Construct StochasticRSI indicator properly

[FatihErdem]

Hello,

I am trying to build Stochastic RSI indicator. In StochasticRSI there are K, D, Rsi and Stochastic parameters.

I want build StochRSI(14, 9, 3, 3):

• RSI = 14
• Stoch = 9
• K = 3
• D = 3

StochasticOscillatorKIndicator k = new StochasticOscillatorKIndicator(series, 3);
StochasticOscillatorDIndicator d = new StochasticOscillatorDIndicator(k);
RSIIndicator rsi = new RSIIndicator(d, 14); // I think it's wrong
StochasticRSIIndicator stochRsi = new StochasticRSIIndicator(rsi, 9);

Result of the stochRsi calculation is totally wrong when compare to result with exchange's graphs.
How can I construct StochasticRSIIndicator properly?

[edlins]

There are many different formulas for RSI in common use. ta4j uses a modified moving average to calculate average gain and average loss. Then a simple ratio of the mma's is used for RS (which is then converted to RSI.) This is the original Wilder RSI published in 1978 and it used mma's because all of his computations were done by hand in pencil on a worksheet, and mma's are easy to calculate by hand.

Take a look at https://github.com/edlins/ta4j/blob/b511e18ead627470f64b13b0b5e71cbd37063e57/ta4j-core/src/test/java/org/ta4j/core/indicators/RSIIndicatorTest.java (unreleased code). The last test case refers to a website that uses a different formula. That site uses a simple moving average to calculate average gain and average loss. Then a modified moving average of the RS ratios to calculate RSI. Do you know exactly what formula your exchange is using?

As a side note @team172011 , should we rename RSIIndicator to WilderRSIIndicator? All of the RSI ambiguity online is very annoying..

[FatihErdem]

Hello @edlins ,

Is there any multiple Stochastic RSI variation? I thought StochasticRSI is the one of the RSI variation but is standard indicator. I am trying to build this version.
https://www.tradingview.com/wiki/Stochastic_RSI_(STOCH_RSI)#CALCULATION

[edlins]

Stoch RSI is just stochastics on the RSI. AFAIK the stochastics formula does not have multiple variations like RSI does. But Stoch on Cutler's RSI will be different than Stoch on Wilder's RSI, so the RSI variation matters. If you're trying to match other people's results, you'll still need to match their RSI variation. https://www.tradingview.com/wiki/Relative_Strength_Index_(RSI) shows:

RS = Average Gain / Average Loss

Which I take to be Cutler's RSI. https://en.wikipedia.org/wiki/Relative_strength_index#Cutler's_RSI
Cutler's will be a little different from Wilder's (which are both different from http://cns.bu.edu/~gsc/CN710/fincast/Technical%20_indicators/Relative%20Strength%20Index%20(RSI).htm).

When I said ta4j uses Wilder's RSI, that's apparently only in the develop branch of unreleased code. It appears that the last master branch, presumably used for release 0.10, uses Cutler's RSI. Here is the source code for both:
Cutler's: https://github.com/ta4j/ta4j/blob/master/ta4j-core/src/main/java/org/ta4j/core/indicators/RSIIndicator.java
Wilder's: https://github.com/ta4j/ta4j/blob/develop/ta4j-core/src/main/java/org/ta4j/core/indicators/RSIIndicator.java

So, you need to know which variation you are trying to match, and you need to know which variation you are using. If the variations don't match, you'll get slightly different results. Which version of ta4j are you using?

[jarjedi]

Hi @FatihErdem ,
Did you find some solution to your problem? I have the same problem. I'm searching some way to implement the same StochasticRSI from tradingview.com and I don't find any choise.

[team172011]

In the TradingView wiki the formular of stochastic RSI is the same as in our StochasticRSIIndicator

Stoch RSI = (RSI - Lowest Low RSI) / (Highest High RSI - Lowest Low RSI)

    /**
     * Constructor.
     * @param rsi the rsi indicator
     * @param timeFrame the time frame
     */
    public StochasticRSIIndicator(RSIIndicator rsi, int timeFrame) {
        super(rsi);
        this.rsi = rsi;
        minRsi = new LowestValueIndicator(rsi, timeFrame);
        maxRsi = new HighestValueIndicator(rsi, timeFrame);
    }

