11月 282019
 

With time series data analysis, we can apply moving average methods to predict data points without seasonality. This includes Simple Average (SA), Simple Moving Average (SMA), Weighted Moving Average (WMA), Exponential Moving Average (EMA), etc. For series with a trend but without seasonality, we can use linear, non-linear and autoregressive prediction models. In practice, we often use these kinds of moving average methods for series data with large variance, e.g. financial and stock market data.

Since introducing the window concept, SMA reflects historical data with a lag problem, and all points in the same window have the same importance to the current point. Weights are introduced to reflect the different importances of points in a window, and it leads to WMA and EMA. WMA gives greater weight to recent observations while less weight to the longer-term observations. The weights decrease in a linear fashion. If the sequence fluctuations are not large, the same weight is given to the observation and WMA degenerates into a SMA and the WMA becomes an EMA when weights decrease exponentially. However, these traditional moving average methods ave a lag problem due to introducing the smooth window concept and its results are less responsive to the current value. To gain a better balance between lag reduction and curve smoothing in a moving average, we must get help from some smarter mathematics algorithm.

What is Hull Moving Average?

Hull Moving Average (HMA) is a fast moving average method with low lag developed by Australia investor Alan Hull in 2005. It’s known to almost eliminate lag altogether and manages to improve smoothing at the same time. Longer period HMAs can be used as an indicator to identify trend, and shorter period HMAs can be used as entry signals in the direction of the prevailing trend. Due to lag problems in all moving average methods, it’s not suggested to be used as a reverse signal alone. Anyway, we can use different window HMAs to build a more robust trading strategy. E.g., 52-week HMA for trend indicator, and 13-week HMA for entry signal. Here is how it works:

  • Long period HMA identifies the trend: rising HMA indicates the prevailing trend is rising, it’s better to enter long positions; falling HMA indicate the prevailing trend is falling, it’s better to enter short positions.
  • Short period HMA identifies the entry signals in the direction of the prevailing trend: When the prevailing trend is rising, HMA goes up to indicate a long entry signal. When the prevailing trend is falling, HMA goes down to indicate a short entry signal.

Both long & short period HMAs in bullish mode generate Long Signal, and both in bearish mode generate Short Signal. So Long Trades get a buy signal after a Long Signal occurs, while Short Trades get a sell signal after a Short Signal occurs. Long Trades get a Sell Signal when Long or Short Trend is no longer bullish and Short Trades get a buy signal when Long or Short trend is no longer bearish. Figure 1 is a sample for this HMA based trading strategy.

In fact, HMA is quite simple. Its calculation only includes three WMA steps:

  1. Calculate WMA of original sequence X using window length n ;
  2. Calculate WMA of X using window length n/2;
  3. Calculate WMA of derived sequence Y using window length Sqrt(n), while Y is the new series of 2 * WMA(x, n/2) – WMA(x, n). So HMA(x, n) formula can be defined as below:

HMA(x, n) = WMA( 2*WMA(x, n/2) − WMA(x, n), sqrt(n) )

HMA makes the moving average line more responsive to current value and keeps the smoothing of the curve line. It almost eliminates lag altogether and manages to improve smoothing at the same time.

How does HMA achieve the perfect balance?

We know SMA can gain the curve line of an average value for historical data falling in the current window, but it has the constant lag of at least a half window. The output may also have poor smoothness. If we embed SMA multiple times to get the average of the averages, it would be smoother but would increase lag at the same time.

Suppose we have this series: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The SMA at the last point is 5.5 with window length n=10, i.e. SUM[1...10]/10
is much different than the real value 10. If we reduce the window length to n=5, the SMA at the last point is 8, i.e. SUM[6…10]/5. If we have the second SMA with half size window and add the difference between these two SMAs, then we can get 8 + (8-5.5)=10.5, which is very close to the real value 10 and will eliminate lag. Then we can use SMA with specific window length again to reduce that slight overcompensation and improve smoothness. HMA uses linear WMA instead of SMA, and the last window length is sqrt(n). This is the key trick with using HMA.

Figure 2 below shows the difference between HMA(x, 4) and embed SMA twice SMA( SMA(x,4),4).

Figure 2: Comparison of SMA( SMA(x, 4), 4) and HMA(x, 4).

In stock trading, there are lots of complicated technical indicators derived from stock historical data, such as Ease of Movement indicators (EMV), momentum indicators (MTM), MACD indicators, Energy indicator CR, KDJ indicators, Bollinger (BOL) and so on. Its essence is like feature engineering extraction before neural network training. The fundamental purpose is to find some intrinsic property from price fluctuation. The Hull moving average itself was discovered and applied to real trading practices, which gives us a glimpse to the complication of building an automatic financial trading strategy under the veil. It’s just the beginning, the reflex in trading and aging of models may both bring uncertainty, so the effectiveness of quantitative trading strategies is a very complex topic which needs to be validated through real practice.

HMA in SAS Code

SAS was created for data analysis however, and my colleagues go into detail about how data analysis connects with HMA. Check out Cindy Wang's How to Calculate Hull Moving Average in SAS Visual Analytics and Rick Wicklin's The Hull moving average: Implement a custom time series smoother in SAS. I would like to share how to implement both HMA/WMA with only 25 lines of code in this article. You can use the macro anywhere to build HMA series with specific window lengths.

First, implement Weighted Moving Average (WMA) with SAS macro as shown below (only 11 lines). The macro has four arguments:

  • WMA_N       WMA window length
  • INPUT          Input variable name
  • OUT             Output variable name, default is input name with _WMA_n suffix
  • OUTLABEL The label of output variable, default is “input name WMA(n)
%macro AddWMA(WMA_N=4, INPUT=close, OUT=&INPUT._WMA_&WMA_N., OUTLABEL=&INPUT. WMA(&WMA_N.));
  retain _sumx_&INPUT._WMA_&WMA_N.;
  _sumx_&INPUT._WMA_&WMA_N.=sum (_sumx_&INPUT._WMA_&WMA_N., &INPUT., -lag&WMA_N.( &INPUT. )) ;
 
  retain _sum_&INPUT._WMA_&WMA_N.;
  _sum_&INPUT._WMA_&WMA_N.=sum (_sum_&INPUT._WMA_&WMA_N., &WMA_N * &INPUT., -lag( _sumx_&INPUT._WMA_&WMA_N.  )) ;
 
  retain _sumw_&INPUT._WMA_&WMA_N.;
  if _N_ <= &WMA_N then _sumw_&INPUT._WMA_&WMA_N. = sum (_sumw_&INPUT._WMA_&WMA_N., &WMA_N, -_N_, 1);
 
  &OUT=_sum_&INPUT._WMA_&WMA_N. / _sumw_&INPUT._WMA_&WMA_N.;
 
  drop _sumx_&INPUT._WMA_&WMA_N. _sumw_&INPUT._WMA_&WMA_N. _sum_&INPUT._WMA_&WMA_N.;
  label &OUT="&OUTLABEL"; 
%mend;

We must watch out for the trick in the upper implementation of WMA with linear weights. In fact, there is no do-loop in the code, and it also doesn’t perform repeat computations for weights and values. We just keep the last sum of all values in the window (See Ln2-3), and the sum for WMA is the last sum for WMA plus window length times the value subtracted from it (See Ln4-5). For weight sum computation, we just need to pay attention to the points less than the window length(See Ln6-7).

