# Bitcoin Analysis and Forecasting through Fuzzy Transform

^{1}

^{2}

^{*}

## Abstract

**:**

_{p}-norm F-transform is a powerful and flexible methodology for data analysis, non-parametric smoothing and for fitting and forecasting. Its capabilities are illustrated by empirical analyses concerning Bitcoin prices and Google Trend scores (six years of daily data): we apply the (inverse) F-transform to both time series and, using clustering techniques, we identify stylized facts for Bitcoin prices, based on (local) smoothing and fitting F-transform, and we study their time evolution in terms of a transition matrix. Finally, we examine the dependence of Bitcoin prices on Google Trend scores and we estimate short-term forecasting models; the Diebold–Mariano (DM) test statistics, applied for their significance, shows that sentiment analysis is useful in short-term forecasting of Bitcoin cryptocurrency.

## 1. Introduction

## 2. Fuzzy-Transform Smoothing

_{2}-norm) operator while the second one is based on the L

_{1}-type direct F-transform and it is obtained by minimizing an L

_{1}-norm operator.

#### 2.1. L_{2}-Norm F-Transform in Expectile Smoothing

**Definition**

**1.**

**Proposition**

**1.**

**Definition**

**2.**

#### 2.2. L_{1}-Norm F-Transform in Quantile Smoothing

_{1}-norm direct and inverse F-transform are defined as follows.

**Definition**

**3.**

**Definition**

**4.**

#### 2.3. General L_{p}-Norm-Based Discrete F-Transform

**Proposition**

**2.**

**Definition**

**5.**

## 3. Analysis of Bitcoin Prices and Google Trends by F-Transform

**Clustering A**. Clusters are based on variable $BitCoin$: the number of clusters is $nCl=20$. The ${L}_{2}$-norm F-transform reconstruction (of order 1) of $BitCoin$ in terms of $GT100$ with pre-clustering A has much higher Kendall rank correlation $\tau =0.9457$ and Spearman correlation $\rho =0.9956$.

**Clustering B**. Clusters are based on both variables $(BitCoin,GT100)$: the number of clusters is $nCl=21$. The ${L}_{2}$-norm F-transform reconstruction (of order 1) of $BitCoin$ in terms of $GT100$ with pre-clustering B has high Kendall correlation $\tau =0.9447$ and Spearman correlation $\rho =0.9954$, similar to clustering A.

**Clustering C**. Clusters are based on variables $(BitCoin,GT100,\Delta BitCoin,\Delta GT100)$; the number of clusters is $nCl=24$. The ${L}_{2}$-norm F-transform reconstruction (of order 1) of $BitCoin$ in terms of $GT100$ with this pre-clustering has high Kendall correlation $\tau =0.9183$ and Spearman correlation $\rho =0.9905$, similar but not better than pre-clustering A and B.

## 4. Stylized Facts of Bitcoin Prices Identified by F-Transform Components

#### 4.1. F-Transform Fitting with Dense r-Partition

P = | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

1 | 0.57 | 0.29 | 0.14 | 0 | 0 | 0 | 0 | 0 | 0 | |

2 | 0.08 | 0.44 | 0.44 | 0 | 0.04 | 0 | 0 | 0 | 0 | |

3 | 0.02 | 0.12 | 0.39 | 0.25 | 0.12 | 0.03 | 0.05 | 0.02 | 0 | |

4 | 0 | 0.02 | 0.10 | 0.50 | 0.31 | 0.04 | 0.01 | 0.01 | 0 | |

5 | 0 | 0 | 0 | 0.03 | 0.93 | 0.03 | 0 | 0 | 0 | |

6 | 0 | 0 | 0.1 | 0.04 | 0.32 | 0.53 | 0.10 | 0.01 | 0 | |

7 | 0 | 0 | 0 | 0.03 | 0.04 | 0.27 | 0.56 | 0.10 | 0 | |

8 | 0 | 0.05 | 0.09 | 0.05 | 0 | 0.09 | 0.18 | 0.50 | 0.05 | |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.50 | 0.50 |

#### 4.2. F-Transform Fitting with Sparse r-Partition

## 5. Forecasting Bitcoin Prices with Gt100 Index

_{p}-norm-based criterion: we assume $p=1.5$ as a good intermediate value between quantile ($p=1$) and expectile ($p=2$) estimators.