    @Override
    protected Decimal calculate(int index) {
        Decimal minRsiValue = minRsi.getValue(index);
        return rsi.getValue(index).minus(minRsiValue)
                .dividedBy(maxRsi.getValue(index).minus(minRsiValue));
    }

But TradingView is not showing the stochastic RSI Inidcator, but the stochastic K and stochastic D oscillator based on the stochastic RSI. So if you want to calculate those two lines, you have to put the StochasticRSIIndicator on a StochasticOscillatorDIndicator and a StochasticOscillatorKIndicator:

TimeSeries series = getTimeSeries();
ClosePriceIndicator cp = new ClosePriceIndicator(series);
RSIIndicator r = new RSIIndicator(cp, 14);
StochasticRSIIndicator rsi = new StochasticRSIIndicator(cp, 14);

StochasticOscillatorKIndicator stochK= new StochasticOscillatorKIndicator(rsi,3,new MaxPriceIndicator(series), new MinPriceIndicator(series));
StochasticOscillatorDIndicator stochD = new StochasticOscillatorDIndicator(stochK); // time frame always 3, maybe we should constructor parameter for this

Would be nice to see if those results are similar to trading view.

[jarjedi]

Hi @team172011,
thanks about fast answer. This is the code that I applied:

        ClosePriceIndicator closePrice = new ClosePriceIndicator(series);
        RSIIndicator rsi = new RSIIndicator(closePrice, 14);
        StochasticRSIIndicator stoRsi = new StochasticRSIIndicator(closePrice, 14);
        MaxPriceIndicator maxPrice = new MaxPriceIndicator(series);
        MinPriceIndicator minPrice = new MinPriceIndicator(series);

        StochasticOscillatorKIndicator stochK= new StochasticOscillatorKIndicator(stoRsi,3,maxPrice, minPrice);
        StochasticOscillatorDIndicator stochD = new StochasticOscillatorDIndicator(stochK);

And this is the output for an altcoin chartdata set:

