## 3 Proposed Novel Matrix Profile for Similarity Search
### Inputting the Time Series (Its)
#### 3.1.2 Data Representation  
Time-series data representation emphasizes the following essential characteristics:  
* Temporal representation: storing a temporal point on the displayed data. * Spectral representation: designing the data in the frequency domain. * Other representations: implementing different modifications not related to the above. This approach is described in a concise way and gains additional advantages, such as efficient storage, speedup of processing, and implicit noise removal. The essential characteristics of data representation require the following for any representation:  
* A significant drop in the data dimensionality. * An emphasis on the essential shape features for both global and local scales. * Lower computational costs for the computational representation. * Better restoration quality for the reduced representation. * Implicit noise handling or insensitivity to noise. By studying several properties of a sequence at the same time, issues such as amplitude, scaling, temporal warping, noise, and outliers can be prevented. The following transformations are considered for a given time series \(T\) with \(n\) data points \(T=\left\{t_{1},...,t_{n}\right\}\): _Amplitude shifting_: The series \(G=\left\{g_{1},...,g_{n}\right\}\) obtained by a linear amplitude shift of the original series is given by:  
\[g_{i}=t_{i}+k \tag{2}\]where \(g_{i}\) is the series of \(T\); \(t_{i}\) is a time series such that \(t_{i}\in T\); and \(k\) is a constant, where \(k\in\mathbb{R}\). _Uniform amplification_: The series \(G\) obtained by multiplying the amplitude of the original series is specified as:  
\[g_{i}=k\times t_{i} \tag{3}\]  
_Uniform time scaling_: The series \(G=\big{\{}g_{1},...,g_{m}\big{\}}\) produced by a uniform change in the time scale of the original series is given as follows:  
\[g_{i}=t_{\left|k\times i\right|} \tag{4}\]  
_Dynamic amplification_: The series \(G\) obtained by multiplying the original series by a dynamic amplification function that is assigned is:  
\[g_{i}=h(i)\times t_{i} \tag{5}\]  
where \(h(i)\) is a function such that \(\forall t\in[1,2,...,e],h^{\prime}(t)=0\) if and only if \(t^{\prime}_{i}=0\). _Dynamic time scaling_: The series \(G\) obtained by a dynamic change in the time scale is generated by:  
\[g_{i}=t_{h(i)} \tag{6}\]  
where \(h(i)\) is a positive, strictly increasing function such that \(h\) is:  
\[\mathbb{N}\longrightarrow[1,...,e] \tag{7}\]  
where \(e\) is the length of \(\mathbb{N}\). _Additive Noise_: The series \(G\) obtained by adding a noisy component to the original series is given by:  
\[g_{i}=t_{i}+\varepsilon_{i} \tag{8}\]  
where \(t_{i}+\varepsilon_{i}\) and \(\varepsilon_{i}\) are independent identically distributed white noise. _Outliers_: The series \(G\) is obtained by adding outliers at random positions. Formally, these are a given set of random time positions expressed by:  
\[\rho=\Big{\{}k\in[1,...,e]\Big{\}} \tag{9}\]  
where \(\rho\) is a random time position and \(g_{k}=\varepsilon_{k}\), where \(\varepsilon_{k}\) is independent identically distributed white noise. Definition 2: The similarity measure \(D(T,U)\) between time series \(T\) and \(U\) is a function that takes two time series as inputs and returns the distance \(d\) between the series. This distance has to be nonnegative; i.e., \(D(T,U)\geq 0\). If this measure satisfies the _additional symmetry property_\(D(T,U)=D(U,T)\) and _subadditivity_\(D(T,V)\leq D(T,U)+D(U,V)\) (also known as the _triangle inequality_), the distance is said to be a _metric_. The similarity measure \(D(T,G)\) should be robust to any combination of these transformations. This property leads to our formalization of four general types of robustness. We introduce properties that express robustness for scaling (amplitude modifications), warping (temporal modifications), noise, and outliers. Let \(S\) be a collection of time series, and let \(H\) be the maximal group of homeomorphisms under which \(S\) is closed.