## 6. Final Comments and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Daily Bitcoin prices (blue color, left scale) and daily Google Trends series (red color, right scale) from 28 April 2013 to 17 June 2019.

**Figure 2.**Scatter-plots of $({f}_{t},{f}_{t}^{r})$ for daily Bitcoin prices (

**left**picture) and daily GT100 trends (

**right**), from April 2013 to June 2019.

**Figure 3.**${L}_{p}$-based smoothing for Bitcoin (

**top**picture, blue points) and GT100 (

**bottom**, blue points) series, obtained with $p=1.5$ and $q=0$. The green curves plot the observed series.

**Figure 4.**Scatterplot $(GT{100}_{t},BitCoi{n}_{t})$ of daily Bitcoin prices vs daily GT100 series. For this visualization, the two time series are normalized to the common range $[0,1000]$.

**Figure 5.**The figure shows (inverse) ${L}_{1}$-norm (on

**top**) and ${L}_{2}$-norm (on

**bottom**) iF-transform functions obtained for the observations $(GT{100}_{t},BitCoi{n}_{t})$; here, the daily Bitcoin prices are considered to be functions in the domain $[0,100]$ of GT100.

**Figure 6.**Fuzzy-valued Espectile F-transform function of the data-set $(BitCoi{n}_{t},GT{100}_{t})$; the five computed $\alpha $-cuts correspond to $\alpha \in \{0.01,0.25,0.5,0.75,1\}$.

**Figure 7.**The fitted $BitCoin$ time series (blue color) is obtained from ${L}_{2}$-norm iF-transform function applied to the data-set $(GT{100}_{t},BitCoi{n}_{t})$; the observed $BitCoi{n}_{t}$ values are green and the observed $GT{100}_{t}$ values are red.

**Figure 8.**Clustering A: the fitted $BitCoin$ time series (blue colour) is obtained from ${L}_{2}$-norm iF-transform function applied to the data-set $(GT{100}_{t},BitCoi{n}_{t})$; the observed $BitCoi{n}_{t}$ values are green and the observed $GT{100}_{t}$ values are red.

**Figure 9.**Clustering A: for each cluster, the fitted $BitCoin$ subseries (blue colour) are obtained from ${L}_{2}$-norm iF-transform function applied to the subset of data assigned to each cluster; the observed $(GT{100}_{t},BitCoi{n}_{t})$ values are green.

**Figure 10.**Clustering B: the fitted $BitCoin$ time series (blue colour) is obtained from ${L}_{2}$-norm iF-transform function applied to the data-set $(GT{100}_{t},BitCoi{n}_{t})$; the observed $BitCoi{n}_{t}$ values are green and the observed $GT{100}_{t}$ values are red.

**Figure 11.**Clustering B: the fitted $BitCoin$ time series is in blue colour; the observed $BitCoi{n}_{t}$ values are green.

**Figure 12.**Clustering C: the fitted $Bitcoin$ time series (blue color) is obtained from ${L}_{2}$-norm iF-transform function applied to the data-set $(GT{100}_{t},Bitcoi{n}_{t})$; the observed $Bitcoi{n}_{t}$ values are green and the observed $GT{100}_{t}$ values are red.

**Figure 13.**The blue points correspond to the observations $(GT{100}_{t},BitCoi{n}_{t})$ assigned to each cluster by clustering C. The red points give the centroid values relative to two variables $GT100$ and $BitCoin$.

**Figure 14.**The fitted $BitCoin$ time series (blue color) is obtained from ${L}_{2}$-norm iF-transform function applied to the data-set $(GT{100}_{t},BitCoi{n}_{t})$; the observed $BitCoi{n}_{t}$ values are green and the observed $GT{100}_{t}$ values are red.