Second, implement HMA with three times WMA invocation as below, (only 9 lines). The macro has four arguments:

  • HMA_N       HMA window length
  • INPUT         Input variable name
  • OUT            Output variable name, default is input name with _HMA_n suffix
  • OUTLABEL The label of output variable, default is “input name HMA(n)
%macro AddHMA(HMA_N=5, INPUT=close, OUT=&INPUT._HMA_&HMA_N., outlabel=&INPUT. HMA(&HMA_N.));
  %AddWMA(WMA_N=&HMA_N, INPUT=&INPUT.);
 
  %AddWMA(WMA_N=%sysfunc(round(&HMA_N./2)), INPUT=&INPUT.);
 
  &INPUT._HMA_&HMA_N._DELTA=2 * &INPUT._WMA_%sysfunc(round(&HMA_N./2))  - &INPUT._WMA_&HMA_N;
  %AddWMA(WMA_N=%sysfunc(round(%sysfunc(sqrt(&HMA_N)))), INPUT=&INPUT._HMA_&HMA_N._DELTA);
 
  rename &INPUT._HMA_&HMA_N._DELTA_WMA_%sysfunc(round(%sysfunc(sqrt(&HMA_N.))))=&OUT;
  label &INPUT._HMA_&HMA_N._DELTA_WMA_%sysfunc(round(%sysfunc(sqrt(&HMA_N.))))="&outlabel"; 
 
  drop &INPUT._WMA_&HMA_N &INPUT._WMA_%sysfunc(round(&HMA_N./2)) &INPUT._HMA_&HMA_N._DELTA;
%mend;

The upper code is very intuitive as the first two lines perform WMA with window n and n/2 for X, then the next two lines generate new series Y and performs WMA with window sqrt(n). Other codes are just to improve macro flexibility and to drop temp variables, etc.

To compare HMA with SMA result, we must also implement SMA with the 7 lines of code as shown below. It has the same argument as WMA/HMA above, but SMA is an unnecessary part of HMA macro %ADDHMA implmentation.

 
%macro AddMA(MA_N=5, INPUT=close, out=&INPUT._MA_&MA_N., outlabel=&INPUT. MA(&MA_N.));
  retain _sum_&INPUT._MA_&MA_N.;
  _sum_&INPUT._MA_&MA_N.=sum (_sum_&INPUT._MA_&MA_N., &INPUT., -lag&MA_N.( &INPUT. )) ;
  &out=_sum_&INPUT._MA_&MA_N. / min(_n_, &MA_N.);
 
  drop _sum_&INPUT._MA_&MA_N.;
  label &out="&outlabel";
%mend;

Now we can test all the above implementations as shown below. First is a simple case we described above:

data a;
  input x @@;
  %AddMA( input=x, MA_N=5, out=SMA, outlabel=%str(SMA)); 
  %AddHMA( input=x, HMA_N=5, out=HMA, outlabel=%str(HMA))  
datalines;
1 2 3 4 5 6 7 8 9 10
;
proc print data=a label;
  format _all_ 8.2;
run;

The output is listed below, the HMA result is much more close to the real value x than SMA. It’s more responsive and less in lag.

Figure 3: HMA output sample

Now we can use the HMA macro to check IBM stock data in the SAS system dataset SASHELP.Stocks and draw trends in the plot to compare HMA(x, 5) and SMA(x, 5) output.

data Stocks_HMA;
  set sashelp.Stocks(where=(stock='IBM'));
  %AddHMA( HMA_N=5, out=HMA, outlabel=%str(HMA))
  %AddMA(MA_N=5, out=SMA, outlabel=%str(SMA));
run; 
ods graphics / width=800px height=300px;
title "Comparison of Simple and Hull Moving Average";
proc sgplot data=Stocks_HMA noautolegend;
   highlow x=Date high=High low=Low / lineattrs=(color=LightGray);
   scatter x=Date y=Close / markerattrs=(color=LightGray);
   series x=Date y=SMA / curvelabel curvelabelattrs=(color=DarkGreen) lineattrs=(color=DarkGreen);
   series x=Date y=HMA / curvelabel curvelabelattrs=(color=DarkOrange) lineattrs=(color=DarkOrange);
   refline '01JUN1992'd / axis=x label='June 1992' labelloc=inside;
   xaxis grid display=(nolabel); 
   yaxis grid max=200 label="IBM Closing Price";
run;

Figure 4: Comparison of SMA and HMA

Summary

We have talked about what Hull Moving Average is, how it works, and worked through a super simple SAS Macro implementation for HMA, WMA and SMA with only 25 lines of SAS code. This HMA implementation makes it quite simple and efficient to calculate HMA in SAS Data step without SAS/IML and Visual Analytics. In fact, it shows how easy process rolling-window computation for time series data can be in SAS, and opens a new window to build complicated technical indicators for stock trading systems yourself.

Learn more

Implementing HMA, WMA & SMA with 25 lines of SAS code was published on SAS Users.

11月 282019
 

With time series data analysis, we can apply moving average methods to predict data points without seasonality. This includes Simple Average (SA), Simple Moving Average (SMA), Weighted Moving Average (WMA), Exponential Moving Average (EMA), etc. For series with a trend but without seasonality, we can use linear, non-linear and autoregressive prediction models. In practice, we often use these kinds of moving average methods for series data with large variance, e.g. financial and stock market data.

Since introducing the window concept, SMA reflects historical data with a lag problem, and all points in the same window have the same importance to the current point. Weights are introduced to reflect the different importances of points in a window, and it leads to WMA and EMA. WMA gives greater weight to recent observations while less weight to the longer-term observations. The weights decrease in a linear fashion. If the sequence fluctuations are not large, the same weight is given to the observation and WMA degenerates into a SMA and the WMA becomes an EMA when weights decrease exponentially. However, these traditional moving average methods ave a lag problem due to introducing the smooth window concept and its results are less responsive to the current value. To gain a better balance between lag reduction and curve smoothing in a moving average, we must get help from some smarter mathematics algorithm.