[INS] 2018-01-31T13:09:59+01:00[Europe/Madrid]: maxPrice[0.35290000] minPrice[0.35180000] rsi[35.620172648796001817355747387552] stoRSI[35.620172648796001817355747387552] K[46321.428571428571428571428571429] D[46321.428571428571428571428571429]
[INS] 2018-01-31T13:14:59+01:00[Europe/Madrid]: maxPrice[0.35290000] minPrice[0.35000000] rsi[13.065829113518364207422601115315] stoRSI[13.065829113518364207422601115315] K[579.64896149451074091183367998207] D[579.64896149451074091183367998207]
[INS] 2018-01-31T13:19:59+01:00[Europe/Madrid]: maxPrice[0.35580000] minPrice[0.35000000] rsi[72.483446881602558480193846200907] stoRSI[72.483446881602558480193846200907] K[11206.896551724137931034482758621] D[11206.896551724137931034482758621]
[INS] 2018-01-31T13:24:59+01:00[Europe/Madrid]: maxPrice[0.35600000] minPrice[0.35580000] rsi[73.176209882473504993557715224963] stoRSI[73.176209882473504993557715224963] K[10833.333333333333333333333333333] D[10833.333333333333333333333333333]
[INS] 2018-01-31T13:29:59+01:00[Europe/Madrid]: maxPrice[0.35710000] minPrice[0.35330000] rsi[55.212563565198127118750222964852] stoRSI[55.212563565198127118750222964852] K[5697.3973621919803644986517131568] D[5697.3973621919803644986517131568]
[INS] 2018-01-31T13:34:59+01:00[Europe/Madrid]: maxPrice[0.35440000] minPrice[0.35200000] rsi[51.789677209321399391187715953974] stoRSI[51.789677209321399391187715953974] K[6975.2797625850910184353593487198] D[6975.2797625850910184353593487198]
[INS] 2018-01-31T13:39:59+01:00[Europe/Madrid]: maxPrice[0.35440000] minPrice[0.35400000] rsi[58.287686326527418478736315720420] stoRSI[58.287686326527418478736315720420] K[8716.4459650880764319994694197684] D[8716.4459650880764319994694197684]
[INS] 2018-01-31T13:44:59+01:00[Europe/Madrid]: maxPrice[0.35550000] minPrice[0.35210000] rsi[47.064457497198871262363983764458] stoRSI[47.064457497198871262363983764458] K[8319.0314185396470328508360208597] D[8319.0314185396470328508360208597]
[INS] 2018-01-31T13:49:59+01:00[Europe/Madrid]: maxPrice[0.35320000] minPrice[0.35000000] rsi[39.572277697129412278888750337253] stoRSI[39.572277697129412278888750337253] K[3468.7389299896697923392684485151] D[3468.7389299896697923392684485151]
[INS] 2018-01-31T13:54:59+01:00[Europe/Madrid]: maxPrice[0.35120000] minPrice[0.35000000] rsi[44.133021789320936953510057141946] stoRSI[44.133021789320936953510057141946] K[4601.9298945286597672503516311793] D[4601.9298945286597672503516311793]
[INS] 2018-01-31T13:59:59+01:00[Europe/Madrid]: maxPrice[0.35200000] minPrice[0.35090000] rsi[47.941229101311101646526364110171] stoRSI[47.941229101311101646526364110171] K[9535.8671205728611131269971276875] D[9535.8671205728611131269971276875]
[INS] 2018-01-31T14:04:59+01:00[Europe/Madrid]: maxPrice[0.35250000] minPrice[0.35200000] rsi[48.362414687711666127254080870824] stoRSI[48.362414687711666127254080870824] K[12436.140797882448606754845447386] D[12436.140797882448606754845447386]
[INS] 2018-01-31T14:09:59+01:00[Europe/Madrid]: maxPrice[0.35380000] minPrice[0.35230000] rsi[51.330775824797406380173852344869] stoRSI[51.330775824797406380173852344869] K[12088.554660346811595783289827843] D[12088.554660346811595783289827843]
[INS] 2018-01-31T14:14:59+01:00[Europe/Madrid]: maxPrice[0.35390000] minPrice[0.35250000] rsi[50.880797736131787479034668588833] stoRSI[50.880797736131787479034668588833] K[14583.797322324084787101702184687] D[14583.797322324084787101702184687]
[INS] 2018-01-31T14:19:59+01:00[Europe/Madrid]: maxPrice[0.35580000] minPrice[0.35110000] rsi[59.328056983281217359299001221050] stoRSI[59.328056983281217359299001221050] K[8904.7079722190996238097809160883] D[8904.7079722190996238097809160883]
[INS] 2018-01-31T14:24:59+01:00[Europe/Madrid]: maxPrice[0.35630000] minPrice[0.35590000] rsi[63.340063096010001968656321009816] stoRSI[63.340063096010001968656321009816] K[6849.8421454119819040835751368902] D[6849.8421454119819040835751368902]
[INS] 2018-01-31T14:29:59+01:00[Europe/Madrid]: maxPrice[0.35880000] minPrice[0.35610000] rsi[65.851499279258385707902160496337] stoRSI[65.851499279258385707902160496337] K[5596.4697365574183540807133498762] D[5596.4697365574183540807133498762]
[INS] 2018-01-31T14:34:59+01:00[Europe/Madrid]: maxPrice[0.35850000] minPrice[0.35680000] rsi[65.316081571361184085397317268519] stoRSI[65.316081571361184085397317268519] K[21507.786742866621476500877958598] D[21507.786742866621476500877958598]
[INS] 2018-01-31T14:39:59+01:00[Europe/Madrid]: maxPrice[0.35830000] minPrice[0.35800000] rsi[69.103544894944911650442469485232] stoRSI[69.103544894944911650442469485232] K[23848.148148148148148148148148148] D[23848.148148148148148148148148148]
[INS] 2018-01-31T14:44:59+01:00[Europe/Madrid]: maxPrice[0.35890000] minPrice[0.35830000] rsi[70.348884963335443336651758777000] stoRSI[70.348884963335443336651758777000] K[30628.571428571428571428571428571] D[30628.571428571428571428571428571]
[INS] 2018-01-31T14:49:59+01:00[Europe/Madrid]: maxPrice[0.35880000] minPrice[0.35440000] rsi[51.551907622256320321345428085164] stoRSI[51.551907622256320321345428085164] K[774.32570112432970901909950607133] D[774.32570112432970901909950607133]
[INS] 2018-01-31T14:54:59+01:00[Europe/Madrid]: maxPrice[0.35590000] minPrice[0.35400000] rsi[52.527634613803610947629617672325] stoRSI[52.527634613803610947629617672325] K[1366.2894549874496833279563400853] D[1366.2894549874496833279563400853]
[INS] 2018-01-31T14:59:59+01:00[Europe/Madrid]: maxPrice[0.35600000] minPrice[0.35380000] rsi[56.024795549555600335181811447651] stoRSI[56.024795549555600335181811447651] K[1996.1969986644874174164952262838] D[1996.1969986644874174164952262838]
[INS] 2018-01-31T15:04:59+01:00[Europe/Madrid]: maxPrice[0.36000000] minPrice[0.35490000] rsi[59.794406852584060941102896269280] stoRSI[59.794406852584060941102896269280] K[2825.4665622084953404462578234637] D[2825.4665622084953404462578234637]
[INS] 2018-01-31T15:09:59+01:00[Europe/Madrid]: maxPrice[0.36000000] minPrice[0.35610000] rsi[55.099181632242727535411664186798] stoRSI[55.099181632242727535411664186798] K[-764.43363542587022042305009763968] D[-764.43363542587022042305009763968]
[INS] 2018-01-31T15:14:59+01:00[Europe/Madrid]: maxPrice[0.35840000] minPrice[0.35640000] rsi[61.361652077990329801821946358063] stoRSI[61.361652077990329801821946358063] K[3597.2699111608113237204727789324] D[3597.2699111608113237204727789324]
[INS] 2018-01-31T15:19:59+01:00[Europe/Madrid]: maxPrice[0.36000000] minPrice[0.35800000] rsi[59.799561296900097247182140174859] stoRSI[59.799561296900097247182140174859] K[2615.9546501650782140913732819708] D[2615.9546501650782140913732819708]

