretched to match the other shape.

  3. If in any dimension the sizes disagree and neither is equal to 1, an error is raised.

Some Broadcasting examples...

In [65]:
x = np.arange(10)
print x ** 2
[ 0  1  4  9 16 25 36 49 64 81]

In [66]:
Y = x * x[:, np.newaxis]
print Y
[[ 0  0  0  0  0  0  0  0  0  0]
 [ 0  1  2  3  4  5  6  7  8  9]
 [ 0  2  4  6  8 10 12 14 16 18]
 [ 0  3  6  9 12 15 18 21 24 27]
 [ 0  4  8 12 16 20 24 28 32 36]
 [ 0  5 10 15 20 25 30 35 40 45]
 [ 0  6 12 18 24 30 36 42 48 54]
 [ 0  7 14 21 28 35 42 49 56 63]
 [ 0  8 16 24 32 40 48 56 64 72]
 [ 0  9 18 27 36 45 54 63 72 81]]

In [67]:
print Y + 10 * x
[[  0  10  20  30  40  50  60  70  80  90]
 [  0  11  22  33  44  55  66  77  88  99]
 [  0  12  24  36  48  60  72  84  96 108]
 [  0  13  26  39  52  65  78  91 104 117]
 [  0  14  28  42  56  70  84  98 112 126]
 [  0  15  30  45  60  75  90 105 120 135]
 [  0  16  32  48  64  80  96 112 128 144]
 [  0  17  34  51  68  85 102 119 136 153]
 [  0  18  36  54  72  90 108 126 144 162]
 [  0  19  38  57  76  95 114 133 152 171]]

In [68]:
print Y + 10 * x[:, np.newaxis]
[[  0   0   0   0   0   0   0   0   0   0]
 [ 10  11  12  13  14  15  16  17  18  19]
 [ 20  22  24  26  28  30  32  34  36  38]
 [ 30  33  36  39  42  45  48  51  54  57]
 [ 40  44  48  52  56  60  64  68  72  76]
 [ 50  55  60  65  70  75  80  85  90  95]
 [ 60  66  72  78  84  90  96 102 108 114]
 [ 70  77  84  91  98 105 112 119 126 133]
 [ 80  88  96 104 112 120 128 136 144 152]
 [ 90  99 108 117 126 135 144 153 162 171]]

In [69]:
Y = np.random.random((2, 3, 4))
x = 10 * np.arange(3)

print Y + x[:, np.newaxis]
[[[  0.86229625   0.64617272   0.81395431   0.57150281]
  [ 10.72496019  10.95945758  10.0420753   10.37791557]
  [ 20.70117792  20.47498435  20.92576569  20.26448722]]

 [[  0.98847073   0.84611754   0.62249531   0.97384757]
  [ 10.29122012  10.81186259  10.9161937   10.14179008]
  [ 20.65886012  20.99372243  20.88986833  20.12829632]]]

Quick Broadcasting Exercise

Now, assume you have \(N\) points in \(D\) dimensions, represented by an array of shape [N, D].

  1. Compute the mean of the distribution of points efficiently using the built-in np.mean aggregate (that is, find the D-dimensional point which is the mean of the rest of the points)
  2. Compute the mean of the distribution of points efficiently using the np.add ufunc.
  3. Compute the standard error of the mean \(\sigma_{mean} = \sigma N^{-1/2}\), where \(\sigma\) is the standard-deviation, using the np.std aggregate.
  4. Compute this again using the np.add ufunc.
  5. Construct the matrix M, the centered and normalized version of the X array: \[ M_{ij} = (X_{ij} - \mu_j) / \sigma_j \] This is one version of whitening the array.
In [70]:
X = np.random.random((1000, 5))  # 1000 points in 5 dimensions
In [71]:
# 1. Compute the mean of the 1000 points in X
In [72]:
# 2. Compute the mean using np.add
In [73]:
# 3. Compute the standard deviation across the 1000 points
In [74]:
# 4. Compute the standard deviation using np.add only
In [75]:
# 5. Compute the whitened version of the array

Strategy 4: Fancy Indexing and Masking

The last strategy we will cover is fancy indexing and masking.

