Kiyasin Bayesian na Rashin Ƙayyadaddun Tsari na Autocovariance na Kuskure a cikin Jerin Lokaci tare da Canjin Ƙarfin Ƙarfi na Lokaci

1. Gabatarwa

Bambancin bambancin (Heteroskedasticity) wata muhimmin sifa ce ta yawancin jerin lokaci na tattalin arziki da na kuɗi, kamar yadda Engle (1982) ya kafa tare da tsarin ARCH. Hanyoyin gargajiya na ƙirar kuskuren autocovariance sau da yawa suna sanya ƙayyadaddun tsarin ƙayyadaddun bayanai, suna haifar da haɗarin kuskuren ƙirar tsarin. Wannan takarda tana ba da shawarar hanyar Bayesian ta rashin ƙayyadaddun tsari don ƙididdige yawan zane-zane na aikin kuskuren autocovariance, yana motsa matsalar yadda ya kamata zuwa yankin mitar don guje wa rikitattun zaɓin bandwidth a cikin hanyoyin kernel na yankin lokaci. An faɗaɗa tsarin don ɗaukar duka ƙarfin kuskure mai tsayi da na canzawa tare da lokaci, tare da aikace-aikacen da ke nuna mafi girman aiki a cikin hasashen musayar kuɗi idan aka kwatanta da ma'auni kamar tsarin tafiya bazuwar.

2. Hanyar Aiki

Babbar hanyar aiki ta ƙunshi tsarin Bayesian mai matakai don haɗin kiyasin sigogin tsarin, canjin ƙarfin ƙarfi na lokaci, da yawan zane-zane na tsarin kuskure.

2.1 Tsarin Tsarin

Tsarin tushe shine tsarin koma baya: $y = X\beta + \epsilon$, inda $\epsilon_t = \sigma_{\epsilon, t} e_t$. A nan, $e_t$ tsari ne na Gaussian mai daidaitawa, mai tsayayye tare da aikin autocorrelation $\gamma(\cdot)$ da yawan zane-zane $\lambda(\cdot)$. Canjin ƙarfin ƙarfi na lokaci $\sigma^2_{\epsilon, t}$ ana ƙirar shi cikin sassauƙa, sau da yawa ana amfani da canjin log wanda ake wakilta shi ta hanyar ayyukan B-spline.

2.2 Kiyasin Bayesian na Rashin Ƙayyadaddun Tsari na Zane-zane

Biyo bayan Dey da sauransu (2018), an sanya fifikon tsarin Gaussian akan yawan zane-zane na log, $\log \lambda(\omega)$. Wannan fifikon yana da sassauƙa kuma yana guje wa ƙayyadaddun zato na ƙayyadaddun bayanai. Ana amfani da kusancin yiwuwar Whittle a cikin yankin mitar don ingantaccen lissafi. Ana gudanar da binciken baya na $\lambda(\omega)$ da saboda haka $\gamma(\cdot)$ ta hanyoyin Markov Chain Monte Carlo (MCMC).

2.3 Ƙirar Canjin Ƙarfin Ƙarfi na Lokaci

Don yanayin canzawa tare da lokaci, $\log(\sigma^2_{\epsilon, t})$ ana ƙirar shi azaman aiki mai santsi na lokaci, yawanci ana amfani da haɗin layi na ayyukan tushe na B-spline: $\log(\sigma^2_{\epsilon, t}) = \sum_{j=1}^J \theta_j B_j(t)$. Ana sanya fifiko akan ƙididdiga $\theta_j$, yana ƙarfafa santsi.

3. Sakamakon Gwaji & Bincike

3.1 Nazarin Kwaikwayo

An tabbatar da hanyar akan bayanan da aka yi kwaikwayo tare da sanannun tsarin autocorrelation (misali, nau'in ARMA) da alamu na ƙarfin ƙarfi na stochastic. Maɓallan ma'auni sun haɗa da daidaito wajen dawo da ainihin yawan zane-zane da ɗaukar kewayon amintattu. Hanyar Bayesian ta rashin ƙayyadaddun tsari ta nuna ingantaccen aiki a cikin nau'ikan hanyoyin samar da bayanai daban-daban, yana ɗaukar duka dogaro na gajeren lokaci da na dogon lokaci ba tare da sanin tsarin jinkiri ba.