What is Hull Moving Average?

Hull Moving Average (HMA) is a fast moving average method with low lag developed by Australia investor Alan Hull in 2005. It’s known to almost eliminate lag altogether and manages to improve smoothing at the same time. Longer period HMAs can be used as an indicator to identify trend, and shorter period HMAs can be used as entry signals in the direction of the prevailing trend. Due to lag problems in all moving average methods, it’s not suggested to be used as a reverse signal alone. Anyway, we can use different window HMAs to build a more robust trading strategy. E.g., 52-week HMA for trend indicator, and 13-week HMA for entry signal. Here is how it works:

  • Long period HMA identifies the trend: rising HMA indicates the prevailing trend is rising, it’s better to enter long positions; falling HMA indicate the prevailing trend is falling, it’s better to enter short positions.
  • Short period HMA identifies the entry signals in the direction of the prevailing trend: When the prevailing trend is rising, HMA goes up to indicate a long entry signal. When the prevailing trend is falling, HMA goes down to indicate a short entry signal.

Both long & short period HMAs in bullish mode generate Long Signal, and both in bearish mode generate Short Signal. So Long Trades get a buy signal after a Long Signal occurs, while Short Trades get a sell signal after a Short Signal occurs. Long Trades get a Sell Signal when Long or Short Trend is no longer bullish and Short Trades get a buy signal when Long or Short trend is no longer bearish. Figure 1 is a sample for this HMA based trading strategy.

In fact, HMA is quite simple. Its calculation only includes three WMA steps:

  1. Calculate WMA of original sequence X using window length n ;
  2. Calculate WMA of X using window length n/2;
  3. Calculate WMA of derived sequence Y using window length Sqrt(n), while Y is the new series of 2 * WMA(x, n/2) – WMA(x, n). So HMA(x, n) formula can be defined as below:

HMA(x, n) = WMA( 2*WMA(x, n/2) − WMA(x, n), sqrt(n) )

HMA makes the moving average line more responsive to current value and keeps the smoothing of the curve line. It almost eliminates lag altogether and manages to improve smoothing at the same time.

How does HMA achieve the perfect balance?

We know SMA can gain the curve line of an average value for historical data falling in the current window, but it has the constant lag of at least a half window. The output may also have poor smoothness. If we embed SMA multiple times to get the average of the averages, it would be smoother but would increase lag at the same time.

Suppose we have this series: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The SMA at the last point is 5.5 with window length n=10, i.e. SUM[1...10]/10
is much different than the real value 10. If we reduce the window length to n=5, the SMA at the last point is 8, i.e. SUM[6…10]/5. If we have the second SMA with half size window and add the difference between these two SMAs, then we can get 8 + (8-5.5)=10.5, which is very close to the real value 10 and will eliminate lag. Then we can use SMA with specific window length again to reduce that slight overcompensation and improve smoothness. HMA uses linear WMA instead of SMA, and the last window length is sqrt(n). This is the key trick with using HMA.

Figure 2 below shows the difference between HMA(x, 4) and embed SMA twice SMA( SMA(x,4),4).

Figure 2: Comparison of SMA( SMA(x, 4), 4) and HMA(x, 4).

In stock trading, there are lots of complicated technical indicators derived from stock historical data, such as Ease of Movement indicators (EMV), momentum indicators (MTM), MACD indicators, Energy indicator CR, KDJ indicators, Bollinger (BOL) and so on. Its essence is like feature engineering extraction before neural network training. The fundamental purpose is to find some intrinsic property from price fluctuation. The Hull moving average itself was discovered and applied to real trading practices, which gives us a glimpse to the complication of building an automatic financial trading strategy under the veil. It’s just the beginning, the reflex in trading and aging of models may both bring uncertainty, so the effectiveness of quantitative trading strategies is a very complex topic which needs to be validated through real practice.

HMA in SAS Code

SAS was created for data analysis however, and my colleagues go into detail about how data analysis connects with HMA. Check out Cindy Wang's How to Calculate Hull Moving Average in SAS Visual Analytics and Rick Wicklin's The Hull moving average: Implement a custom time series smoother in SAS. I would like to share how to implement both HMA/WMA with only 25 lines of code in this article. You can use the macro anywhere to build HMA series with specific window lengths.

First, implement Weighted Moving Average (WMA) with SAS macro as shown below (only 11 lines). The macro has four arguments:

  • WMA_N       WMA window length
  • INPUT          Input variable name
  • OUT             Output variable name, default is input name with _WMA_n suffix
  • OUTLABEL The label of output variable, default is “input name WMA(n)
%macro AddWMA(WMA_N=4, INPUT=close, OUT=&INPUT._WMA_&WMA_N., OUTLABEL=&INPUT. WMA(&WMA_N.));
  retain _sumx_&INPUT._WMA_&WMA_N.;
  _sumx_&INPUT._WMA_&WMA_N.=sum (_sumx_&INPUT._WMA_&WMA_N., &INPUT., -lag&WMA_N.( &INPUT. )) ;
 
  retain _sum_&INPUT._WMA_&WMA_N.;
  _sum_&INPUT._WMA_&WMA_N.=sum (_sum_&INPUT._WMA_&WMA_N., &WMA_N * &INPUT., -lag( _sumx_&INPUT._WMA_&WMA_N.  )) ;
 
  retain _sumw_&INPUT._WMA_&WMA_N.;
  if _N_ <= &WMA_N then _sumw_&INPUT._WMA_&WMA_N. = sum (_sumw_&INPUT._WMA_&WMA_N., &WMA_N, -_N_, 1);
 
  &OUT=_sum_&INPUT._WMA_&WMA_N. / _sumw_&INPUT._WMA_&WMA_N.;
 
  drop _sumx_&INPUT._WMA_&WMA_N. _sumw_&INPUT._WMA_&WMA_N. _sum_&INPUT._WMA_&WMA_N.;
  label &OUT="&OUTLABEL"; 
%mend;

We must watch out for the trick in the upper implementation of WMA with linear weights. In fact, there is no do-loop in the code, and it also doesn’t perform repeat computations for weights and values. We just keep the last sum of all values in the window (See Ln2-3), and the sum for WMA is the last sum for WMA plus window length times the value subtracted from it (See Ln4-5). For weight sum computation, we just need to pay attention to the points less than the window length(See Ln6-7).