Seems like StochasticOscillatorKIndicator is not prepared to receive an indicator that don't get real values. I'm not sure If I misspelled something :-s.

Thanks in advance!.

[team172011]

Yes, at a first look I think that the StochasticOscillatorKIndicator just works for prices, because it expects MinPrice/MaxPriceIndicator. Maybe it works if we change these parameters to LowestValue/HighestValueIndicator and let the stoch. K Indicator work with highest/lowest values of rsi instead of price?

[jarjedi]

I create a CustomStochasticOscillatorKIndicator that accepts a Lowest/HighestValueIndicator instead of max/min indicators, but I'm not sure about params highest/lowest indicators:

        ClosePriceIndicator closePrice = new ClosePriceIndicator(series);
        RSIIndicator rsi = new RSIIndicator(closePrice, 14);
        StochasticRSIIndicator stoRsi = new StochasticRSIIndicator(closePrice, 14);
        MaxPriceIndicator maxPrice = new MaxPriceIndicator(series);
        MinPriceIndicator minPrice = new MinPriceIndicator(series);
        HighestValueIndicator highestValue = new HighestValueIndicator(stoRsi, 14);
        LowestValueIndicator lowestValue = new LowestValueIndicator(stoRsi, 14);

        CustomStochasticOscillatorKIndicator customStoK = new CustomStochasticOscillatorKIndicator(stoRsi, 3, highestValue, lowestValue);

This is the output:

