Algorithm to adjust the impact of quantity

Question

I have the problem specified here. After some research I have found this approach to overcome the issue.

What about having a scaling factor to adjust the impact of quantity (total) of individual ratings as needed? Rough draft:

sort_score = (pos / total) + (W * log(total))

Here, W is the weighting (scaling) factor. Total = positive + negative

For example, in this case I have three ratings for the same thing. My implementation for a rating scale of 1 star to 4 stars is:

counter_positive=0
counter_negative=0
total_rating = 0.0

reviews = [(340, Decimal('4.0')), (340, Decimal('4.0')), (340, Decimal('4.0'))]

for s in reviews:
    if s[1] > 2.0:
        counter_positive+=1
    else:
        counter_negative+=1

    total_rating+=float(s[1])

counter_total = counter_positive + counter_negative

tup = (counter_positive / counter_total) + ((total_rating/len(reviews)) * log(counter_total))
print tup  #5.39444915467

Since the result is 5.39444915467 when the scale is <= 4 , certainly I am doing the given implementation wrong. I don't have a math background, so, please be patient.

DEMO

Answer 1

I came up with this possible solution. Compute your arithmetic average star rating $A$ of a given product, then multiply it by this scaling function,

$$1-\exp\left(-\frac{n^2}{h^2}\right),$$

where $\exp$ is the exponent function, e.g. Math.exp() in many languages. A product rating $R$ would be calculated as

$$R=(1-e^{-n^2/h^2})\times A$$

where $n$ is the number of reviews for that product, and $h$ is a parameter of your choosing, say $h=10$.

The scaling function looks like this:

So as you can see, the lower the number of reviews, the lower the scale factor to $A$. As soon as the number of reviews surpasses a certain number the scaling factor is very close to 1 forever, which will give you the straight arithmetic average.

For example, with $h=10$, for product $P_1$ suppose we have the following product reviews: $$P_1=\{4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4\}$$ with 16 reviews at 4 stars each, giving an arithmetic mean of $A=4$, then $$R=(1-\exp(16^2/10^2))\times 4=3.69078.$$ On the other hand a product $P_2$ with one 5-star review, $P_2=\{5\}$, has an arithmetic mean of $A=5$, and would have the rating $$R=(1-\exp(1^2/10^2))\times 5=0.0497508.$$

You can always round up or down to get a whole number answer if you like.

In code you can use:

Rating = (1.0 - Math.exp(n*n / (h*h))) * a

where n is number of product reviews, h is a fixed parameter specifying how sensitive the rating system is, and a is the arithmetic average of the product. Use Math.floor or Math.ceil to round down or up !