Second, implement HMA with three times WMA invocation as below, (only 9 lines). The macro has four arguments:

  • HMA_N       HMA window length
  • INPUT         Input variable name
  • OUT            Output variable name, default is input name with _HMA_n suffix
  • OUTLABEL The label of output variable, default is “input name HMA(n)
%macro AddHMA(HMA_N=5, INPUT=close, OUT=&INPUT._HMA_&HMA_N., outlabel=&INPUT. HMA(&HMA_N.));
  %AddWMA(WMA_N=&HMA_N, INPUT=&INPUT.);
 
  %AddWMA(WMA_N=%sysfunc(round(&HMA_N./2)), INPUT=&INPUT.);
 
  &INPUT._HMA_&HMA_N._DELTA=2 * &INPUT._WMA_%sysfunc(round(&HMA_N./2))  - &INPUT._WMA_&HMA_N;
  %AddWMA(WMA_N=%sysfunc(round(%sysfunc(sqrt(&HMA_N)))), INPUT=&INPUT._HMA_&HMA_N._DELTA);
 
  rename &INPUT._HMA_&HMA_N._DELTA_WMA_%sysfunc(round(%sysfunc(sqrt(&HMA_N.))))=&OUT;
  label &INPUT._HMA_&HMA_N._DELTA_WMA_%sysfunc(round(%sysfunc(sqrt(&HMA_N.))))="&outlabel"; 
 
  drop &INPUT._WMA_&HMA_N &INPUT._WMA_%sysfunc(round(&HMA_N./2)) &INPUT._HMA_&HMA_N._DELTA;
%mend;

The upper code is very intuitive as the first two lines perform WMA with window n and n/2 for X, then the next two lines generate new series Y and performs WMA with window sqrt(n). Other codes are just to improve macro flexibility and to drop temp variables, etc.

To compare HMA with SMA result, we must also implement SMA with the 7 lines of code as shown below. It has the same argument as WMA/HMA above, but SMA is an unnecessary part of HMA macro %ADDHMA implmentation.

 
%macro AddMA(MA_N=5, INPUT=close, out=&INPUT._MA_&MA_N., outlabel=&INPUT. MA(&MA_N.));
  retain _sum_&INPUT._MA_&MA_N.;
  _sum_&INPUT._MA_&MA_N.=sum (_sum_&INPUT._MA_&MA_N., &INPUT., -lag&MA_N.( &INPUT. )) ;
  &out=_sum_&INPUT._MA_&MA_N. / min(_n_, &MA_N.);
 
  drop _sum_&INPUT._MA_&MA_N.;
  label &out="&outlabel";
%mend;

Now we can test all the above implementations as shown below. First is a simple case we described above:

data a;
  input x @@;
  %AddMA( input=x, MA_N=5, out=SMA, outlabel=%str(SMA)); 
  %AddHMA( input=x, HMA_N=5, out=HMA, outlabel=%str(HMA))  
datalines;
1 2 3 4 5 6 7 8 9 10
;
proc print data=a label;
  format _all_ 8.2;
run;

The output is listed below, the HMA result is much more close to the real value x than SMA. It’s more responsive and less in lag.

Figure 3: HMA output sample

Now we can use the HMA macro to check IBM stock data in the SAS system dataset SASHELP.Stocks and draw trends in the plot to compare HMA(x, 5) and SMA(x, 5) output.

data Stocks_HMA;
  set sashelp.Stocks(where=(stock='IBM'));
  %AddHMA( HMA_N=5, out=HMA, outlabel=%str(HMA))
  %AddMA(MA_N=5, out=SMA, outlabel=%str(SMA));
run; 
ods graphics / width=800px height=300px;
title "Comparison of Simple and Hull Moving Average";
proc sgplot data=Stocks_HMA noautolegend;
   highlow x=Date high=High low=Low / lineattrs=(color=LightGray);
   scatter x=Date y=Close / markerattrs=(color=LightGray);
   series x=Date y=SMA / curvelabel curvelabelattrs=(color=DarkGreen) lineattrs=(color=DarkGreen);
   series x=Date y=HMA / curvelabel curvelabelattrs=(color=DarkOrange) lineattrs=(color=DarkOrange);
   refline '01JUN1992'd / axis=x label='June 1992' labelloc=inside;
   xaxis grid display=(nolabel); 
   yaxis grid max=200 label="IBM Closing Price";
run;

Figure 4: Comparison of SMA and HMA

Summary

We have talked about what Hull Moving Average is, how it works, and worked through a super simple SAS Macro implementation for HMA, WMA and SMA with only 25 lines of SAS code. This HMA implementation makes it quite simple and efficient to calculate HMA in SAS Data step without SAS/IML and Visual Analytics. In fact, it shows how easy process rolling-window computation for time series data can be in SAS, and opens a new window to build complicated technical indicators for stock trading systems yourself.

Learn more

Implementing HMA, WMA & SMA with 25 lines of SAS code was published on SAS Users.

11月 282019
 

With time series data analysis, we can apply moving average methods to predict data points without seasonality. This includes Simple Average (SA), Simple Moving Average (SMA), Weighted Moving Average (WMA), Exponential Moving Average (EMA), etc. For series with a trend but without seasonality, we can use linear, non-linear and autoregressive prediction models. In practice, we often use these kinds of moving average methods for series data with large variance, e.g. financial and stock market data.

Since introducing the window concept, SMA reflects historical data with a lag problem, and all points in the same window have the same importance to the current point. Weights are introduced to reflect the different importances of points in a window, and it leads to WMA and EMA. WMA gives greater weight to recent observations while less weight to the longer-term observations. The weights decrease in a linear fashion. If the sequence fluctuations are not large, the same weight is given to the observation and WMA degenerates into a SMA and the WMA becomes an EMA when weights decrease exponentially. However, these traditional moving average methods ave a lag problem due to introducing the smooth window concept and its results are less responsive to the current value. To gain a better balance between lag reduction and curve smoothing in a moving average, we must get help from some smarter mathematics algorithm.

What is Hull Moving Average?

Hull Moving Average (HMA) is a fast moving average method with low lag developed by Australia investor Alan Hull in 2005. It’s known to almost eliminate lag altogether and manages to improve smoothing at the same time. Longer period HMAs can be used as an indicator to identify trend, and shorter period HMAs can be used as entry signals in the direction of the prevailing trend. Due to lag problems in all moving average methods, it’s not suggested to be used as a reverse signal alone. Anyway, we can use different window HMAs to build a more robust trading strategy. E.g., 52-week HMA for trend indicator, and 13-week HMA for entry signal. Here is how it works:

  • Long period HMA identifies the trend: rising HMA indicates the prevailing trend is rising, it’s better to enter long positions; falling HMA indicate the prevailing trend is falling, it’s better to enter short positions.
  • Short period HMA identifies the entry signals in the direction of the prevailing trend: When the prevailing trend is rising, HMA goes up to indicate a long entry signal. When the prevailing trend is falling, HMA goes down to indicate a short entry signal.

