# Speed of PLS Algorithms

##### Nov 1, 2010

Previously I wrote about accuracy of PLS algorithms, and compared SIMPLS, NIPALS, BIDIAG2 and the new DSPLS. I now turn to speed of the algorithms. In the paragraphs that follow I’ll compare SIMPLS, NIPALS and DSPLS as implemented in PLS_Toolbox 6.0. It should be noted that the code I’ll test is our standard code (PLS_Toolbox functions simpls.m, dspls.m and nippls.m). These are not stripped down algorithms. They include all the error trapping (dimension checks, ranks checks, etc.) required to use these algorithms with real data. I didn’t include BIDIAG2 here because we don’t support it, and as such, I don’t have production code for it, just the research code (provided by Infometrix) I used to investigate BIDIAG2 accuracy. The SIMPLS and DSPLS code used here includes the re-orthoginalization step investigated previously.

The tests were performed on my (4 year old!) MacBook Pro laptop, with a 2.16GHz Intel Core Duo, and 2GB RAM running MATLAB 2009b (32 bit). The first figure, below, shows straight computation time as a function of number of samples for SIMPLS to calculate a 20 Latent Variable (LV) model, for data with 10 to 1,000,000 variables (legend gives number of variables for each line). The maximum size of **X** for each run was 30 million elements, so the lines all terminate at this point. Times range from a minimum of ~0.003 seconds to just over 10 seconds.

It is interesting to note that, for the largest **X** matrices the times vary from 4 to 12 seconds, with the faster times being for the more square **X**‘s. It is fairly impressive, (at least to me), that problems of this size are feasible on an outdated laptop! A 10-way split cross-validation could be done in less than a minute for most of the large cases.

The second figure shows the ratio of the computation time of NIPALS to SIMPLS as a function of number of samples. Each line is a fixed number of variables (indicated by the legend). Maximum size of **X** here is 9 million elements (I just didn’t want to wait for NIPALS on the big cases). Note that SIMPLS is *always* faster than NIPALS. The difference is relatively small (around a factor of 2) for the tall skinny cases (many samples, few variables) but considerable for the short fat cases (few samples, many variables). For the case of 100 samples and 100,000 variables, SIMPLS is faster than NIPALS by more than a factor of 10.

The ratio of computation time for DSPLS to SIMPLS is shown in the third figure. These two methods are quite comparable, with the difference always less than a factor of 2. Thus I’ve chosen to display the results as a map. Note that each method has its sweet spot. SIMPLS is faster (red squares) for the many variable problems while DSPLS is faster (light blue squares) for the many sample problems. (Dark blue area represents models not computed for **X** larger than 30 million elements). Overall, SIMPLS retains a slight time advantage over DSPLS, but for the most part they are equivalent.

Given the results of these tests, and the previous work on accuracy, it is easy to see why SIMPLS remains our default algorithm, and why we found it useful to include DSPLS in our latest releases.

BMW