3.2 Aikace-aikacen Hasashen Musayar Kuɗi

Babban aikace-aikacen na zahiri ya haɗa da hasashen manyan ƙimar musayar kuɗi (misali, USD/EUR, USD/JPY).

Taƙaitaccen Aikin Hasashe

Ma'auni: Tafiya Bazuwar ba tare da Drift ba, GARCH(1,1), ARIMA mai ƙayyadaddun bayanai.

Ma'auni: Tushen Matsakaicin Kuskuren Hasashe (RMSEF) da Matsakaicin Kuskuren Hasashe na Cikakke (MAFE) akan lokutan da ba a cikin samfurin ba.

Sakamako: Tsarin Bayesian na rashin ƙayyadaddun tsari da aka ba da shawara ya ci gaba da fi dacewa fiye da ma'aunin tafiya bazuwar kuma ya yi gasa da kyau, kuma sau da yawa ya doke, daidaitattun tsarin GARCH da jerin lokaci na ƙayyadaddun bayanai. An ƙara gagarumin ci gaba musamman a lokutan da kasuwa ke da ƙarfin ƙarfi, inda ƙirar ƙarfin ƙarfi mai sassauƙa ta tabbatar da fa'ida.

Bayanin Ginshiƙi: Zanen layi zai nuna hanyoyin hasashe na waje da ba a cikin samfurin ba na tsarin da aka ba da shawara da tafiya bazuwar da GARCH. Hasashen tsarin da aka ba da shawara zai rungumi ainihin hanyar ƙimar musayar kuɗi da aka gano, musamman a kusa da wuraren juyawa da matakan canzawa. Zanen mashaya zai kwatanta RMSEF/MAFE a cikin tsare-tsare, tare da hanyar da aka ba da shawara tana da mashaya mafi gajere.

4. Fahimtar Jigo & Ra'ayi na Manazarta

Fahimtar Jigo: Wannan takarda tana ba da ingantacciyar haɓaka, wacce ake yawan yin watsi da ita, ga ƙirar jerin lokaci: ɗaukar dogaron kuskure a matsayin ɗan ƙasa na farko da za a koya, ba a zato ba. Ta hanyar ƙididdige cikakken tsarin autocovariance ba tare da ƙayyadaddun tsari ba ta hanyar yawan zane-zane, yana kai hari kai tsaye ga ƙafar Achilles na yawancin tsare-tsare—kuskuren ƙirar kuskure. Ƙarin canjin ƙarfin ƙarfi na lokaci ba kawai ƙarin fasali ba ne; yana da matakin gaskiya na gaskiya don bayanan kuɗi, yana mai da tsarin kayan aiki mai ƙarfi don yanayin da ƙarfin ƙarfi ke taruwa, kamar kasuwannin kuɗi.

Kwararar Ma'ana: Hujja tana da kyau. Mataki na 1: Yardawa cewa ƙayyadaddun tsarin kuskure abu ne na alhaki. Mataki na 2: Matsa zuwa yankin mitar don ɗaukar ƙididdiga mara ƙayyadaddun tsari yadda ya kamata (karkatar da la'anar zaɓin bandwidth). Mataki na 3: Yi amfani da fifikon tsarin Gaussian akan log-spectrum—zaɓi mai ma'ana na lissafi kuma mai sassauƙa. Mataki na 4: Haɗa wannan tare da ƙirar ƙarfin ƙarfi mai canzawa tare da lokaci, sanin cewa ma'auni da dogaro suna haɗuwa a cikin bayanan gaske. Mataki na 5: Tabbatar ta hanyar doke mafi ƙarfin ma'auni a cikin kuɗi: tafiya bazuwar don ƙimar musayar kuɗi. Kwararar daga gano matsalar zuwa mafita ta fasaha zuwa tabbacin gwaji yana da haɗin kai kuma mai gamsarwa.