Both long & short period HMAs in bullish mode generate Long Signal, and both in bearish mode generate Short Signal. So Long Trades get a buy signal after a Long Signal occurs, while Short Trades get a sell signal after a Short Signal occurs. Long Trades get a Sell Signal when Long or Short Trend is no longer bullish and Short Trades get a buy signal when Long or Short trend is no longer bearish. Figure 1 is a sample for this HMA based trading strategy.

In fact, HMA is quite simple. Its calculation only includes three WMA steps:

  1. Calculate WMA of original sequence X using window length n ;
  2. Calculate WMA of X using window length n/2;
  3. Calculate WMA of derived sequence Y using window length Sqrt(n), while Y is the new series of 2 * WMA(x, n/2) – WMA(x, n). So HMA(x, n) formula can be defined as below:

HMA(x, n) = WMA( 2*WMA(x, n/2) − WMA(x, n), sqrt(n) )

HMA makes the moving average line more responsive to current value and keeps the smoothing of the curve line. It almost eliminates lag altogether and manages to improve smoothing at the same time.

How does HMA achieve the perfect balance?

We know SMA can gain the curve line of an average value for historical data falling in the current window, but it has the constant lag of at least a half window. The output may also have poor smoothness. If we embed SMA multiple times to get the average of the averages, it would be smoother but would increase lag at the same time.

Suppose we have this series: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The SMA at the last point is 5.5 with window length n=10, i.e. SUM[1...10]/10
is much different than the real value 10. If we reduce the window length to n=5, the SMA at the last point is 8, i.e. SUM[6…10]/5. If we have the second SMA with half size window and add the difference between these two SMAs, then we can get 8 + (8-5.5)=10.5, which is very close to the real value 10 and will eliminate lag. Then we can use SMA with specific window length again to reduce that slight overcompensation and improve smoothness. HMA uses linear WMA instead of SMA, and the last window length is sqrt(n). This is the key trick with using HMA.

Figure 2 below shows the difference between HMA(x, 4) and embed SMA twice SMA( SMA(x,4),4).

Figure 2: Comparison of SMA( SMA(x, 4), 4) and HMA(x, 4).

In stock trading, there are lots of complicated technical indicators derived from stock historical data, such as Ease of Movement indicators (EMV), momentum indicators (MTM), MACD indicators, Energy indicator CR, KDJ indicators, Bollinger (BOL) and so on. Its essence is like feature engineering extraction before neural network training. The fundamental purpose is to find some intrinsic property from price fluctuation. The Hull moving average itself was discovered and applied to real trading practices, which gives us a glimpse to the complication of building an automatic financial trading strategy under the veil. It’s just the beginning, the reflex in trading and aging of models may both bring uncertainty, so the effectiveness of quantitative trading strategies is a very complex topic which needs to be validated through real practice.

HMA in SAS Code

SAS was created for data analysis however, and my colleagues go into detail about how data analysis connects with HMA. Check out Cindy Wang's How to Calculate Hull Moving Average in SAS Visual Analytics and Rick Wicklin's The Hull moving average: Implement a custom time series smoother in SAS. I would like to share how to implement both HMA/WMA with only 25 lines of code in this article. You can use the macro anywhere to build HMA series with specific window lengths.

First, implement Weighted Moving Average (WMA) with SAS macro as shown below (only 11 lines). The macro has four arguments:

  • WMA_N       WMA window length
  • INPUT          Input variable name
  • OUT             Output variable name, default is input name with _WMA_n suffix
  • OUTLABEL The label of output variable, default is “input name WMA(n)
%macro AddWMA(WMA_N=4, INPUT=close, OUT=&INPUT._WMA_&WMA_N., OUTLABEL=&INPUT. WMA(&WMA_N.));
  retain _sumx_&INPUT._WMA_&WMA_N.;
  _sumx_&INPUT._WMA_&WMA_N.=sum (_sumx_&INPUT._WMA_&WMA_N., &INPUT., -lag&WMA_N.( &INPUT. )) ;
 
  retain _sum_&INPUT._WMA_&WMA_N.;
  _sum_&INPUT._WMA_&WMA_N.=sum (_sum_&INPUT._WMA_&WMA_N., &WMA_N * &INPUT., -lag( _sumx_&INPUT._WMA_&WMA_N.  )) ;
 
  retain _sumw_&INPUT._WMA_&WMA_N.;
  if _N_ <= &WMA_N then _sumw_&INPUT._WMA_&WMA_N. = sum (_sumw_&INPUT._WMA_&WMA_N., &WMA_N, -_N_, 1);
 
  &OUT=_sum_&INPUT._WMA_&WMA_N. / _sumw_&INPUT._WMA_&WMA_N.;
 
  drop _sumx_&INPUT._WMA_&WMA_N. _sumw_&INPUT._WMA_&WMA_N. _sum_&INPUT._WMA_&WMA_N.;
  label &OUT="&OUTLABEL"; 
%mend;

We must watch out for the trick in the upper implementation of WMA with linear weights. In fact, there is no do-loop in the code, and it also doesn’t perform repeat computations for weights and values. We just keep the last sum of all values in the window (See Ln2-3), and the sum for WMA is the last sum for WMA plus window length times the value subtracted from it (See Ln4-5). For weight sum computation, we just need to pay attention to the points less than the window length(See Ln6-7).

Second, implement HMA with three times WMA invocation as below, (only 9 lines). The macro has four arguments:

  • HMA_N       HMA window length
  • INPUT         Input variable name
  • OUT            Output variable name, default is input name with _HMA_n suffix
  • OUTLABEL The label of output variable, default is “input name HMA(n)
%macro AddHMA(HMA_N=5, INPUT=close, OUT=&INPUT._HMA_&HMA_N., outlabel=&INPUT. HMA(&HMA_N.));
  %AddWMA(WMA_N=&HMA_N, INPUT=&INPUT.);
 
  %AddWMA(WMA_N=%sysfunc(round(&HMA_N./2)), INPUT=&INPUT.);
 
  &INPUT._HMA_&HMA_N._DELTA=2 * &INPUT._WMA_%sysfunc(round(&HMA_N./2))  - &INPUT._WMA_&HMA_N;
  %AddWMA(WMA_N=%sysfunc(round(%sysfunc(sqrt(&HMA_N)))), INPUT=&INPUT._HMA_&HMA_N._DELTA);
 
  rename &INPUT._HMA_&HMA_N._DELTA_WMA_%sysfunc(round(%sysfunc(sqrt(&HMA_N.))))=&OUT;
  label &INPUT._HMA_&HMA_N._DELTA_WMA_%sysfunc(round(%sysfunc(sqrt(&HMA_N.))))="&outlabel"; 
 
  drop &INPUT._WMA_&HMA_N &INPUT._WMA_%sysfunc(round(&HMA_N./2)) &INPUT._HMA_&HMA_N._DELTA;
%mend;

The upper code is very intuitive as the first two lines perform WMA with window n and n/2 for X, then the next two lines generate new series Y and performs WMA with window sqrt(n). Other codes are just to improve macro flexibility and to drop temp variables, etc.

