What's the meaning of rolling_slope, rolling_rsquare, and rolling_resi?

[Bowen905]

In the document, slope means regression slope, but in rolling_slope function, there is only one parameter without window size, i just confuse about the regression for what, the parameter autoregression? And, the same question for rolling_rsquare and rolling_resi.

[bxdd]

Hi, What you said "there is only one parameter without window size" where can I find the related code?
And, the regression is an linear regression for the (series.index, series.value) in window, not autoregression. Please refer to code, obviously, the following code is the slope of linear regression:

return (N*self.xy_sum - self.x_sum*self.y_sum) / (N*self.x2_sum - self.x_sum*self.x_sum)

[Bowen905]

The slope function code is:
`cdef class Slope(Rolling):
"""1-D array rolling slope"""
cdef double i_sum # can be used as i2_sum
cdef double x_sum
cdef double x2_sum
cdef double y_sum
cdef double xy_sum
def init(self, int window):
super(Slope, self).init(window)
self.i_sum = 0
self.x_sum = 0
self.x2_sum = 0
self.y_sum = 0
self.xy_sum = 0

cdef double update(self, double val):
    self.barv.push_back(val)
    self.xy_sum = self.xy_sum - self.y_sum
    self.x2_sum = self.x2_sum + self.i_sum - 2*self.x_sum
    self.x_sum = self.x_sum - self.i_sum
    cdef double _val
    _val = self.barv.front()
    if not isnan(_val):
        self.i_sum -= 1
        self.y_sum -= _val
    else:
        self.na_count -= 1
    self.barv.pop_front()
    if isnan(val):
        self.na_count += 1
        # return NAN
    else:
        self.i_sum  += 1
        self.x_sum  += self.window
        self.x2_sum += self.window * self.window
        self.y_sum  += val
        self.xy_sum += self.window * val
    cdef int N = self.window - self.na_count
    return (N*self.xy_sum - self.x_sum*self.y_sum) / \
        (N*self.x2_sum - self.x_sum*self.x_sum)`

in this slope function, the x means window size, and y means update value, right? My question is, if the inputs are update value and window size, what's the meaning for this regression?

[bxdd]

x means series.index rather than window size, y means series.value. window size is passed in Slope.__init__.
By the way, Slope is not the rolling_slope function, but this https://github.com/microsoft/qlib/blob/main/qlib/data/_libs/rolling.pyx#L197

[Bowen905]

Thank you for answering my question, but I'm still confusing about the input x, and y. So, could you give me an example for Slope function? What does the input data set look like?

[bxdd]

The return value of rolling_slope(np.array([1,2,7,10]), 2) is np.array([nan, 1, 5, 3])

[Bowen905]

For the example, x, the series.index, is range(4), and y, the series.value, is array([1,2,7,10]), is this right? Is there any economic meaning for this regression?

[bxdd]

Hi, what you said is almost right (in fact, there are still some small details that you must read the code to know).

These operators are just basic operations

• rolling_slope: the linear regression slope in the given window
• rolling_rsquare: the R-square R^2 of linear regression in the given window
• rolling_resi: the last linear regression residual between value_{last} and slope*index_{last}+interp in the given window

And they may be used when calculating some alpha factors, such as Idiosyncratic volatility

[Bowen905]

Thank you. I get the basic knowledge for those operators, but I'm still not clear about how to use those operators. For example, the alpha factor 'Beta5', which is 'Slope($close, 5)/$close'. In my knowledge, the beta means the correlation between the returns of benchmark index and the returns of a stock. But, for the explanation you give me, i can't find the relationship.

[bxdd]

Wow, I’m not sure about the meaning of Beta5, but I guess that Beta5 means the coefficient standardized by close price in linear regression .
Is it right? @you-n-g

[you-n-g]

@bxdd it is right.
@Bowen905 It is different from the Beta you explained.

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