**Figure 15.**${L}_{2}$-norm F-transform components of order 1 for Bitcoin series. The fuzzy r-partition is ${P}_{a}$ with $r=4$ (

**top**picture) and $r=7$ (

**bottom**).

**Figure 16.**${L}_{2}$-norm F-transform components of order 2 for Bitcoin series. The fuzzy r-partition is ${P}_{a}$ with $r=4$ (

**top**picture) and $r=7$ (

**bottom**).

**Figure 17.**Clustering of ${L}_{2}$-norm F-transform components (polynomials) of order 1 for Bitcoin series. The clusters correspond to the fuzzy r-partition ${P}_{a}$ with $r=4$ (

**bottom**picture) and $r=7$ (

**top**).

**Figure 18.**Clustering of ${L}_{2}$-norm F-transform components (polynomials) of order 2 for Bitcoin series. The clusters correspond to the fuzzy r-partition ${P}_{a}$ with $r=4$ (

**bottom**picture) and $r=7$ (

**top**).

**Figure 19.**${L}_{2}$-norm modified iF-transform values ${f}_{c,({\mathbb{P}}_{a},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ (

**top**: $q=1$,

**bottom**: $q=2$) for Bitcoin series. The fuzzy r-partition is ${P}_{a}$ with $r=4$.

**Figure 20.**${L}_{2}$-norm modified iF-transform values ${f}_{c,({\mathbb{P}}_{a},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ (

**top**: $q=1$,

**bottom**: $q=2$) for Bitcoin series. The fuzzy r-partition is ${P}_{a}$ with $r=7$.

**Figure 21.**Time evolution of clusters obtained with the ${L}_{2}$-norm F-transform smoothing of order 1 for Bitcoin series. The fuzzy r-partition is ${P}_{a}$ with $r=4$.

**Figure 22.**${L}_{2}$-norm F-transform components of order 1 for Bitcoin series. The fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 23.**${L}_{2}$-norm F-transform components of order 2 for Bitcoin series. The fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 24.**Clustering of ${L}_{2}$-norm F-transform components (polynomials) of order 1 for Bitcoin series. The fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 25.**Clustering of ${L}_{2}$-norm F-transform components (polynomials) of order 2 for Bitcoin series. The fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 26.**${L}_{2}$-norm modified iF-transform values ${f}_{c,({\mathbb{P}}_{b},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ for Bitcoin series. Here, $q=1$ and the fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 27.**${L}_{2}$-norm modified iF-transform values ${f}_{c,({\mathbb{P}}_{b},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ for Bitcoin series. Here, $q=2$ and the fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 28.**Pairwise scatter-plots of values ${f}_{j}$, ${f}_{({\mathbb{P}}_{b},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ and ${f}_{c,({\mathbb{P}}_{b},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ using ${L}_{2}$-norm F-transform smoothing of order $q=1$ for Bitcoin series. The fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 29.**Pairwise scatter-plots of values ${f}_{j}$, ${f}_{({\mathbb{P}}_{b},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ and ${f}_{c,({\mathbb{P}}_{b},\mathbb{A},r)}^{q}\left({t}_{j}\right)$ using ${L}_{2}$-norm F-transform smoothing of order $q=2$ for Bitcoin series. The fuzzy r-partition is ${P}_{b}$ with $r=3$.