To compare HMA with SMA result, we must also implement SMA with the 7 lines of code as shown below. It has the same argument as WMA/HMA above, but SMA is an unnecessary part of HMA macro %ADDHMA implmentation.

 
%macro AddMA(MA_N=5, INPUT=close, out=&INPUT._MA_&MA_N., outlabel=&INPUT. MA(&MA_N.));
  retain _sum_&INPUT._MA_&MA_N.;
  _sum_&INPUT._MA_&MA_N.=sum (_sum_&INPUT._MA_&MA_N., &INPUT., -lag&MA_N.( &INPUT. )) ;
  &out=_sum_&INPUT._MA_&MA_N. / min(_n_, &MA_N.);
 
  drop _sum_&INPUT._MA_&MA_N.;
  label &out="&outlabel";
%mend;

Now we can test all the above implementations as shown below. First is a simple case we described above:

data a;
  input x @@;
  %AddMA( input=x, MA_N=5, out=SMA, outlabel=%str(SMA)); 
  %AddHMA( input=x, HMA_N=5, out=HMA, outlabel=%str(HMA))  
datalines;
1 2 3 4 5 6 7 8 9 10
;
proc print data=a label;
  format _all_ 8.2;
run;

The output is listed below, the HMA result is much more close to the real value x than SMA. It’s more responsive and less in lag.

Figure 3: HMA output sample

Now we can use the HMA macro to check IBM stock data in the SAS system dataset SASHELP.Stocks and draw trends in the plot to compare HMA(x, 5) and SMA(x, 5) output.

data Stocks_HMA;
  set sashelp.Stocks(where=(stock='IBM'));
  %AddHMA( HMA_N=5, out=HMA, outlabel=%str(HMA))
  %AddMA(MA_N=5, out=SMA, outlabel=%str(SMA));
run; 
ods graphics / width=800px height=300px;
title "Comparison of Simple and Hull Moving Average";
proc sgplot data=Stocks_HMA noautolegend;
   highlow x=Date high=High low=Low / lineattrs=(color=LightGray);
   scatter x=Date y=Close / markerattrs=(color=LightGray);
   series x=Date y=SMA / curvelabel curvelabelattrs=(color=DarkGreen) lineattrs=(color=DarkGreen);
   series x=Date y=HMA / curvelabel curvelabelattrs=(color=DarkOrange) lineattrs=(color=DarkOrange);
   refline '01JUN1992'd / axis=x label='June 1992' labelloc=inside;
   xaxis grid display=(nolabel); 
   yaxis grid max=200 label="IBM Closing Price";
run;

Figure 4: Comparison of SMA and HMA

Summary

We have talked about what Hull Moving Average is, how it works, and worked through a super simple SAS Macro implementation for HMA, WMA and SMA with only 25 lines of SAS code. This HMA implementation makes it quite simple and efficient to calculate HMA in SAS Data step without SAS/IML and Visual Analytics. In fact, it shows how easy process rolling-window computation for time series data can be in SAS, and opens a new window to build complicated technical indicators for stock trading systems yourself.

Learn more

Implementing HMA, WMA & SMA with 25 lines of SAS code was published on SAS Users.

11月 272019
 
Visualization of a quadratic function and a linear subspace

This article discusses how to restrict a multivariate function to a linear subspace. This is a useful technique in many situations, including visualizing an objective function that is constrained by linear equalities. For example, the graph to the right is from a previous article about how to evaluate quadratic polynomials. The graph shows a heat map for a quadratic polynomial of two variables. The diagonal line represents a linear constraint between the X and Y variables. If you restrict the polynomial to the diagonal line, you obtain a one-dimensional function that you can easily graph and visualize. By repeating this process for other lines, you can construct a "stack" of lower-dimensional slices that enable you to understand the graph of the original bivariate function.

This example generalizes. For a function of many variables, you can restrict the function to a lower-dimensional subspace. By visualizing the restricted function, you can understand the high-dimensional function better. I previously demonstrated this technique for multidimensional regression models by using the SLICEFIT option in the EFFECTPLOT statement in SAS. This article shows how to use ideas from vector calculus to restrict a function to a parameterized linear subspace.

Visualize high-dimensional data and functions

There are basically two techniques for visualizing high-dimensional objects: projection and slicing. Many methods in multivariate statistics compute a linear subspace such that the projection of the data onto the subspace has a desirable property. One example is principal component analysis, which endeavors to find a low dimensional subspace that captures most of the variation in the data. Another example is linear discriminant analysis, which aims to find a linear subspace that maximally separates groups in the data. In both cases, the data are projected onto the computed subspaces to reveal characteristics of the data. There are also statistical methods that attempt to find nonlinear subspaces.

Whereas projection is used for data, slicing is often used to visualize the graphs of functions. In regression, you model a response variable as a function of the explanatory variables. Slicing the function along a linear subspace of the domain can reveal important characteristics about the function. That is the method used by the SLICEFIT option in the EFFECTPLOT statement, which enables you to visualize multidimensional regression models. The most common "slice" for a regression model is to specify constant values for all but one continuous variable in the model.

When visualizing an objective function that is constrained by a set of linear equalities, the idea is similar. The main difference is that the "slice" is usually determined by a general linear combination of variables.

Restrict a function to a one-dimensional subspace

A common exercise in multivariate calculus is to restrict a bivariate function to a line. (This is equivalent to a "slice" of the bivariate function.) Suppose that you choose a point, x0, and a unit vector, u, which represents the "direction". Then a parametric line through x0 in the direction of u is given by the vector expression v(t) = x0 + t u, where t is any real number. If you restrict a multivariate function to the image of v, you obtain a one-dimensional function.

For example, consider the quadratic polynomial in two variables:
f(x,y) = (9*x##2 + x#y + 4*y##2) - 12*x - 4*y + 6
The function is visualized by the heat map at the top of this article. Suppose x0 = {0, -1} and choose u to be the unit vector in the direction of the vector {3, 1}. (This choice corresponds to a linear constraint of the form x – 3y = 3.)

