# summation syntax for defining conditions and solving

I am trying to generate datasets for which specific conditions related to the grand mean, group means, and differences among these values hold. I had tried in R, but was hoping that using Mathematica summation notation and Reduce[] would give more explicit outputs, versus my guess-and-check in R.

For example, for a polynomial of four additive terms: x^4 + 6*x^2*y^2 + 4*x*y^3 + y^4, (A+B+C+D), I want to generate a dataset with 10 groups of 100 individuals each, such that the third term predominates. I had done Reduce[C/A > 1 & C/B > 1 & C/D > 1] to get general conditions on x and y. However, as I mentioned, the terms are actually sums, and the third and fourth terms are, respectively, for example:

$$4 \sum_{i=1}^k \sum_{j=1}^{n_{i}} (\overline{y_{i \bullet}} - \overline{y_{\bullet \bullet}}) (y_{ij} - \overline{y_{i \bullet}})^3$$

and

$$\sum_{i=1}^k \sum_{j=1}^{n_{i}} (y_{ij} - \overline{y_{i \bullet}})^4$$

where $$y_{ij}$$ is individual $$i$$ in group $$j$$, $$\overline{y_{\bullet \bullet}}$$ is the grand mean, and $$\overline{y_{i \bullet}}$$ is the mean of a group. I'd like to make use of this summation notation to output a set of values that satisfy the condition that the third term predominates. For example, C/D > 1:

Reduce[$$\frac{4 \sum_{i=1}^k \sum_{j=1}^{n_{i}} (\overline{y_{i \bullet}} - \overline{y_{\bullet \bullet}}) (y_{ij} - \overline{y_{i \bullet}})^3}{\sum_{i=1}^k \sum_{j=1}^{n_{i}} (y_{ij} - \overline{y_{i \bullet}})^4} > 1$$]

but generating a list of $$i$$ values that satisfy the condition. Is there syntax for this kind of Reduce[], and data generation, in Mathematica? I am not familiar with how to transform the dot notation into something interpretable by Mathematica.

Thanks!

Only for the "dot" and summation notation: If you have Y = Array[y, {10, 100}]; then the list of group means $$(y_{1,\bullet},\dotsc,y_{k,\bullet})$$ can be obtained by

groupmeans = Mean /@ Y


And

mean = Mean[groupmeans]


as well as Mean[Mean[Y]] equals the overall mean.

$$4 \sum_{i=1}^k \sum_{j=1}^{n_{i}} (\overline{y_{i \bullet}} - \overline{y_{\bullet \bullet}}) (y_{ij} - \overline{y_{i \bullet}})^3$$

Can be expressed as

c = 4 (groupmeans - mean).Total[(Y - groupmeans^3), {2}]


and

$$\sum_{i=1}^k \sum_{j=1}^{n_{i}} (y_{ij} - \overline{y_{i \bullet}})^4$$ is obtained by

d = Total[(Y - Mean[Mean[Y]])^4, {2}]


But I don't know what you want to do with Reduce. With so many unknowns, you are bound to make yourself unhappy. FindInstance might be more helpful. A possible syntax would be

FindInstance[c > d, Flatten[Y]]


but I did not wait for an answer to be returned.

• I was using Reduce to find the general conditions on x and y for which C/D & C/B & C/A. The only constraints I have been thinking of for the $i$ would be that they are positive and real. Very interesting about the FindInstance syntax. I assume it could work then as FindInstance[c>d && c>a && c>b, Flatten[Y], Reals], assuming $a$ and $b$ are also formulated as you suggest, above. I will try this on my computer that has Mathematica. Thanks very much for these suggestions.
– J.D.
Oct 24 '18 at 16:56