**Figure 30.**Model A: ${L}_{p}$-norm F-transform forecast (denoted as fFor) for the last 1200 days of available data for Bitcoin series (observed time series is denoted as fSer). Here, $p=1.5$, $q=1$ and two lags $l=1$ (

**top**two pictures), $l=2$ (

**bottom**). The percent errors $100\frac{fSe{r}_{t}-fFo{r}_{t}}{fSe{r}_{t}}$ are also plotted.

**Figure 31.**Model B: ${L}_{p}$-norm F-transform forecast (denoted as fFor) for the last 1200 days of available data for Bitcoin series (observed time series is denoted as fSer). Here, $p=1.5$, $q=1$ and two lags $l=4$ (

**top**two pictures), $l=5$ (

**bottom**). The percent errors $100\frac{fSe{r}_{t}-fFo{r}_{t}}{fSe{r}_{t}}$ are also plotted.

**Table 1.**${L}_{p}$-norm F-transform of Bitcoin and GT100 time series For five values p and three orders $q\in \left\{0,1,3\right\}$ the table shows ratios $L(f,{f}^{\left(r\right)})$ with $f\in \left\{Bitcoin,GT100\right\}$.

p | q | Bitcoin | GT100 |
---|---|---|---|

1 | 0 | 0.3526 | 0.1646 |

1 | 1 | 0.4298 | 0.2643 |

1 | 3 | 0.5813 | 0.4922 |

1.25 | 0 | 0.3493 | 0.1628 |

1.25 | 1 | 0.4315 | 0.2677 |

1.25 | 3 | 0.5693 | 0.4813 |

1.5 | 0 | 0.3465 | 0.1649 |

1.5 | 1 | 0.4282 | 0.2702 |

1.5 | 3 | 0.5558 | 0.4785 |

1.75 | 0 | 0.3448 | 0.1777 |

1.75 | 1 | 0.4250 | 0.2783 |

1.75 | 3 | 0.5483 | 0.4798 |

2 | 0 | 0.3428 | 0.1715 |

2 | 1 | 0.4221 | 0.2859 |

2 | 3 | 0.5432 | 0.4815 |

**Table 2.**Approximation indices for ${L}_{p}$-norm F-transform of Bitcoin and GT100 For five values $p\in \left\{1,1.25,1.5,1.75,2\right\}$ and three orders $q\in \left\{0,1,3\right\}$ the table shows indices $MSE$, $\%MAE$, and Kendall $\tau $ correlation for the pairs $(f,{f}^{\left(r\right)})$ with $f\in \left\{Bitcoin,GT100\right\}$.