Ƙarfi & Kurakurai: Ƙarfinsa shine cikakken sassauƙansa. Ba ya tilasta bayanai cikin akwatin ARMA ko GARCH. Amfani da yiwuwar Whittle da MCMC daidai ne amma yana da tasiri. Kuskuren, kamar yadda yake da yawancin hanyoyin Bayesian na rashin ƙayyadaddun tsari, shine farashin lissafi. MCMC don tsarin Gaussian da splines ba abu ne mai sauƙi ba don dogayen jerin. Takardar kuma ta dogara sosai akan misalin ƙimar musayar kuɗi; ƙarin aikace-aikace iri-iri (misali, tattalin arziki, makamashi) zai ƙarfafa hujjar gabaɗaya. Bugu da ƙari, yayin da yake ambaton Dey da sauransu (2018), bambanci mafi bayyananne na gudummawar sa—haɗin kai tare da canjin ƙarfin ƙarfi na lokaci—zai iya zama mafi kaifi.

Fahimta Mai Aiki: Ga masu ƙididdiga da masana tattalin arziki: Wannan tsari ne da aka shirya don hasashe mai matuƙar mahimmanci inda daidaitattun tsare-tsare suka gaza. Kasancewar lambar a GitHub babbar fa'ida ce. Aikin nan take shine gwada shi akan bayanan mallakar mallakar inda tsarin kuskure yake da shakku. Ga masu bincike: Hanyar aiki samfuri ce. Tunanin GP-on-spectrum za a iya canza shi zuwa wasu tsare-tsaren masu canji na ɓoye. Mataki na gaba na ma'ana shine magance saitunan masu girma ko haɗa wasu fifikon marasa ƙayyadaddun tsari, kamar waɗanda suka dogara da hanyoyin sadarwar jijiyoyi kamar yadda aka gani a cikin ilmantarwa mai zurfi na zamani don jerin lokaci (misali, gine-ginen da aka yi wahayi zuwa gare su ta Hanyoyin Haɗin Lokaci). Fannin yana tafiya zuwa ga tsare-tsaren gauraye waɗanda ke haɗa rashin ƙayyadaddun tsari na Bayesian tare da ilmantarwa mai zurfi, kamar yadda aka lura a cikin bita daga wurare kamar Cibiyar Nazarin Alan Turing, kuma wannan aikin yana kan mahadar mai albarka.

5. Cikakkun Bayanai na Fasaha

Mahimman Tsarin Lissafi:

Tsarin: $y_t = x_t'\beta + \epsilon_t, \quad \epsilon_t = \sigma_{\epsilon, t} e_t$.
Tsarin Kuskure: $e_t \sim \text{GP}(0, \gamma)$, tare da $\text{Cov}(e_t, e_{t-k}) = \gamma(k)$.
Yawan Zane-zane: $\lambda(\omega) = \frac{1}{2\pi} \sum_{k=-\infty}^{\infty} \gamma(k) e^{-i k \omega}, \quad \omega \in [-\pi, \pi]$.
Fifiko don Spectrum: $\log \lambda(\omega) \sim \text{GP}(\mu(\omega), C(\omega, \omega'))$, inda $C$ kernel ɗin da ya dace na haɗin kai.
Tsarin Ƙarfin Ƙarfi: $\log(\sigma^2_{\epsilon, t}) = \sum_{j=1}^J \theta_j B_j(t), \quad \theta \sim N(0, \tau^2 I)$.
Yiwuwa (Kusancin Whittle): $p(I(\omega_j) | \lambda(\omega_j)) \approx \frac{1}{\lambda(\omega_j)} \exp\left(-\frac{I(\omega_j)}{\lambda(\omega_j)}\right)$, inda $I(\omega_j)$ shine periodogram a mitar Fourier $\omega_j$.

6. Misalin Tsarin Bincike

Yanayi: Nazarin dawowar yau da kullun na cryptocurrency (misali, Bitcoin) don hasashen ƙarfin ƙarfi da tsarin dogaro.