For example, imagine you have an array of data where negative values indicate some kind of error.

In [76]:
x = np.array([1, 2, 3, -999, 2, 4, -999])

How might you clean this array, setting all negative values to, say, zero?

In [77]:
for i in range(len(x)):
    if x[i] < 0:
        x[i] = 0
print x
[1 2 3 0 2 4 0]

A faster way is to construct a boolean mask:

In [78]:
x = np.array([1, 2, 3, -999, 2, 4, -999])

mask = (x < 0)
print mask
[False False False  True False False  True]

And the mask can be used directly to set the value you desire:

In [79]:
x[mask] = 0
print x
[1 2 3 0 2 4 0]

Typically this is done directly:

In [80]:
x = np.array([1, 2, 3, -999, 2, 4, -999])
x[x < 0] = 0
print x
[1 2 3 0 2 4 0]

Useful masking functions

In [81]:
x = np.random.random(5)
print x
[ 0.97053187  0.72813759  0.95240005  0.04301341  0.77007516]

In [82]:
x[x > 0.5] = np.nan
print x
[        nan         nan         nan  0.04301341         nan]

In [83]:
x[np.isnan(x)] = np.inf
print x
[        inf         inf         inf  0.04301341         inf]

In [84]:
np.nan == np.nan
Out[84]:
False
In [85]:
x[np.isinf(x)] = 0
print x
[ 0.          0.          0.          0.04301341  0.        ]

In [86]:
x = np.array([1, 0, -np.inf, np.inf, np.nan])
print "input   ", x
print "x < 0   ", (x < 0)
print "x > 0   ", (x > 0)
print "isinf   ", np.isinf(x)
print "isnan   ", np.isnan(x)
print "isposinf", np.isposinf(x)
print "isneginf", np.isneginf(x)
input    [  1.   0. -inf  inf  nan]
x < 0    [False False  True False False]
x > 0    [ True False False  True False]
isinf    [False False  True  True False]
isnan    [False False False False  True]
isposinf [False False False  True False]
isneginf [False False  True False False]

Boolean Operations on Masks

Use bitwise operators (and make sure to use parentheses!)

In [87]:
x = np.arange(16).reshape((4, 4))
print x
[[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]
 [12 13 14 15]]

In [88]:
print (x < 5)
[[ True  True  True  True]
 [ True False False False]
 [False False False False]
 [False False False False]]

In [89]:
print ~(x < 5)
[[False False False False]
 [False  True  True  True]
 [ True  True  True  True]
 [ True  True  True  True]]

In [90]:
print (x < 10) & (x % 2 == 0)
[[ True False  True False]
 [ True False  True False]
 [ True False False False]
 [False False False False]]

In [91]:
print (x > 3) & (x < 8)
[[False False False False]
 [ True  True  True  True]
 [False False False False]
 [False False False False]]

Counting elements with a mask

Sum over a mask to find the number of True elements:

In [92]:
x = np.random.random(100)
print "array is length", len(x), "and has"
print (x > 0.5).sum(), "elements are greater than 0.5"
array is length 100 and has
50 elements are greater than 0.5

In [93]:
# clip is a useful function:
x = np.clip(x, 0.3, 0.6)

print np.sum(x < 0.3)
print np.sum(x > 0.6)
0
0

In [94]:
# works for 2D arrays as well
X = np.random.random((10, 10))
print (X < 0.1).sum()
9

where function: Turning a mask into indices

In [95]:
x = np.random.random((3, 3))
print x
[[ 0.82518495  0.49805524  0.00840461]
 [ 0.85186368  0.98583616  0.77922605]
 [ 0.52795826  0.20786051  0.49627606]]

In [96]:
print np.where(x < 0.3)
(array([0, 2]), array([2, 1]))

In [97]:
print x[x < 0.3]
[ 0.00840461  0.20786051]

In [98]:
print x[np.where(x < 0.3)]
[ 0.00840461  0.20786051]