Bitcoin | GT100 | ||||||
---|---|---|---|---|---|---|---|

p | q | $MSE$ | $\%MAE$ | $\tau $ | $MSE$ | $\%MAE$ | $\tau $ |

1 | 0 | 363.98 | 2.5173 | 0.9766 | 2.7670 | 7.8219 | 0.8897 |

1 | 1 | 201.66 | 2.0152 | 0.9815 | 1.8610 | 6.7139 | 0.9049 |

1 | 3 | 148.75 | 1.4739 | 0.9860 | 1.5614 | 4.8293 | 0.9331 |

1.25 | 0 | 219.72 | 2.5171 | 0.9768 | 2.3939 | 7.9307 | 0.8911 |

1.25 | 1 | 191.58 | 2.0315 | 0.9817 | 1.8211 | 6.8342 | 0.9074 |

1.25 | 3 | 139.34 | 1.4874 | 0.9863 | 1.4556 | 4.9210 | 0.9345 |

1.5 | 0 | 217.61 | 2.5641 | 0.9765 | 2.3339 | 8.1374 | 0.8910 |

1.5 | 1 | 190.20 | 2.0805 | 0.9814 | 1.8027 | 7.0240 | 0.9068 |

1.5 | 3 | 137.72 | 1.5284 | 0.9861 | 1.4169 | 5.1172 | 0.9331 |

1.75 | 0 | 217.79 | 2.6232 | 0.9762 | 2.3020 | 8.3932 | 0.8896 |

1.75 | 1 | 191.26 | 2.1298 | 0.9810 | 1.7920 | 7.2505 | 0.9057 |

1.75 | 3 | 137.75 | 1.5734 | 0.9858 | 1.4031 | 5.3430 | 0.9311 |

2 | 0 | 219.41 | 2.6859 | 0.9757 | 2.2855 | 8.6858 | 0.8877 |

2 | 1 | 192.74 | 2.1778 | 0.9708 | 1.7964 | 7.4879 | 0.9038 |

2 | 3 | 138.46 | 1.6155 | 0.9856 | 1.4040 | 5.5762 | 0.9289 |

**Table 3.**Approximation indices for ${L}_{p}$-norm F-transform of Bitcoin and GT100 For five values $p\in \left\{1,1.25,1.5,1.75,2\right\}$ and three orders $q\in \left\{0,1,3\right\}$ the table shows indices $MSE$, $\%MAE$, and Kendall $\tau $ correlation for the pairs $(f,{f}^{\left(r\right)})$ with $f\in \left\{Bitcoin,GT100\right\}$.

Bitcoin | GT100 | ||||||
---|---|---|---|---|---|---|---|

p | q | $MSE$ | $\%MAE$ | $\tau $ | $MSE$ | $\%MAE$ | $\tau $ |

1 | 0 | 363.98 | 2.5173 | 0.9766 | 2.7670 | 7.8219 | 0.8897 |

1 | 1 | 201.66 | 2.0152 | 0.9815 | 1.8610 | 6.7139 | 0.9049 |

1 | 3 | 148.75 | 1.4739 | 0.9860 | 1.5614 | 4.8293 | 0.9331 |

1.25 | 0 | 219.72 | 2.5171 | 0.9768 | 2.3939 | 7.9307 | 0.8911 |

1.25 | 1 | 191.58 | 2.0315 | 0.9817 | 1.8211 | 6.8342 | 0.9074 |

1.25 | 3 | 139.34 | 1.4874 | 0.9863 | 1.4556 | 4.9210 | 0.9345 |

1.5 | 0 | 217.61 | 2.5641 | 0.9765 | 2.3339 | 8.1374 | 0.8910 |

1.5 | 1 | 190.20 | 2.0805 | 0.9814 | 1.8027 | 7.0240 | 0.9068 |

1.5 | 3 | 137.72 | 1.5284 | 0.9861 | 1.4169 | 5.1172 | 0.9331 |

1.75 | 0 | 217.79 | 2.6232 | 0.9762 | 2.3020 | 8.3932 | 0.8896 |

1.75 | 1 | 191.26 | 2.1298 | 0.9810 | 1.7920 | 7.2505 | 0.9057 |

1.75 | 3 | 137.75 | 1.5734 | 0.9858 | 1.4031 | 5.3430 | 0.9311 |

2 | 0 | 219.41 | 2.6859 | 0.9757 | 2.2855 | 8.6858 | 0.8877 |

2 | 1 | 192.74 | 2.1778 | 0.9708 | 1.7964 | 7.4879 | 0.9038 |

2 | 3 | 138.46 | 1.6155 | 0.9856 | 1.4040 | 5.5762 | 0.9289 |

**Table 4.**Model A: ${L}_{p}$-norm Fitting and Forecasting results For $p=1.5$, the table shows indices $\%MAE$ and Kendall $\tau $ correlation for the fitting (columns 4,5 with label $fit$) and the forecasting (columns 6,7 with label $for$) corresponding to five lags $l\in \left\{1,2,3,4,5\right\}$ and three pairs $(q,r)$ of order q and bandwidth r.

q | r | l | $\%\mathit{MAE}$-fit | $\mathit{\tau}$-fit | $\%\mathit{MAE}$-for | $\mathit{\tau}$-for |
---|---|---|---|---|---|---|

