Python2.7の両対数スケールに最適な線

Data that falls along a straight line on a log-log scale follows a power relationship of the form y = c*x^(m). By taking the logarithm of both sides, you get the linear equation that you are fitting:

log(y) = m*log(x) + c

Calling np.polyfit(log(x), log(y), 1) provides the values of m and c. You can then use these values to calculate the fitted values of log_y_fit as:

log_y_fit = m*log(x) + c

and the fitted values that you want to plot against your original data are:

y_fit = exp(log_y_fit) = exp(m*log(x) + c)

So, the two problems you are having are that:

1. you are calculating the fitted values using the original x coordinates, not the log(x) coordinates

2. you are plotting the logarithm of the fitted y values without transforming them back to the original scale

I've addressed both of these in the code below by replacing plt.plot(z, np.poly1d(np.polyfit(logA, logB, 1))(z)) with:

m, c = np.polyfit(logA, logB, 1) # fit log(y) = m*log(x) + c
y_fit = np.exp(m*logA + c) # calculate the fitted values of y 
plt.plot(z, y_fit, ':')

This could be placed on one line as: plt.plot(z, np.exp(np.poly1d(np.polyfit(logA, logB, 1))(logA))), but I find that makes it harder to debug.

A few other things that are different in the code below:

• You are using a list comprehension when you calculate logA from z to filter out any values < 1, but z is a linear range and only the first value is < 1. It seems easier to just create z starting at 1 and this is how I've coded it.

• I'm not sure why you have the term x*log(x) in your list comprehension for logA. This looked like an error to me, so I didn't include it in the answer.

This code should work correctly for you:

fig=plt.figure()
ax = fig.add_subplot(111)

z=np.arange(1, len(x)+1) #start at 1, to avoid error from log(0)

logA = np.log(z) #no need for list comprehension since all z values >= 1
logB = np.log(y)

m, c = np.polyfit(logA, logB, 1) # fit log(y) = m*log(x) + c
y_fit = np.exp(m*logA + c) # calculate the fitted values of y 

plt.plot(z, y, color = 'r')
plt.plot(z, y_fit, ':')

ax.set_yscale('symlog')
ax.set_xscale('symlog')
#slope, intercept = np.polyfit(logA, logB, 1)
plt.xlabel("Pre_referer")
plt.ylabel("Popularity")
ax.set_title('Pre Referral URL Popularity distribution')
plt.show()

When I run it on simulated data, I get the following graph:

Notes:

• 行の左端と右端の「ねじれ」は、「log」と「symlog」の違いは何ですか？の回答で説明されているように、非常に小さい値を線形化する「symlog」を使用した結果です。。このデータが「両対数」軸にプロットされた場合、近似されたデータは直線になります。

• https://stackoverflow.com/a/3433503/7517724もお読みください。これは、重み付けを使用して対数変換されたデータに「より適切に」適合する方法を説明しています。