Matakan Tsarin (Ra'ayi):

Gyara Kafin Aiki: Sami dawowar log. Zaɓi, cire duk wani yanayi mai ƙarancin mitar.
Ƙayyadaddun Tsarin:
- Ma'anar lissafi: Wataƙila madaidaicin madaidaici ko lokaci na AR(1): $r_t = \mu + \phi r_{t-1} + \epsilon_t$.
- Rarraba kuskure: $\epsilon_t = \sigma_t e_t$.
- Ƙayyade tushen B-spline don $\log(\sigma^2_t)$ (misali, 20 knots a cikin lokacin samfurin).
- Ƙayyade fifikon tsarin Gaussian don $\log \lambda(\omega)$ (misali, tare da kernel ɗin haɗin kai na Matern).
Fitar da Fifiko: Saita hyperparameters don santsin GP, bambancin ƙididdiga na spline ($\tau^2$), da sigogin koma baya ($\beta$). Yi amfani da fifikon bayanai marasa ƙarfi.
Lissafin Baya: Aiwatar da mai tattara MCMC (misali, Hamiltonian Monte Carlo a cikin Stan ko mai tattara Gibbs na al'ada) don zana samfurori daga haɗin baya na $(\beta, \theta, \lambda(\cdot))$.
Bincike & Hasashe:
- Bincika matsakaicin baya/tsaka-tsaki na $\sigma_t$ don ganin juyin halitta na ƙarfin ƙarfi.
- Zana matsakaicin baya na $\lambda(\omega)$ don fahimtar tsarin mitar na dogaro.
- Canza $\lambda(\omega)$ komawa yankin lokaci don samun ƙididdiga na aikin autocorrelation $\gamma(k)$.
- Samar da rarraba hasashe don dawowar gaba ta amfani da samfurorin baya.

Lura: Ma'ajiyar lambar marubutan akan GitHub tana ba da madaidaicin farawa don aiwatarwa.

7. Aikace-aikace na Gaba & Jagorori

Kuɗi Mai Girma: Daidaita tsarin don ɗaukar bayanan cikin rana tare da hayaniyar tsari da ƙididdiga mai girma na zane-zane.
Faɗaɗa Multivariate: Haɓaka ƙirar Bayesian mara ƙayyadaddun tsari don matrix ɗin yawan zane-zane na giciye na tsarin kuskure na vector, mai mahimmanci don nazarin fayil da nazarin zubewa.
Haɗin kai tare da Ilmantarwa Mai Zurfi: Maye gurbin fifikon GP tare da ƙirar samarwa mai zurfi (misali, Mai Canza kai da kai na Bambance-bambance akan yankin zane-zane) don ɗaukar rikitattun alamu na dogaro, masu canzawa, bin ruhin ƙirƙira a cikin takardu kamar "CycleGAN" don canja wurin salo amma ana amfani da su ga jerin lokaci.
Tsarin Hasashe na Ainihin Lokaci: Ƙirƙirar sigogin kusancin bincike masu iya aunawa (misali, ta amfani da Binciken Bambance-bambance na Stochastic) don tsarin sarrafa haɗari na ainihin lokaci da dandamalin ciniki na algorithm.
Kuɗi-Macro: Yin amfani da tsarin don ƙirar tsarin kuskure a cikin manyan VARs na Bayesian waɗanda bankunan tsakiya da cibiyoyin manufofi ke amfani da su, inda kuskuren ƙirar girgiza zai iya haifar da kurakurai na manufofi.

8. Nassoshi

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007.
Kim, K., & Kim, K. (2016). Time-varying volatility and macroeconomic uncertainty. Economics Letters, 149, 24-28.
Dey, D., Kim, K., & Roy, A. (2018). Bayesian nonparametric spectral density estimation for irregularly spaced time series. Journal of the American Statistical Association, 113(524), 1551-1564.
Kim, K. (2011). Hierarchical Bayesian analysis of structural instability in macroeconomic time series. Studies in Nonlinear Dynamics & Econometrics, 15(4).
Whittle, P. (1953). Estimation and information in stationary time series. Arkiv för Matematik, 2(5), 423-434.
Zhu, J. Y., Park, T., Isola, P., & Efros, A. A. (2017). Unpaired image-to-image translation using cycle-consistent adversarial networks. Proceedings of the IEEE international conference on computer vision (Takardar CycleGAN a matsayin misali na ƙirar samarwa mai sassauƙa, mai zurfi).
Alan Turing Institute. (2023). Jigogin Bincike: Injiniyanci Mai Maida Hankali kan Bayanai da AI don Kimiyya. (Don mahallin hanyoyin haɗin AI/ƙididdiga).