0 | 1 | 1 | 1.65 | 0.98 | 3.30 | 0.96 |

2 | 1.67 | 0.98 | 4.41 | 0.95 | ||

3 | 1.67 | 0.98 | 5.35 | 0.94 | ||

4 | 1.58 | 0.98 | 6.17 | 0.93 | ||

5 | 1.58 | 0.98 | 6.88 | 0.92 | ||

1 | 1 | 1 | 0.62 | 0.99 | 3.15 | 0.96 |

2 | 0.59 | 0.99 | 5.08 | 0.94 | ||

3 | 0.60 | 0.99 | 6.59 | 0.93 | ||

4 | 0.60 | 0.99 | 8.09 | 0.91 | ||

5 | 0.60 | 0.99 | 10.13 | 0.89 | ||

0 | 2 | 1 | 2.24 | 0.97 | 3.70 | 0.95 |

2 | 2.28 | 0.97 | 4.77 | 0.94 | ||

3 | 2.31 | 0.97 | 5.65 | 0.93 | ||

4 | 2.22 | 0.97 | 6.43 | 0.92 | ||

5 | 2.21 | 0.97 | 7.11 | 0.92 |

**Table 5.**Model B: ${L}_{p}$-norm Fitting and Forecasting results For $p=1.5$, the table shows indices $\%MAE$ and Kendall $\tau $ correlation for the fitting (columns 4,5 with label $fit$) and the forecasting (columns 6,7 with label $for$) corresponding to five lags $l\in \left\{1,2,3,4,5\right\}$ and three pairs of order q and bandwidth r.

q | r | l | $\%\mathit{MAE}$-fit | $\mathit{\tau}$-fit | $\%\mathit{MAE}$-for | $\mathit{\tau}$-for |
---|---|---|---|---|---|---|

0 | 1 | 1 | 1.56 | 0.98 | 3.21 | 0.96 |

2 | 1.57 | 0.98 | 4.33 | 0.95 | ||

3 | 1.56 | 0.98 | 5.31 | 0.93 | ||

4 | 1.57 | 0.98 | 6.12 | 0.93 | ||

5 | 1.57 | 0.98 | 6.83 | 0.92 | ||

1 | 1 | 1 | 1.17 | 0.98 | 3.16 | 0.96 |

2 | 1.20 | 0.98 | 4.54 | 0.94 | ||

3 | 1.17 | 0.98 | 5.84 | 0.93 | ||

4 | 1.16 | 0.98 | 6.98 | 0.92 | ||

5 | 1.18 | 0.98 | 7.89 | 0.91 | ||

0 | 2 | 1 | 1.56 | 0.98 | 3.21 | 0.96 |

2 | 1.57 | 0.98 | 4.34 | 0.95 | ||

3 | 1.56 | 0.98 | 5.31 | 0.93 | ||

4 | 1.57 | 0.98 | 6.12 | 0.93 | ||

5 | 1.57 | 0.98 | 6.83 | 0.92 |

**Table 6.**Diebold–Mariano test statistics for ${L}_{p}$-norm forecasting results The table shows indexes $sMSE=\sqrt{MSE}$ and Kendall rank correlation $\tau $, corresponding to lag $l=1$ forecasts for Models A and B, obtained with $p=1.5$, $q=1$ and bandwidth $r=1$.

$\mathit{sMSE}$ | DM-stat | p-Value | $\mathit{\tau}$ | |
---|---|---|---|---|

Model A | 488.2 | −2.9468 | 0.0016 | 0.9265 |

Model B | 487.0 | −3.8301 | 0.000064 | 0.9186 |

Random Walk | 585.1 | - | - | 0.9078 |

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**MDPI and ACS Style**

Guerra, M.L.; Sorini, L.; Stefanini, L.
Bitcoin Analysis and Forecasting through Fuzzy Transform. *Axioms* **2020**, *9*, 139.
https://doi.org/10.3390/axioms9040139

**AMA Style**

Guerra ML, Sorini L, Stefanini L.
Bitcoin Analysis and Forecasting through Fuzzy Transform. *Axioms*. 2020; 9(4):139.
https://doi.org/10.3390/axioms9040139

**Chicago/Turabian Style**

Guerra, Maria Letizia, Laerte Sorini, and Luciano Stefanini.
2020. "Bitcoin Analysis and Forecasting through Fuzzy Transform" *Axioms* 9, no. 4: 139.
https://doi.org/10.3390/axioms9040139