每天涂点鸦 (http://www.myfootprints.cn)
求 MA(2) 模型的自相关系数
Google 文件可以插入公式，而且编辑起来比 Word 中的公式编辑器还要方便得多，因为可以使用键盘输入，而不一定非要使用鼠标来点选。
这是使用 Google 文件输入的文章。
MA(2) 模型形式为：
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=R_%7Bt%7D%20%3Da_%7Bt%7D%20-%5Ctheta_%7B1%7D%20a_%7Bt-1%7D-%5Ctheta_%7B2%7Da_%7Bt-2%7D. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=R_%7Bt%7D%20%3Da_%7Bt%7D%20-%5Ctheta_%7B1%7D%20a_%7Bt-1%7D-%5Ctheta_%7B2%7Da_%7Bt-2%7D --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:57:00 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
两边乘以Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=R_%7Bt-l%7D. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=R_%7Bt-l%7D --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:57:44 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---，得
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=R_%7Bt%7DR_%7Bt-l%7D%20%3D%20a_%7Bt%7DR_%7Bt-l%7D%20-%20%5Ctheta_%7B1%7Da_%7Bt-1%7DR_%7Bt-l%7D%20-%20%5Ctheta_%7B2%7Da_%7Bt-2%7DR_%7Bt-l%7D. Unable to fetch:
https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=R_%7Bt%7DR_%7Bt-l%7D%20%3D%20a_%7Bt%7DR_%7Bt-l%7D%20-%20%5Ctheta_%7B1%7Da_%7Bt-1%7DR_%7Bt-l%7D%20-%20%5Ctheta_%7B2%7Da_%7Bt-2%7DR_%7Bt-l%7D
---
Reason: Response code 404 (Not Found)
---
Fetch details:
{
"attempt": 3,
"method": "GET",
"errorCode": "ERR_NON_2XX_3XX_RESPONSE",
"responseStatusCode": 404,
"responseStatusMessage": "Not Found",
"requestHeaders": {
"user-agent": "got (https://github.com/sindresorhus/got)",
"accept-encoding": "gzip, deflate, br"
},
"responseHeaders": {
"date": "Mon, 15 Apr 2024 04:57:53 GMT",
"content-type": "text/html",
"server": "HTTP server (unknown)",
"content-length": "49",
"x-xss-protection": "0",
"alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000",
"connection": "close"
}
}
---
再取期望，得到
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7Bl%7D%20%3D%20-%5Ctheta_%7B1%7DE(a_%7Bt-1%7DR_%7Bt-l%7D)%20-%20%5Ctheta_%7B2%7DE(a_%7Bt-2%7DR_%7Bt-l%7D). Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7Bl%7D%20%3D%20-%5Ctheta_%7B1%7DE(a_%7Bt-1%7DR_%7Bt-l%7D)%20-%20%5Ctheta_%7B2%7DE(a_%7Bt-2%7DR_%7Bt-l%7D) --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:00 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=l%3D1. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=l%3D1 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:05 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } --- 时，
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7B1%7D%20%3D%20-%5Ctheta_%7B1%7D%5Csigma_%7Ba%7D%5E%7B2%7D%20-%20%5Ctheta_%7B2%7DE(a_%7Bt-2%7D%5Ccdot%20(a_%7Bt-1%7D-%5Ctheta_%7B1%7Da_%7Bt-2%7D-%5Ctheta_%7B2%7Da_%7Bt-3%7D))%20%3D%20-%5Ctheta_%7B1%7D%5Csigma_%7Ba%7D%5E%7B2%7D%20%2B%20%5Ctheta_%7B1%7D%5Ctheta_%7B2%7D%5Csigma_%7Ba%7D%5E%7B2%7D. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7B1%7D%20%3D%20-%5Ctheta_%7B1%7D%5Csigma_%7Ba%7D%5E%7B2%7D%20-%20%5Ctheta_%7B2%7DE(a_%7Bt-2%7D%5Ccdot%20(a_%7Bt-1%7D-%5Ctheta_%7B1%7Da_%7Bt-2%7D-%5Ctheta_%7B2%7Da_%7Bt-3%7D))%20%3D%20-%5Ctheta_%7B1%7D%5Csigma_%7Ba%7D%5E%7B2%7D%20%2B%20%5Ctheta_%7B1%7D%5Ctheta_%7B2%7D%5Csigma_%7Ba%7D%5E%7B2%7D --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:09 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=l%3D2. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=l%3D2 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:13 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } --- 时，
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7B2%7D%20%3D%20-%5Ctheta_%7B2%7D%5Csigma_%7Ba%7D%5E%7B2%7D. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7B2%7D%20%3D%20-%5Ctheta_%7B2%7D%5Csigma_%7Ba%7D%5E%7B2%7D --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:18 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=l%3E2. Unable to fetch:
https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=l%3E2
---
Reason: Response code 404 (Not Found)
---