The following SAS/IML program constructs the parameterized line v(t) for a uniform set of t values. The quadratic function is evaluated at this set of values and the resulting one-dimensional function is graphed:

proc iml;
/* Evaluate the quadratic function at each column of X and return a row vector. */
start Func(XY);
   x = XY[1,]; y = XY[2,];
   return (9*x##2  + x#y + 4*y##2) - 12*x - 4*y + 6;
finish;
 
x0 = {0, -1};          /* evaluate polynomial at this point */
d  = {3, 1};           /* vector that determines direction */
u  = d / norm(d);      /* unit vector (direction) */
 
t = do(-2, 2, 0.1);    /* parameter values */
v = x0 + t @ u;        /* evenly spaced points along the linear subspace */
f = Func(v);           /* evaluate the 2-D function along the 1-D subspace */
title "Quadratic Form Evaluated on Linear Subspace";
call series(t, f) grid={x y};
Quadratic function restricted to linear subspace

The graph shows the restriction of the 2-D function to the 1-D subspace. According to the graph, the restricted function reaches a minimum value for t ≈ 0.92. The corresponding (x, y) values and the corresponding function value are shown below:

tMin = 0.92;           /* t* = parameter near the minimum */
vMin = x0 + tMin * u;  /* corresponding (x(t*), y(t*)) */
fMin = Func(vMin);     /* f(x(t*), y(t*)) */
print tMin vMin fMin;

If you look back to the heat map at the top of this article, you will see that the value (x,y) = (0.87,-0.71) correspond to the location for which the function achieves a minimum value when restricted to the linear subspace. The value of the function at that (x,y) value is approximately 6.6.

The SAS/IML program uses the Kronecker direct-product operator (@) to create points along the line. The operation is v = x0 + t @ u. The symbols x0 and u are both column vectors with two elements. The Kronecker product creates a 2 x k matrix, where k is the number of elements in the row vector t. The Kronecker product enables you to vectorize the program instead of looping over the elements of t and forming the individual vectors x0 + t[i]*u.

You can generalize this example to evaluate a multivariate function on a two-dimensional subspace. If u1 and u2 are two orthogonal, p-dimensional, unit vectors, the expression x0 + s*u1 + t*u2 spans a two-dimensional subspace of Rp. You can use the ExpandGrid function in SAS/IML to generate ordered pairs (s, t) on a two-dimensional grid.

Summary

In summary, you can slice a multivariate function along a linear subspace. Usually, 1-D or 2-D subspaces are used and the resulting (restricted) function is graphed. This helps you to understand the function. You can choose a series of slices (often parallel to each other) and "stack" the slices in order to visualize the multivariate function. Typically, this technique works best for functions that have three or four continuous variables.

You can download the SAS program that creates the graphs in this article.

The post Evaluate a function on a linear subspace appeared first on The DO Loop.

11月 252019
 

The SAS Global Certification Program started in 1999 and has issued over 150,000 credentials to SAS users. Today, the program offers 23 different credentials across seven categories. According to Medium.com, “SAS Certification gives recognition of competency, shows commitment to the profession, and helps with job advancement.”

The SAS Global Certification Program and SAS Documentation have partnered together to work on SAS® Certified Specialist Prep Guide: Base Programming Using SAS® 9.4 and SAS® Certified Professional Prep Guide: Advanced Programming Using SAS® 9.4. This partnership has improved the prep guides and offers additional assets to support their use.

Changes to the Prep Guides

We have implemented changes to the certification guide based on the changes to the exam. We have taken the opportunity to make numerous general improvements to the certification guides. Both prep guides have been streamlined, shortened (significantly), and include more easy-to-use examples. For example, through reorganization and removing redundant topics, we have shortened SAS® Certified Professional Prep Guide: Advanced Programming Using SAS® 9.4 by more than 400 pages (428 pages to be exact).

Other changes to the prep guides include:

  • streamlining the examples and making them easier to read
  • aligning the quiz questions so that they are more like what the user will see on the exam
  • developing a workbook to provide hands-on practice for the programming portion of the exam

Tip Sheets


In 2018, I attended my first SAS Global Forum and I spoke with professors who use the certification guides in their class. They expressed interest in having a downloadable document with syntax that students can use for quick reference. Based on this feedback, I worked with SAS Training to develop tip sheets that summarize key concepts and syntax covered in the certification guides.

You can download the tip sheets from each guide’s book page. The tip sheets are expected to be incorporated into the extended learning pages for the related classes offered through Training.

We also made changes to how you can download sample data.

Sample Data Changes

In the past, we received feedback that the sample data that users were trying to download and import into SAS was generating errors. This was happening because of the copy and paste functionality that we had before. Due to some changes internal to SAS, we decided to streamline the process of downloading sample data.

We wanted to make the sample data easy to access.

The sample data now is “environment agnostic” and can run on any SAS environment. These changes are particularly helpful to customers who use SAS University Edition, SAS Studio, and SAS On-Demand. Incorporating these changes required modification to virtually all the previous examples in both prep guides.

The sample data now lives on the sas-cert-prep-data repository, which is on the SAS Software page in the GitHub repository. The repository is organized by the name of each prep guide. This enables our users to provide feedback by simply creating an issue on GitHub. The GitHub repository itself contains a lot of additional resources such as other SAS Press books, links to SAS documentation, Base SAS Glossary, and the SAS User YouTube channel with How-To videos.

With all of these changes and more, we hope that you find the new prep guides easier to use on the path to becoming SAS Certified.

Are you a SAS certified professional? was published on SAS Users.

11月 252019
 

What is an efficient way to evaluate a multivariate quadratic polynomial in p variables? The answer is to use matrix computations! A multivariate quadratic polynomial can be written as the sum of a purely quadratic term (degree 2), a purely linear term (degree 1), and a constant term (degree 0). The purely quadratic term is called a quadratic form. This article shows how to use matrix computations to efficiently evaluate a multivariate quadratic polynomial.

Quadratic polynomials and matrix expressions

Visualization of a quadratic polynomial in SAS

As I wrote in a previous article about optimizing a quadratic function, the matrix of second derivatives and the gradient of first derivatives appear in the matrix representation of a quadratic polynomial. I will use the same example as in my previous article. Namely, the following quadratic polynomial in two variables:
f(x,y) = (9*x##2 + x#y + 4*y##2) - 12*x - 4*y + 6
       = (1/2) x` Q x + L` x + 6
where Q = {18  1, 1  8} is a 2x2 symmetric matrix, L = {-12  -4} is a column vector, and x = {x, y} is a column vector that represents the coordinates at which to evaluate the quadratic function.