[ADA] 2018-01-31T15:04:59+01:00[Europe/Madrid]: maxPrice[0.04852000] minPrice[0.04845000] rsi[14.150807062531654514823260856282] stoRSI[14.150807062531654514823260856282] customStoK[0]
[ADA] 2018-01-31T15:09:59+01:00[Europe/Madrid]: maxPrice[0.04876000] minPrice[0.04845000] rsi[47.369620750957588663462047433981] stoRSI[47.369620750957588663462047433981] customStoK[38.694380869278725439464555973735]
[ADA] 2018-01-31T15:14:59+01:00[Europe/Madrid]: maxPrice[0.04889000] minPrice[0.04874000] rsi[55.344603465070894502235743289833] stoRSI[55.344603465070894502235743289833] customStoK[47.983906421280361451369277461389]
[ADA] 2018-01-31T15:19:59+01:00[Europe/Madrid]: maxPrice[0.04889000] minPrice[0.04879000] rsi[55.943710753464511272409314399050] stoRSI[55.943710753464511272409314399050] customStoK[48.681766550064565928284524096959]
[ADA] 2018-01-31T15:24:59+01:00[Europe/Madrid]: maxPrice[0.04892000] minPrice[0.04879000] rsi[55.943710753464511272409314399050] stoRSI[55.943710753464511272409314399050] customStoK[48.681766550064565928284524096959]
[ADA] 2018-01-31T15:29:59+01:00[Europe/Madrid]: maxPrice[0.04890000] minPrice[0.04879000] rsi[56.618708765975194015262119828500] stoRSI[56.618708765975194015262119828500] customStoK[41.138674053457147097087487579535]
[ADA] 2018-01-31T15:34:59+01:00[Europe/Madrid]: maxPrice[0.04892000] minPrice[0.04879000] rsi[58.004543879026688422779232760705] stoRSI[58.004543879026688422779232760705] customStoK[43.019025928126785051493658887442]
[ADA] 2018-01-31T15:39:59+01:00[Europe/Madrid]: maxPrice[0.04895000] minPrice[0.04884000] rsi[54.270434267184134548279557482901] stoRSI[54.270434267184134548279557482901] customStoK[61.850899659254731863593743390536]
[ADA] 2018-01-31T15:44:59+01:00[Europe/Madrid]: maxPrice[0.04890000] minPrice[0.04857000] rsi[36.969838584947926319612804066248] stoRSI[36.969838584947926319612804066248] customStoK[87.615282495752114150809499152289]
[ADA] 2018-01-31T15:49:59+01:00[Europe/Madrid]: maxPrice[0.04881000] minPrice[0.04865000] rsi[48.708896099539152522345925423813] stoRSI[48.708896099539152522345925423813] customStoK[100]
[ADA] 2018-01-31T15:54:59+01:00[Europe/Madrid]: maxPrice[0.04879000] minPrice[0.04850000] rsi[37.125674328622002891378060982279] stoRSI[37.125674328622002891378060982279] customStoK[59.145578635689647896517212877713]
[ADA] 2018-01-31T15:59:59+01:00[Europe/Madrid]: maxPrice[0.04873000] minPrice[0.04840000] rsi[36.789187604650746746755835849177] stoRSI[36.789187604650746746755835849177] customStoK[57.958778393365563142757990578505]
[ADA] 2018-01-31T16:04:59+01:00[Europe/Madrid]: maxPrice[0.04851000] minPrice[0.04840000] rsi[36.084766814117594114356612639536] stoRSI[36.084766814117594114356612639536] customStoK[55.474262050384128100929935886798]
[ADA] 2018-01-31T16:09:59+01:00[Europe/Madrid]: maxPrice[0.04849000] minPrice[0.04842000] rsi[36.737020136605604630084190711027] stoRSI[36.737020136605604630084190711027] customStoK[57.774781940290946803769374263713]
[ADA] 2018-01-31T16:14:59+01:00[Europe/Madrid]: maxPrice[0.04882000] minPrice[0.04836000] rsi[39.400947804987043390733514232620] stoRSI[39.400947804987043390733514232620] customStoK[0]
[ADA] 2018-01-31T16:19:59+01:00[Europe/Madrid]: maxPrice[0.04860000] minPrice[0.04843000] rsi[42.651729499106923728972982949505] stoRSI[42.651729499106923728972982949505] customStoK[23.290947673292397829971607196155]
[ADA] 2018-01-31T16:24:59+01:00[Europe/Madrid]: maxPrice[0.04858000] minPrice[0.04851000] rsi[40.322287917351249383653799782236] stoRSI[40.322287917351249383653799782236] customStoK[6.6011459290531837054443195535767]
[ADA] 2018-01-31T16:29:59+01:00[Europe/Madrid]: maxPrice[0.04853000] minPrice[0.04843000] rsi[37.254622023258975825826830177845] stoRSI[37.254622023258975825826830177845] customStoK[0.0]
[ADA] 2018-01-31T16:34:59+01:00[Europe/Madrid]: maxPrice[0.04851000] minPrice[0.04835000] rsi[38.689820995621403145090701553655] stoRSI[38.689820995621403145090701553655] customStoK[8.9122859058344468293496997817419]
[ADA] 2018-01-31T16:39:59+01:00[Europe/Madrid]: maxPrice[0.04851000] minPrice[0.04845000] rsi[40.163759477059393969344233119490] stoRSI[40.163759477059393969344233119490] customStoK[18.065136073058221915272583682535]
[ADA] 2018-01-31T16:44:59+01:00[Europe/Madrid]: maxPrice[0.04851000] minPrice[0.04846000] rsi[40.928439778836358647756476652047] stoRSI[40.928439778836358647756476652047] customStoK[28.989674304939342035792046691652]

It seems the results are not good comparing with the same period of altcoin ADA in tradingview.