Fetch details:
{
"attempt": 3,
"method": "GET",
"errorCode": "ERR_NON_2XX_3XX_RESPONSE",
"responseStatusCode": 404,
"responseStatusMessage": "Not Found",
"requestHeaders": {
"user-agent": "got (https://github.com/sindresorhus/got)",
"accept-encoding": "gzip, deflate, br"
},
"responseHeaders": {
"date": "Mon, 15 Apr 2024 04:58:22 GMT",
"content-type": "text/html",
"server": "HTTP server (unknown)",
"content-length": "49",
"x-xss-protection": "0",
"alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000",
"connection": "close"
}
}
--- 时，
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7Bl%7D%3D0. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cgamma_%7Bl%7D%3D0 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:27 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
又
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=Var(R_%7Bt%7D)%20%3D%20%5Csigma_%7Ba%7D%5E%7B2%7D%20%2B%20%5Ctheta_%7B1%7D%5Csigma_%7Ba%7D%5E%7B2%7D%20%2B%20%5Ctheta_%7B2%7D%5Csigma_%7Ba%7D%5E%7B2%7D%0A%3D%20(1%2B%5Ctheta_%7B1%7D%20%2B%20%5Ctheta_%7B2%7D)%5Csigma_%7Ba%7D%5E%7B2%7D. Unable to fetch:
https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=Var(R_%7Bt%7D)%20%3D%20%5Csigma_%7Ba%7D%5E%7B2%7D%20%2B%20%5Ctheta_%7B1%7D%5Csigma_%7Ba%7D%5E%7B2%7D%20%2B%20%5Ctheta_%7B2%7D%5Csigma_%7Ba%7D%5E%7B2%7D%0A%3D%20(1%2B%5Ctheta_%7B1%7D%20%2B%20%5Ctheta_%7B2%7D)%5Csigma_%7Ba%7D%5E%7B2%7D
---
Reason: Response code 404 (Not Found)
---
Fetch details:
{
"attempt": 3,
"method": "GET",
"errorCode": "ERR_NON_2XX_3XX_RESPONSE",
"responseStatusCode": 404,
"responseStatusMessage": "Not Found",
"requestHeaders": {
"user-agent": "got (https://github.com/sindresorhus/got)",
"accept-encoding": "gzip, deflate, br"
},
"responseHeaders": {
"date": "Mon, 15 Apr 2024 04:58:31 GMT",
"content-type": "text/html",
"server": "HTTP server (unknown)",
"content-length": "49",
"x-xss-protection": "0",
"alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000",
"connection": "close"
}
}
---
于是
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Crho_%7B1%7D%20%3D%20%5Cfrac%7B%5Cgamma_%7B1%7D%7D%7B%5Cgamma_%7B0%7D%7D%20%3D%20%5Cfrac%7B-%5Ctheta_%7B1%7D%2B%5Ctheta_%7B1%7D%5Ctheta_%7B2%7D%7D%7B1%2B%5Ctheta_%7B1%7D%2B%5Ctheta_%7B2%7D%7D. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Crho_%7B1%7D%20%3D%20%5Cfrac%7B%5Cgamma_%7B1%7D%7D%7B%5Cgamma_%7B0%7D%7D%20%3D%20%5Cfrac%7B-%5Ctheta_%7B1%7D%2B%5Ctheta_%7B1%7D%5Ctheta_%7B2%7D%7D%7B1%2B%5Ctheta_%7B1%7D%2B%5Ctheta_%7B2%7D%7D --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:35 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Crho_%7B2%7D%3D%5Cfrac%7B%5Cgamma_%7B2%7D%7D%7B%5Cgamma_%7B0%7D%7D%3D%5Cfrac%7B-%5Ctheta_%7B2%7D%7D%7B1%2B%5Ctheta_%7B1%7D%2B%5Ctheta_%7B2%7D%7D. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Crho_%7B2%7D%3D%5Cfrac%7B%5Cgamma_%7B2%7D%7D%7B%5Cgamma_%7B0%7D%7D%3D%5Cfrac%7B-%5Ctheta_%7B2%7D%7D%7B1%2B%5Ctheta_%7B1%7D%2B%5Ctheta_%7B2%7D%7D --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:39 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
Error creating node with https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Crho_%7Bl%7D%20%3D%200%2C%20l%20%3E%202. Unable to fetch: https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Crho_%7Bl%7D%20%3D%200%2C%20l%20%3E%202 --- Reason: Response code 404 (Not Found) --- Fetch details: { "attempt": 3, "method": "GET", "errorCode": "ERR_NON_2XX_3XX_RESPONSE", "responseStatusCode": 404, "responseStatusMessage": "Not Found", "requestHeaders": { "user-agent": "got (https://github.com/sindresorhus/got)", "accept-encoding": "gzip, deflate, br" }, "responseHeaders": { "date": "Mon, 15 Apr 2024 04:58:44 GMT", "content-type": "text/html", "server": "HTTP server (unknown)", "content-length": "49", "x-xss-protection": "0", "alt-svc": "h3=\":443\"; ma=2592000,h3-29=\":443\"; ma=2592000", "connection": "close" } } ---
可以看到，MA(2)的自相关系数是在2处截尾的。