# Choosing dimensionality reduction Technique

I am working with daily sales of products in a supermarket. The purpose is that of accurate forecasting. For each product sales I have many predictors, which include price,stockout,promotions of the product itself and of the other products related to this.

I am not interested in fitting a different model for each product, since in such a case I would include all regressors for each regression curve and then apply ridge regression or some sort of variable selection method.

Instead I would like to have a smaller representation of my regressors: I expect that there is an intelligent being behind the price shifts policy and inventory level and promotion choices.

Stockouts and Promotions are $0/1$ flags, while prices are real. I tend to exclude Principal Component Analysis as a technique (since I have binary variables) and I would instead use Independent Component Analysis.

Which technique do you think suits best my problem?

EDIT:

Let me be clearer, this is the kind of data I would like to reduce dimensionally:

priceP1  priceP2   priceP3   priceP4   promP1   promP2   promP3   promP4
0        0          0         0         1        0        1        0
-1       -2         0         0         1        0        0        1
0        0         +1         0         1        0        1        1


Where -1 denotes a minor downshift in price (less than 15%), -2 denotes a big downshift in price (over 15%), and as regards promotions I have 0/1 flags.

Right now I'm looking at products correlated (for example all product Coca-Cola in pet bottles), since I expect that there are latent factors behind them ore than considering random products (like water shifts in prices and red wine ones).

I tested in Python both PCA and Factor Analysis over 13 products, considering first just shifts in prices:

from sklearn.decomposition import PCA
from sklearn.decomposition import FactorAnalysis
factor = FactorAnalysis(n_components=8, , random_state=101).fit(X)
pca = PCA().fit(X)


PCA results are:

print pca.explained_variance_ratio_
[  2.76790694e-01   2.37085044e-01   1.43677025e-01   1.30717864e-01
6.70670210e-02   5.33406116e-02   4.09484124e-02   1.76802848e-02
1.52392222e-02   1.23767594e-02   5.07706129e-03   4.85649858e-34
1.87735378e-35]


I would say that 6 of my components explain almost 80% of the variance. Now let me output the score of 5 products over each component found:

    COCA COLA PET 500X4    ML 2000  COCA COLA S/CAFF 1750  ML 1750  \
0                     3.745765e-02                    2.716548e-01
1                    -1.435577e-01                   -1.940115e-01
2                    -7.634235e-03                   -6.941557e-01
3                    -6.643485e-01                    3.019650e-03
4                    -9.205165e-03                   -1.247160e-01
5                    -4.726620e-02                   -5.909807e-01
6                     5.509576e-03                   -1.119261e-01
7                     4.843525e-01                   -1.656424e-01
8                     2.050570e-01                   -2.948274e-02
9                     5.075194e-01                    3.249049e-02
10                   -0.000000e+00                   -0.000000e+00
11                   -1.158277e-16                    3.832722e-17
12                   -2.253293e-16                    3.507935e-17
COCA S/CAF.PET 500X4   ML 2000  COCACOLA S/CAFF.500    ML  500  \
0                    -2.132371e-02                   -6.776264e-21
1                    -1.031520e-01                    0.000000e+00
2                    -2.274758e-03                   -0.000000e+00
3                    -7.223773e-01                    0.000000e+00
4                     7.432427e-02                    5.551115e-17
5                     1.130822e-01                   -2.775558e-16
6                     3.091060e-02                    5.464379e-17
7                    -4.619952e-01                    2.220446e-16
8                    -1.486847e-01                    1.166059e-16
9                    -4.607135e-01                    0.000000e+00
10                   -0.000000e+00                   -0.000000e+00
11                    2.139575e-16                    5.861627e-02
12                    1.699868e-16                    9.982806e-01

COCA SENZA CAFF.LT 1,5 ML 1500  \
0                             -0.0
1                              0.0
2                             -0.0
3                              0.0
4                              0.0
5                             -0.0
6                              0.0
7                              0.0
8                              0.0
9                              0.0
10                            -1.0
11                            -0.0
12                            -0.0


As you can see there are some components which basically describe just one product, for example component 10 for product 5 and component 12 for product 4.

Thank you!

• If you truly believe price, stockout, and promotions are indicators of some underlying factor, and you expect that as that factor increases, the indicators change in a systematic (not necessarily deterministic) way, then you might look into factor analysis. – Noah Mar 13 '17 at 1:51