This is the code of my custom stoK:

public class CustomStochasticOscillatorKIndicator extends CachedIndicator<Decimal> {
    private final Indicator<Decimal> indicator;
    private final int timeFrame;
    private HighestValueIndicator maxPriceIndicator;
    private LowestValueIndicator minPriceIndicator;

    public CustomStochasticOscillatorKIndicator(Indicator<Decimal> indicator, int timeFrame, HighestValueIndicator maxPriceIndicator, LowestValueIndicator minPriceIndicator) {
        super(indicator);
        this.indicator = indicator;
        this.timeFrame = timeFrame;
        this.maxPriceIndicator = maxPriceIndicator;
        this.minPriceIndicator = minPriceIndicator;
    }

    protected Decimal calculate(int index) {
        Decimal highestHighPrice = (Decimal)maxPriceIndicator.getValue(index);
        Decimal lowestLowPrice = (Decimal)minPriceIndicator.getValue(index);
        return ((Decimal)this.indicator.getValue(index)).minus(lowestLowPrice).dividedBy(highestHighPrice.minus(lowestLowPrice)).multipliedBy(Decimal.HUNDRED);
    }

    public String toString() {
        return this.getClass().getSimpleName() + " timeFrame: " + this.timeFrame;
    }
}

[edlins]

Build a spreadsheet for XLS Testing? I might be able to work on that tomorrow.

[edlins]

I don't know tradingview well at all, but I don't see an obvious way to export their OHLC and indicator data. If that can't be done, I'm not inclined to manually transcribe enough data to do a comparative analysis.

I've made a spreadsheet AAPL_StochRSI.xlsx and spot-checked a few values and it appears that tradingview is using the chartschool algorithm. This does not use high or low prices. It simply uses the highest RSI(close) and lowest RSI(close) over the stochastic period. Also, to get the chartschool-style Stoch RSI on tradingview I turned off D, set the K param to 1, and pulled a few %K values for comparison. tradingview matched my chartschool-style excel calculations to < 0.0002.

Aside, I pulled the AAPL data from Yahoo and the tradingview data matches Yahoo's Close values, and does not match Yahoo's Adjusted Close. I removed Adjusted Close for clarity.

Please remember, too, that Wilder's RSI does not settle for quite a while due to the MMA. So, I used one full year of close prices and spot-checked half a dozen values at the very end of the year of data. I'm guessing tradingview calculates RSI's from an older start point to ensure they are stable on the chart..? If so, expect your values not to match for up to several hundred bars. Also, if you're calculating a 14-bar RSI and using a 14-bar stochastic, your first 13 values will be invalid unless you padded the front end with older data.

I'll make a unit test to check the spreadsheet against ta4j calculations tomorrow (running a huge backtest all day today).

[edlins]

ta4j matches tradingview to two decimal points (on 100-pt scale) after about 140 bars with my testing setup. They match to three decimal points after about 165 bars. They match almost to four after 250 bars. This is because Wilder's modified moving average on gain or loss takes several hundred bars to settle. This is called data length dependency and is a "feature" of Wilder's RSI.

The correct form to match tradingview %K where K = 1 after 250 bars is

stochRsi = new StochasticRSIIndicator(new ClosePriceIndicator(series), stochPeriod);
stochRsi.getValue(index);  // for index >= 250.

I would apply an SMAIndicator to stochRsi to get different D values. I have not evaluated StochasticOscillatorKIndicator or StochasticOscillatorDIndicator yet.

To run my unit test I did have to modify StochasticRSIIndicator to handle divide-by-zero differently, and I had to modify XlsTestsUtils to handle spreadsheets with daily data so you can't run my unit test out-of-the-box. But you can use my updated spreadsheet AAPL_StochRSI.xlsx to see the ta4j-style calculation and compare its results to tradingview.

@team172011 Can we make the MMA complain more loudly when it's not settled? I thought there used to be warnings printed to stderr when it was unsettled. And curse Wilder for his "optimization".