The graph at the right visualizes this quadratic function by using a heat map. Small values of the function are shown in white. Larger values of the function are shown in blues and greens. The largest values are shown in reds and blacks. The global minimum of this function is approximately (x, y) = (0.64, 0.42), and that point is indicated by a star.

This example generalizes. Every quadratic function in p variables can be written as the sum of a quadratic form (1/2 x` Q x), a linear term (L` x) and a constant. Here Q is a p x p symmetric matrix of second derivatives (the Hessian matrix of f) and L and x are p-dimensional column vectors.

Evaluate quadratic polynomial in SAS

Because the computation involves vectors and matrices, the SAS/IML language is the natural place to use matrices to evaluate a quadratic function. The following SAS/IML statements implement a simple function to evaluate a quadratic polynomial (given in terms of Q, L, and a constant) at an arbitrary two-dimensional vector, x:

proc iml;
/* Evaluate  f(x) = 0.5 * x` * Q * x + L`*x + const, where
   Q is p x p symmetric matrix, 
   L and x are col vector with p elements.
   This version evaluates ONE vector x and returns a scalar value. */
start EvalQuad(x, Q, L, const=0);
   return 0.5 * x`*Q*x + L`*x + const;
finish;
 
/* compute Q and L for f(x,y)= 9*x##2 + x#y + 4*y##2) - 12*x - 4*y + 6 */
Q = {18 1,             /* matrix of second derivatives */
      1 8};
L = { -12, -4};        /* use column vectors for L and x */
const = 6;
x0 = {0, -1};          /* evaluate polynomial at this point */
f = EvalQuad(x0, Q, L, const);
print f[L="f(0,-1)"];

Evaluate a quadratic polynomial at multiple points

When I implement a function in the SAS/IML language, I try to "vectorize" it so that it can evaluate multiple points in a single call. Often you can use matrix operations to vectorize a function evaluation, but I don't see how to make the math work for this problem. The natural way to evaluate a quadratic polynomial at k vectors X1, X2, ..., Xk, is to pack those vectors into a p x k matrix X such that each column of X is a point at which to evaluate the polynomial. Unfortunately, the matrix computation of the quadratic form M = 0.5 * X`*Q*X results in a k x k matrix. Only the k diagonal elements are needed for evaluating the polynomial on the k input vectors, so although it is possible to compute M, doing so would be very inefficient.

In this case, it seems more efficient to loop over the columns of X. The following function implements a SAS/IML module that evaluates a quadratic polynomial at every column of X and returns a row vector of the results. The module is demonstrated by calling it on a matrix that has 5 columns.

/* Evaluate the quadratic function at each column of X and return a row vector. */
start EvalQuadVec(X, Q, L, const=0);
   f = j(1, ncol(X), .);
   do i = 1 to ncol(X);
      v = X[,i];
      f[i] = 0.5 * v`*Q*v + L`*v + const;
   end;
   return f;
finish;
 
/*    X1  X2  X3 X4  X5  */
vx = {-1 -0.5 0  0.5 1 ,
      -3 -2  -1  0   1 };
f = EvalQuadVec(vx, Q, L, const=0);
print (vx // f)[r={'x' 'y' 'f(x,y)'} c=('X1':'X5')];

Evaluate a quadratic polynomial on a uniform grid of points

You can use the EvalQuadVec function to evaluate a quadratic polynomial on any set of multiple points. In particular, you can use the ExpandGrid function to construct a regular 2-D grid of points. By evaluating the function at each point on the grid, you can visualize the function. The following statements create a heat map of the function on a regular grid. The heat map is shown at the top of this article.

x = do(-1, 1, 0.1); 
y = do(-3, 1.5, 0.1);
xy = expandgrid(x, y);             /* 966 x 2 matrix */
f = EvalQuadVec(xy`, Q, L, const); /* evaluate polynomial at all points */
 
/* write results to a SAS data set and visualize the function by using a heat map */
M = xy || f`;
create Heatmap from M[c={'x' 'y' 'f'}];  append from M;  close;
QUIT;
 
data optimal;
   xx=0.64; yy=0.42;  /* optional: add the optimal (x,y) value */
run;
data All;  set Heatmap optimal; run;
 
title "Heat Map of Quadratic Function";
proc sgplot data=All;
   heatmapparm x=x y=y colorresponse=f / colormodel= (WHITE CYAN YELLOW RED BLACK);
   scatter x=xx y=yy / markerattrs=(symbol=StarFilled);
   xaxis offsetmin=0 offsetmax=0;
   yaxis offsetmin=0 offsetmax=0;
run;

Quadratic approximations

Perhaps you do not often use quadratic polynomials. This technique is useful even for general nonlinear functions because it enables you to find the best quadratic approximation to a multivariate function at any point x0. The multivariate Taylor series at the point x0, truncated at second order, is
f(x) ≈ f(x0) + L` · (xx0) + (1/2) (xx0)` · Q · (xx0)
where L = ∇f(x0) is the gradient of f evaluate at x0 and Q = D2f(x0) is the symmetric Hessian matrix of second derivatives of f evaluated at x0.

Summary

In summary, you can use matrix computations to evaluate a multivariate quadratic polynomial. This article shows how to evaluate a quadratic polynomial at multiple points. For a polynomial of two variables, you can use this technique to visualize quadratic functions.

The post Evaluate a quadratic polynomial in SAS appeared first on The DO Loop.

11月 232019
 

“Ocean acidification is sometimes referred to as global warming's equally evil twin.” ~ Elizabeth Kolbert This is the second post in my two-part series about climate change. You can read part 1 of this series here. When engaging in data exploration for insights, it’s good practice to start with a [...]

Climate change: Hurricanes, acidification and extinction was published on SAS Voices by Leo Sadovy

11月 222019
 

sxlion

大家有一个普通的印象:SAS的更新很慢,很老很落后。可能跟它的版本命名有关,SAS9.0是2004年出来的,到现在都快20年了,版本号 还停留在9字头,并且还没有继续更新的迹象。当然这个与SAS公司的明面

原创文章: ”难道是SAS10?云分析服务时代的到来“,转载请注明: 转自SAS资源资讯列表

本文链接地址: http://saslist.net/archives/454

11月 222019
 

sxlion

大家有一个普通的印象:SAS的更新很慢,很老很落后。可能跟它的版本命名有关,SAS9.0是2004年出来的,到现在都快20年了,版本号 还停留在9字头,并且还没有继续更新的迹象。当然这个与SAS公司的明面

原创文章: ”难道是SAS10?云分析服务时代的到来“,转载请注明: 转自SAS资源资讯列表

本文链接地址: http://saslist.net/archives/454