**4.2 Multidimensional scaling**

10 Will-be-set-by-IN-TECH

**Synergy 1 Synergy 2 Synergy 3**


> -0.1 0 0.1

> > 0 0.1 0.2


> 0 0.1 0.2


> 0 0.1 0.2


> 0 0.1 0.2

> 0 0.1


> 0 0.1 0.2








> 0 0.1

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> -0.1

**Samples Samples Samples**

**Synergy 4 Synergy 5 Synergy 6**

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> -0.1

**Samples Samples Samples**

Fig. 4. Six kinematic synergies obtained for subject 1 using PCA. Each synergy is about 0.45 s in duration (39 samples at 86 Hz). Abbreviations: T, thumb; I, index finger; M, middle finger; R, ring finger; P, pinky finger; MCP, metacarpophalangeal joint; IP, interphalangeal joint; PIP,


> -0.1 0 0.1

> -0.1 0 0.1

> -0.2 0 0.2

> -0.2 0 0.2

> -0.2 0 0.2

> -0.1 0 0.1

> -0.1 0 0.1

> -0.1 0 0.1

> > 0 0.2










> 0 0.2

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> -0.2

**T\_MCP**

**IͲMCP IͲPIP**

**MͲMCP MͲPIP RͲMCP RͲPIP**

**PͲMCP PͲPIP**

**T\_MCP**

**IͲMCP IͲPIP**

**MͲMCP MͲPIP RͲMCP RͲPIP**

**PͲMCP PͲPIP**

**TͲIP**

**TͲIP**

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> -0.2

0 0.01 0.02

> -0.1 0 0.1

**(radians/sample)**



0 0.05 0.1

> -0.2 0 0.2

0 0.05 0.1

> -0.2 0 0.2

0 0.05 0.1

**Angular**

**velocities of ten joints**

> 0 0.2


> -0.1 0 0.1

**(radians/sample)**








**Angular**

**velocities of ten joints**

> 0 0.2

proximal IP joint.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> -0.2

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> -0.2

Classical Multidimensional Scaling (MDS) can still be grouped under linear methods. This was introduced here to the reader to give a different perspective of dimensionality reduction in a slightly different analytical approach when compared to PCA discussed previously. The two methods PCA and MDS are unique as they perform dimensionality reduction in different ways. PCA operates on covariance matrix where as MDS operates on distance matrix. In MDS, a Euclidean distance matrix is calculated from the original matrix. This is nothing but a pairwise distance matrix between the variables in the input matrix. This method tries to preserve these pairwise distances in a low dimensional space, thus allowing for dimensionality reduction and preserving the inherent structure of the data simultaneously. PCA and MDS were compared using a simple example in MATLAB below.

```
load hald; %Load sample dataset in MATLAB
[pc,score,latent,tsquare] = princomp(ingredients); %Peform PCA
D = pdist(ingredients); %Calculate pairwise distances between ingredients
[Y, e] = cmdscale(D); %Perform Classical MDS
```
score in PCA represented the data that was projected in the PC space. Compare this to Y calculated in MDS. These are same. Similarly in place of sample dataset when the posture matrix *V* was used as input to MDS, it yielded the same synergies as PCA. This was introduced here because we build upon this method for the nonlinear methods coming up in the next section.