When you index with the result of a where function, you are using what is called fancy indexing: indexing with tuples

Fancy Indexing (indexing with sequences)

In [99]:
X = np.arange(16).reshape((4, 4))
print X
[[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]
 [12 13 14 15]]

In [100]:
X[(0, 1), (1, 0)]
Out[100]:
array([1, 4])
In [101]:
X[range(4), range(4)]
Out[101]:
array([ 0,  5, 10, 15])
In [102]:
X.diagonal()
Out[102]:
array([ 0,  5, 10, 15])
In [103]:
X.diagonal() = 100
  File "<ipython-input-103-3064a6b8dbd8>", line 1
    X.diagonal() = 100
SyntaxError: can't assign to function call
In [104]:
X[range(4), range(4)] = 100
In [105]:
print X
[[100   1   2   3]
 [  4 100   6   7]
 [  8   9 100  11]
 [ 12  13  14 100]]

Randomizing the rows

In [106]:
X = np.arange(24).reshape((6, 4))
print X
[[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]
 [12 13 14 15]
 [16 17 18 19]
 [20 21 22 23]]

In [107]:
i = np.arange(6)
np.random.shuffle(i)
print i
[0 5 3 1 4 2]

In [108]:
print X[i]  # X[i, :] is identical
[[ 0  1  2  3]
 [20 21 22 23]
 [12 13 14 15]
 [ 4  5  6  7]
 [16 17 18 19]
 [ 8  9 10 11]]

Fancy indexing also works for multi-dimensional index arrays

In [109]:
i2 = i.reshape(3, 2)
print X[i2]
[[[ 0  1  2  3]
  [20 21 22 23]]

 [[12 13 14 15]
  [ 4  5  6  7]]

 [[16 17 18 19]
  [ 8  9 10 11]]]

In [110]:
print X[i2].shape
(3, 2, 4)

Summary: Speeding up NumPy

It's all about moving loops into compiled code:

  1. Use Numpy ufuncs to your advantage (eliminate loops!)

  2. Use Numpy aggregates to your advantage (eliminate loops!)

  3. Use Numpy broadcasting to your advantage (eliminate loops!)

  4. Use Numpy slicing and masking to your advantage (eliminate loops!)

  5. Use a tool like SWIG, cython or f2py to interface to compiled code.

Homework: asteroid data

In the github repository, there is a file containing measurements of 5000 asteroid orbits, at notebooks/data/asteroids5000.csv.

These are compiled from a query at http://ssd.jpl.nasa.gov/sbdb_query.cgi

Part 1: loading and exploring the data

  1. Use np.genfromtxt to load the data from the file. This is like loadtxt, but can handle missing data.

    • Remember to set the appropriate delimiter keyword.
  2. genfromtxt sets all missing values to np.nan. Use the operations we discussed here to answer these questions:

    • How many values are missing in this data?
    • How many complete rows are there? i.e. how many objects have no missing values?
  3. Create a new array containing only the rows with no missing values.

  4. Compute the maximum, minimum, mean, and standard deviation of the values in each column.

In []:

Part 2: Plotting the data

  1. Use the bash head command to display the first line of the data file: this lists the names of the columns in the dataset. (remember that bash commands in the notebook are indicated by !, and that head -n displays the first n lines of a file)

  2. Invoke the matplotlib inline magic to make figures appear inline in the notebook

  3. Use plt.scatter to plot the semi-major axis versus the sine of the inclination angle (note that the inclination angle is listed in degrees -- you'll have to convert it to radians to compute the sine). What do you notice about the distribution? What do you think could explain this?

  4. Use plt.scatter to plot a color-magnitude diagram of the asteroids (H vs B-V). You should see two distinct "families" of asteroids indicated in this plot. Over-plot a line that divides these.

  5. Repeat the orbital parameter plot from above, but plot the two "families" from #4 in different colors. Note that this magnitude is undefined for many of the asteroids. Do you see any correlation between color and orbit?

  6. Compare what you found to plots in Parker et al. 2008. Note that we're not using the same data here, but we're looking at a similar collection of objects.

In []: